DISCRETE AND CONTINUOUS doi103934dcdsb2014193133DYNAMICAL SYSTEMS SERIES BVolume 19 Number 10 December 2014 pp 3133ndash3145

A PERIODIC ROSS-MACDONALD MODEL IN A

PATCHY ENVIRONMENT

Daozhou Gao

Francis I Proctor Foundation for Research in OphthalmologyUniversity of California San Francisco

San Francisco CA 94143 USA

Yijun Lou

Department of Applied Mathematics

The Hong Kong Polytechnic University

Hung Hom Kowloon Hong Kong China

Shigui Ruan

Department of Mathematics

University of MiamiCoral Gables FL 33124 USA

Dedicated to Professor Chris Cosner on the occasion of his 60th birthday

Abstract Based on the classical Ross-Macdonald model in this paper wepropose a periodic malaria model to incorporate the effects of temporal and

spatial heterogeneity on disease transmission The temporal heterogeneity is

described by assuming that some model coefficients are time-periodic while thespatial heterogeneity is modeled by using a multi-patch structure and assuming

that individuals travel among patches We calculate the basic reproduction

number R0 and show that either the disease-free periodic solution is globallyasymptotically stable if R0 le 1 or the positive periodic solution is globally

asymptotically stable if R0 gt 1 Numerical simulations are conducted to

confirm the analytical results and explore the effect of travel control on thedisease prevalence

1 Introduction Malaria a widely prevalent vector-borne disease in tropical andsubtropical areas is caused by a parasite that is transmitted to humans and manyother animals by the Anopheles mosquito Once infected people may experience avariety of symptoms ranging from absent or very mild symptoms to severe compli-cations and even death It is one of the most deadly infectious diseases that causesmajor economic loss due to illness and death in humans

The Ross-Macdonald model is the earliest and also simplest mathematical modeldescribing malaria transmission between human and mosquito populations It wasinitially proposed by Ross [31] in 1911 and later extended by Macdonald [23 24 25]in 1950s The modeling framework is now widely used for malaria and some other

2010 Mathematics Subject Classification Primary 92D30 Secondary 34D23Key words and phrases Malaria patch model seasonality threshold dynamics basic repro-

duction numberThe work was supported in part by the Models of Infectious Disease Agent Study (MIDAS)

(UCSF 1 U01 GM087728) NSFC (11301442) NIH (R01GM093345) and NSF (DMS-1022728

DMS-1412454)

3133

3134 DAOZHOU GAO YIJUN LOU AND SHIGUI RUAN

mosquito-borne diseases It captures the essential features of malaria transmissionprocess but ignores many factors of real-world ecology and epidemiology (see RuanXiao and Beier [32])

One omission in the classical Ross-Macdonald model is the temporal heterogene-ity in the number of both populations and the human feeding rate of mosquitoes Inmany nations like Niger seasonal human migration has a long history and destina-tions and reasons vary by community and ethnic group (Rain [29]) It is a commonsense that there are more mosquitoes in the summer and fewer in the winter Thebiological activity and geographic distribution of a malaria parasite and its vectorare greatly influenced by climatic factors such as rainfall temperature and humid-ity (Craig et al [9] Gao et al [12]) These influences can be investigated byassuming some parameters to be time dependent In recent years epidemic modelswith seasonal fluctuations have been proposed and explored by many researchersA vector-borne model for epidemics of cutaneous leishmaniasis with a seasonallyvarying vector population and a distributed delay in humans was proposed andstudied by Bacaer and Guernaoui [4] Dembele Friedman and Yakubu [10] intro-duced a new malaria model with periodic mosquito birth and death rates Bai andZhou [6] developed a SIRS model for bacillary dysentery with periodic coefficientsUsing an ODE model with periodic transmission rates Zhang et al [38] consideredthe transmission dynamics of rabies from dogs to humans in China Liu Zhao andZhou [17] used a compartmental model with periodic infection rate and reactivationrate to describe the TB transmission in China For a detailed discussion of seasonalinfectious diseases we refer the reader to two excellent survey papers (Altizer etal [1] Grassly and Fraser [13]) and the introduction of a recent paper (Lou Louand Wu [19]) Periodicity mainly occurs in the contact rate birth or death ratevaccination rate etc

