Research ArticleGlobal Stability of a Variation Epidemic Spreading Model onComplex Networks

De-gang Xu Xi-yang Xu Chun-hua Yang and Wei-hua Gui

College of Information Science and Engineering Central South University Hunan Changsha 410083 China

Correspondence should be addressed to De-gang Xu dgxucsueducn

Received 20 September 2015 Accepted 18 November 2015

Academic Editor Xinguang Zhang

Copyright copy 2015 De-gang Xu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Epidemic spreading on networks becomes a hot issue of nonlinear systems which has attracted many researchersrsquo attention inrecent years A novel epidemic spreading model with variant factors in complex networks is proposed and investigated in thispaper One main feature of this model is that virus variation is investigated in the process of epidemic dynamical spreading Theglobal dynamics of this model involving an endemic equilibrium and a disease-free equilibrium are respectively discussed Somesufficient conditions are given for the existence of the endemic equilibrium In addition the global asymptotic stability problemsof the disease-free equilibrium and the endemic equilibrium are also investigated by the Routh-Hurwitz stability criterion andLyapunov stability criterion And the uniform persistence condition of the new system is studied Finally numerical simulationsare provided to illustrate obtained theoretical results

1 Introduction

The research of infectious disease has always been a hotissue of nonlinear systems with applications The populardynamics on complex network is the epidemic spreadingwhich describes how infections spread throughout a network[1] In recent years much research work has been done aboutthe viral dynamics of epidemic spreading [2 3] These resultsare helpful for preventing and controlling most emerginginfectious diseases like SARS HIVAIDS H5N1 and H1N1They are also meaningful to provide important informationfor the research in the field of rumor spreading [4ndash7]traffic dynamics [8ndash10] computer viruses [11 12] biologymechanism [13] and medicine developing [14ndash16]

In real world the population size is large enough suchthat the mixing of individuals can be considered to behomogeneous Social and biological systems can be properlydescribed as complex networks with nodes representing indi-viduals and links mimicking the interactions [17 18] Suitablemathematical models of the infectious disease spreading incomplex homogeneous networks are of great practical valueto analyze the detailed spreading process Because epidemicspreading usually brings great harm to society it is very

urgent to establish accurate propagation models consideringthe infection contagion spreading problems In past decadescomplicated 119878119868119877 models were formulated from differentperspectives of epidemiology [19 20] In these models 119878119868 and 119877 denote respectively the number of individualssusceptible to the disease the number of infectious indi-viduals and the number of individuals who are recoveredfrom being infectiousThe process of epidemic spreading canbe further modeled with differential equations such as 119878119868119878119878119868119877 and 119878119868119877119878 model [21ndash26] In these research works thenetwork topological structure is simplified presumptively toregular network or sufficient mixing homogeneous networkwhere the relationship between the network structure andthe epidemic spreading is discussed Kephart et al [27]established a virus spreadingmodel based on a homogeneousnetwork by characterizing the average degree as the networkmetrics and obtained a virus spreading threshold 120591

119888= 1⟨119896⟩

while Pastor-Satorras [28ndash31] studied the epidemic outbreaksin complex heterogeneous network which chose the degreeand average degree as the network metrics and obtained theepidemic spreading threshold 120591

119888= ⟨119896⟩⟨119896

2

⟩ Moore andNewman [32] applied the percolation theory to analyze theepidemic spreading behaviors in small-world network and

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 365049 8 pageshttpdxdoiorg1011552015365049

2 Mathematical Problems in Engineering

showed the differences of spreading action between small-world network and regular network These research resultsillustrate that the different network topological structure canaffect the epidemic spreading

Apart from the network topological structure one of themost important characteristics of epidemic spreadingmodelsis the dynamical stability which can reflect the developmentof the spreading behaviors of infectious disease Hence thestability problems of these epidemic models need to beinvestigated Kuniya [21] applied a discretization methodto prove the global asymptotic stability of the 119878119868119877 modelwith the age structure Zhang and Feng [22] deal with theglobal analysis of a dynamical model describing the spreadof tuberculosis with isolation and incomplete treatmentLahrouz et al [23] studied a nonlinear 119878119868119877119878 model withsaturated birth and death rates and the global asymptoticstability of the model is also discussed Xu et al [24] analyzeda time-delayed 119878119868119877119878 model with temporary immunity andsome conditions for the globally asymptotically stability ofthe disease-free equilibrium and the endemic equilibriumare given Besides for the epidemic model with time-delayKang and Fu [25] presented a new 119878119868119878 model with aninfective vector on scale-free networks and the global stabilityof equilibrium is proved The influences of treatment andvaccination efforts on a dynamic disease model in presenceof incubation delays and relapse are studied and sufficientconditions for the local stability of the equilibriumare derived[26]

However few papers are available in the literature toconsider variant factors in the epidemic spreading from asystematic framework In real world certain variants existin the infectious disease transmitting resulting from somefactors including gene mutation and cell division environ-ment Viruses evolve rapidly because they have strong abilityof propensity for genetic variation and short generation timewhich leads to evading human immunization response andobtaining drug resistance For example influenza viruses canbe classified into three major types (A B and C) Thereare many different virus forms because of mutation typeA infects many animal species including humans whiletype B and type C viruses are mainly human pathogens Ifindividuals are affected by viruses not all infected individualscan be recovered Some of them may suffer from otherdiseases because virus variation or the infectious individualscontacted with variants In fact some infectious personswho may be infected by some diseases would have certainprobability to become variantmembers of another group It isnecessary to propose a newmodel considering this conditionHow to build models with variant factors in the epidemicsspreading becomes a challengeTherefore the paper presentsa novel 119878119868119881119877119878 epidemic spreadingmodel considering variantfactors where 119878 stands for the susceptible and 119868 119881 and119877 stand for the infectious the variant and the recoveredrespectively

Given the mechanism of the 119878119868119881119877119878 model in a homo-geneous network which is only composed by blank nodesinitially the entire population can be divided into four groupsdescribed by the symbols of 119878 119868 119881 and 119877 respectively Theydenote four epidemiological statuses susceptible infectious

variant and recovered All new individuals are supposed tobe blank nodes in complex networks When a susceptibleindividual contacts the other infected individual this indi-vidual may be infectious with certain probability Then aninfectious individual would have only three states includinginfectious variant and recovered The infectious individualswould become the variants with certain probability affectedby some factors such as genemutation and the indeterminacyof cell division Similarly an infectious person may becomea variant with certain probability after contacting with avariant Usually human body can be protected by onersquosimmune system Some infectious individuals with recoveryprobability may become the recovered while others will keepthe infectious statusWe assume that the four groups have thesame mortality rate

In this paper a novel 119878119868119881119877119878 epidemic spreading modelwith virus variation in complex homogeneous network isproposed and investigatedThe rest of this paper is organizedas follows In Section 2 the propagation mechanism ofthe 119878119868119881119877119878 model in complex networks is presented andmean-field equations are used to describe the dynamics ofepidemic spreadingmodel with virus variationThe existenceof endemic equilibrium is considered Section 3 is devotedto discuss the global stability of the disease-free equilibriumwhich is followed by the discussion of the system uniformpersistence in Section 4 Then the proofs of global stabilityof an endemic equilibrium are presented in Section 5 InSection 6 numerical simulations are performed to illustrateobtained theoretical results Finally conclusions are given inSection 7

2 Epidemic Spreading Model and Its Property

21The SIVRSModel As described above in the paper a new119878119868119881119877119878model is establishedThemodel involves a new variantgroupwhich is caused by the infectious variation Assume thenumber of nodes is 119873 in a closed complex network whichincludes four statuses susceptible 119878 infectious 119868 variant 119881and recovered 119881 as well as some initial blank nodes All newnodes produced from blank nodes are susceptible

The flow chart of epidemic spreading is shown in Figure 1Assume in a homogeneous network only composed by

blank nodes initially the susceptible individuals are producedby the blank nodes the probability is characterized as 120575Others come from the recovered group with the probability120601 A susceptible individual will become infectious withprobability 120572 if heshe contacts the infected individual Thenan infectious individual may perhaps become one variantwhen heshe has tight relation with variants or is affectedby other factors We assume that the variant probability is 120574when contacting with a variant In the process of epidemicspreading an infectious individual may become the variantwith internal probability 120578 Some infectious individuals withrecovery probability 120573 may recover and others will keep theinfectious status In this paper the four groups are supposedto have the same mortality rate 120583 Then the 119878119868119881119877119878 epidemicspreading rules can be summarized as follows

Mathematical Problems in Engineering 3

120583120583 120583

120583

120575

120573I120601R

120574IV

120572IS120578I

S I V

R

Figure 1 A schematic representation of 119878119868119881119877119878 epidemic spreadingmodel

(1) Apart from the four groups in a closed networkthere are blank nodes which exist in initial networkThe blank nodes may become susceptible ones withprobability 120575 namely crude birth rate

(2) The susceptible individual becomes infectious withprobability 120572when contacting with an infectious onenamely infection rate

(3) An infectious individual can be recovered with prob-ability 120573 namely recovery rate

(4) The variants coming from some of the infectiousnodes at a variation rate 120578 (internal variation rate)can reflect the variation factors When an infectiousindividual contacts with a variant the individual maybecome a variant with probability 120574 ( contact variantrate)

(5) The recovered node turns into susceptible with prob-ability 120601 after a period of time due to the loss ofimmunity For the four groups in the network allindividuals will become blank with probability 120583namely natural mortality rate

A closed and homogeneous network consisting of 119873individuals is investigated in this paper Individuals in thenetwork can be represented with nodes and the contactbetween different individuals can be denoted by edgesThen the network can be described by an undirected graph119866 = (119881 119864) where 119881 and 119864 denote the set of nodes andedges respectively Therefore a differential equation modelis derived based on the aforementioned rules and the basicassumptions

119889119878

119889119905= 120575 (1 minus 119873) minus 120572 ⟨119896⟩ 119878119868 + 120601119877 minus 120583119878

119889119868

119889119905= 120572 ⟨119896⟩ 119878119868 minus 120574 ⟨119896⟩ 119868119881 minus (120578 + 120573 + 120583) 119868

119889119881

119889119905= 120574 ⟨119896⟩ 119868119881 + 120578119868 minus 120583119881

119889119877

119889119905= 120573119868 minus 120601119877 minus 120583119877

(1)

where ⟨119896⟩ denotes the average degree of the network

The total population satisfies119873 = 119878 + 119868 + 119881 + 119877 and thefollowing equation is obtained

119889119873

119889119905= 120575 minus (120575 + 120583)119873 (2)

which is derived by adding the four equations in (1)In (2) 119873 will eventually tend to 119873

0= 120575(120575 + 120583) with the

exponential decay Therefore assume that 119873(0) = 1198730 The

closed and positively invariant set for (1) is Σ = (119878 119868 119881 119877) isinR4+ 0 le 119878 + 119868 + 119881 + 119877 = 119873

0le 1 where R4

+denotes the

nonnegative cone ofR4 with its lower dimensional faces Use120597Σ and Σ∘ to denote the boundary and interior of Σ in R4

+

respectively

22 Existence of Equilibrium The system (1) has a disease-free equilibrium (DFE) 119864

0 where

1198640= (1198780 1198680 1198810 1198770) = (

120575

120575 + 120583 0 0 0) (3)

Denote the basic reproduction number parameter as

1198770=120572 ⟨119896⟩ 119878

0minus 120574 ⟨119896⟩119881

0

120578 + 120573 + 120583=

120572 ⟨119896⟩ 120575

(120575 + 120583) (120578 + 120573 + 120583) (4)

The following theorem summarizes the parameter restric-tions on the existence of equilibrium

Theorem 1 If 1198770gt 1 and the inequality

1 gt 120578 gt120572 (1 minus 120574 ⟨119896⟩)

120572 + 120574(5)

is satisfied there are two endemic equilibria for system (1)

Proof Assume that 119864lowast = (119878lowast

119868lowast

119881lowast

119877lowast

) is an endemicequilibrium (EE) of system (1) According to system (1) wehave

120575 (1 minus 119873lowast

) minus 120572 ⟨119896⟩ 119878lowast

119868lowast

+ 120601119877lowast

minus 120583119878lowast

= 0

120572 ⟨119896⟩ 119878lowast

119868lowast

minus 120574 ⟨119896⟩ 119868lowast

119881lowast

minus (120578 + 120573 + 120583) 119868lowast

= 0

120574 ⟨119896⟩ 119868lowast

119881lowast

+ 120578119868lowast

minus 120583119881lowast

= 0

120573119868lowast

minus 120601119877lowast

minus 120583119877lowast

= 0

(6)

For (6) a straightforward calculation leads to

119878lowast

=120578 + 120573 + 120583

120572 ⟨119896⟩+

120574120578119868lowast

120572 (120583 minus 120574 ⟨119896⟩ 119868lowast)

119881lowast

=120578119868lowast

120583 minus 120574 ⟨119896⟩ 119868lowast

119877lowast

=120573

120583 + 120601119868lowast

(7)

From (7) 120583 minus 120574⟨119896⟩119868lowast gt 0 which implies that

0 lt 119868lowast

lt min1120583

(120574 ⟨119896⟩) (8)

4 Mathematical Problems in Engineering

the component 119868lowast is a positive solution of

119901 (119868lowast

) = 119860119868lowast2

+ 119861119868lowast

+ 119862 = 0 (9)

where

119860 = 120572120574 ⟨119896⟩2

(120583 + 120601 + 120573)

119861 = minus120578 ⟨119896⟩ (120583 + 120601) (120572 + 120574) minus 120572 ⟨119896⟩ 120583 (120583 + 120601 + 120573)

+ (120583 + 120601) (120578 + 120573 + 120583) (1 minus 120574 ⟨119896⟩ 1198770)

119862 = 120583 (120578 + 120573 + 120583) (120583 + 120601) (1198770minus 1)

(10)

Therefore consider the following

(1) If 1198770lt 1 we have 119862 lt 0 (9) has only one positive

solution

(2) If 1198770= 1 we have 119862 = 0 (9) has only one positive

solution minus119861119860

(3) If 1198770gt 1 we have 119862 gt 0

(i) if 119861 gt 0 the positive solution of (9) does notexist

(ii) if 119861 lt 0 and 119861 gt minus2radic119860119862 the solution of (9)does not exist

(iii) if 119861 lt 0 and 119861 = minus2radic119860119862 (9) has only onepositive solution

(iv) if 119861 lt 0 and 119861 lt minus2radic119860119862 (9) has two positivesolutions

Therefore if 1198770gt 1 the solution of (9) exists only when

119861 le minus2radic119860119862According to the inequality 119886 + 119887 ge 2radic119886119887 forall119886 119887 isin 119885

+assume 119877

0gt 1 choose 119886 = 120572⟨119896⟩120583(120583 + 120601 + 120573) 119887 = 120574⟨119896⟩(120578 +

120573 + 120583)(120583 + 120601)(1198770minus 1) and then

2radic119860119862

= 2radic120572120574 ⟨119896⟩2

120583 (120583 + 120601 + 120573) (120578 + 120573 + 120583) (120583 + 120601) (1198770minus 1)

lt 120572 ⟨119896⟩ 120583 (120583 + 120601 + 120573)

+ 120574 ⟨119896⟩ (120578 + 120573 + 120583) (120583 + 120601) (1198770minus 1)

= 120572 ⟨119896⟩ 120583 (120583 + 120601 + 120573) + 120574 ⟨119896⟩ 1198770(120578 + 120573 + 120583) (120583 + 120601)

minus 120574 ⟨119896⟩ (120578 + 120573 + 120583) (120583 + 120601)

(11)

If the inequality 120574⟨119896⟩(120578 + 120573 + 120583)(120583 + 120601) gt (120583 + 120601)[120578 + 120573 + 120583 minus120578⟨119896⟩(120572 + 120574)] is satisfied we have

120578 + 120573 + 120583 lt120578 ⟨119896⟩ (120572 + 120574)

1 minus 120574 ⟨119896⟩ (12)

Then the following equation is obtained

2radic119860119862 lt 120572 ⟨119896⟩ 120583 (120583 + 120601 + 120573)

+ 120574 ⟨119896⟩ 1198770(120578 + 120573 + 120583) (120583 + 120601)

minus 120574 ⟨119896⟩ (120578 + 120573 + 120583) (120583 + 120601)

lt 120572 ⟨119896⟩ 120583 (120583 + 120601 + 120573)

+ 120574 ⟨119896⟩ 1198770(120578 + 120573 + 120583) (120583 + 120601)

minus [120578 + 120573 + 120583 minus 120578 ⟨119896⟩ (120572 + 120574)] (120583 + 120601)

lt 120572 ⟨119896⟩ 120583 (120583 + 120601 + 120573)

+ (120578 + 120573 + 120583) (120583 + 120601) (120574 ⟨119896⟩ 1198770minus 1)

+ 120578 ⟨119896⟩ (120583 + 120601) (120572 + 120574) lt minus119861

(13)

which implies system (1) has two endemic equilibriaWhen 119877

0gt 1

120572 ⟨119896⟩

120578 + 120573 + 120583gt 1198770=

120572 ⟨119896⟩ 120575

(120575 + 120583) (120578 + 120573 + 120583)gt 1 (14)

which can transfer to inequality 120572⟨119896⟩ gt 120578 + 120573 + 120583If120572 lt 120578(120572+120574)(1minus120574⟨119896⟩) we have 1 gt 120578 gt 120572(1minus120574⟨119896⟩)(120572+

120574) Then inequality (12) is satisfied

Remark 2 According to this theorem if the reproductionnumber parameter is above the threshold then the endemicequilibrium is globally asymptotically stable which will bediscussed further in Section 5

3 Global Stability of the Disease-FreeEquilibrium

Definition 3 If the equilibrium is stable under the meaningof Lyapunov for 120575(120576 119905

0) and forall120583 gt 0 there is real number

119879(120583 120575 1199050) gt 0 which makes any initial value 119909

0of inequality

1199090minus 119909119890 le 120575(120576 119905

0) 119905 ge 119905

0 satisfy the following inequality

1003817100381710038171003817120601 (119905 1199090 1199050 minus 119909119890)1003817100381710038171003817 le 120583 forall119905 ge 119905

0+ 119879 (120583 120575 119905

0) (15)

then the equilibrium is asymptotically stable

The Jacobian matrix at the disease-free equilibrium 1198640of

system (1) is

119869 (1198640)

= (

minus120575 minus 120583 minus120575 minus 120572 ⟨119896⟩ 1198780

minus120575 minus120575 + 120601

0 120572 ⟨119896⟩ 1198780minus (120578 + 120573 + 120583) 0 0

0 120578 minus120583 0

0 120573 0 minus120583 minus 120601

)

(16)

Obviously if 1198770

lt 1 all eigenvalues of matrix (16)are negative Then the disease-free equilibrium 119864

0is locally

asymptotically stable in Σ Moreover if 1198770gt 1 there is one

positive eigenvalue and 1198640is unstable

Mathematical Problems in Engineering 5

Theorem 4 If 1198770lt 1 the disease-free equilibrium (DFE) 119864

0

is globally asymptotically stable in Σ and if 1198770gt 1 the disease-

free equilibrium (DFE) 1198640is unstable in Σ

Proof Let 119871(119878 119868 119881 119877) = 119868 gt 0 as a Lyapunov function then119871(1198640) = 0 When 119877

0lt 1

119889119871

119889119905(119878 119868 119881 119877) = 120572 ⟨119896⟩ 119878119868 minus 120574 ⟨119896⟩ 119868119881 minus (120578 + 120573 + 120583) 119868

lt 120572 ⟨119896⟩ 119878119868 minus (120578 + 120573 + 120583) 119868

lt 119868 (120572 ⟨119896⟩1198730minus (120578 + 120573 + 120583))

lt 119868 (120578 + 120573 + 120583) (1198770minus 1) lt 0

(17)

119871 is positive definite and is negative definite Therefore thedisease-free equilibrium (DFE) 119864

0is globally asymptotically

stable in Σ the following result can be given

4 Uniform Persistence

In this section the uniform persistence of system (1) will bediscussed when the basic reproduction number 119877

0gt 1

Definition 5 (see [33]) System (1) is said to be uniformly per-sistent if there exists a constant 0 lt 119888 lt 1 which makes anysolution (119878(119905) 119868(119905) 119881(119905) 119877(119905)) with (119878(0) 119868(0) 119881(0) 119877(0)) isinΣ∘ satisfy

min lim lim119905rarrinfin

119878 (119905) lim119905rarrinfin

119868 (119905) lim119905rarrinfin

119881 (119905) lim119905rarrinfin

119877 (119905)

ge 119888

(18)