Another omission is the spatial movements of hosts between different patchesThe migration of humans can influence disease spread in a complicated way (Gaoand Ruan [11]) To have a better understanding of the disease dynamics it isnecessary to incorporate periodic variations and population dispersal into epidemicmodels These two issues are considered in (Lou and Zhao [20]) via a periodicreaction-diffusion-advection model In this paper we will formulate a periodic Ross-Macdonald model in a patchy environment and establish the threshold dynamicsof the model in Section 2 and Section 3 respectively The last section gives somenumerical simulations and a brief discussion of our main results and future researchdirections

2 Model formulation Most mosquitoes can only travel a couple of kilometersthroughout their lifetime (Costantini et al [8] and Midega et al [26]) so we as-sume no movement for vector populations between patches (see Auger et al [3])For simplicity the human vital dynamics are ignored and the population model forvectors is the same as that studied in Smith Dushoff and Mckenzie [33] Follow-ing the classical Ross-Macdonald model we divide the adult female mosquito andhuman populations into two classes in each patch susceptible and infectious Thetotal number of patches is p We assume that the total populations of humans andmosquitoes at time t in patch i are Hi(t) and Vi(t) respectively Let hi(t) andvi(t) denote the numbers of infectious humans and infectious mosquitoes in patchi respectively The interactions between humans and mosquitoes in patch i can bedescribed by the following periodic system with non-negative initial conditions

PERIODIC ROSS-MACDONALD MODEL 3135

Table 1 The parameters in model (21) and their descriptions

Symbol Descriptionεi(t) gt 0 Recruitment rate of mosquitoesdi(t) gt 0 Mortality rate of mosquitoesai(t) gt 0 Mosquito biting ratebi gt 0 Transmission probability from infectious mosquitoes

to susceptible humansci gt 0 Transmission probability from infectious humans

to susceptible mosquitoes1ri gt 0 Human infectious periodmij(t) ge 0 Human immigration rate from patch j to patch i for i 6= jminusmii(t) ge 0 Human emigration rate from patch i

dHi(t)

dt=

psumj=1

mij(t)Hj(t) 1 le i le p (21a)

dVi(t)

dt= εi(t)minus di(t)Vi(t) 1 le i le p (21b)

dhi(t)

dt= biai(t)

Hi(t)minus hi(t)Hi(t)

vi(t)minus rihi(t) +

psumj=1

mij(t)hj(t) 1 le i le p (21c)

dvi(t)

dt= ciai(t)

hi(t)

Hi(t)(Vi(t)minus vi(t))minus di(t)vi(t) 1 le i le p (21d)

The model parameters and their descriptions are listed in Table 1 To incor-porate the seasonal mosquito dynamics and human migration all time-dependentparameters in system (21) are continuous and periodic functions with the sameperiod ω = 365 days We assume that there is no death or birth during travel sothe emigration rate of humans in patch i minusmii(t) ge 0 satisfies

psumj=1

mji(t) = 0 for i = 1 p and t isin [0 ω]

For travel rates between two patches we assume there must be some travelersbetween different patches at some specific seasons Mathematically we assume thatthe matrix (

int ω0mij(t)dt)ptimesp which is related to the travel rates between different

patches is irreducible The notation H(t) will mean (H1(t) Hp(t)) with similarnotations for other vectors We will denote both zero value and zero vector by 0but this should cause no confusion The following theorem indicates that model(21) is mathematically and epidemiologically well-posed

Theorem 21 For any initial value z in

Γ = (HV h v) isin R4p+ hi le Hi vi le Vi i = 1 p

system (21) has a unique nonnegative bounded solution through z for all t ge 0

Proof Let G(t z) be the vector field described by (21) with z(t) isin Γ Then G(t z)is continuous and Lipschitzian in z on each compact subset of R1 times Γ ClearlyGk(t z) ge 0 whenever z ge 0 and zk = 0 k = 1 4p It follows from Theorem521 in Smith [34] that there exists a unique nonnegative solution for system (21)

3136 DAOZHOU GAO YIJUN LOU AND SHIGUI RUAN

through z isin Γ in its maximal interval of existence The total numbers for hosts and

vectors Nh(t) =psumi=1

Hi(t) and Nv(t) =psumi=1

Vi(t) satisfy

dNh(t)

dt=

psumi=1

psumj=1

mij(t)Hj(t) =

psumi=1

psumj=1

mji(t)Hi(t) = 0

and

dNv(t)

dt=

psumi=1

(εi(t)minus di(t)Vi(t)) le εminus dNv(t)

respectively where ε = max0letleω

( psumi=1

εi(t))

and d = min0letleω

(min

1leilepdi(t)