Let119883 be a locally compactmetric space withmetric 120597 andlet Γ be a closed nonempty subset of119883with boundary 120597Γ andinterior Γ∘ Obviously 120597Γ is a closed subset of Γ Let Φ

119905be

a dynamical system defined on Γ A set 119861 in 119883 is said to beinvariant if Φ

119905(119861 119905) = 119861 Define 119872

120597fl 119909 isin 120597Γ Φ

119905119909 isin

120597Γ forall119905 ge 0

Lemma 6 (see [34]) Assume the following

(H1) Φ119905has a global attractor

(H2) There exists an 119872 = 1198721 119872

119896 of pair-wise

disjoint compact and isolated invariant set on 120597Γ suchthat

(a) ⋃119909isin119872120597

120596(119909) sub ⋃119896

119895=1119872119895

(b) no subsets of119872 form a cycle on 120597Γ(c) each119872

119895is also isolated in Γ

(d) 119882119904(119872119895) cap Γ∘

= 120601 for each 1 le 119895 le 119896 where119882119904

(119872119895) is the stable manifold of119872

119895 Then Φ

119905is

uniformly persistent with respect to Γ∘

According to Lemma 6 the following result is obtained

Theorem 7 When 1198770gt 1 system (1) is uniformly persistent

Proof Let

Γ = Σ = (119878 119868 119881 119877) isin R+

4| 0 le 119878 + 119868 + 119881 + 119877 le 1

Γ∘

= (119878 119868 119881 119877) isin 119864 119868 119881 gt 0

120597Γ =Γ

Γ∘

(19)

Obviously119872120597= 120597Γ

Choose 119872 = 1198640 120596(119909) = 119864

0 for all 119909 isin 119872

120597 On 120597Γ

system (1) reduces to 1198781015840 = 120575minus(120575+120583)119878 in which 119878(119905) rarr 120575(120575+

120583) as 119905 rarr infin It is concluded that119872 = 1198640 120596(119909) = 119864

0 for

all 119909 isin 119872120597 which indicates that hypotheses (a) and (b) hold

When 1198770gt 1 the disease-free equilibrium 119864

0is unstable

according to Theorem 4 119882119904(119872) = 120597Γ Hypotheses (c) and(d) are then satisfied Due to the ultimate boundedness of allsolutions of system (1) there is a global attractormaking (H1)true

5 Global Dynamics of Endemic Equilibrium

From the previous analysis the disease dies out when 1198770gt 1

then the disease becomes endemic In this section Lyapunovasymptotic stability theorem is used to investigate the globallyasymptotic stability of the endemic equilibrium 119864

lowast when1198770gt 1

Theorem8 The endemic equilibrium119864lowast is globally asymptot-

ically stable in Σ whenever 1198770gt 1

Proof Consider the following function

1198811= ln [(119878 minus 119878lowast) + (119868 minus 119868lowast) + (119881 minus 119881

lowast

) + (119877 minus 119877lowast

)

+ 1]

(20)

Then the derivative of 1198811along the solution of (1) is given by

1=1205971198811

120597119878

119889119878

119889119905+1205971198811

120597119868

119889119868

119889119905+1205971198811

120597119881

119889119881

119889119905+1205971198811

120597119877

119889119877

119889119905

=(119889119878 + 119889119868 + 119889119881 + 119889119877) (1119889119905)

(119878 minus 119878lowast) + (119868 minus 119868lowast) + (119881 minus 119881lowast) + (119877 minus 119877lowast) + 1

(21)

From (2) all solutions of (6) satisfy the equality

119873lowast

= 119878lowast

+ 119868lowast

+ 119881lowast

+ 119877lowast

=120575

120575 + 120583(22)

and 119873 = 119890minus(120575+120583)119905+119862

+ 120575(120575 + 120583) le 120575(120575 + 120583) where 119862 is thevalue that makes119873

0= 120575(120575 + 120583) satisfied

Hence 1198811= ln(119873 minus 119873

lowast

+ 1) ge 0 then

1=

1

119873 minus 120575 (120575 + 120583) + 1

119889119873

119889119905

=120575 + 120583

119873 minus 120575 (120575 + 120583) + 1(

120575

120575 + 120583minus 119873) le 0

(23)

If and only if119873 = 120575(120575 + 120583) 1198811= 0 and

1= 0 are satisfied

6 Mathematical Problems in Engineering

08

07

06

05

04

03

02

01

0050454035302520151050

S(t)I(t)

V(t)R(t)

t time step

The v

alue

s ofS

(t)I(t)

V(t

)R

(t)

(a) The states of disease-free equilibrium

0204

0608

1

002

0406

0

01

02

03

IV (08 02 0 0)IV (06 04 0 0)

IV (04 06 0 0)

minus02

minus01

R(t

)

S(t)I(t) + V(t)

(b) The evolution for the disease-free system states

Figure 2 (a) and (b) showed that the disease-free equilibrium 1198640of the system (1) is globally asymptotically stable with different initial

conditions (08 02 0 0) (06 03 01 0) and (04 035 02 005) and the parameters ⟨119896⟩ = 10 120572 = 08 120573 = 05 120578 = 02 120575 = 05 120574 = 02120601 = 001 and 120583 = 03 119877

0= 05 lt 1 The value of DFE is 119864

0= (0625 0 0 0)

1198811is positive definite and

1is negative definite There-

fore the function1198811is a Lyapunov function for system (1) and

the endemic equilibrium 119864lowast is globally asymptotically stable

by Lyapunov asymptotic stability theorem [35] The proof iscompleted

6 Numerical Simulation

To demonstrate the theoretical results obtained in this papersome numerical simulations will be discussed In this paperthe hypothetical set of initial values (IV) and parametervalues will be given as follows

Consider the initial values of (119878(0) 119868(0) 119881(0) 119877(0)) areset as (08 02 0 0) (06 03 01 0) and (04 035 02 005)respectively

(1) The disease-free equilibrium Set ⟨119896⟩ = 10 120572 = 008120573 = 05 120578 = 02 120575 = 05 120574 = 002 120601 = 001 and120583 = 03119877

0= 05 lt 1 and the disease-free equilibrium

1198640= (0625 0 0 0) from the parameter values above

through the calculation According toTheorem 4 thedisease-free equilibrium 119864

0of system (1) is globally

asymptotically stable in Σ in this case The simulationresults are shown in Figures 2(a) and 2(b)

(2) The endemic equilibrium Set ⟨119896⟩ = 10 120572 = 008120573 = 008 120578 = 03 120575 = 02 120574 = 001 120601 = 025 and120583 = 002 By direct computation 119877

0= 1818 gt 1

and the endemic equilibrium 119864lowast

= (05435 00201

03395 000596) can be obtained from the parametervalues above According to Theorem 4 the positiveendemic equilibrium 119864

lowastof system (1) is globallyasymptotically stable in Σ∘ The simulation results areshown in Figures 3(a) and 3(b)

Figure 2 shows that if 1198770lt 1 all solutions in Σ would

be attracted to the disease-free equilibrium 1198640regardless of

the initial values of system (1) which illustrates the validityof Theorem 4 Similarly it can be seen from Figure 3 that allsolutions inΣ∘ would be attracted to the endemic equilibrium119864lowast regardless of the initial values of system (1) if 119877

0gt 1 and

the conditions of Theorem 8 are satisfied which is obviouslythe content ofTheorem 4Moreover the relationship betweenthe values of equilibrium can be verified as shown in (24)which is coincident with the theoretical results

1198780+ 1198680+ 1198810+ 1198770= 119878lowast

+ 119868lowast

+ 119881lowast

+ 119877lowast

= minus120575

120575 + 120583 (24)

7 Conclusion

The stability of the 119878119868119881119877119878 epidemic spreading model withvirus variation in complex networks has been discussed inthis paper The model involves a new variant group which iscaused by the infectious variation By analyzing the modelthe disease-free equilibrium 119864

0is proved to exist when the

basic reproduction number 1198770is less than 1 The analysis

result reveals that the infectious disease dies out when 1198770is

more than 1 and it becomes endemicThe existing conditionsof endemic equilibrium related with the variation rate andthe network nodes degree are obtained Besides the globalasymptotically stability condition of the disease-free equilib-rium is obtained by the Routh-Hurwitz stability criterion andthe Lyapunov stability criterion And the condition of thesystem uniform persistence is also given The proof of thestability of endemic equilibrium is also illustrated Finallya numerical simulation is given to illustrate the correctnessof the disease-free equilibrium and the endemic equilibriumresults

Mathematical Problems in Engineering 7

08

07

06

05

04

03

02

01

0028024020016012080400

S(t)I(t)

V(t)R(t)

t time step

The v

alue

s ofS

(t)I(t)

V(t

)R

(t)

(a) The system states of endemic equilibrium

002

0406

08

0204

0608

10

002

004

006

008

IV (08 02 0 0)IV (06 04 0 0)

IV (04 06 0 0)

R(t

)

S(t)I(t) + V(t)

(b) The evolution for the endemic system states

Figure 3 (a) and (b) showed that the disease-free equilibrium119864lowast of system (1) is globally asymptotically stable with different initial conditions

(08 02 0 0) (06 03 01 0) and (04 035 02 005) and the parameter values 120572 = 008 120573 = 008 120578 = 03 120575 = 02 120574 = 01 120601 = 025120583 = 002 and ⟨119896⟩ = 10 119877

0= 1818 gt 1 The value of EE is 119864lowast = (05435 00201 03395 000596)

Conflict of Interests

The authors have declared that no competing interests exist

Acknowledgments

This research was partially supported by the National NaturalScience Foundation of China (Grant nos 61473319 61104135)the Science Fund for Creative Research Groups of NationalNatural Science of China (Grant 61321003) and the ScienceFund for Creative Research Project (2015CX007)

References

[1] R Albert and A-L Barabasi ldquoStatistical mechanics of complexnetworksrdquo Reviews of Modern Physics vol 74 no 1 pp 47ndash972002

[2] M E J Newman ldquoThe structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[3] S N Dorogovtsev A V Goltsev and J F F Mendes ldquoCriticalphenomena in complex networksrdquo Reviews of Modern Physicsvol 80 no 4 pp 1275ndash1335 2008

[4] L Zhao W Xie H O Gao X Qiu X Wang and S Zhang ldquoArumor spreading model with variable forgetting raterdquo PhysicaA vol 392 no 23 pp 6146ndash6154 2013

[5] F Chierichetti S Lattanzi andA Panconesi ldquoRumor spreadingin social networksrdquo Theoretical Computer Science vol 412 no24 pp 2602ndash2610 2011

[6] P Manshour and A Montakhab ldquoContagion spreading oncomplex networks with local deterministic dynamicsrdquoCommu-nications in Nonlinear Science andNumerical Simulation vol 19no 7 pp 2414ndash2422 2014

[7] C Liu and Z-K Zhang ldquoInformation spreading on dynamicsocial networksrdquo Communications in Nonlinear Science andNumerical Simulation vol 19 no 4 pp 896ndash904 2014

[8] K Qiao P Zhao and X-M Yao ldquoPerformance analysis ofurban rail transit networkrdquo Journal of Transportation SystemsEngineering and Information Technology vol 12 no 4 pp 115ndash121 2012

[9] O Lordan J M Sallan and P Simo ldquoStudy of the topology androbustness of airline route networks from the complex networkapproach a survey and research agendardquo Journal of TransportGeography vol 37 pp 112ndash120 2014

[10] X Ling M-B Hu W-B Du R Jiang Y-H Wu and Q-S WuldquoBandwidth allocation strategy for traffic systems of scale-freenetworkrdquoPhysics Letters A vol 374 no 48 pp 4825ndash4830 2010

[11] L-X Yang andX-F Yang ldquoThe spread of computer viruses overa reduced scale-free networkrdquo Physica A Statistical Mechanicsand Its Applications vol 396 pp 173ndash184 2014

[12] C Gan X Yang W Liu Q Zhu and X Zhang ldquoAn epidemicmodel of computer viruses with vaccination and generalizednonlinear incidence raterdquo Applied Mathematics and Computa-tion vol 222 pp 265ndash274 2013

[13] R Steuer ldquoComputational approaches to the topology stabilityand dynamics of metabolic networksrdquo Phytochemistry vol 68no 16ndash18 pp 2139ndash2151 2007

[14] A Li J Li and Y Pan ldquoDiscovering natural communities innetworksrdquo Physica A vol 436 no 15 pp 878ndash896 2015

[15] Y Maeno ldquoDiscovering network behind infectious diseaseoutbreakrdquo Physica A vol 389 no 21 pp 4755ndash4768 2010

[16] R Franke and G Ivanova ldquoFALCON or how to compute mea-sures time efficiently on dynamically evolving dense complexnetworksrdquo Journal of Biomedical Informatics vol 47 no 2 pp62ndash70 2014

[17] J-H Li X-Y Shao Y-M Long H-P Zhu and B R Sch-lessman ldquoGlobal optimization by small-world optimizationalgorithm based on social relationship networkrdquo Journal ofCentral South University vol 19 no 8 pp 2247ndash2265 2012

[18] X-H Yang B Wang and B Sun ldquoA novel weighted evolvingnetwork model based on clique overlapping growthrdquo Journal of

8 Mathematical Problems in Engineering

Central South University of Technology vol 17 no 4 pp 830ndash835 2010

[19] O Diekmann and J A P Heesterbeek Mathematical Epi-demiology of Infectious Diseases Model Building Analysis andInterpretation vol 251 John Wiley amp Sons Chichester UK2000

[20] H R Thieme Mathematics in Population Biology vol 4Princeton University Press 2003

[21] T Kuniya ldquoGlobal stability analysis with a discretizationapproach for an age-structured multigroup SIR epidemicmodelrdquoNonlinear Analysis Real World Applications vol 12 no5 pp 2640ndash2655 2011

[22] J Zhang and G Feng ldquoGlobal stability for a tuberculosis modelwith isolation and incomplete treatmentrdquo Computational andApplied Mathematics vol 34 no 3 pp 1237ndash1249 2015

[23] A Lahrouz L Omari and D Kiouach ldquoGlobal analysis of adeterministic and stochastic nonlinear SIRS epidemic modelrdquoNonlinear Analysis Modelling and Control vol 16 no 1 pp 59ndash76 2011

[24] R Xu Z Ma and Z Wang ldquoGlobal stability of a delayedSIRS epidemic model with saturation incidence and temporaryimmunityrdquoComputersampMathematics with Applications vol 59no 9 pp 3211ndash3221 2010

[25] H Kang and X Fu ldquoEpidemic spreading and global stability ofan SIS model with an infective vector on complex networksrdquoCommunications in Nonlinear Science amp Numerical Simulationvol 27 no 1ndash3 pp 30ndash39 2015

[26] P Raja Sekhara Rao and M Naresh Kumar ldquoA dynamic modelfor infectious diseases the role of vaccination and treatmentrdquoChaos Solitons amp Fractals vol 75 pp 34ndash49 2015

[27] J O Kephart S R White and D M Chess ldquoComputers andepidemiologyrdquo IEEE Spectrum vol 30 no 5 pp 20ndash26 1993

[28] Y Moreno R Pastor-Satorras and A Vespignani ldquoEpidemicoutbreaks in complex heterogeneous networksrdquo European Phys-ical Journal B vol 26 no 4 pp 521ndash529 2002

[29] R Pastor-Satorras and A Vespignani ldquoEpidemic dynamics andendemic states in complex networksrdquo Physical Review E vol 63Article ID 066117 2001

[30] R Pastor-Satorras and A Vespignani ldquoEpidemic spreading inscale-free networksrdquo Physical Review Letters vol 86 no 14 pp3200ndash3203 2001

[31] R Pastor-Satorras and A Vespignani ldquoEpidemic dynamics infinite size scale-free networksrdquo Physical Review E vol 65 no 3Article ID 035108 2002

[32] C Moore and M E J Newman ldquoEpidemics and percolationin small-world networksrdquo Physical Review E vol 61 no 5 pp5678ndash5682 2000

[33] G Butler H I Freedman and P Waltman ldquoUniformly persis-tent systemsrdquoProceedings of the AmericanMathematical Societyvol 96 no 3 pp 425ndash430 1986

[34] X-Q ZhaoDynamical Systems in Population Biology CanadianMathematical Society Springer 2003

[35] A M LyapunovThe General Problem of the Stability of MotionTaylor amp Francis London UK 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

showed the differences of spreading action between small-world network and regular network These research resultsillustrate that the different network topological structure canaffect the epidemic spreading

Apart from the network topological structure one of themost important characteristics of epidemic spreadingmodelsis the dynamical stability which can reflect the developmentof the spreading behaviors of infectious disease Hence thestability problems of these epidemic models need to beinvestigated Kuniya [21] applied a discretization methodto prove the global asymptotic stability of the 119878119868119877 modelwith the age structure Zhang and Feng [22] deal with theglobal analysis of a dynamical model describing the spreadof tuberculosis with isolation and incomplete treatmentLahrouz et al [23] studied a nonlinear 119878119868119877119878 model withsaturated birth and death rates and the global asymptoticstability of the model is also discussed Xu et al [24] analyzeda time-delayed 119878119868119877119878 model with temporary immunity andsome conditions for the globally asymptotically stability ofthe disease-free equilibrium and the endemic equilibriumare given Besides for the epidemic model with time-delayKang and Fu [25] presented a new 119878119868119878 model with aninfective vector on scale-free networks and the global stabilityof equilibrium is proved The influences of treatment andvaccination efforts on a dynamic disease model in presenceof incubation delays and relapse are studied and sufficientconditions for the local stability of the equilibriumare derived[26]

However few papers are available in the literature toconsider variant factors in the epidemic spreading from asystematic framework In real world certain variants existin the infectious disease transmitting resulting from somefactors including gene mutation and cell division environ-ment Viruses evolve rapidly because they have strong abilityof propensity for genetic variation and short generation timewhich leads to evading human immunization response andobtaining drug resistance For example influenza viruses canbe classified into three major types (A B and C) Thereare many different virus forms because of mutation typeA infects many animal species including humans whiletype B and type C viruses are mainly human pathogens Ifindividuals are affected by viruses not all infected individualscan be recovered Some of them may suffer from otherdiseases because virus variation or the infectious individualscontacted with variants In fact some infectious personswho may be infected by some diseases would have certainprobability to become variantmembers of another group It isnecessary to propose a newmodel considering this conditionHow to build models with variant factors in the epidemicsspreading becomes a challengeTherefore the paper presentsa novel 119878119868119881119877119878 epidemic spreadingmodel considering variantfactors where 119878 stands for the susceptible and 119868 119881 and119877 stand for the infectious the variant and the recoveredrespectively

Given the mechanism of the 119878119868119881119877119878 model in a homo-geneous network which is only composed by blank nodesinitially the entire population can be divided into four groupsdescribed by the symbols of 119878 119868 119881 and 119877 respectively Theydenote four epidemiological statuses susceptible infectious

variant and recovered All new individuals are supposed tobe blank nodes in complex networks When a susceptibleindividual contacts the other infected individual this indi-vidual may be infectious with certain probability Then aninfectious individual would have only three states includinginfectious variant and recovered The infectious individualswould become the variants with certain probability affectedby some factors such as genemutation and the indeterminacyof cell division Similarly an infectious person may becomea variant with certain probability after contacting with avariant Usually human body can be protected by onersquosimmune system Some infectious individuals with recoveryprobability may become the recovered while others will keepthe infectious statusWe assume that the four groups have thesame mortality rate

In this paper a novel 119878119868119881119877119878 epidemic spreading modelwith virus variation in complex homogeneous network isproposed and investigatedThe rest of this paper is organizedas follows In Section 2 the propagation mechanism ofthe 119878119868119881119877119878 model in complex networks is presented andmean-field equations are used to describe the dynamics ofepidemic spreadingmodel with virus variationThe existenceof endemic equilibrium is considered Section 3 is devotedto discuss the global stability of the disease-free equilibriumwhich is followed by the discussion of the system uniformpersistence in Section 4 Then the proofs of global stabilityof an endemic equilibrium are presented in Section 5 InSection 6 numerical simulations are performed to illustrateobtained theoretical results Finally conclusions are given inSection 7

2 Epidemic Spreading Model and Its Property

21The SIVRSModel As described above in the paper a new119878119868119881119877119878model is establishedThemodel involves a new variantgroupwhich is caused by the infectious variation Assume thenumber of nodes is 119873 in a closed complex network whichincludes four statuses susceptible 119878 infectious 119868 variant 119881and recovered 119881 as well as some initial blank nodes All newnodes produced from blank nodes are susceptible

The flow chart of epidemic spreading is shown in Figure 1Assume in a homogeneous network only composed by

blank nodes initially the susceptible individuals are producedby the blank nodes the probability is characterized as 120575Others come from the recovered group with the probability120601 A susceptible individual will become infectious withprobability 120572 if heshe contacts the infected individual Thenan infectious individual may perhaps become one variantwhen heshe has tight relation with variants or is affectedby other factors We assume that the variant probability is 120574when contacting with a variant In the process of epidemicspreading an infectious individual may become the variantwith internal probability 120578 Some infectious individuals withrecovery probability 120573 may recover and others will keep theinfectious status In this paper the four groups are supposedto have the same mortality rate 120583 Then the 119878119868119881119877119878 epidemicspreading rules can be summarized as follows

Mathematical Problems in Engineering 3

120583120583 120583

120583

120575

120573I120601R

120574IV

120572IS120578I

S I V

R

Figure 1 A schematic representation of 119878119868119881119877119878 epidemic spreadingmodel

(1) Apart from the four groups in a closed networkthere are blank nodes which exist in initial networkThe blank nodes may become susceptible ones withprobability 120575 namely crude birth rate

(2) The susceptible individual becomes infectious withprobability 120572when contacting with an infectious onenamely infection rate

(3) An infectious individual can be recovered with prob-ability 120573 namely recovery rate

(4) The variants coming from some of the infectiousnodes at a variation rate 120578 (internal variation rate)can reflect the variation factors When an infectiousindividual contacts with a variant the individual maybecome a variant with probability 120574 ( contact variantrate)

(5) The recovered node turns into susceptible with prob-ability 120601 after a period of time due to the loss ofimmunity For the four groups in the network allindividuals will become blank with probability 120583namely natural mortality rate

A closed and homogeneous network consisting of 119873individuals is investigated in this paper Individuals in thenetwork can be represented with nodes and the contactbetween different individuals can be denoted by edgesThen the network can be described by an undirected graph119866 = (119881 119864) where 119881 and 119864 denote the set of nodes andedges respectively Therefore a differential equation modelis derived based on the aforementioned rules and the basicassumptions

119889119878

119889119905= 120575 (1 minus 119873) minus 120572 ⟨119896⟩ 119878119868 + 120601119877 minus 120583119878

119889119868

119889119905= 120572 ⟨119896⟩ 119878119868 minus 120574 ⟨119896⟩ 119868119881 minus (120578 + 120573 + 120583) 119868

119889119881

119889119905= 120574 ⟨119896⟩ 119868119881 + 120578119868 minus 120583119881

119889119877

119889119905= 120573119868 minus 120601119877 minus 120583119877

(1)

where ⟨119896⟩ denotes the average degree of the network

The total population satisfies119873 = 119878 + 119868 + 119881 + 119877 and thefollowing equation is obtained

119889119873

119889119905= 120575 minus (120575 + 120583)119873 (2)

which is derived by adding the four equations in (1)In (2) 119873 will eventually tend to 119873

0= 120575(120575 + 120583) with the

exponential decay Therefore assume that 119873(0) = 1198730 The

closed and positively invariant set for (1) is Σ = (119878 119868 119881 119877) isinR4+ 0 le 119878 + 119868 + 119881 + 119877 = 119873

0le 1 where R4

+denotes the

nonnegative cone ofR4 with its lower dimensional faces Use120597Σ and Σ∘ to denote the boundary and interior of Σ in R4

+

respectively

22 Existence of Equilibrium The system (1) has a disease-free equilibrium (DFE) 119864

0 where

1198640= (1198780 1198680 1198810 1198770) = (

120575

120575 + 120583 0 0 0) (3)

Denote the basic reproduction number parameter as

1198770=120572 ⟨119896⟩ 119878

0minus 120574 ⟨119896⟩119881

0

120578 + 120573 + 120583=

120572 ⟨119896⟩ 120575

(120575 + 120583) (120578 + 120573 + 120583) (4)

The following theorem summarizes the parameter restric-tions on the existence of equilibrium

Theorem 1 If 1198770gt 1 and the inequality

1 gt 120578 gt120572 (1 minus 120574 ⟨119896⟩)

120572 + 120574(5)

is satisfied there are two endemic equilibria for system (1)

Proof Assume that 119864lowast = (119878lowast

119868lowast

119881lowast

119877lowast

) is an endemicequilibrium (EE) of system (1) According to system (1) wehave

120575 (1 minus 119873lowast

) minus 120572 ⟨119896⟩ 119878lowast

119868lowast

+ 120601119877lowast

minus 120583119878lowast

= 0

120572 ⟨119896⟩ 119878lowast

119868lowast

minus 120574 ⟨119896⟩ 119868lowast

119881lowast

minus (120578 + 120573 + 120583) 119868lowast

= 0

120574 ⟨119896⟩ 119868lowast

119881lowast

+ 120578119868lowast

minus 120583119881lowast

= 0

120573119868lowast

minus 120601119877lowast

minus 120583119877lowast

= 0

(6)

For (6) a straightforward calculation leads to

119878lowast

=120578 + 120573 + 120583

120572 ⟨119896⟩+

120574120578119868lowast

120572 (120583 minus 120574 ⟨119896⟩ 119868lowast)

119881lowast

=120578119868lowast

120583 minus 120574 ⟨119896⟩ 119868lowast

119877lowast

=120573

120583 + 120601119868lowast

(7)

From (7) 120583 minus 120574⟨119896⟩119868lowast gt 0 which implies that

0 lt 119868lowast

lt min1120583

(120574 ⟨119896⟩) (8)

4 Mathematical Problems in Engineering

the component 119868lowast is a positive solution of

119901 (119868lowast

) = 119860119868lowast2

+ 119861119868lowast

+ 119862 = 0 (9)

where

119860 = 120572120574 ⟨119896⟩2

(120583 + 120601 + 120573)

119861 = minus120578 ⟨119896⟩ (120583 + 120601) (120572 + 120574) minus 120572 ⟨119896⟩ 120583 (120583 + 120601 + 120573)

+ (120583 + 120601) (120578 + 120573 + 120583) (1 minus 120574 ⟨119896⟩ 1198770)

119862 = 120583 (120578 + 120573 + 120583) (120583 + 120601) (1198770minus 1)

(10)

Therefore consider the following

(1) If 1198770lt 1 we have 119862 lt 0 (9) has only one positive

solution

(2) If 1198770= 1 we have 119862 = 0 (9) has only one positive

solution minus119861119860

(3) If 1198770gt 1 we have 119862 gt 0

(i) if 119861 gt 0 the positive solution of (9) does notexist

(ii) if 119861 lt 0 and 119861 gt minus2radic119860119862 the solution of (9)does not exist

(iii) if 119861 lt 0 and 119861 = minus2radic119860119862 (9) has only onepositive solution

(iv) if 119861 lt 0 and 119861 lt minus2radic119860119862 (9) has two positivesolutions

Therefore if 1198770gt 1 the solution of (9) exists only when

119861 le minus2radic119860119862According to the inequality 119886 + 119887 ge 2radic119886119887 forall119886 119887 isin 119885

+assume 119877

0gt 1 choose 119886 = 120572⟨119896⟩120583(120583 + 120601 + 120573) 119887 = 120574⟨119896⟩(120578 +

120573 + 120583)(120583 + 120601)(1198770minus 1) and then

2radic119860119862

= 2radic120572120574 ⟨119896⟩2

120583 (120583 + 120601 + 120573) (120578 + 120573 + 120583) (120583 + 120601) (1198770minus 1)

lt 120572 ⟨119896⟩ 120583 (120583 + 120601 + 120573)

+ 120574 ⟨119896⟩ (120578 + 120573 + 120583) (120583 + 120601) (1198770minus 1)

= 120572 ⟨119896⟩ 120583 (120583 + 120601 + 120573) + 120574 ⟨119896⟩ 1198770(120578 + 120573 + 120583) (120583 + 120601)

minus 120574 ⟨119896⟩ (120578 + 120573 + 120583) (120583 + 120601)

(11)

If the inequality 120574⟨119896⟩(120578 + 120573 + 120583)(120583 + 120601) gt (120583 + 120601)[120578 + 120573 + 120583 minus120578⟨119896⟩(120572 + 120574)] is satisfied we have

120578 + 120573 + 120583 lt120578 ⟨119896⟩ (120572 + 120574)

1 minus 120574 ⟨119896⟩ (12)

Then the following equation is obtained

2radic119860119862 lt 120572 ⟨119896⟩ 120583 (120583 + 120601 + 120573)

+ 120574 ⟨119896⟩ 1198770(120578 + 120573 + 120583) (120583 + 120601)

minus 120574 ⟨119896⟩ (120578 + 120573 + 120583) (120583 + 120601)

lt 120572 ⟨119896⟩ 120583 (120583 + 120601 + 120573)

+ 120574 ⟨119896⟩ 1198770(120578 + 120573 + 120583) (120583 + 120601)

minus [120578 + 120573 + 120583 minus 120578 ⟨119896⟩ (120572 + 120574)] (120583 + 120601)

lt 120572 ⟨119896⟩ 120583 (120583 + 120601 + 120573)

+ (120578 + 120573 + 120583) (120583 + 120601) (120574 ⟨119896⟩ 1198770minus 1)

+ 120578 ⟨119896⟩ (120583 + 120601) (120572 + 120574) lt minus119861

(13)

which implies system (1) has two endemic equilibriaWhen 119877

0gt 1

120572 ⟨119896⟩

120578 + 120573 + 120583gt 1198770=

120572 ⟨119896⟩ 120575

(120575 + 120583) (120578 + 120573 + 120583)gt 1 (14)

which can transfer to inequality 120572⟨119896⟩ gt 120578 + 120573 + 120583If120572 lt 120578(120572+120574)(1minus120574⟨119896⟩) we have 1 gt 120578 gt 120572(1minus120574⟨119896⟩)(120572+

120574) Then inequality (12) is satisfied

Remark 2 According to this theorem if the reproductionnumber parameter is above the threshold then the endemicequilibrium is globally asymptotically stable which will bediscussed further in Section 5

3 Global Stability of the Disease-FreeEquilibrium

Definition 3 If the equilibrium is stable under the meaningof Lyapunov for 120575(120576 119905

0) and forall120583 gt 0 there is real number

119879(120583 120575 1199050) gt 0 which makes any initial value 119909

0of inequality

1199090minus 119909119890 le 120575(120576 119905

0) 119905 ge 119905

0 satisfy the following inequality

1003817100381710038171003817120601 (119905 1199090 1199050 minus 119909119890)1003817100381710038171003817 le 120583 forall119905 ge 119905

0+ 119879 (120583 120575 119905

0) (15)

then the equilibrium is asymptotically stable

The Jacobian matrix at the disease-free equilibrium 1198640of

system (1) is

119869 (1198640)

= (

minus120575 minus 120583 minus120575 minus 120572 ⟨119896⟩ 1198780

minus120575 minus120575 + 120601

0 120572 ⟨119896⟩ 1198780minus (120578 + 120573 + 120583) 0 0

0 120578 minus120583 0

0 120573 0 minus120583 minus 120601

)

(16)

Obviously if 1198770

lt 1 all eigenvalues of matrix (16)are negative Then the disease-free equilibrium 119864

0is locally

asymptotically stable in Σ Moreover if 1198770gt 1 there is one

positive eigenvalue and 1198640is unstable

Mathematical Problems in Engineering 5

Theorem 4 If 1198770lt 1 the disease-free equilibrium (DFE) 119864

0

is globally asymptotically stable in Σ and if 1198770gt 1 the disease-

free equilibrium (DFE) 1198640is unstable in Σ

Proof Let 119871(119878 119868 119881 119877) = 119868 gt 0 as a Lyapunov function then119871(1198640) = 0 When 119877

0lt 1

119889119871

119889119905(119878 119868 119881 119877) = 120572 ⟨119896⟩ 119878119868 minus 120574 ⟨119896⟩ 119868119881 minus (120578 + 120573 + 120583) 119868

lt 120572 ⟨119896⟩ 119878119868 minus (120578 + 120573 + 120583) 119868

lt 119868 (120572 ⟨119896⟩1198730minus (120578 + 120573 + 120583))

lt 119868 (120578 + 120573 + 120583) (1198770minus 1) lt 0

(17)

119871 is positive definite and is negative definite Therefore thedisease-free equilibrium (DFE) 119864

0is globally asymptotically

stable in Σ the following result can be given

4 Uniform Persistence

In this section the uniform persistence of system (1) will bediscussed when the basic reproduction number 119877

0gt 1

Definition 5 (see [33]) System (1) is said to be uniformly per-sistent if there exists a constant 0 lt 119888 lt 1 which makes anysolution (119878(119905) 119868(119905) 119881(119905) 119877(119905)) with (119878(0) 119868(0) 119881(0) 119877(0)) isinΣ∘ satisfy

min lim lim119905rarrinfin

119878 (119905) lim119905rarrinfin

119868 (119905) lim119905rarrinfin

119881 (119905) lim119905rarrinfin

119877 (119905)

ge 119888

(18)

Let119883 be a locally compactmetric space withmetric 120597 andlet Γ be a closed nonempty subset of119883with boundary 120597Γ andinterior Γ∘ Obviously 120597Γ is a closed subset of Γ Let Φ

119905be

a dynamical system defined on Γ A set 119861 in 119883 is said to beinvariant if Φ

119905(119861 119905) = 119861 Define 119872

120597fl 119909 isin 120597Γ Φ

119905119909 isin

120597Γ forall119905 ge 0

Lemma 6 (see [34]) Assume the following

(H1) Φ119905has a global attractor

(H2) There exists an 119872 = 1198721 119872

119896 of pair-wise

disjoint compact and isolated invariant set on 120597Γ suchthat

(a) ⋃119909isin119872120597

120596(119909) sub ⋃119896

119895=1119872119895

(b) no subsets of119872 form a cycle on 120597Γ(c) each119872

119895is also isolated in Γ

(d) 119882119904(119872119895) cap Γ∘

= 120601 for each 1 le 119895 le 119896 where119882119904

(119872119895) is the stable manifold of119872

119895 Then Φ

119905is

uniformly persistent with respect to Γ∘

According to Lemma 6 the following result is obtained

Theorem 7 When 1198770gt 1 system (1) is uniformly persistent

Proof Let

Γ = Σ = (119878 119868 119881 119877) isin R+

4| 0 le 119878 + 119868 + 119881 + 119877 le 1

Γ∘

= (119878 119868 119881 119877) isin 119864 119868 119881 gt 0

120597Γ =Γ

Γ∘

(19)

Obviously119872120597= 120597Γ

Choose 119872 = 1198640 120596(119909) = 119864

0 for all 119909 isin 119872

120597 On 120597Γ

system (1) reduces to 1198781015840 = 120575minus(120575+120583)119878 in which 119878(119905) rarr 120575(120575+

120583) as 119905 rarr infin It is concluded that119872 = 1198640 120596(119909) = 119864

0 for

all 119909 isin 119872120597 which indicates that hypotheses (a) and (b) hold

When 1198770gt 1 the disease-free equilibrium 119864

0is unstable

according to Theorem 4 119882119904(119872) = 120597Γ Hypotheses (c) and(d) are then satisfied Due to the ultimate boundedness of allsolutions of system (1) there is a global attractormaking (H1)true

5 Global Dynamics of Endemic Equilibrium

From the previous analysis the disease dies out when 1198770gt 1

then the disease becomes endemic In this section Lyapunovasymptotic stability theorem is used to investigate the globallyasymptotic stability of the endemic equilibrium 119864

lowast when1198770gt 1

Theorem8 The endemic equilibrium119864lowast is globally asymptot-

ically stable in Σ whenever 1198770gt 1

Proof Consider the following function

1198811= ln [(119878 minus 119878lowast) + (119868 minus 119868lowast) + (119881 minus 119881

lowast

) + (119877 minus 119877lowast

)

+ 1]

(20)

Then the derivative of 1198811along the solution of (1) is given by

1=1205971198811

120597119878

119889119878

119889119905+1205971198811

120597119868

119889119868

119889119905+1205971198811

120597119881

119889119881

119889119905+1205971198811

120597119877

119889119877

119889119905

=(119889119878 + 119889119868 + 119889119881 + 119889119877) (1119889119905)

(119878 minus 119878lowast) + (119868 minus 119868lowast) + (119881 minus 119881lowast) + (119877 minus 119877lowast) + 1

(21)

From (2) all solutions of (6) satisfy the equality

119873lowast

= 119878lowast

+ 119868lowast

+ 119881lowast

+ 119877lowast

=120575

120575 + 120583(22)

and 119873 = 119890minus(120575+120583)119905+119862

+ 120575(120575 + 120583) le 120575(120575 + 120583) where 119862 is thevalue that makes119873

0= 120575(120575 + 120583) satisfied

Hence 1198811= ln(119873 minus 119873

lowast

+ 1) ge 0 then

1=

1

119873 minus 120575 (120575 + 120583) + 1

119889119873

119889119905

=120575 + 120583

119873 minus 120575 (120575 + 120583) + 1(

120575

120575 + 120583minus 119873) le 0

(23)

If and only if119873 = 120575(120575 + 120583) 1198811= 0 and

1= 0 are satisfied

6 Mathematical Problems in Engineering

08

07

06

05

04

03

02

01

0050454035302520151050

S(t)I(t)

V(t)R(t)

t time step

The v

alue

s ofS

(t)I(t)

V(t

)R

(t)

(a) The states of disease-free equilibrium

0204

0608

1

002

0406

0

01

02

03

IV (08 02 0 0)IV (06 04 0 0)

IV (04 06 0 0)

minus02

minus01

R(t

)

S(t)I(t) + V(t)

(b) The evolution for the disease-free system states

Figure 2 (a) and (b) showed that the disease-free equilibrium 1198640of the system (1) is globally asymptotically stable with different initial

conditions (08 02 0 0) (06 03 01 0) and (04 035 02 005) and the parameters ⟨119896⟩ = 10 120572 = 08 120573 = 05 120578 = 02 120575 = 05 120574 = 02120601 = 001 and 120583 = 03 119877

0= 05 lt 1 The value of DFE is 119864

0= (0625 0 0 0)

1198811is positive definite and

1is negative definite There-

fore the function1198811is a Lyapunov function for system (1) and

the endemic equilibrium 119864lowast is globally asymptotically stable

by Lyapunov asymptotic stability theorem [35] The proof iscompleted

6 Numerical Simulation

To demonstrate the theoretical results obtained in this papersome numerical simulations will be discussed In this paperthe hypothetical set of initial values (IV) and parametervalues will be given as follows

Consider the initial values of (119878(0) 119868(0) 119881(0) 119877(0)) areset as (08 02 0 0) (06 03 01 0) and (04 035 02 005)respectively

(1) The disease-free equilibrium Set ⟨119896⟩ = 10 120572 = 008120573 = 05 120578 = 02 120575 = 05 120574 = 002 120601 = 001 and120583 = 03119877

0= 05 lt 1 and the disease-free equilibrium

1198640= (0625 0 0 0) from the parameter values above

through the calculation According toTheorem 4 thedisease-free equilibrium 119864

0of system (1) is globally

asymptotically stable in Σ in this case The simulationresults are shown in Figures 2(a) and 2(b)

(2) The endemic equilibrium Set ⟨119896⟩ = 10 120572 = 008120573 = 008 120578 = 03 120575 = 02 120574 = 001 120601 = 025 and120583 = 002 By direct computation 119877

0= 1818 gt 1

and the endemic equilibrium 119864lowast

= (05435 00201

03395 000596) can be obtained from the parametervalues above According to Theorem 4 the positiveendemic equilibrium 119864

lowastof system (1) is globallyasymptotically stable in Σ∘ The simulation results areshown in Figures 3(a) and 3(b)

Figure 2 shows that if 1198770lt 1 all solutions in Σ would

be attracted to the disease-free equilibrium 1198640regardless of

the initial values of system (1) which illustrates the validityof Theorem 4 Similarly it can be seen from Figure 3 that allsolutions inΣ∘ would be attracted to the endemic equilibrium119864lowast regardless of the initial values of system (1) if 119877

0gt 1 and

the conditions of Theorem 8 are satisfied which is obviouslythe content ofTheorem 4Moreover the relationship betweenthe values of equilibrium can be verified as shown in (24)which is coincident with the theoretical results