)are positive

constants Thus Nh(t) equiv Nh(0) The comparison principle (see Theorem B1 inSmith and Waltman [35]) implies that Nv(t) is ultimately bounded Hence everysolution of (21) exists globally

3 Mathematical analysis In this section we first show the existence of a uniquedisease-free periodic solution and then evaluate the basic reproduction number forthe periodic system By using the theory of monotone dynamical systems (Smith[34] Zhao [40]) and internally chain transitive sets (Hirsch Smith and Zhao [15]Zhao [40]) we establish the global dynamics of the system The following result isanalogous to Lemma 1 in Cosner et al [7] for the autonomous case

Lemma 31 The human migration model (21a) with Hi(0) ge 0 for i = 1 pand Nh(0) gt 0 has a unique positive ω-periodic solution Hlowast(t) equiv (Hlowast1 (t) Hlowastp (t))which is globally asymptotically stable The mosquito growth model (21b) withVi(0) ge 0 for i = 1 p has a unique positive ω-periodic solution V lowast(t) equiv(V lowast1 (t) V lowastp (t)) which is globally asymptotically stable

Proof Note that the travel rate matrix M(t) equiv (mij(t))ptimesp has nonnegative off-diagonal entries and the matrix (

int ω0mij(t)dt)ptimesp is irreducible Let φ(t) be the fun-

damental matrix of system (21a) which satisfies dφ(t)dt = M(t)φ(t) and φ(0) = IpThen the pω period fundamental matrix φ(pω) is a strongly positive operator (Smith[34]) By using the pω period fundamental matrix φ(pω) in the proof of Theorem 1in Aronsson and Kellogg [2] we see that the pω periodic system (21a) has a positivepω periodic solution Hlowast(t) which is globally asymptotically stable with respect toall nonzero solutions By using the global attractivity of the pω periodic solutionwe can further show it is also ω periodic by using a similar argument in Weng and

Zhao [37] Moreover it is easy to see thatpsumi=1

Hlowasti (t) = Nh(0)

The i-th equation of (21b) has a unique positive periodic solution

V lowasti (t) = eminus

tint0

di(s)ds(V lowasti (0)+

tint0

e

sint0

di(τ)dτεi(s)ds

)with V lowasti (0) =

ωint0

e

sint0

di(τ)dτεi(s)ds

e

ωint0

di(s)dsminus 1

which is globally asymptotically stable

Remark 31 We can relax the positivity assumption on εi(t) to [εi] gt 0 where[εi] = 1

ω

int ω0εi(t)dt is the average value of εi(t) over [0 ω] or equivalently εi(t) gt 0

PERIODIC ROSS-MACDONALD MODEL 3137

for some t isin [0 2π) See Lemma 21 in Zhang and Teng [39] for results on a generalnonautonomous system of the form dVi(t)dt = εi(t)minus di(t)Vi(t)

The above result guarantees that system (21) admits a unique disease-free peri-odic solution

E0(t) = (Hlowast1 (t) Hlowastp (t) V lowast1 (t) V lowastp (t) 0 0 0 0)

Biologically both human and mosquito populations in each patch are seasonallyforced due to periodically changing human migration and seasonal birth deathand biting rates of mosquitoes respectively Now we consider the asymptoticallyperiodic system for malaria transmission

dhi(t)

dt= biai(t)

Hlowasti (t)minus hi(t)Hlowasti (t)

vi(t)minus rihi(t) +

psumj=1

mij(t)hj(t) 1 le i le p

dvi(t)

dt= ciai(t)

hi(t)

Hlowasti (t)(V lowasti (t)minus vi(t))minus di(t)vi(t) 1 le i le p

(31)

In what follows we use the definition of Bacaer and Guernaoui [4] (see alsoBacaer [5]) and the general calculation method in Wang and Zhao [36] to evaluatethe basic reproduction number R0 for system (31) Then we analyze the thresholddynamics of system (31) Finally we study global dynamics for the whole system(21) by applying the theory of internally chain transitive sets (Hirsch Smith andZhao [15] and Zhao [40])

Let x = (h1 hp v1 vp) be the vector of all infectious class variables Thelinearization of system (31) at the disease-free equilibrium P0 = (0 0 0 0)is

dx

dt= (F (t)minus V (t))x (32)

where

F (t) =

[0 AB 0

]and V (t) =

[C 00 D

]