1198780+ 1198680+ 1198810+ 1198770= 119878lowast

+ 119868lowast

+ 119881lowast

+ 119877lowast

= minus120575

120575 + 120583 (24)

7 Conclusion

The stability of the 119878119868119881119877119878 epidemic spreading model withvirus variation in complex networks has been discussed inthis paper The model involves a new variant group which iscaused by the infectious variation By analyzing the modelthe disease-free equilibrium 119864

0is proved to exist when the

basic reproduction number 1198770is less than 1 The analysis

result reveals that the infectious disease dies out when 1198770is

more than 1 and it becomes endemicThe existing conditionsof endemic equilibrium related with the variation rate andthe network nodes degree are obtained Besides the globalasymptotically stability condition of the disease-free equilib-rium is obtained by the Routh-Hurwitz stability criterion andthe Lyapunov stability criterion And the condition of thesystem uniform persistence is also given The proof of thestability of endemic equilibrium is also illustrated Finallya numerical simulation is given to illustrate the correctnessof the disease-free equilibrium and the endemic equilibriumresults

Mathematical Problems in Engineering 7

08

07

06

05

04

03

02

01

0028024020016012080400

S(t)I(t)

V(t)R(t)

t time step

The v

alue

s ofS

(t)I(t)

V(t

)R

(t)

(a) The system states of endemic equilibrium

002

0406

08

0204

0608

10

002

004

006

008

IV (08 02 0 0)IV (06 04 0 0)

IV (04 06 0 0)

R(t

)

S(t)I(t) + V(t)

(b) The evolution for the endemic system states

Figure 3 (a) and (b) showed that the disease-free equilibrium119864lowast of system (1) is globally asymptotically stable with different initial conditions

(08 02 0 0) (06 03 01 0) and (04 035 02 005) and the parameter values 120572 = 008 120573 = 008 120578 = 03 120575 = 02 120574 = 01 120601 = 025120583 = 002 and ⟨119896⟩ = 10 119877

0= 1818 gt 1 The value of EE is 119864lowast = (05435 00201 03395 000596)

Conflict of Interests

The authors have declared that no competing interests exist

Acknowledgments

This research was partially supported by the National NaturalScience Foundation of China (Grant nos 61473319 61104135)the Science Fund for Creative Research Groups of NationalNatural Science of China (Grant 61321003) and the ScienceFund for Creative Research Project (2015CX007)

References

[1] R Albert and A-L Barabasi ldquoStatistical mechanics of complexnetworksrdquo Reviews of Modern Physics vol 74 no 1 pp 47ndash972002

[2] M E J Newman ldquoThe structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[3] S N Dorogovtsev A V Goltsev and J F F Mendes ldquoCriticalphenomena in complex networksrdquo Reviews of Modern Physicsvol 80 no 4 pp 1275ndash1335 2008

[4] L Zhao W Xie H O Gao X Qiu X Wang and S Zhang ldquoArumor spreading model with variable forgetting raterdquo PhysicaA vol 392 no 23 pp 6146ndash6154 2013

[5] F Chierichetti S Lattanzi andA Panconesi ldquoRumor spreadingin social networksrdquo Theoretical Computer Science vol 412 no24 pp 2602ndash2610 2011

[6] P Manshour and A Montakhab ldquoContagion spreading oncomplex networks with local deterministic dynamicsrdquoCommu-nications in Nonlinear Science andNumerical Simulation vol 19no 7 pp 2414ndash2422 2014

[7] C Liu and Z-K Zhang ldquoInformation spreading on dynamicsocial networksrdquo Communications in Nonlinear Science andNumerical Simulation vol 19 no 4 pp 896ndash904 2014

[8] K Qiao P Zhao and X-M Yao ldquoPerformance analysis ofurban rail transit networkrdquo Journal of Transportation SystemsEngineering and Information Technology vol 12 no 4 pp 115ndash121 2012

[9] O Lordan J M Sallan and P Simo ldquoStudy of the topology androbustness of airline route networks from the complex networkapproach a survey and research agendardquo Journal of TransportGeography vol 37 pp 112ndash120 2014

[10] X Ling M-B Hu W-B Du R Jiang Y-H Wu and Q-S WuldquoBandwidth allocation strategy for traffic systems of scale-freenetworkrdquoPhysics Letters A vol 374 no 48 pp 4825ndash4830 2010

[11] L-X Yang andX-F Yang ldquoThe spread of computer viruses overa reduced scale-free networkrdquo Physica A Statistical Mechanicsand Its Applications vol 396 pp 173ndash184 2014

[12] C Gan X Yang W Liu Q Zhu and X Zhang ldquoAn epidemicmodel of computer viruses with vaccination and generalizednonlinear incidence raterdquo Applied Mathematics and Computa-tion vol 222 pp 265ndash274 2013

[13] R Steuer ldquoComputational approaches to the topology stabilityand dynamics of metabolic networksrdquo Phytochemistry vol 68no 16ndash18 pp 2139ndash2151 2007

[14] A Li J Li and Y Pan ldquoDiscovering natural communities innetworksrdquo Physica A vol 436 no 15 pp 878ndash896 2015

[15] Y Maeno ldquoDiscovering network behind infectious diseaseoutbreakrdquo Physica A vol 389 no 21 pp 4755ndash4768 2010

[16] R Franke and G Ivanova ldquoFALCON or how to compute mea-sures time efficiently on dynamically evolving dense complexnetworksrdquo Journal of Biomedical Informatics vol 47 no 2 pp62ndash70 2014

[17] J-H Li X-Y Shao Y-M Long H-P Zhu and B R Sch-lessman ldquoGlobal optimization by small-world optimizationalgorithm based on social relationship networkrdquo Journal ofCentral South University vol 19 no 8 pp 2247ndash2265 2012

[18] X-H Yang B Wang and B Sun ldquoA novel weighted evolvingnetwork model based on clique overlapping growthrdquo Journal of

8 Mathematical Problems in Engineering

Central South University of Technology vol 17 no 4 pp 830ndash835 2010

[19] O Diekmann and J A P Heesterbeek Mathematical Epi-demiology of Infectious Diseases Model Building Analysis andInterpretation vol 251 John Wiley amp Sons Chichester UK2000

[20] H R Thieme Mathematics in Population Biology vol 4Princeton University Press 2003

[21] T Kuniya ldquoGlobal stability analysis with a discretizationapproach for an age-structured multigroup SIR epidemicmodelrdquoNonlinear Analysis Real World Applications vol 12 no5 pp 2640ndash2655 2011

[22] J Zhang and G Feng ldquoGlobal stability for a tuberculosis modelwith isolation and incomplete treatmentrdquo Computational andApplied Mathematics vol 34 no 3 pp 1237ndash1249 2015

[23] A Lahrouz L Omari and D Kiouach ldquoGlobal analysis of adeterministic and stochastic nonlinear SIRS epidemic modelrdquoNonlinear Analysis Modelling and Control vol 16 no 1 pp 59ndash76 2011

[24] R Xu Z Ma and Z Wang ldquoGlobal stability of a delayedSIRS epidemic model with saturation incidence and temporaryimmunityrdquoComputersampMathematics with Applications vol 59no 9 pp 3211ndash3221 2010

[25] H Kang and X Fu ldquoEpidemic spreading and global stability ofan SIS model with an infective vector on complex networksrdquoCommunications in Nonlinear Science amp Numerical Simulationvol 27 no 1ndash3 pp 30ndash39 2015

[26] P Raja Sekhara Rao and M Naresh Kumar ldquoA dynamic modelfor infectious diseases the role of vaccination and treatmentrdquoChaos Solitons amp Fractals vol 75 pp 34ndash49 2015

[27] J O Kephart S R White and D M Chess ldquoComputers andepidemiologyrdquo IEEE Spectrum vol 30 no 5 pp 20ndash26 1993

[28] Y Moreno R Pastor-Satorras and A Vespignani ldquoEpidemicoutbreaks in complex heterogeneous networksrdquo European Phys-ical Journal B vol 26 no 4 pp 521ndash529 2002

[29] R Pastor-Satorras and A Vespignani ldquoEpidemic dynamics andendemic states in complex networksrdquo Physical Review E vol 63Article ID 066117 2001

[30] R Pastor-Satorras and A Vespignani ldquoEpidemic spreading inscale-free networksrdquo Physical Review Letters vol 86 no 14 pp3200ndash3203 2001

[31] R Pastor-Satorras and A Vespignani ldquoEpidemic dynamics infinite size scale-free networksrdquo Physical Review E vol 65 no 3Article ID 035108 2002

[32] C Moore and M E J Newman ldquoEpidemics and percolationin small-world networksrdquo Physical Review E vol 61 no 5 pp5678ndash5682 2000

[33] G Butler H I Freedman and P Waltman ldquoUniformly persis-tent systemsrdquoProceedings of the AmericanMathematical Societyvol 96 no 3 pp 425ndash430 1986

[34] X-Q ZhaoDynamical Systems in Population Biology CanadianMathematical Society Springer 2003

[35] A M LyapunovThe General Problem of the Stability of MotionTaylor amp Francis London UK 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

120583120583 120583

120583

120575

120573I120601R

120574IV

120572IS120578I

S I V

R

Figure 1 A schematic representation of 119878119868119881119877119878 epidemic spreadingmodel

(1) Apart from the four groups in a closed networkthere are blank nodes which exist in initial networkThe blank nodes may become susceptible ones withprobability 120575 namely crude birth rate

(2) The susceptible individual becomes infectious withprobability 120572when contacting with an infectious onenamely infection rate

(3) An infectious individual can be recovered with prob-ability 120573 namely recovery rate

(4) The variants coming from some of the infectiousnodes at a variation rate 120578 (internal variation rate)can reflect the variation factors When an infectiousindividual contacts with a variant the individual maybecome a variant with probability 120574 ( contact variantrate)

(5) The recovered node turns into susceptible with prob-ability 120601 after a period of time due to the loss ofimmunity For the four groups in the network allindividuals will become blank with probability 120583namely natural mortality rate

A closed and homogeneous network consisting of 119873individuals is investigated in this paper Individuals in thenetwork can be represented with nodes and the contactbetween different individuals can be denoted by edgesThen the network can be described by an undirected graph119866 = (119881 119864) where 119881 and 119864 denote the set of nodes andedges respectively Therefore a differential equation modelis derived based on the aforementioned rules and the basicassumptions

119889119878

119889119905= 120575 (1 minus 119873) minus 120572 ⟨119896⟩ 119878119868 + 120601119877 minus 120583119878

119889119868

119889119905= 120572 ⟨119896⟩ 119878119868 minus 120574 ⟨119896⟩ 119868119881 minus (120578 + 120573 + 120583) 119868

119889119881

119889119905= 120574 ⟨119896⟩ 119868119881 + 120578119868 minus 120583119881

119889119877

119889119905= 120573119868 minus 120601119877 minus 120583119877

(1)

where ⟨119896⟩ denotes the average degree of the network

The total population satisfies119873 = 119878 + 119868 + 119881 + 119877 and thefollowing equation is obtained

119889119873

119889119905= 120575 minus (120575 + 120583)119873 (2)

which is derived by adding the four equations in (1)In (2) 119873 will eventually tend to 119873

0= 120575(120575 + 120583) with the

exponential decay Therefore assume that 119873(0) = 1198730 The

closed and positively invariant set for (1) is Σ = (119878 119868 119881 119877) isinR4+ 0 le 119878 + 119868 + 119881 + 119877 = 119873

0le 1 where R4

+denotes the

nonnegative cone ofR4 with its lower dimensional faces Use120597Σ and Σ∘ to denote the boundary and interior of Σ in R4

+

respectively

22 Existence of Equilibrium The system (1) has a disease-free equilibrium (DFE) 119864

0 where

1198640= (1198780 1198680 1198810 1198770) = (

120575

120575 + 120583 0 0 0) (3)

Denote the basic reproduction number parameter as

1198770=120572 ⟨119896⟩ 119878

0minus 120574 ⟨119896⟩119881

0

120578 + 120573 + 120583=

120572 ⟨119896⟩ 120575

(120575 + 120583) (120578 + 120573 + 120583) (4)

The following theorem summarizes the parameter restric-tions on the existence of equilibrium

Theorem 1 If 1198770gt 1 and the inequality

1 gt 120578 gt120572 (1 minus 120574 ⟨119896⟩)

120572 + 120574(5)

is satisfied there are two endemic equilibria for system (1)

Proof Assume that 119864lowast = (119878lowast

119868lowast

119881lowast

119877lowast

) is an endemicequilibrium (EE) of system (1) According to system (1) wehave

120575 (1 minus 119873lowast

) minus 120572 ⟨119896⟩ 119878lowast

119868lowast

+ 120601119877lowast

minus 120583119878lowast

= 0

120572 ⟨119896⟩ 119878lowast

119868lowast

minus 120574 ⟨119896⟩ 119868lowast

119881lowast

minus (120578 + 120573 + 120583) 119868lowast

= 0

120574 ⟨119896⟩ 119868lowast

119881lowast

+ 120578119868lowast

minus 120583119881lowast

= 0

120573119868lowast

minus 120601119877lowast

minus 120583119877lowast

= 0

(6)

For (6) a straightforward calculation leads to

119878lowast

=120578 + 120573 + 120583

120572 ⟨119896⟩+

120574120578119868lowast

120572 (120583 minus 120574 ⟨119896⟩ 119868lowast)

119881lowast

=120578119868lowast

120583 minus 120574 ⟨119896⟩ 119868lowast

119877lowast

=120573

120583 + 120601119868lowast

(7)

From (7) 120583 minus 120574⟨119896⟩119868lowast gt 0 which implies that

0 lt 119868lowast

lt min1120583

(120574 ⟨119896⟩) (8)

4 Mathematical Problems in Engineering

the component 119868lowast is a positive solution of

119901 (119868lowast

) = 119860119868lowast2

+ 119861119868lowast

+ 119862 = 0 (9)

where

119860 = 120572120574 ⟨119896⟩2

(120583 + 120601 + 120573)

119861 = minus120578 ⟨119896⟩ (120583 + 120601) (120572 + 120574) minus 120572 ⟨119896⟩ 120583 (120583 + 120601 + 120573)

+ (120583 + 120601) (120578 + 120573 + 120583) (1 minus 120574 ⟨119896⟩ 1198770)

119862 = 120583 (120578 + 120573 + 120583) (120583 + 120601) (1198770minus 1)

(10)

Therefore consider the following

(1) If 1198770lt 1 we have 119862 lt 0 (9) has only one positive

solution

(2) If 1198770= 1 we have 119862 = 0 (9) has only one positive

solution minus119861119860

(3) If 1198770gt 1 we have 119862 gt 0

(i) if 119861 gt 0 the positive solution of (9) does notexist

(ii) if 119861 lt 0 and 119861 gt minus2radic119860119862 the solution of (9)does not exist

(iii) if 119861 lt 0 and 119861 = minus2radic119860119862 (9) has only onepositive solution

(iv) if 119861 lt 0 and 119861 lt minus2radic119860119862 (9) has two positivesolutions

Therefore if 1198770gt 1 the solution of (9) exists only when

119861 le minus2radic119860119862According to the inequality 119886 + 119887 ge 2radic119886119887 forall119886 119887 isin 119885

+assume 119877

0gt 1 choose 119886 = 120572⟨119896⟩120583(120583 + 120601 + 120573) 119887 = 120574⟨119896⟩(120578 +

120573 + 120583)(120583 + 120601)(1198770minus 1) and then

2radic119860119862

= 2radic120572120574 ⟨119896⟩2

120583 (120583 + 120601 + 120573) (120578 + 120573 + 120583) (120583 + 120601) (1198770minus 1)

lt 120572 ⟨119896⟩ 120583 (120583 + 120601 + 120573)

+ 120574 ⟨119896⟩ (120578 + 120573 + 120583) (120583 + 120601) (1198770minus 1)

= 120572 ⟨119896⟩ 120583 (120583 + 120601 + 120573) + 120574 ⟨119896⟩ 1198770(120578 + 120573 + 120583) (120583 + 120601)

minus 120574 ⟨119896⟩ (120578 + 120573 + 120583) (120583 + 120601)

(11)

If the inequality 120574⟨119896⟩(120578 + 120573 + 120583)(120583 + 120601) gt (120583 + 120601)[120578 + 120573 + 120583 minus120578⟨119896⟩(120572 + 120574)] is satisfied we have

120578 + 120573 + 120583 lt120578 ⟨119896⟩ (120572 + 120574)

1 minus 120574 ⟨119896⟩ (12)

Then the following equation is obtained

2radic119860119862 lt 120572 ⟨119896⟩ 120583 (120583 + 120601 + 120573)

+ 120574 ⟨119896⟩ 1198770(120578 + 120573 + 120583) (120583 + 120601)

minus 120574 ⟨119896⟩ (120578 + 120573 + 120583) (120583 + 120601)

lt 120572 ⟨119896⟩ 120583 (120583 + 120601 + 120573)

+ 120574 ⟨119896⟩ 1198770(120578 + 120573 + 120583) (120583 + 120601)

minus [120578 + 120573 + 120583 minus 120578 ⟨119896⟩ (120572 + 120574)] (120583 + 120601)

lt 120572 ⟨119896⟩ 120583 (120583 + 120601 + 120573)

+ (120578 + 120573 + 120583) (120583 + 120601) (120574 ⟨119896⟩ 1198770minus 1)

+ 120578 ⟨119896⟩ (120583 + 120601) (120572 + 120574) lt minus119861

(13)

which implies system (1) has two endemic equilibriaWhen 119877

0gt 1

120572 ⟨119896⟩

120578 + 120573 + 120583gt 1198770=

120572 ⟨119896⟩ 120575

(120575 + 120583) (120578 + 120573 + 120583)gt 1 (14)

which can transfer to inequality 120572⟨119896⟩ gt 120578 + 120573 + 120583If120572 lt 120578(120572+120574)(1minus120574⟨119896⟩) we have 1 gt 120578 gt 120572(1minus120574⟨119896⟩)(120572+

120574) Then inequality (12) is satisfied

Remark 2 According to this theorem if the reproductionnumber parameter is above the threshold then the endemicequilibrium is globally asymptotically stable which will bediscussed further in Section 5

3 Global Stability of the Disease-FreeEquilibrium

Definition 3 If the equilibrium is stable under the meaningof Lyapunov for 120575(120576 119905

0) and forall120583 gt 0 there is real number

119879(120583 120575 1199050) gt 0 which makes any initial value 119909

0of inequality

1199090minus 119909119890 le 120575(120576 119905

0) 119905 ge 119905

0 satisfy the following inequality

1003817100381710038171003817120601 (119905 1199090 1199050 minus 119909119890)1003817100381710038171003817 le 120583 forall119905 ge 119905

0+ 119879 (120583 120575 119905

0) (15)

then the equilibrium is asymptotically stable

The Jacobian matrix at the disease-free equilibrium 1198640of

system (1) is

119869 (1198640)

= (

minus120575 minus 120583 minus120575 minus 120572 ⟨119896⟩ 1198780

minus120575 minus120575 + 120601

0 120572 ⟨119896⟩ 1198780minus (120578 + 120573 + 120583) 0 0

0 120578 minus120583 0

0 120573 0 minus120583 minus 120601

)

(16)

Obviously if 1198770

lt 1 all eigenvalues of matrix (16)are negative Then the disease-free equilibrium 119864

0is locally

asymptotically stable in Σ Moreover if 1198770gt 1 there is one

positive eigenvalue and 1198640is unstable

Mathematical Problems in Engineering 5

Theorem 4 If 1198770lt 1 the disease-free equilibrium (DFE) 119864

0

is globally asymptotically stable in Σ and if 1198770gt 1 the disease-

free equilibrium (DFE) 1198640is unstable in Σ

Proof Let 119871(119878 119868 119881 119877) = 119868 gt 0 as a Lyapunov function then119871(1198640) = 0 When 119877

0lt 1

119889119871

119889119905(119878 119868 119881 119877) = 120572 ⟨119896⟩ 119878119868 minus 120574 ⟨119896⟩ 119868119881 minus (120578 + 120573 + 120583) 119868

lt 120572 ⟨119896⟩ 119878119868 minus (120578 + 120573 + 120583) 119868

lt 119868 (120572 ⟨119896⟩1198730minus (120578 + 120573 + 120583))

lt 119868 (120578 + 120573 + 120583) (1198770minus 1) lt 0

(17)

119871 is positive definite and is negative definite Therefore thedisease-free equilibrium (DFE) 119864