Here A = (δijai(t)bi)ptimesp B = (δijai(t)ciVlowasti (t)Hlowasti (t))ptimesp C = (δijri minusmij(t))ptimesp

D = (δijdi(t))ptimesp and δij denotes the Kronecker delta function (ie 1 when i = jand 0 elsewhere)

Let Y (t s) t ge s be the evolution operator of the linear ω-periodic system

dy

dt= minusV (t)y

That is for each s isin R1 the 2ptimes 2p matrix Y (t s) satisfies

dY (t s)

dt= minusV (t)Y (t s) forallt ge s Y (s s) = I2p

where I2p is the 2ptimes 2p identity matrixLet Cω be the ordered Banach space of all ω-periodic functions from R1 to R2p

equipped with the maximum norm We define a linear operator L Cω rarr Cω by

(Lφ)(t) =

int t

minusinfinY (t s)F (s)φ(s)ds

=

int infin0

Y (t tminus a)F (tminus a)φ(tminus a)da forallt isin R1 φ isin Cω(33)

It is easy to verify that conditions (A1)-(A7) in Wang and Zhao [36] are satisfiedTherefore the basic reproduction number of the periodic system (31) is then defined

3138 DAOZHOU GAO YIJUN LOU AND SHIGUI RUAN

as R0 = ρ(L) the spectral radius of L The following lemma shows that the basicreproduction numberR0 is the threshold parameter for local stability of the disease-free steady state

Lemma 32 (Theorem 22 in Wang and Zhao [36]) Let ΦFminusV (t) and ρ(ΦFminusV (ω))be the monodromy matrix of system (32) and the spectral radius of ΦFminusV (ω) re-spectively The following statements are valid

(i) R0 = 1 if and only if ρ(ΦFminusV (ω)) = 1(ii) R0 gt 1 if and only if ρ(ΦFminusV (ω)) gt 1

(iii) R0 lt 1 if and only if ρ(ΦFminusV (ω)) lt 1

Thus the disease-free equilibrium P0 of system (31) is locally asymptotically stableif R0 lt 1 and unstable if R0 gt 1

Lou and Zhao [20] presented a global qualitative analysis for system (31) with asingle patch We will show that the system with multiple patches does not exhibitmore complicated dynamics

Lemma 33 Let Dt = (h v) 0 le h le Hlowast(t) 0 le v le V lowast(t) For each x(0) =

(h(0) v(0)) isin R2p+ system (31) admits a unique solution x(t) = (h(t) v(t)) isin R2p

+

through x(0) for all t ge 0 Moreover x(t) isin Dt forallt ge 0 provided that x(0) isin D0

Proof Let K(t x) denote the vector field described by (31) Since K(t x) is con-tinuous and locally Lipschitizan in x in any bounded set By Theorem 521 inSmith [34] and Theorem 21 system (31) has a unique solution (h(t) v(t)) through

x(0) isin R2p+ which exists globally

To prove the second statement we let u(t) = Hlowast(t)minush(t) and w(t) = V lowast(t)minusv(t)where (h(t) v(t)) is the solution through x(0) isin D0 Then (u(t) w(t)) satisfies thefollowing system with u(0) = Hlowast(0)minus h(0) ge 0 and w(0) = V lowast(0)minus v(0) ge 0

dui(t)

dt= minusai(t)bi

ui(t)

Hlowasti (t)(V lowasti (t)minus wi(t))minus riui(t) + riH

lowasti (t) +

psumj=1

mij(t)uj(t)

dwi(t)

dt= εi(t)minus ai(t)ci

Hlowasti (t)minus ui(t)Hlowasti (t)

wi(t)minus di(t)wi(t)

for i = 1 p It then follows from Theorem 511 in Smith [34] that (u(t) w(t)) ge0 That is the solution for system (31) x(t) = (h(t) v(t)) le (Hlowast(t) V lowast(t)) holdsfor all t ge 0 whenever x(0) isin D0

Theorem 31 System (31) admits a unique positive ω-periodic solution de-noted by P lowast(t) = (hlowast(t) vlowast(t)) = (hlowast1(t) hlowastp(t) v

lowast1(t) vlowastp(t)) which is globally

asymptotically stable with initial values in D00 if R0 gt 1 and the disease-freeequilibrium P0 is globally asymptotically stable in D0 if R0 le 1

Proof To show the global asymptotic stability of P lowast(t) or P0 it suffices to verifythat K(t x) R1

+ times D0 rarr R2p satisfies (A1)-(A3) in Theorem 312 in Zhao [40]For every x = (h v) ge 0 with hi = 0 or vi = 0 t isin R1

+ we have

Ki(t x) = ai(t)bivi +sum

1lejlepj 6=i

mij(t)hj ge 0

or

Kp+i(t x) = ai(t)cihi

Hlowasti (t)V lowasti (t) ge 0

for i = 1 p So (A1) is satisfied

PERIODIC ROSS-MACDONALD MODEL 3139

By Lemma 33 the system (31) is cooperative in Dt and the ω-periodic semi-flow is monotone We can further prove that the pω-periodic semiflow is stronglymonotone in IntD0 the interior of D0 due to the irreducibility of (

int ω0mij(t)dt)ptimesp

For each t ge 0 i = 1 p there hold

Ki(t αx) = ai(t)biHlowasti (t)minus αhi

Hlowasti (t)αvi minus riαhi +

psumj=1

mij(t)αhj

gt α

(ai(t)bi

Hlowasti (t)minus hiHlowasti (t)

vi minus rihi +

psumj=1

mij(t)hj

)= αKi(t x)

and

Kp+i(t αx) = ai(t)ciαhiHlowasti (t)

(V lowasti (t)minus αvi)minus di(t)αvi

gt α

(ai(t)ci

hiHlowasti (t)

(V lowasti (t)minus vi)minus di(t)vi)

= αKp+i(t x)

for each x 0 α isin (0 1) That is K(t middot) is strictly subhomogeneous on R2p+

Moreover K(t 0) equiv 0 and all solutions are ultimately bounded Therefore byTheorem 312 in Zhao [40] or Theorem 234 in Zhao [40] as applied to the pωsolution map associated with (31) the proof is complete

Using the theory of internally chain transitive sets (Hirsch Smith and Zhao [15]or Zhao [40]) as argued in Lou and Zhao [22] we find that the disease either diesout or persists and stabilizes at a periodic state

Theorem 32 For system (21) if R0 gt 1 then there is a unique positive ω-periodic solution denoted by Elowast(t) = (Hlowast(t) V lowast(t) hlowast(t) vlowast(t)) which is globallyasymptotically stable if R0 le 1 then the disease-free periodic solution E0(t) isglobally asymptotically stable

Proof Let P be the Poincare map of the system (21) on R4p+ that is

P (H(0) V (0) h(0) v(0)) = (H(ω) V (ω) h(ω) v(ω))

Then P is a compact map Let Ω =Ω(H(0) V (0) h(0) v(0)) be the omega limitset of Pn(H(0) V (0) h(0) v(0)) It then follows from Lemma 21prime in Hirsch Smithand Zhao [15] (see also Lemma 121prime in Zhao [40]) that Ω is an internally chaintransitive set for P Based on Lemma 31 we have

limtrarrinfin

(H(t) V (t)) = (Hlowast(t) V lowast(t))

for any H(0) gt 0 and V (0) gt 0 Thus Ω = (Hlowast(0) V lowast(0)) times Ω1 for some

Ω1 sub R2p+ and

Pn|Ω(Hlowast(0) V lowast(0) h(0) v(0)) = (Hlowast(0) V lowast(0) Pn1 (h(0) v(0)))

where P1 is the Poincare map associated with the system (31) Since Ω is aninternally chain transitive set for Pn then Ω1 is an internally chain transitive setfor Pn1 Next we will discuss different scenarios when R0 le 1 and R0 gt 1

Case 1 R0 le 1 In this case the zero equilibrium P0 is globally asymptoticallystable for system (31) according to Theorem 31 It then follows from Theorem 32and Remark 46 of Hirsch Smith and Zhao [15] that Ω1 = (0 0) which further

3140 DAOZHOU GAO YIJUN LOU AND SHIGUI RUAN

implies that Ω = (Hlowast(0) V lowast(0) 0 0) Therefore the disease free periodic solution(Hlowast(t) V lowast(t) 0 0) is globally asymptotically stable

Case 2 R0 gt 1 In this case there are two fixed points for the map P1 (0 0)and (hlowast(0) vlowast(0)) We claim that Ω1 6= (0 0) for all h(0) gt 0 or v(0) gt 0Assume that by contradiction Ω1 = (0 0) for some h(0) gt 0 or v(0) gt 0 ThenΩ = (Hlowast(0) V lowast(0) 0 0) and we have

limtrarrinfin

(H(t) V (t) h(t) v(t)) = (Hlowast(t) V lowast(t) 0 0)