0is globally asymptotically

stable in Σ the following result can be given

4 Uniform Persistence

In this section the uniform persistence of system (1) will bediscussed when the basic reproduction number 119877

0gt 1

Definition 5 (see [33]) System (1) is said to be uniformly per-sistent if there exists a constant 0 lt 119888 lt 1 which makes anysolution (119878(119905) 119868(119905) 119881(119905) 119877(119905)) with (119878(0) 119868(0) 119881(0) 119877(0)) isinΣ∘ satisfy

min lim lim119905rarrinfin

119878 (119905) lim119905rarrinfin

119868 (119905) lim119905rarrinfin

119881 (119905) lim119905rarrinfin

119877 (119905)

ge 119888

(18)

Let119883 be a locally compactmetric space withmetric 120597 andlet Γ be a closed nonempty subset of119883with boundary 120597Γ andinterior Γ∘ Obviously 120597Γ is a closed subset of Γ Let Φ

119905be

a dynamical system defined on Γ A set 119861 in 119883 is said to beinvariant if Φ

119905(119861 119905) = 119861 Define 119872

120597fl 119909 isin 120597Γ Φ

119905119909 isin

120597Γ forall119905 ge 0

Lemma 6 (see [34]) Assume the following

(H1) Φ119905has a global attractor

(H2) There exists an 119872 = 1198721 119872

119896 of pair-wise

disjoint compact and isolated invariant set on 120597Γ suchthat

(a) ⋃119909isin119872120597

120596(119909) sub ⋃119896

119895=1119872119895

(b) no subsets of119872 form a cycle on 120597Γ(c) each119872

119895is also isolated in Γ

(d) 119882119904(119872119895) cap Γ∘

= 120601 for each 1 le 119895 le 119896 where119882119904

(119872119895) is the stable manifold of119872

119895 Then Φ

119905is

uniformly persistent with respect to Γ∘

According to Lemma 6 the following result is obtained

Theorem 7 When 1198770gt 1 system (1) is uniformly persistent

Proof Let

Γ = Σ = (119878 119868 119881 119877) isin R+

4| 0 le 119878 + 119868 + 119881 + 119877 le 1

Γ∘

= (119878 119868 119881 119877) isin 119864 119868 119881 gt 0

120597Γ =Γ

Γ∘

(19)

Obviously119872120597= 120597Γ

Choose 119872 = 1198640 120596(119909) = 119864

0 for all 119909 isin 119872

120597 On 120597Γ

system (1) reduces to 1198781015840 = 120575minus(120575+120583)119878 in which 119878(119905) rarr 120575(120575+

120583) as 119905 rarr infin It is concluded that119872 = 1198640 120596(119909) = 119864

0 for

all 119909 isin 119872120597 which indicates that hypotheses (a) and (b) hold

When 1198770gt 1 the disease-free equilibrium 119864

0is unstable

according to Theorem 4 119882119904(119872) = 120597Γ Hypotheses (c) and(d) are then satisfied Due to the ultimate boundedness of allsolutions of system (1) there is a global attractormaking (H1)true

5 Global Dynamics of Endemic Equilibrium

From the previous analysis the disease dies out when 1198770gt 1

then the disease becomes endemic In this section Lyapunovasymptotic stability theorem is used to investigate the globallyasymptotic stability of the endemic equilibrium 119864

lowast when1198770gt 1

Theorem8 The endemic equilibrium119864lowast is globally asymptot-

ically stable in Σ whenever 1198770gt 1

Proof Consider the following function

1198811= ln [(119878 minus 119878lowast) + (119868 minus 119868lowast) + (119881 minus 119881

lowast

) + (119877 minus 119877lowast

)

+ 1]

(20)

Then the derivative of 1198811along the solution of (1) is given by

1=1205971198811

120597119878

119889119878

119889119905+1205971198811

120597119868

119889119868

119889119905+1205971198811

120597119881

119889119881

119889119905+1205971198811

120597119877

119889119877

119889119905

=(119889119878 + 119889119868 + 119889119881 + 119889119877) (1119889119905)

(119878 minus 119878lowast) + (119868 minus 119868lowast) + (119881 minus 119881lowast) + (119877 minus 119877lowast) + 1

(21)

From (2) all solutions of (6) satisfy the equality

119873lowast

= 119878lowast

+ 119868lowast

+ 119881lowast

+ 119877lowast

=120575

120575 + 120583(22)

and 119873 = 119890minus(120575+120583)119905+119862

+ 120575(120575 + 120583) le 120575(120575 + 120583) where 119862 is thevalue that makes119873

0= 120575(120575 + 120583) satisfied

Hence 1198811= ln(119873 minus 119873

lowast

+ 1) ge 0 then

1=

1

119873 minus 120575 (120575 + 120583) + 1

119889119873

119889119905

=120575 + 120583

119873 minus 120575 (120575 + 120583) + 1(

120575

120575 + 120583minus 119873) le 0

(23)

If and only if119873 = 120575(120575 + 120583) 1198811= 0 and

1= 0 are satisfied

6 Mathematical Problems in Engineering

08

07

06

05

04

03

02

01

0050454035302520151050

S(t)I(t)

V(t)R(t)

t time step

The v

alue

s ofS

(t)I(t)

V(t

)R

(t)

(a) The states of disease-free equilibrium

0204

0608

1

002

0406

0

01

02

03

IV (08 02 0 0)IV (06 04 0 0)

IV (04 06 0 0)

minus02

minus01

R(t

)

S(t)I(t) + V(t)

(b) The evolution for the disease-free system states

Figure 2 (a) and (b) showed that the disease-free equilibrium 1198640of the system (1) is globally asymptotically stable with different initial

conditions (08 02 0 0) (06 03 01 0) and (04 035 02 005) and the parameters ⟨119896⟩ = 10 120572 = 08 120573 = 05 120578 = 02 120575 = 05 120574 = 02120601 = 001 and 120583 = 03 119877

0= 05 lt 1 The value of DFE is 119864

0= (0625 0 0 0)

1198811is positive definite and

1is negative definite There-

fore the function1198811is a Lyapunov function for system (1) and

the endemic equilibrium 119864lowast is globally asymptotically stable

by Lyapunov asymptotic stability theorem [35] The proof iscompleted

6 Numerical Simulation

To demonstrate the theoretical results obtained in this papersome numerical simulations will be discussed In this paperthe hypothetical set of initial values (IV) and parametervalues will be given as follows

Consider the initial values of (119878(0) 119868(0) 119881(0) 119877(0)) areset as (08 02 0 0) (06 03 01 0) and (04 035 02 005)respectively

(1) The disease-free equilibrium Set ⟨119896⟩ = 10 120572 = 008120573 = 05 120578 = 02 120575 = 05 120574 = 002 120601 = 001 and120583 = 03119877

0= 05 lt 1 and the disease-free equilibrium

1198640= (0625 0 0 0) from the parameter values above

through the calculation According toTheorem 4 thedisease-free equilibrium 119864

0of system (1) is globally

asymptotically stable in Σ in this case The simulationresults are shown in Figures 2(a) and 2(b)

(2) The endemic equilibrium Set ⟨119896⟩ = 10 120572 = 008120573 = 008 120578 = 03 120575 = 02 120574 = 001 120601 = 025 and120583 = 002 By direct computation 119877

0= 1818 gt 1

and the endemic equilibrium 119864lowast

= (05435 00201

03395 000596) can be obtained from the parametervalues above According to Theorem 4 the positiveendemic equilibrium 119864

lowastof system (1) is globallyasymptotically stable in Σ∘ The simulation results areshown in Figures 3(a) and 3(b)

Figure 2 shows that if 1198770lt 1 all solutions in Σ would

be attracted to the disease-free equilibrium 1198640regardless of

the initial values of system (1) which illustrates the validityof Theorem 4 Similarly it can be seen from Figure 3 that allsolutions inΣ∘ would be attracted to the endemic equilibrium119864lowast regardless of the initial values of system (1) if 119877

0gt 1 and

the conditions of Theorem 8 are satisfied which is obviouslythe content ofTheorem 4Moreover the relationship betweenthe values of equilibrium can be verified as shown in (24)which is coincident with the theoretical results

1198780+ 1198680+ 1198810+ 1198770= 119878lowast

+ 119868lowast

+ 119881lowast

+ 119877lowast

= minus120575

120575 + 120583 (24)

7 Conclusion

The stability of the 119878119868119881119877119878 epidemic spreading model withvirus variation in complex networks has been discussed inthis paper The model involves a new variant group which iscaused by the infectious variation By analyzing the modelthe disease-free equilibrium 119864

0is proved to exist when the

basic reproduction number 1198770is less than 1 The analysis

result reveals that the infectious disease dies out when 1198770is

more than 1 and it becomes endemicThe existing conditionsof endemic equilibrium related with the variation rate andthe network nodes degree are obtained Besides the globalasymptotically stability condition of the disease-free equilib-rium is obtained by the Routh-Hurwitz stability criterion andthe Lyapunov stability criterion And the condition of thesystem uniform persistence is also given The proof of thestability of endemic equilibrium is also illustrated Finallya numerical simulation is given to illustrate the correctnessof the disease-free equilibrium and the endemic equilibriumresults

Mathematical Problems in Engineering 7

08

07

06

05

04

03

02

01

0028024020016012080400

S(t)I(t)

V(t)R(t)

t time step

The v

alue

s ofS

(t)I(t)

V(t

)R

(t)

(a) The system states of endemic equilibrium

002

0406

08

0204

0608

10

002

004

006

008

IV (08 02 0 0)IV (06 04 0 0)

IV (04 06 0 0)

R(t

)

S(t)I(t) + V(t)

(b) The evolution for the endemic system states

Figure 3 (a) and (b) showed that the disease-free equilibrium119864lowast of system (1) is globally asymptotically stable with different initial conditions

(08 02 0 0) (06 03 01 0) and (04 035 02 005) and the parameter values 120572 = 008 120573 = 008 120578 = 03 120575 = 02 120574 = 01 120601 = 025120583 = 002 and ⟨119896⟩ = 10 119877

0= 1818 gt 1 The value of EE is 119864lowast = (05435 00201 03395 000596)

Conflict of Interests

The authors have declared that no competing interests exist

Acknowledgments

This research was partially supported by the National NaturalScience Foundation of China (Grant nos 61473319 61104135)the Science Fund for Creative Research Groups of NationalNatural Science of China (Grant 61321003) and the ScienceFund for Creative Research Project (2015CX007)

References

[1] R Albert and A-L Barabasi ldquoStatistical mechanics of complexnetworksrdquo Reviews of Modern Physics vol 74 no 1 pp 47ndash972002

[2] M E J Newman ldquoThe structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[3] S N Dorogovtsev A V Goltsev and J F F Mendes ldquoCriticalphenomena in complex networksrdquo Reviews of Modern Physicsvol 80 no 4 pp 1275ndash1335 2008

[4] L Zhao W Xie H O Gao X Qiu X Wang and S Zhang ldquoArumor spreading model with variable forgetting raterdquo PhysicaA vol 392 no 23 pp 6146ndash6154 2013

[5] F Chierichetti S Lattanzi andA Panconesi ldquoRumor spreadingin social networksrdquo Theoretical Computer Science vol 412 no24 pp 2602ndash2610 2011

[6] P Manshour and A Montakhab ldquoContagion spreading oncomplex networks with local deterministic dynamicsrdquoCommu-nications in Nonlinear Science andNumerical Simulation vol 19no 7 pp 2414ndash2422 2014

[7] C Liu and Z-K Zhang ldquoInformation spreading on dynamicsocial networksrdquo Communications in Nonlinear Science andNumerical Simulation vol 19 no 4 pp 896ndash904 2014

[8] K Qiao P Zhao and X-M Yao ldquoPerformance analysis ofurban rail transit networkrdquo Journal of Transportation SystemsEngineering and Information Technology vol 12 no 4 pp 115ndash121 2012

[9] O Lordan J M Sallan and P Simo ldquoStudy of the topology androbustness of airline route networks from the complex networkapproach a survey and research agendardquo Journal of TransportGeography vol 37 pp 112ndash120 2014

[10] X Ling M-B Hu W-B Du R Jiang Y-H Wu and Q-S WuldquoBandwidth allocation strategy for traffic systems of scale-freenetworkrdquoPhysics Letters A vol 374 no 48 pp 4825ndash4830 2010

[11] L-X Yang andX-F Yang ldquoThe spread of computer viruses overa reduced scale-free networkrdquo Physica A Statistical Mechanicsand Its Applications vol 396 pp 173ndash184 2014

[12] C Gan X Yang W Liu Q Zhu and X Zhang ldquoAn epidemicmodel of computer viruses with vaccination and generalizednonlinear incidence raterdquo Applied Mathematics and Computa-tion vol 222 pp 265ndash274 2013

[13] R Steuer ldquoComputational approaches to the topology stabilityand dynamics of metabolic networksrdquo Phytochemistry vol 68no 16ndash18 pp 2139ndash2151 2007

[14] A Li J Li and Y Pan ldquoDiscovering natural communities innetworksrdquo Physica A vol 436 no 15 pp 878ndash896 2015

[15] Y Maeno ldquoDiscovering network behind infectious diseaseoutbreakrdquo Physica A vol 389 no 21 pp 4755ndash4768 2010

[16] R Franke and G Ivanova ldquoFALCON or how to compute mea-sures time efficiently on dynamically evolving dense complexnetworksrdquo Journal of Biomedical Informatics vol 47 no 2 pp62ndash70 2014

[17] J-H Li X-Y Shao Y-M Long H-P Zhu and B R Sch-lessman ldquoGlobal optimization by small-world optimizationalgorithm based on social relationship networkrdquo Journal ofCentral South University vol 19 no 8 pp 2247ndash2265 2012

[18] X-H Yang B Wang and B Sun ldquoA novel weighted evolvingnetwork model based on clique overlapping growthrdquo Journal of

8 Mathematical Problems in Engineering

Central South University of Technology vol 17 no 4 pp 830ndash835 2010

[19] O Diekmann and J A P Heesterbeek Mathematical Epi-demiology of Infectious Diseases Model Building Analysis andInterpretation vol 251 John Wiley amp Sons Chichester UK2000

[20] H R Thieme Mathematics in Population Biology vol 4Princeton University Press 2003

[21] T Kuniya ldquoGlobal stability analysis with a discretizationapproach for an age-structured multigroup SIR epidemicmodelrdquoNonlinear Analysis Real World Applications vol 12 no5 pp 2640ndash2655 2011

[22] J Zhang and G Feng ldquoGlobal stability for a tuberculosis modelwith isolation and incomplete treatmentrdquo Computational andApplied Mathematics vol 34 no 3 pp 1237ndash1249 2015

[23] A Lahrouz L Omari and D Kiouach ldquoGlobal analysis of adeterministic and stochastic nonlinear SIRS epidemic modelrdquoNonlinear Analysis Modelling and Control vol 16 no 1 pp 59ndash76 2011

[24] R Xu Z Ma and Z Wang ldquoGlobal stability of a delayedSIRS epidemic model with saturation incidence and temporaryimmunityrdquoComputersampMathematics with Applications vol 59no 9 pp 3211ndash3221 2010

[25] H Kang and X Fu ldquoEpidemic spreading and global stability ofan SIS model with an infective vector on complex networksrdquoCommunications in Nonlinear Science amp Numerical Simulationvol 27 no 1ndash3 pp 30ndash39 2015

[26] P Raja Sekhara Rao and M Naresh Kumar ldquoA dynamic modelfor infectious diseases the role of vaccination and treatmentrdquoChaos Solitons amp Fractals vol 75 pp 34ndash49 2015

[27] J O Kephart S R White and D M Chess ldquoComputers andepidemiologyrdquo IEEE Spectrum vol 30 no 5 pp 20ndash26 1993

[28] Y Moreno R Pastor-Satorras and A Vespignani ldquoEpidemicoutbreaks in complex heterogeneous networksrdquo European Phys-ical Journal B vol 26 no 4 pp 521ndash529 2002

[29] R Pastor-Satorras and A Vespignani ldquoEpidemic dynamics andendemic states in complex networksrdquo Physical Review E vol 63Article ID 066117 2001

[30] R Pastor-Satorras and A Vespignani ldquoEpidemic spreading inscale-free networksrdquo Physical Review Letters vol 86 no 14 pp3200ndash3203 2001

[31] R Pastor-Satorras and A Vespignani ldquoEpidemic dynamics infinite size scale-free networksrdquo Physical Review E vol 65 no 3Article ID 035108 2002

[32] C Moore and M E J Newman ldquoEpidemics and percolationin small-world networksrdquo Physical Review E vol 61 no 5 pp5678ndash5682 2000

[33] G Butler H I Freedman and P Waltman ldquoUniformly persis-tent systemsrdquoProceedings of the AmericanMathematical Societyvol 96 no 3 pp 425ndash430 1986

[34] X-Q ZhaoDynamical Systems in Population Biology CanadianMathematical Society Springer 2003

[35] A M LyapunovThe General Problem of the Stability of MotionTaylor amp Francis London UK 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

the component 119868lowast is a positive solution of

119901 (119868lowast

) = 119860119868lowast2

+ 119861119868lowast

+ 119862 = 0 (9)

where

119860 = 120572120574 ⟨119896⟩2

(120583 + 120601 + 120573)

119861 = minus120578 ⟨119896⟩ (120583 + 120601) (120572 + 120574) minus 120572 ⟨119896⟩ 120583 (120583 + 120601 + 120573)

+ (120583 + 120601) (120578 + 120573 + 120583) (1 minus 120574 ⟨119896⟩ 1198770)

119862 = 120583 (120578 + 120573 + 120583) (120583 + 120601) (1198770minus 1)

(10)

Therefore consider the following

(1) If 1198770lt 1 we have 119862 lt 0 (9) has only one positive

solution

(2) If 1198770= 1 we have 119862 = 0 (9) has only one positive

solution minus119861119860

(3) If 1198770gt 1 we have 119862 gt 0

(i) if 119861 gt 0 the positive solution of (9) does notexist

(ii) if 119861 lt 0 and 119861 gt minus2radic119860119862 the solution of (9)does not exist

(iii) if 119861 lt 0 and 119861 = minus2radic119860119862 (9) has only onepositive solution

(iv) if 119861 lt 0 and 119861 lt minus2radic119860119862 (9) has two positivesolutions

Therefore if 1198770gt 1 the solution of (9) exists only when

119861 le minus2radic119860119862According to the inequality 119886 + 119887 ge 2radic119886119887 forall119886 119887 isin 119885

+assume 119877

0gt 1 choose 119886 = 120572⟨119896⟩120583(120583 + 120601 + 120573) 119887 = 120574⟨119896⟩(120578 +

120573 + 120583)(120583 + 120601)(1198770minus 1) and then

2radic119860119862

= 2radic120572120574 ⟨119896⟩2

120583 (120583 + 120601 + 120573) (120578 + 120573 + 120583) (120583 + 120601) (1198770minus 1)

lt 120572 ⟨119896⟩ 120583 (120583 + 120601 + 120573)

+ 120574 ⟨119896⟩ (120578 + 120573 + 120583) (120583 + 120601) (1198770minus 1)

= 120572 ⟨119896⟩ 120583 (120583 + 120601 + 120573) + 120574 ⟨119896⟩ 1198770(120578 + 120573 + 120583) (120583 + 120601)

minus 120574 ⟨119896⟩ (120578 + 120573 + 120583) (120583 + 120601)

(11)

If the inequality 120574⟨119896⟩(120578 + 120573 + 120583)(120583 + 120601) gt (120583 + 120601)[120578 + 120573 + 120583 minus120578⟨119896⟩(120572 + 120574)] is satisfied we have

120578 + 120573 + 120583 lt120578 ⟨119896⟩ (120572 + 120574)

1 minus 120574 ⟨119896⟩ (12)

Then the following equation is obtained

2radic119860119862 lt 120572 ⟨119896⟩ 120583 (120583 + 120601 + 120573)

+ 120574 ⟨119896⟩ 1198770(120578 + 120573 + 120583) (120583 + 120601)

minus 120574 ⟨119896⟩ (120578 + 120573 + 120583) (120583 + 120601)

lt 120572 ⟨119896⟩ 120583 (120583 + 120601 + 120573)

+ 120574 ⟨119896⟩ 1198770(120578 + 120573 + 120583) (120583 + 120601)

minus [120578 + 120573 + 120583 minus 120578 ⟨119896⟩ (120572 + 120574)] (120583 + 120601)

lt 120572 ⟨119896⟩ 120583 (120583 + 120601 + 120573)

+ (120578 + 120573 + 120583) (120583 + 120601) (120574 ⟨119896⟩ 1198770minus 1)

+ 120578 ⟨119896⟩ (120583 + 120601) (120572 + 120574) lt minus119861

(13)

which implies system (1) has two endemic equilibriaWhen 119877

0gt 1

120572 ⟨119896⟩

120578 + 120573 + 120583gt 1198770=

120572 ⟨119896⟩ 120575

(120575 + 120583) (120578 + 120573 + 120583)gt 1 (14)

which can transfer to inequality 120572⟨119896⟩ gt 120578 + 120573 + 120583If120572 lt 120578(120572+120574)(1minus120574⟨119896⟩) we have 1 gt 120578 gt 120572(1minus120574⟨119896⟩)(120572+

120574) Then inequality (12) is satisfied

Remark 2 According to this theorem if the reproductionnumber parameter is above the threshold then the endemicequilibrium is globally asymptotically stable which will bediscussed further in Section 5

3 Global Stability of the Disease-FreeEquilibrium

Definition 3 If the equilibrium is stable under the meaningof Lyapunov for 120575(120576 119905

0) and forall120583 gt 0 there is real number

119879(120583 120575 1199050) gt 0 which makes any initial value 119909

0of inequality

1199090minus 119909119890 le 120575(120576 119905

0) 119905 ge 119905

0 satisfy the following inequality

1003817100381710038171003817120601 (119905 1199090 1199050 minus 119909119890)1003817100381710038171003817 le 120583 forall119905 ge 119905

0+ 119879 (120583 120575 119905

0) (15)

then the equilibrium is asymptotically stable

The Jacobian matrix at the disease-free equilibrium 1198640of

system (1) is

119869 (1198640)

= (

minus120575 minus 120583 minus120575 minus 120572 ⟨119896⟩ 1198780

minus120575 minus120575 + 120601

0 120572 ⟨119896⟩ 1198780minus (120578 + 120573 + 120583) 0 0

0 120578 minus120583 0

0 120573 0 minus120583 minus 120601

)

(16)

Obviously if 1198770

lt 1 all eigenvalues of matrix (16)are negative Then the disease-free equilibrium 119864

0is locally

asymptotically stable in Σ Moreover if 1198770gt 1 there is one

positive eigenvalue and 1198640is unstable

Mathematical Problems in Engineering 5

Theorem 4 If 1198770lt 1 the disease-free equilibrium (DFE) 119864

0

is globally asymptotically stable in Σ and if 1198770gt 1 the disease-

free equilibrium (DFE) 1198640is unstable in Σ

Proof Let 119871(119878 119868 119881 119877) = 119868 gt 0 as a Lyapunov function then119871(1198640) = 0 When 119877

0lt 1

119889119871

119889119905(119878 119868 119881 119877) = 120572 ⟨119896⟩ 119878119868 minus 120574 ⟨119896⟩ 119868119881 minus (120578 + 120573 + 120583) 119868

lt 120572 ⟨119896⟩ 119878119868 minus (120578 + 120573 + 120583) 119868

lt 119868 (120572 ⟨119896⟩1198730minus (120578 + 120573 + 120583))

lt 119868 (120578 + 120573 + 120583) (1198770minus 1) lt 0

(17)

119871 is positive definite and is negative definite Therefore thedisease-free equilibrium (DFE) 119864

0is globally asymptotically

stable in Σ the following result can be given

4 Uniform Persistence

In this section the uniform persistence of system (1) will bediscussed when the basic reproduction number 119877

0gt 1

Definition 5 (see [33]) System (1) is said to be uniformly per-sistent if there exists a constant 0 lt 119888 lt 1 which makes anysolution (119878(119905) 119868(119905) 119881(119905) 119877(119905)) with (119878(0) 119868(0) 119881(0) 119877(0)) isinΣ∘ satisfy

min lim lim119905rarrinfin

119878 (119905) lim119905rarrinfin

119868 (119905) lim119905rarrinfin

119881 (119905) lim119905rarrinfin

119877 (119905)

ge 119888

(18)

Let119883 be a locally compactmetric space withmetric 120597 andlet Γ be a closed nonempty subset of119883with boundary 120597Γ andinterior Γ∘ Obviously 120597Γ is a closed subset of Γ Let Φ

119905be

a dynamical system defined on Γ A set 119861 in 119883 is said to beinvariant if Φ

119905(119861 119905) = 119861 Define 119872

120597fl 119909 isin 120597Γ Φ

119905119909 isin

120597Γ forall119905 ge 0

Lemma 6 (see [34]) Assume the following

(H1) Φ119905has a global attractor

(H2) There exists an 119872 = 1198721 119872

119896 of pair-wise

disjoint compact and isolated invariant set on 120597Γ suchthat

(a) ⋃119909isin119872120597

120596(119909) sub ⋃119896

119895=1119872119895

(b) no subsets of119872 form a cycle on 120597Γ(c) each119872

119895is also isolated in Γ

(d) 119882119904(119872119895) cap Γ∘

= 120601 for each 1 le 119895 le 119896 where119882119904

(119872119895) is the stable manifold of119872

119895 Then Φ

119905is

uniformly persistent with respect to Γ∘

According to Lemma 6 the following result is obtained

Theorem 7 When 1198770gt 1 system (1) is uniformly persistent

Proof Let

Γ = Σ = (119878 119868 119881 119877) isin R+

4| 0 le 119878 + 119868 + 119881 + 119877 le 1

Γ∘

= (119878 119868 119881 119877) isin 119864 119868 119881 gt 0

120597Γ =Γ

Γ∘

(19)

Obviously119872120597= 120597Γ

Choose 119872 = 1198640 120596(119909) = 119864

0 for all 119909 isin 119872

120597 On 120597Γ

system (1) reduces to 1198781015840 = 120575minus(120575+120583)119878 in which 119878(119905) rarr 120575(120575+

120583) as 119905 rarr infin It is concluded that119872 = 1198640 120596(119909) = 119864

0 for

all 119909 isin 119872120597 which indicates that hypotheses (a) and (b) hold

When 1198770gt 1 the disease-free equilibrium 119864

0is unstable

according to Theorem 4 119882119904(119872) = 120597Γ Hypotheses (c) and(d) are then satisfied Due to the ultimate boundedness of allsolutions of system (1) there is a global attractormaking (H1)true

5 Global Dynamics of Endemic Equilibrium

From the previous analysis the disease dies out when 1198770gt 1

then the disease becomes endemic In this section Lyapunovasymptotic stability theorem is used to investigate the globallyasymptotic stability of the endemic equilibrium 119864

lowast when1198770gt 1

Theorem8 The endemic equilibrium119864lowast is globally asymptot-

ically stable in Σ whenever 1198770gt 1

Proof Consider the following function

1198811= ln [(119878 minus 119878lowast) + (119868 minus 119868lowast) + (119881 minus 119881

lowast

) + (119877 minus 119877lowast

)

+ 1]

(20)

Then the derivative of 1198811along the solution of (1) is given by

1=1205971198811

120597119878

119889119878

119889119905+1205971198811

120597119868

119889119868

119889119905+1205971198811

120597119881

119889119881

119889119905+1205971198811

120597119877

119889119877

119889119905

=(119889119878 + 119889119868 + 119889119881 + 119889119877) (1119889119905)

(119878 minus 119878lowast) + (119868 minus 119868lowast) + (119881 minus 119881lowast) + (119877 minus 119877lowast) + 1

(21)

From (2) all solutions of (6) satisfy the equality

119873lowast

= 119878lowast

+ 119868lowast

+ 119881lowast

+ 119877lowast

=120575

120575 + 120583(22)

and 119873 = 119890minus(120575+120583)119905+119862

+ 120575(120575 + 120583) le 120575(120575 + 120583) where 119862 is thevalue that makes119873

0= 120575(120575 + 120583) satisfied

Hence 1198811= ln(119873 minus 119873

lowast

+ 1) ge 0 then

1=

1

119873 minus 120575 (120575 + 120583) + 1

119889119873

119889119905

=120575 + 120583

119873 minus 120575 (120575 + 120583) + 1(

120575

120575 + 120583minus 119873) le 0

(23)

If and only if119873 = 120575(120575 + 120583) 1198811= 0 and

1= 0 are satisfied

6 Mathematical Problems in Engineering

08

07

06

05

04

03

02

01

0050454035302520151050

S(t)I(t)

V(t)R(t)

t time step

The v

alue

s ofS

(t)I(t)

V(t

)R

(t)

(a) The states of disease-free equilibrium

0204

0608

1

002

0406

0

01

02

03

IV (08 02 0 0)IV (06 04 0 0)

IV (04 06 0 0)

minus02

minus01

R(t

)

S(t)I(t) + V(t)

(b) The evolution for the disease-free system states

Figure 2 (a) and (b) showed that the disease-free equilibrium 1198640of the system (1) is globally asymptotically stable with different initial

conditions (08 02 0 0) (06 03 01 0) and (04 035 02 005) and the parameters ⟨119896⟩ = 10 120572 = 08 120573 = 05 120578 = 02 120575 = 05 120574 = 02120601 = 001 and 120583 = 03 119877

0= 05 lt 1 The value of DFE is 119864

0= (0625 0 0 0)

1198811is positive definite and

1is negative definite There-

fore the function1198811is a Lyapunov function for system (1) and

the endemic equilibrium 119864lowast is globally asymptotically stable

by Lyapunov asymptotic stability theorem [35] The proof iscompleted

6 Numerical Simulation

To demonstrate the theoretical results obtained in this papersome numerical simulations will be discussed In this paperthe hypothetical set of initial values (IV) and parametervalues will be given as follows

Consider the initial values of (119878(0) 119868(0) 119881(0) 119877(0)) areset as (08 02 0 0) (06 03 01 0) and (04 035 02 005)respectively

(1) The disease-free equilibrium Set ⟨119896⟩ = 10 120572 = 008120573 = 05 120578 = 02 120575 = 05 120574 = 002 120601 = 001 and120583 = 03119877

0= 05 lt 1 and the disease-free equilibrium

1198640= (0625 0 0 0) from the parameter values above

through the calculation According toTheorem 4 thedisease-free equilibrium 119864

0of system (1) is globally

asymptotically stable in Σ in this case The simulationresults are shown in Figures 2(a) and 2(b)

(2) The endemic equilibrium Set ⟨119896⟩ = 10 120572 = 008120573 = 008 120578 = 03 120575 = 02 120574 = 001 120601 = 025 and120583 = 002 By direct computation 119877

0= 1818 gt 1

and the endemic equilibrium 119864lowast

= (05435 00201

03395 000596) can be obtained from the parametervalues above According to Theorem 4 the positiveendemic equilibrium 119864

lowastof system (1) is globallyasymptotically stable in Σ∘ The simulation results areshown in Figures 3(a) and 3(b)

Figure 2 shows that if 1198770lt 1 all solutions in Σ would

be attracted to the disease-free equilibrium 1198640regardless of

the initial values of system (1) which illustrates the validityof Theorem 4 Similarly it can be seen from Figure 3 that allsolutions inΣ∘ would be attracted to the endemic equilibrium119864lowast regardless of the initial values of system (1) if 119877

0gt 1 and

the conditions of Theorem 8 are satisfied which is obviouslythe content ofTheorem 4Moreover the relationship betweenthe values of equilibrium can be verified as shown in (24)which is coincident with the theoretical results

1198780+ 1198680+ 1198810+ 1198770= 119878lowast

+ 119868lowast

+ 119881lowast

+ 119877lowast

= minus120575

120575 + 120583 (24)

7 Conclusion

The stability of the 119878119868119881119877119878 epidemic spreading model withvirus variation in complex networks has been discussed inthis paper The model involves a new variant group which iscaused by the infectious variation By analyzing the modelthe disease-free equilibrium 119864

0is proved to exist when the

basic reproduction number 1198770is less than 1 The analysis

result reveals that the infectious disease dies out when 1198770is

more than 1 and it becomes endemicThe existing conditionsof endemic equilibrium related with the variation rate andthe network nodes degree are obtained Besides the globalasymptotically stability condition of the disease-free equilib-rium is obtained by the Routh-Hurwitz stability criterion andthe Lyapunov stability criterion And the condition of thesystem uniform persistence is also given The proof of thestability of endemic equilibrium is also illustrated Finallya numerical simulation is given to illustrate the correctnessof the disease-free equilibrium and the endemic equilibriumresults

Mathematical Problems in Engineering 7

08

07

06

05

04

03

02

01

0028024020016012080400

S(t)I(t)

V(t)R(t)

t time step

The v

alue

s ofS

(t)I(t)

V(t

)R

(t)

(a) The system states of endemic equilibrium

002

0406

08

0204

0608

10

002

004

006

008

IV (08 02 0 0)IV (06 04 0 0)

IV (04 06 0 0)

R(t

)

S(t)I(t) + V(t)

(b) The evolution for the endemic system states

Figure 3 (a) and (b) showed that the disease-free equilibrium119864lowast of system (1) is globally asymptotically stable with different initial conditions

(08 02 0 0) (06 03 01 0) and (04 035 02 005) and the parameter values 120572 = 008 120573 = 008 120578 = 03 120575 = 02 120574 = 01 120601 = 025120583 = 002 and ⟨119896⟩ = 10 119877

0= 1818 gt 1 The value of EE is 119864lowast = (05435 00201 03395 000596)

Conflict of Interests

The authors have declared that no competing interests exist

Acknowledgments

This research was partially supported by the National NaturalScience Foundation of China (Grant nos 61473319 61104135)the Science Fund for Creative Research Groups of NationalNatural Science of China (Grant 61321003) and the ScienceFund for Creative Research Project (2015CX007)

References

[1] R Albert and A-L Barabasi ldquoStatistical mechanics of complexnetworksrdquo Reviews of Modern Physics vol 74 no 1 pp 47ndash972002

[2] M E J Newman ldquoThe structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[3] S N Dorogovtsev A V Goltsev and J F F Mendes ldquoCriticalphenomena in complex networksrdquo Reviews of Modern Physicsvol 80 no 4 pp 1275ndash1335 2008

[4] L Zhao W Xie H O Gao X Qiu X Wang and S Zhang ldquoArumor spreading model with variable forgetting raterdquo PhysicaA vol 392 no 23 pp 6146ndash6154 2013

[5] F Chierichetti S Lattanzi andA Panconesi ldquoRumor spreadingin social networksrdquo Theoretical Computer Science vol 412 no24 pp 2602ndash2610 2011

[6] P Manshour and A Montakhab ldquoContagion spreading oncomplex networks with local deterministic dynamicsrdquoCommu-nications in Nonlinear Science andNumerical Simulation vol 19no 7 pp 2414ndash2422 2014

[7] C Liu and Z-K Zhang ldquoInformation spreading on dynamicsocial networksrdquo Communications in Nonlinear Science andNumerical Simulation vol 19 no 4 pp 896ndash904 2014

[8] K Qiao P Zhao and X-M Yao ldquoPerformance analysis ofurban rail transit networkrdquo Journal of Transportation SystemsEngineering and Information Technology vol 12 no 4 pp 115ndash121 2012

[9] O Lordan J M Sallan and P Simo ldquoStudy of the topology androbustness of airline route networks from the complex networkapproach a survey and research agendardquo Journal of TransportGeography vol 37 pp 112ndash120 2014

[10] X Ling M-B Hu W-B Du R Jiang Y-H Wu and Q-S WuldquoBandwidth allocation strategy for traffic systems of scale-freenetworkrdquoPhysics Letters A vol 374 no 48 pp 4825ndash4830 2010

[11] L-X Yang andX-F Yang ldquoThe spread of computer viruses overa reduced scale-free networkrdquo Physica A Statistical Mechanicsand Its Applications vol 396 pp 173ndash184 2014

[12] C Gan X Yang W Liu Q Zhu and X Zhang ldquoAn epidemicmodel of computer viruses with vaccination and generalizednonlinear incidence raterdquo Applied Mathematics and Computa-tion vol 222 pp 265ndash274 2013

[13] R Steuer ldquoComputational approaches to the topology stabilityand dynamics of metabolic networksrdquo Phytochemistry vol 68no 16ndash18 pp 2139ndash2151 2007

[14] A Li J Li and Y Pan ldquoDiscovering natural communities innetworksrdquo Physica A vol 436 no 15 pp 878ndash896 2015

[15] Y Maeno ldquoDiscovering network behind infectious diseaseoutbreakrdquo Physica A vol 389 no 21 pp 4755ndash4768 2010

[16] R Franke and G Ivanova ldquoFALCON or how to compute mea-sures time efficiently on dynamically evolving dense complexnetworksrdquo Journal of Biomedical Informatics vol 47 no 2 pp62ndash70 2014

[17] J-H Li X-Y Shao Y-M Long H-P Zhu and B R Sch-lessman ldquoGlobal optimization by small-world optimizationalgorithm based on social relationship networkrdquo Journal ofCentral South University vol 19 no 8 pp 2247ndash2265 2012

[18] X-H Yang B Wang and B Sun ldquoA novel weighted evolvingnetwork model based on clique overlapping growthrdquo Journal of

8 Mathematical Problems in Engineering

Central South University of Technology vol 17 no 4 pp 830ndash835 2010

[19] O Diekmann and J A P Heesterbeek Mathematical Epi-demiology of Infectious Diseases Model Building Analysis andInterpretation vol 251 John Wiley amp Sons Chichester UK2000

[20] H R Thieme Mathematics in Population Biology vol 4Princeton University Press 2003

[21] T Kuniya ldquoGlobal stability analysis with a discretizationapproach for an age-structured multigroup SIR epidemicmodelrdquoNonlinear Analysis Real World Applications vol 12 no5 pp 2640ndash2655 2011

[22] J Zhang and G Feng ldquoGlobal stability for a tuberculosis modelwith isolation and incomplete treatmentrdquo Computational andApplied Mathematics vol 34 no 3 pp 1237ndash1249 2015

[23] A Lahrouz L Omari and D Kiouach ldquoGlobal analysis of adeterministic and stochastic nonlinear SIRS epidemic modelrdquoNonlinear Analysis Modelling and Control vol 16 no 1 pp 59ndash76 2011

[24] R Xu Z Ma and Z Wang ldquoGlobal stability of a delayedSIRS epidemic model with saturation incidence and temporaryimmunityrdquoComputersampMathematics with Applications vol 59no 9 pp 3211ndash3221 2010

[25] H Kang and X Fu ldquoEpidemic spreading and global stability ofan SIS model with an infective vector on complex networksrdquoCommunications in Nonlinear Science amp Numerical Simulationvol 27 no 1ndash3 pp 30ndash39 2015

[26] P Raja Sekhara Rao and M Naresh Kumar ldquoA dynamic modelfor infectious diseases the role of vaccination and treatmentrdquoChaos Solitons amp Fractals vol 75 pp 34ndash49 2015

[27] J O Kephart S R White and D M Chess ldquoComputers andepidemiologyrdquo IEEE Spectrum vol 30 no 5 pp 20ndash26 1993

[28] Y Moreno R Pastor-Satorras and A Vespignani ldquoEpidemicoutbreaks in complex heterogeneous networksrdquo European Phys-ical Journal B vol 26 no 4 pp 521ndash529 2002

[29] R Pastor-Satorras and A Vespignani ldquoEpidemic dynamics andendemic states in complex networksrdquo Physical Review E vol 63Article ID 066117 2001

[30] R Pastor-Satorras and A Vespignani ldquoEpidemic spreading inscale-free networksrdquo Physical Review Letters vol 86 no 14 pp3200ndash3203 2001

[31] R Pastor-Satorras and A Vespignani ldquoEpidemic dynamics infinite size scale-free networksrdquo Physical Review E vol 65 no 3Article ID 035108 2002

[32] C Moore and M E J Newman ldquoEpidemics and percolationin small-world networksrdquo Physical Review E vol 61 no 5 pp5678ndash5682 2000

[33] G Butler H I Freedman and P Waltman ldquoUniformly persis-tent systemsrdquoProceedings of the AmericanMathematical Societyvol 96 no 3 pp 425ndash430 1986

[34] X-Q ZhaoDynamical Systems in Population Biology CanadianMathematical Society Springer 2003

[35] A M LyapunovThe General Problem of the Stability of MotionTaylor amp Francis London UK 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Theorem 4 If 1198770lt 1 the disease-free equilibrium (DFE) 119864

0

is globally asymptotically stable in Σ and if 1198770gt 1 the disease-

free equilibrium (DFE) 1198640is unstable in Σ

Proof Let 119871(119878 119868 119881 119877) = 119868 gt 0 as a Lyapunov function then119871(1198640) = 0 When 119877

0lt 1

119889119871

119889119905(119878 119868 119881 119877) = 120572 ⟨119896⟩ 119878119868 minus 120574 ⟨119896⟩ 119868119881 minus (120578 + 120573 + 120583) 119868

lt 120572 ⟨119896⟩ 119878119868 minus (120578 + 120573 + 120583) 119868

lt 119868 (120572 ⟨119896⟩1198730minus (120578 + 120573 + 120583))

lt 119868 (120578 + 120573 + 120583) (1198770minus 1) lt 0

(17)

119871 is positive definite and is negative definite Therefore thedisease-free equilibrium (DFE) 119864

0is globally asymptotically

stable in Σ the following result can be given

4 Uniform Persistence

In this section the uniform persistence of system (1) will bediscussed when the basic reproduction number 119877

0gt 1

Definition 5 (see [33]) System (1) is said to be uniformly per-sistent if there exists a constant 0 lt 119888 lt 1 which makes anysolution (119878(119905) 119868(119905) 119881(119905) 119877(119905)) with (119878(0) 119868(0) 119881(0) 119877(0)) isinΣ∘ satisfy

min lim lim119905rarrinfin

119878 (119905) lim119905rarrinfin

119868 (119905) lim119905rarrinfin

119881 (119905) lim119905rarrinfin

119877 (119905)

ge 119888

(18)

Let119883 be a locally compactmetric space withmetric 120597 andlet Γ be a closed nonempty subset of119883with boundary 120597Γ andinterior Γ∘ Obviously 120597Γ is a closed subset of Γ Let Φ

119905be

a dynamical system defined on Γ A set 119861 in 119883 is said to beinvariant if Φ

119905(119861 119905) = 119861 Define 119872

120597fl 119909 isin 120597Γ Φ

119905119909 isin

120597Γ forall119905 ge 0

Lemma 6 (see [34]) Assume the following

(H1) Φ119905has a global attractor

(H2) There exists an 119872 = 1198721 119872

119896 of pair-wise

disjoint compact and isolated invariant set on 120597Γ suchthat

(a) ⋃119909isin119872120597

120596(119909) sub ⋃119896

119895=1119872119895

(b) no subsets of119872 form a cycle on 120597Γ(c) each119872

119895is also isolated in Γ

(d) 119882119904(119872119895) cap Γ∘

= 120601 for each 1 le 119895 le 119896 where119882119904

(119872119895) is the stable manifold of119872

119895 Then Φ

119905is

uniformly persistent with respect to Γ∘

According to Lemma 6 the following result is obtained

Theorem 7 When 1198770gt 1 system (1) is uniformly persistent

Proof Let

Γ = Σ = (119878 119868 119881 119877) isin R+

4| 0 le 119878 + 119868 + 119881 + 119877 le 1

Γ∘

= (119878 119868 119881 119877) isin 119864 119868 119881 gt 0

120597Γ =Γ

Γ∘

(19)

Obviously119872120597= 120597Γ

Choose 119872 = 1198640 120596(119909) = 119864

0 for all 119909 isin 119872

120597 On 120597Γ

system (1) reduces to 1198781015840 = 120575minus(120575+120583)119878 in which 119878(119905) rarr 120575(120575+

120583) as 119905 rarr infin It is concluded that119872 = 1198640 120596(119909) = 119864

0 for

all 119909 isin 119872120597 which indicates that hypotheses (a) and (b) hold

When 1198770gt 1 the disease-free equilibrium 119864

0is unstable

according to Theorem 4 119882119904(119872) = 120597Γ Hypotheses (c) and(d) are then satisfied Due to the ultimate boundedness of allsolutions of system (1) there is a global attractormaking (H1)true

5 Global Dynamics of Endemic Equilibrium

From the previous analysis the disease dies out when 1198770gt 1

then the disease becomes endemic In this section Lyapunovasymptotic stability theorem is used to investigate the globallyasymptotic stability of the endemic equilibrium 119864

lowast when1198770gt 1

Theorem8 The endemic equilibrium119864lowast is globally asymptot-

ically stable in Σ whenever 1198770gt 1

Proof Consider the following function

1198811= ln [(119878 minus 119878lowast) + (119868 minus 119868lowast) + (119881 minus 119881

lowast

) + (119877 minus 119877lowast

)

+ 1]

(20)

Then the derivative of 1198811along the solution of (1) is given by

1=1205971198811

120597119878

119889119878

119889119905+1205971198811

120597119868

119889119868

119889119905+1205971198811

120597119881

119889119881

119889119905+1205971198811

120597119877

119889119877

119889119905

=(119889119878 + 119889119868 + 119889119881 + 119889119877) (1119889119905)

(119878 minus 119878lowast) + (119868 minus 119868lowast) + (119881 minus 119881lowast) + (119877 minus 119877lowast) + 1

(21)

From (2) all solutions of (6) satisfy the equality

119873lowast

= 119878lowast

+ 119868lowast

+ 119881lowast

+ 119877lowast

=120575

120575 + 120583(22)

and 119873 = 119890minus(120575+120583)119905+119862

+ 120575(120575 + 120583) le 120575(120575 + 120583) where 119862 is thevalue that makes119873

0= 120575(120575 + 120583) satisfied

Hence 1198811= ln(119873 minus 119873

lowast

+ 1) ge 0 then

1=

1

119873 minus 120575 (120575 + 120583) + 1

119889119873

119889119905

=120575 + 120583

119873 minus 120575 (120575 + 120583) + 1(

120575

120575 + 120583minus 119873) le 0

(23)

If and only if119873 = 120575(120575 + 120583) 1198811= 0 and

1= 0 are satisfied

6 Mathematical Problems in Engineering

08

07

06

05

04

03

02

01

0050454035302520151050

S(t)I(t)

V(t)R(t)

t time step

The v

alue

s ofS

(t)I(t)

V(t

)R

(t)

(a) The states of disease-free equilibrium

0204

0608

1

002

0406

0

01

02

03

IV (08 02 0 0)IV (06 04 0 0)

IV (04 06 0 0)

minus02

minus01

R(t

)

S(t)I(t) + V(t)

(b) The evolution for the disease-free system states

Figure 2 (a) and (b) showed that the disease-free equilibrium 1198640of the system (1) is globally asymptotically stable with different initial

conditions (08 02 0 0) (06 03 01 0) and (04 035 02 005) and the parameters ⟨119896⟩ = 10 120572 = 08 120573 = 05 120578 = 02 120575 = 05 120574 = 02120601 = 001 and 120583 = 03 119877

0= 05 lt 1 The value of DFE is 119864

0= (0625 0 0 0)

1198811is positive definite and

1is negative definite There-

fore the function1198811is a Lyapunov function for system (1) and

the endemic equilibrium 119864lowast is globally asymptotically stable

by Lyapunov asymptotic stability theorem [35] The proof iscompleted

6 Numerical Simulation

To demonstrate the theoretical results obtained in this papersome numerical simulations will be discussed In this paperthe hypothetical set of initial values (IV) and parametervalues will be given as follows

Consider the initial values of (119878(0) 119868(0) 119881(0) 119877(0)) areset as (08 02 0 0) (06 03 01 0) and (04 035 02 005)respectively

(1) The disease-free equilibrium Set ⟨119896⟩ = 10 120572 = 008120573 = 05 120578 = 02 120575 = 05 120574 = 002 120601 = 001 and120583 = 03119877

0= 05 lt 1 and the disease-free equilibrium

1198640= (0625 0 0 0) from the parameter values above

through the calculation According toTheorem 4 thedisease-free equilibrium 119864

0of system (1) is globally

asymptotically stable in Σ in this case The simulationresults are shown in Figures 2(a) and 2(b)

(2) The endemic equilibrium Set ⟨119896⟩ = 10 120572 = 008120573 = 008 120578 = 03 120575 = 02 120574 = 001 120601 = 025 and120583 = 002 By direct computation 119877

0= 1818 gt 1

and the endemic equilibrium 119864lowast

= (05435 00201

03395 000596) can be obtained from the parametervalues above According to Theorem 4 the positiveendemic equilibrium 119864

lowastof system (1) is globallyasymptotically stable in Σ∘ The simulation results areshown in Figures 3(a) and 3(b)

Figure 2 shows that if 1198770lt 1 all solutions in Σ would

be attracted to the disease-free equilibrium 1198640regardless of

the initial values of system (1) which illustrates the validityof Theorem 4 Similarly it can be seen from Figure 3 that allsolutions inΣ∘ would be attracted to the endemic equilibrium119864lowast regardless of the initial values of system (1) if 119877

0gt 1 and

the conditions of Theorem 8 are satisfied which is obviouslythe content ofTheorem 4Moreover the relationship betweenthe values of equilibrium can be verified as shown in (24)which is coincident with the theoretical results

1198780+ 1198680+ 1198810+ 1198770= 119878lowast

+ 119868lowast

+ 119881lowast

+ 119877lowast

= minus120575

120575 + 120583 (24)

7 Conclusion

The stability of the 119878119868119881119877119878 epidemic spreading model withvirus variation in complex networks has been discussed inthis paper The model involves a new variant group which iscaused by the infectious variation By analyzing the modelthe disease-free equilibrium 119864

0is proved to exist when the

basic reproduction number 1198770is less than 1 The analysis

result reveals that the infectious disease dies out when 1198770is

more than 1 and it becomes endemicThe existing conditionsof endemic equilibrium related with the variation rate andthe network nodes degree are obtained Besides the globalasymptotically stability condition of the disease-free equilib-rium is obtained by the Routh-Hurwitz stability criterion andthe Lyapunov stability criterion And the condition of thesystem uniform persistence is also given The proof of thestability of endemic equilibrium is also illustrated Finallya numerical simulation is given to illustrate the correctnessof the disease-free equilibrium and the endemic equilibriumresults

Mathematical Problems in Engineering 7

08

07

06

05

04

03

02

01

0028024020016012080400

S(t)I(t)

V(t)R(t)

t time step

The v

alue

s ofS

(t)I(t)

V(t

)R

(t)

(a) The system states of endemic equilibrium

002

0406

08

0204

0608

10

002

004

006

008

IV (08 02 0 0)IV (06 04 0 0)

IV (04 06 0 0)

R(t

)

S(t)I(t) + V(t)

(b) The evolution for the endemic system states

Figure 3 (a) and (b) showed that the disease-free equilibrium119864lowast of system (1) is globally asymptotically stable with different initial conditions

(08 02 0 0) (06 03 01 0) and (04 035 02 005) and the parameter values 120572 = 008 120573 = 008 120578 = 03 120575 = 02 120574 = 01 120601 = 025120583 = 002 and ⟨119896⟩ = 10 119877

0= 1818 gt 1 The value of EE is 119864lowast = (05435 00201 03395 000596)

Conflict of Interests

The authors have declared that no competing interests exist

Acknowledgments

This research was partially supported by the National NaturalScience Foundation of China (Grant nos 61473319 61104135)the Science Fund for Creative Research Groups of NationalNatural Science of China (Grant 61321003) and the ScienceFund for Creative Research Project (2015CX007)

References

[1] R Albert and A-L Barabasi ldquoStatistical mechanics of complexnetworksrdquo Reviews of Modern Physics vol 74 no 1 pp 47ndash972002

[2] M E J Newman ldquoThe structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[3] S N Dorogovtsev A V Goltsev and J F F Mendes ldquoCriticalphenomena in complex networksrdquo Reviews of Modern Physicsvol 80 no 4 pp 1275ndash1335 2008

[4] L Zhao W Xie H O Gao X Qiu X Wang and S Zhang ldquoArumor spreading model with variable forgetting raterdquo PhysicaA vol 392 no 23 pp 6146ndash6154 2013

[5] F Chierichetti S Lattanzi andA Panconesi ldquoRumor spreadingin social networksrdquo Theoretical Computer Science vol 412 no24 pp 2602ndash2610 2011

[6] P Manshour and A Montakhab ldquoContagion spreading oncomplex networks with local deterministic dynamicsrdquoCommu-nications in Nonlinear Science andNumerical Simulation vol 19no 7 pp 2414ndash2422 2014

[7] C Liu and Z-K Zhang ldquoInformation spreading on dynamicsocial networksrdquo Communications in Nonlinear Science andNumerical Simulation vol 19 no 4 pp 896ndash904 2014

[8] K Qiao P Zhao and X-M Yao ldquoPerformance analysis ofurban rail transit networkrdquo Journal of Transportation SystemsEngineering and Information Technology vol 12 no 4 pp 115ndash121 2012

[9] O Lordan J M Sallan and P Simo ldquoStudy of the topology androbustness of airline route networks from the complex networkapproach a survey and research agendardquo Journal of TransportGeography vol 37 pp 112ndash120 2014

[10] X Ling M-B Hu W-B Du R Jiang Y-H Wu and Q-S WuldquoBandwidth allocation strategy for traffic systems of scale-freenetworkrdquoPhysics Letters A vol 374 no 48 pp 4825ndash4830 2010

[11] L-X Yang andX-F Yang ldquoThe spread of computer viruses overa reduced scale-free networkrdquo Physica A Statistical Mechanicsand Its Applications vol 396 pp 173ndash184 2014

[12] C Gan X Yang W Liu Q Zhu and X Zhang ldquoAn epidemicmodel of computer viruses with vaccination and generalizednonlinear incidence raterdquo Applied Mathematics and Computa-tion vol 222 pp 265ndash274 2013

[13] R Steuer ldquoComputational approaches to the topology stabilityand dynamics of metabolic networksrdquo Phytochemistry vol 68no 16ndash18 pp 2139ndash2151 2007

[14] A Li J Li and Y Pan ldquoDiscovering natural communities innetworksrdquo Physica A vol 436 no 15 pp 878ndash896 2015

[15] Y Maeno ldquoDiscovering network behind infectious diseaseoutbreakrdquo Physica A vol 389 no 21 pp 4755ndash4768 2010

[16] R Franke and G Ivanova ldquoFALCON or how to compute mea-sures time efficiently on dynamically evolving dense complexnetworksrdquo Journal of Biomedical Informatics vol 47 no 2 pp62ndash70 2014

[17] J-H Li X-Y Shao Y-M Long H-P Zhu and B R Sch-lessman ldquoGlobal optimization by small-world optimizationalgorithm based on social relationship networkrdquo Journal ofCentral South University vol 19 no 8 pp 2247ndash2265 2012

[18] X-H Yang B Wang and B Sun ldquoA novel weighted evolvingnetwork model based on clique overlapping growthrdquo Journal of

8 Mathematical Problems in Engineering

Central South University of Technology vol 17 no 4 pp 830ndash835 2010

[19] O Diekmann and J A P Heesterbeek Mathematical Epi-demiology of Infectious Diseases Model Building Analysis andInterpretation vol 251 John Wiley amp Sons Chichester UK2000

[20] H R Thieme Mathematics in Population Biology vol 4Princeton University Press 2003

[21] T Kuniya ldquoGlobal stability analysis with a discretizationapproach for an age-structured multigroup SIR epidemicmodelrdquoNonlinear Analysis Real World Applications vol 12 no5 pp 2640ndash2655 2011

[22] J Zhang and G Feng ldquoGlobal stability for a tuberculosis modelwith isolation and incomplete treatmentrdquo Computational andApplied Mathematics vol 34 no 3 pp 1237ndash1249 2015

[23] A Lahrouz L Omari and D Kiouach ldquoGlobal analysis of adeterministic and stochastic nonlinear SIRS epidemic modelrdquoNonlinear Analysis Modelling and Control vol 16 no 1 pp 59ndash76 2011

[24] R Xu Z Ma and Z Wang ldquoGlobal stability of a delayedSIRS epidemic model with saturation incidence and temporaryimmunityrdquoComputersampMathematics with Applications vol 59no 9 pp 3211ndash3221 2010

[25] H Kang and X Fu ldquoEpidemic spreading and global stability ofan SIS model with an infective vector on complex networksrdquoCommunications in Nonlinear Science amp Numerical Simulationvol 27 no 1ndash3 pp 30ndash39 2015

[26] P Raja Sekhara Rao and M Naresh Kumar ldquoA dynamic modelfor infectious diseases the role of vaccination and treatmentrdquoChaos Solitons amp Fractals vol 75 pp 34ndash49 2015

[27] J O Kephart S R White and D M Chess ldquoComputers andepidemiologyrdquo IEEE Spectrum vol 30 no 5 pp 20ndash26 1993

[28] Y Moreno R Pastor-Satorras and A Vespignani ldquoEpidemicoutbreaks in complex heterogeneous networksrdquo European Phys-ical Journal B vol 26 no 4 pp 521ndash529 2002

[29] R Pastor-Satorras and A Vespignani ldquoEpidemic dynamics andendemic states in complex networksrdquo Physical Review E vol 63Article ID 066117 2001

[30] R Pastor-Satorras and A Vespignani ldquoEpidemic spreading inscale-free networksrdquo Physical Review Letters vol 86 no 14 pp3200ndash3203 2001

[31] R Pastor-Satorras and A Vespignani ldquoEpidemic dynamics infinite size scale-free networksrdquo Physical Review E vol 65 no 3Article ID 035108 2002

[32] C Moore and M E J Newman ldquoEpidemics and percolationin small-world networksrdquo Physical Review E vol 61 no 5 pp5678ndash5682 2000

[33] G Butler H I Freedman and P Waltman ldquoUniformly persis-tent systemsrdquoProceedings of the AmericanMathematical Societyvol 96 no 3 pp 425ndash430 1986

[34] X-Q ZhaoDynamical Systems in Population Biology CanadianMathematical Society Springer 2003

[35] A M LyapunovThe General Problem of the Stability of MotionTaylor amp Francis London UK 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

08

07

06

05

04

03

02

01

0050454035302520151050

S(t)I(t)

V(t)R(t)

t time step

The v

alue

s ofS

(t)I(t)

V(t

)R

(t)

(a) The states of disease-free equilibrium

0204

0608

1

002

0406

0

01

02

03

IV (08 02 0 0)IV (06 04 0 0)

IV (04 06 0 0)

minus02

minus01

R(t

)

S(t)I(t) + V(t)

(b) The evolution for the disease-free system states

Figure 2 (a) and (b) showed that the disease-free equilibrium 1198640of the system (1) is globally asymptotically stable with different initial

conditions (08 02 0 0) (06 03 01 0) and (04 035 02 005) and the parameters ⟨119896⟩ = 10 120572 = 08 120573 = 05 120578 = 02 120575 = 05 120574 = 02120601 = 001 and 120583 = 03 119877

0= 05 lt 1 The value of DFE is 119864

0= (0625 0 0 0)

1198811is positive definite and

1is negative definite There-

fore the function1198811is a Lyapunov function for system (1) and

the endemic equilibrium 119864lowast is globally asymptotically stable

by Lyapunov asymptotic stability theorem [35] The proof iscompleted

6 Numerical Simulation

To demonstrate the theoretical results obtained in this papersome numerical simulations will be discussed In this paperthe hypothetical set of initial values (IV) and parametervalues will be given as follows

Consider the initial values of (119878(0) 119868(0) 119881(0) 119877(0)) areset as (08 02 0 0) (06 03 01 0) and (04 035 02 005)respectively

(1) The disease-free equilibrium Set ⟨119896⟩ = 10 120572 = 008120573 = 05 120578 = 02 120575 = 05 120574 = 002 120601 = 001 and120583 = 03119877

0= 05 lt 1 and the disease-free equilibrium

1198640= (0625 0 0 0) from the parameter values above

through the calculation According toTheorem 4 thedisease-free equilibrium 119864

0of system (1) is globally

asymptotically stable in Σ in this case The simulationresults are shown in Figures 2(a) and 2(b)

(2) The endemic equilibrium Set ⟨119896⟩ = 10 120572 = 008120573 = 008 120578 = 03 120575 = 02 120574 = 001 120601 = 025 and120583 = 002 By direct computation 119877

0= 1818 gt 1

and the endemic equilibrium 119864lowast

= (05435 00201

03395 000596) can be obtained from the parametervalues above According to Theorem 4 the positiveendemic equilibrium 119864

lowastof system (1) is globallyasymptotically stable in Σ∘ The simulation results areshown in Figures 3(a) and 3(b)

Figure 2 shows that if 1198770lt 1 all solutions in Σ would

be attracted to the disease-free equilibrium 1198640regardless of

the initial values of system (1) which illustrates the validityof Theorem 4 Similarly it can be seen from Figure 3 that allsolutions inΣ∘ would be attracted to the endemic equilibrium119864lowast regardless of the initial values of system (1) if 119877

0gt 1 and

the conditions of Theorem 8 are satisfied which is obviouslythe content ofTheorem 4Moreover the relationship betweenthe values of equilibrium can be verified as shown in (24)which is coincident with the theoretical results

1198780+ 1198680+ 1198810+ 1198770= 119878lowast

+ 119868lowast

+ 119881lowast

+ 119877lowast

= minus120575

120575 + 120583 (24)

7 Conclusion

The stability of the 119878119868119881119877119878 epidemic spreading model withvirus variation in complex networks has been discussed inthis paper The model involves a new variant group which iscaused by the infectious variation By analyzing the modelthe disease-free equilibrium 119864

0is proved to exist when the

basic reproduction number 1198770is less than 1 The analysis

result reveals that the infectious disease dies out when 1198770is

more than 1 and it becomes endemicThe existing conditionsof endemic equilibrium related with the variation rate andthe network nodes degree are obtained Besides the globalasymptotically stability condition of the disease-free equilib-rium is obtained by the Routh-Hurwitz stability criterion andthe Lyapunov stability criterion And the condition of thesystem uniform persistence is also given The proof of thestability of endemic equilibrium is also illustrated Finallya numerical simulation is given to illustrate the correctnessof the disease-free equilibrium and the endemic equilibriumresults

Mathematical Problems in Engineering 7

08

07

06

05

04

03

02

01

0028024020016012080400

S(t)I(t)

V(t)R(t)

t time step

The v

alue

s ofS

(t)I(t)

V(t

)R

(t)

(a) The system states of endemic equilibrium

002

0406

08

0204

0608

10

002

004

006

008

IV (08 02 0 0)IV (06 04 0 0)

IV (04 06 0 0)

R(t

)

S(t)I(t) + V(t)

(b) The evolution for the endemic system states

Figure 3 (a) and (b) showed that the disease-free equilibrium119864lowast of system (1) is globally asymptotically stable with different initial conditions

(08 02 0 0) (06 03 01 0) and (04 035 02 005) and the parameter values 120572 = 008 120573 = 008 120578 = 03 120575 = 02 120574 = 01 120601 = 025120583 = 002 and ⟨119896⟩ = 10 119877

0= 1818 gt 1 The value of EE is 119864lowast = (05435 00201 03395 000596)

Conflict of Interests

The authors have declared that no competing interests exist

Acknowledgments

This research was partially supported by the National NaturalScience Foundation of China (Grant nos 61473319 61104135)the Science Fund for Creative Research Groups of NationalNatural Science of China (Grant 61321003) and the ScienceFund for Creative Research Project (2015CX007)

References

[1] R Albert and A-L Barabasi ldquoStatistical mechanics of complexnetworksrdquo Reviews of Modern Physics vol 74 no 1 pp 47ndash972002

[2] M E J Newman ldquoThe structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[3] S N Dorogovtsev A V Goltsev and J F F Mendes ldquoCriticalphenomena in complex networksrdquo Reviews of Modern Physicsvol 80 no 4 pp 1275ndash1335 2008

[4] L Zhao W Xie H O Gao X Qiu X Wang and S Zhang ldquoArumor spreading model with variable forgetting raterdquo PhysicaA vol 392 no 23 pp 6146ndash6154 2013

[5] F Chierichetti S Lattanzi andA Panconesi ldquoRumor spreadingin social networksrdquo Theoretical Computer Science vol 412 no24 pp 2602ndash2610 2011

[6] P Manshour and A Montakhab ldquoContagion spreading oncomplex networks with local deterministic dynamicsrdquoCommu-nications in Nonlinear Science andNumerical Simulation vol 19no 7 pp 2414ndash2422 2014

[7] C Liu and Z-K Zhang ldquoInformation spreading on dynamicsocial networksrdquo Communications in Nonlinear Science andNumerical Simulation vol 19 no 4 pp 896ndash904 2014

[8] K Qiao P Zhao and X-M Yao ldquoPerformance analysis ofurban rail transit networkrdquo Journal of Transportation SystemsEngineering and Information Technology vol 12 no 4 pp 115ndash121 2012

[9] O Lordan J M Sallan and P Simo ldquoStudy of the topology androbustness of airline route networks from the complex networkapproach a survey and research agendardquo Journal of TransportGeography vol 37 pp 112ndash120 2014

[10] X Ling M-B Hu W-B Du R Jiang Y-H Wu and Q-S WuldquoBandwidth allocation strategy for traffic systems of scale-freenetworkrdquoPhysics Letters A vol 374 no 48 pp 4825ndash4830 2010

[11] L-X Yang andX-F Yang ldquoThe spread of computer viruses overa reduced scale-free networkrdquo Physica A Statistical Mechanicsand Its Applications vol 396 pp 173ndash184 2014

[12] C Gan X Yang W Liu Q Zhu and X Zhang ldquoAn epidemicmodel of computer viruses with vaccination and generalizednonlinear incidence raterdquo Applied Mathematics and Computa-tion vol 222 pp 265ndash274 2013

[13] R Steuer ldquoComputational approaches to the topology stabilityand dynamics of metabolic networksrdquo Phytochemistry vol 68no 16ndash18 pp 2139ndash2151 2007

[14] A Li J Li and Y Pan ldquoDiscovering natural communities innetworksrdquo Physica A vol 436 no 15 pp 878ndash896 2015

[15] Y Maeno ldquoDiscovering network behind infectious diseaseoutbreakrdquo Physica A vol 389 no 21 pp 4755ndash4768 2010

[16] R Franke and G Ivanova ldquoFALCON or how to compute mea-sures time efficiently on dynamically evolving dense complexnetworksrdquo Journal of Biomedical Informatics vol 47 no 2 pp62ndash70 2014

[17] J-H Li X-Y Shao Y-M Long H-P Zhu and B R Sch-lessman ldquoGlobal optimization by small-world optimizationalgorithm based on social relationship networkrdquo Journal ofCentral South University vol 19 no 8 pp 2247ndash2265 2012

[18] X-H Yang B Wang and B Sun ldquoA novel weighted evolvingnetwork model based on clique overlapping growthrdquo Journal of

8 Mathematical Problems in Engineering

Central South University of Technology vol 17 no 4 pp 830ndash835 2010

[19] O Diekmann and J A P Heesterbeek Mathematical Epi-demiology of Infectious Diseases Model Building Analysis andInterpretation vol 251 John Wiley amp Sons Chichester UK2000

[20] H R Thieme Mathematics in Population Biology vol 4Princeton University Press 2003

[21] T Kuniya ldquoGlobal stability analysis with a discretizationapproach for an age-structured multigroup SIR epidemicmodelrdquoNonlinear Analysis Real World Applications vol 12 no5 pp 2640ndash2655 2011

[22] J Zhang and G Feng ldquoGlobal stability for a tuberculosis modelwith isolation and incomplete treatmentrdquo Computational andApplied Mathematics vol 34 no 3 pp 1237ndash1249 2015

[23] A Lahrouz L Omari and D Kiouach ldquoGlobal analysis of adeterministic and stochastic nonlinear SIRS epidemic modelrdquoNonlinear Analysis Modelling and Control vol 16 no 1 pp 59ndash76 2011

[24] R Xu Z Ma and Z Wang ldquoGlobal stability of a delayedSIRS epidemic model with saturation incidence and temporaryimmunityrdquoComputersampMathematics with Applications vol 59no 9 pp 3211ndash3221 2010

[25] H Kang and X Fu ldquoEpidemic spreading and global stability ofan SIS model with an infective vector on complex networksrdquoCommunications in Nonlinear Science amp Numerical Simulationvol 27 no 1ndash3 pp 30ndash39 2015

[26] P Raja Sekhara Rao and M Naresh Kumar ldquoA dynamic modelfor infectious diseases the role of vaccination and treatmentrdquoChaos Solitons amp Fractals vol 75 pp 34ndash49 2015

[27] J O Kephart S R White and D M Chess ldquoComputers andepidemiologyrdquo IEEE Spectrum vol 30 no 5 pp 20ndash26 1993

[28] Y Moreno R Pastor-Satorras and A Vespignani ldquoEpidemicoutbreaks in complex heterogeneous networksrdquo European Phys-ical Journal B vol 26 no 4 pp 521ndash529 2002

[29] R Pastor-Satorras and A Vespignani ldquoEpidemic dynamics andendemic states in complex networksrdquo Physical Review E vol 63Article ID 066117 2001

[30] R Pastor-Satorras and A Vespignani ldquoEpidemic spreading inscale-free networksrdquo Physical Review Letters vol 86 no 14 pp3200ndash3203 2001

[31] R Pastor-Satorras and A Vespignani ldquoEpidemic dynamics infinite size scale-free networksrdquo Physical Review E vol 65 no 3Article ID 035108 2002

[32] C Moore and M E J Newman ldquoEpidemics and percolationin small-world networksrdquo Physical Review E vol 61 no 5 pp5678ndash5682 2000

[33] G Butler H I Freedman and P Waltman ldquoUniformly persis-tent systemsrdquoProceedings of the AmericanMathematical Societyvol 96 no 3 pp 425ndash430 1986

[34] X-Q ZhaoDynamical Systems in Population Biology CanadianMathematical Society Springer 2003

[35] A M LyapunovThe General Problem of the Stability of MotionTaylor amp Francis London UK 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

08

07

06

05

04

03

02

01

0028024020016012080400

S(t)I(t)

V(t)R(t)

t time step

The v

alue

s ofS

(t)I(t)

V(t

)R

(t)

(a) The system states of endemic equilibrium

002

0406

08

0204

0608

10

002

004

006

008

IV (08 02 0 0)IV (06 04 0 0)

IV (04 06 0 0)

R(t

)

S(t)I(t) + V(t)

(b) The evolution for the endemic system states

Figure 3 (a) and (b) showed that the disease-free equilibrium119864lowast of system (1) is globally asymptotically stable with different initial conditions

(08 02 0 0) (06 03 01 0) and (04 035 02 005) and the parameter values 120572 = 008 120573 = 008 120578 = 03 120575 = 02 120574 = 01 120601 = 025120583 = 002 and ⟨119896⟩ = 10 119877

0= 1818 gt 1 The value of EE is 119864lowast = (05435 00201 03395 000596)

Conflict of Interests

The authors have declared that no competing interests exist

Acknowledgments

This research was partially supported by the National NaturalScience Foundation of China (Grant nos 61473319 61104135)the Science Fund for Creative Research Groups of NationalNatural Science of China (Grant 61321003) and the ScienceFund for Creative Research Project (2015CX007)

References

[1] R Albert and A-L Barabasi ldquoStatistical mechanics of complexnetworksrdquo Reviews of Modern Physics vol 74 no 1 pp 47ndash972002

[2] M E J Newman ldquoThe structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[3] S N Dorogovtsev A V Goltsev and J F F Mendes ldquoCriticalphenomena in complex networksrdquo Reviews of Modern Physicsvol 80 no 4 pp 1275ndash1335 2008

[4] L Zhao W Xie H O Gao X Qiu X Wang and S Zhang ldquoArumor spreading model with variable forgetting raterdquo PhysicaA vol 392 no 23 pp 6146ndash6154 2013

[5] F Chierichetti S Lattanzi andA Panconesi ldquoRumor spreadingin social networksrdquo Theoretical Computer Science vol 412 no24 pp 2602ndash2610 2011

[6] P Manshour and A Montakhab ldquoContagion spreading oncomplex networks with local deterministic dynamicsrdquoCommu-nications in Nonlinear Science andNumerical Simulation vol 19no 7 pp 2414ndash2422 2014

[7] C Liu and Z-K Zhang ldquoInformation spreading on dynamicsocial networksrdquo Communications in Nonlinear Science andNumerical Simulation vol 19 no 4 pp 896ndash904 2014

[8] K Qiao P Zhao and X-M Yao ldquoPerformance analysis ofurban rail transit networkrdquo Journal of Transportation SystemsEngineering and Information Technology vol 12 no 4 pp 115ndash121 2012

[9] O Lordan J M Sallan and P Simo ldquoStudy of the topology androbustness of airline route networks from the complex networkapproach a survey and research agendardquo Journal of TransportGeography vol 37 pp 112ndash120 2014

[10] X Ling M-B Hu W-B Du R Jiang Y-H Wu and Q-S WuldquoBandwidth allocation strategy for traffic systems of scale-freenetworkrdquoPhysics Letters A vol 374 no 48 pp 4825ndash4830 2010

[11] L-X Yang andX-F Yang ldquoThe spread of computer viruses overa reduced scale-free networkrdquo Physica A Statistical Mechanicsand Its Applications vol 396 pp 173ndash184 2014

[12] C Gan X Yang W Liu Q Zhu and X Zhang ldquoAn epidemicmodel of computer viruses with vaccination and generalizednonlinear incidence raterdquo Applied Mathematics and Computa-tion vol 222 pp 265ndash274 2013

[13] R Steuer ldquoComputational approaches to the topology stabilityand dynamics of metabolic networksrdquo Phytochemistry vol 68no 16ndash18 pp 2139ndash2151 2007

[14] A Li J Li and Y Pan ldquoDiscovering natural communities innetworksrdquo Physica A vol 436 no 15 pp 878ndash896 2015

[15] Y Maeno ldquoDiscovering network behind infectious diseaseoutbreakrdquo Physica A vol 389 no 21 pp 4755ndash4768 2010

[16] R Franke and G Ivanova ldquoFALCON or how to compute mea-sures time efficiently on dynamically evolving dense complexnetworksrdquo Journal of Biomedical Informatics vol 47 no 2 pp62ndash70 2014

[17] J-H Li X-Y Shao Y-M Long H-P Zhu and B R Sch-lessman ldquoGlobal optimization by small-world optimizationalgorithm based on social relationship networkrdquo Journal ofCentral South University vol 19 no 8 pp 2247ndash2265 2012

[18] X-H Yang B Wang and B Sun ldquoA novel weighted evolvingnetwork model based on clique overlapping growthrdquo Journal of

8 Mathematical Problems in Engineering

Central South University of Technology vol 17 no 4 pp 830ndash835 2010

[19] O Diekmann and J A P Heesterbeek Mathematical Epi-demiology of Infectious Diseases Model Building Analysis andInterpretation vol 251 John Wiley amp Sons Chichester UK2000

[20] H R Thieme Mathematics in Population Biology vol 4Princeton University Press 2003

[21] T Kuniya ldquoGlobal stability analysis with a discretizationapproach for an age-structured multigroup SIR epidemicmodelrdquoNonlinear Analysis Real World Applications vol 12 no5 pp 2640ndash2655 2011

[22] J Zhang and G Feng ldquoGlobal stability for a tuberculosis modelwith isolation and incomplete treatmentrdquo Computational andApplied Mathematics vol 34 no 3 pp 1237ndash1249 2015

[23] A Lahrouz L Omari and D Kiouach ldquoGlobal analysis of adeterministic and stochastic nonlinear SIRS epidemic modelrdquoNonlinear Analysis Modelling and Control vol 16 no 1 pp 59ndash76 2011

[24] R Xu Z Ma and Z Wang ldquoGlobal stability of a delayedSIRS epidemic model with saturation incidence and temporaryimmunityrdquoComputersampMathematics with Applications vol 59no 9 pp 3211ndash3221 2010

[25] H Kang and X Fu ldquoEpidemic spreading and global stability ofan SIS model with an infective vector on complex networksrdquoCommunications in Nonlinear Science amp Numerical Simulationvol 27 no 1ndash3 pp 30ndash39 2015

[26] P Raja Sekhara Rao and M Naresh Kumar ldquoA dynamic modelfor infectious diseases the role of vaccination and treatmentrdquoChaos Solitons amp Fractals vol 75 pp 34ndash49 2015

[27] J O Kephart S R White and D M Chess ldquoComputers andepidemiologyrdquo IEEE Spectrum vol 30 no 5 pp 20ndash26 1993

[28] Y Moreno R Pastor-Satorras and A Vespignani ldquoEpidemicoutbreaks in complex heterogeneous networksrdquo European Phys-ical Journal B vol 26 no 4 pp 521ndash529 2002

[29] R Pastor-Satorras and A Vespignani ldquoEpidemic dynamics andendemic states in complex networksrdquo Physical Review E vol 63Article ID 066117 2001

[30] R Pastor-Satorras and A Vespignani ldquoEpidemic spreading inscale-free networksrdquo Physical Review Letters vol 86 no 14 pp3200ndash3203 2001

[31] R Pastor-Satorras and A Vespignani ldquoEpidemic dynamics infinite size scale-free networksrdquo Physical Review E vol 65 no 3Article ID 035108 2002

[32] C Moore and M E J Newman ldquoEpidemics and percolationin small-world networksrdquo Physical Review E vol 61 no 5 pp5678ndash5682 2000

[33] G Butler H I Freedman and P Waltman ldquoUniformly persis-tent systemsrdquoProceedings of the AmericanMathematical Societyvol 96 no 3 pp 425ndash430 1986

[34] X-Q ZhaoDynamical Systems in Population Biology CanadianMathematical Society Springer 2003

[35] A M LyapunovThe General Problem of the Stability of MotionTaylor amp Francis London UK 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Central South University of Technology vol 17 no 4 pp 830ndash835 2010

[19] O Diekmann and J A P Heesterbeek Mathematical Epi-demiology of Infectious Diseases Model Building Analysis andInterpretation vol 251 John Wiley amp Sons Chichester UK2000

[20] H R Thieme Mathematics in Population Biology vol 4Princeton University Press 2003

[21] T Kuniya ldquoGlobal stability analysis with a discretizationapproach for an age-structured multigroup SIR epidemicmodelrdquoNonlinear Analysis Real World Applications vol 12 no5 pp 2640ndash2655 2011

[22] J Zhang and G Feng ldquoGlobal stability for a tuberculosis modelwith isolation and incomplete treatmentrdquo Computational andApplied Mathematics vol 34 no 3 pp 1237ndash1249 2015

[23] A Lahrouz L Omari and D Kiouach ldquoGlobal analysis of adeterministic and stochastic nonlinear SIRS epidemic modelrdquoNonlinear Analysis Modelling and Control vol 16 no 1 pp 59ndash76 2011

[24] R Xu Z Ma and Z Wang ldquoGlobal stability of a delayedSIRS epidemic model with saturation incidence and temporaryimmunityrdquoComputersampMathematics with Applications vol 59no 9 pp 3211ndash3221 2010

[25] H Kang and X Fu ldquoEpidemic spreading and global stability ofan SIS model with an infective vector on complex networksrdquoCommunications in Nonlinear Science amp Numerical Simulationvol 27 no 1ndash3 pp 30ndash39 2015

[26] P Raja Sekhara Rao and M Naresh Kumar ldquoA dynamic modelfor infectious diseases the role of vaccination and treatmentrdquoChaos Solitons amp Fractals vol 75 pp 34ndash49 2015

[27] J O Kephart S R White and D M Chess ldquoComputers andepidemiologyrdquo IEEE Spectrum vol 30 no 5 pp 20ndash26 1993

[28] Y Moreno R Pastor-Satorras and A Vespignani ldquoEpidemicoutbreaks in complex heterogeneous networksrdquo European Phys-ical Journal B vol 26 no 4 pp 521ndash529 2002

[29] R Pastor-Satorras and A Vespignani ldquoEpidemic dynamics andendemic states in complex networksrdquo Physical Review E vol 63Article ID 066117 2001

[30] R Pastor-Satorras and A Vespignani ldquoEpidemic spreading inscale-free networksrdquo Physical Review Letters vol 86 no 14 pp3200ndash3203 2001

[31] R Pastor-Satorras and A Vespignani ldquoEpidemic dynamics infinite size scale-free networksrdquo Physical Review E vol 65 no 3Article ID 035108 2002

[32] C Moore and M E J Newman ldquoEpidemics and percolationin small-world networksrdquo Physical Review E vol 61 no 5 pp5678ndash5682 2000

[33] G Butler H I Freedman and P Waltman ldquoUniformly persis-tent systemsrdquoProceedings of the AmericanMathematical Societyvol 96 no 3 pp 425ndash430 1986

[34] X-Q ZhaoDynamical Systems in Population Biology CanadianMathematical Society Springer 2003

[35] A M LyapunovThe General Problem of the Stability of MotionTaylor amp Francis London UK 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of