Since R0 gt 1 there exists some δ gt 0 such that the following linear system isunstable

dhi(t)

dt= biai(t)(1minus δ)vi(t)minus rihi(t) +

psumj=1

mij(t)hj(t) 1 le i le p

dvi(t)

dt= ciai(t)(1minus δ)

V lowasti (t)

Hlowasti (t)hi(t)minus di(t)vi(t) 1 le i le p

(34)

Moreover there exists some T0 gt 0 such that

|(H(t) V (t) h(t) v(t))minus (Hlowast(t) V lowast(t) 0 0)| lt δ min0letleω

Hlowast(t) V lowast(t) forallt gt T0

Hence we have

dhi(t)

dtge biai(t)(1minus δ)vi(t)minus rihi(t) +

psumj=1

mij(t)hj(t) 1 le i le p

dvi(t)

dtge ciai(t)(1minus δ)

V lowasti (t)

Hlowasti (t)hi(t)minus di(t)vi(t) 1 le i le p

when t gt T0 According to the instability of system (34) and the comparison princi-ple hi(t) and vi(t) will go unbounded as trarrinfin which contradicts the boundednessof solutions (Theorem 21)

Since Ω1 6= (0 0) and (hlowast(0) vlowast(0)) is globally asymptotically stable for Pn1 in

R2p+ (0 0) it follows that Ω1

⋂W s((hlowast(0) vlowast(0))) 6= empty where W s((hlowast(0) vlowast(0)))

is the stable set for (hlowast(0) vlowast(0)) By Theorem 31 and Remark 46 in Hirsch Smithand Zhao [15] we then get Ω1 = (hlowast(0) vlowast(0)) Thus Ω = (Hlowast(0) V lowast(0) hlowast(0)vlowast(0)) and hence the statement for the case when R0 gt 1 is valid

4 Simulations and discussions In this section we provide some numerical sim-ulations for the two-patch case to support our analytical conclusions The followingmatrix and lemma will be used in numerical computation for the basic reproduc-tion number R0 Let W (t λ) be the monodromy matrix (see Hale [14]) of thehomogeneous linear ω-periodic system

dx

dt=(minus V (t) +

F (t)

λ

)x t isin R1

with parameter λ isin (0infin)

Lemma 41 (Theorem 21 in Wang and Zhao [36]) The following statements arevalid

(i) If ρ(W (ω λ)) = 1 has a positive solution λ0 then λ0 is an eigenvalue of Land hence R0 gt 0 where L is the next generation operator defined in (33)

(ii) If R0 gt 0 then λ = R0 is the unique solution of ρ(W (ω λ)) = 1(iii) R0 = 0 if and only if ρ(W (ω λ)) lt 1 for all λ gt 0

PERIODIC ROSS-MACDONALD MODEL 3141

According to Lemma 41(ii) we can show that the basic reproduction number

for (31) is proportional to the biting rate that is R0 = kR0 if βi(t) = kβi(t) fork gt 0 and 1 le i le p (see Lou and Zhao [21] or Liu and Zhao [18])

To implement numerical simulations we choose the baseline parameter values asfollows bi = 02 ri = 002 ci = 03 di(t) = 01 for i = 1 2 For the first patch

ε1(t) = 125minus(5 cos( 2πt365 )+25 sin( 2πt

365 ))minus(5 cos( 4πt365 )minus25 sin( 4πt

365 )) ge 125minus5radic

5 gt 0

a1(t) = 0028ε1(t) gt 0 In the second patch ε2(t) = 15 minus 12 cos( 2πt365 ) a2(t) =

018 minus 012 cos( 2πt365 ) Then using Lemma 41 we can numerically compute that

the respective reproduction numbers of the disease in patches 1 and 2 are R10 =17571 gt 1 and R20 = 08470 lt 1 and the disease persists in the first patch but diesout in the second patch (see Figure 1(a)) When humans migrate between these twopatches with m12(t) = 0006 minus 0004 cos( 2πt

365 ) and m21(t) = 0006 + 0004 cos( 2πt365 )

the basic reproduction number is R0 = 13944 gt 1 Thus the disease becomesendemic in both patches (see Figure 1(b))

0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

time t

h 1amph 2

(a)

h

1

h2

0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

time t

h 1amph 2

(b)

h

1

h2

Figure 1 Numerical solution of system (31) with initial con-ditions H1(0) = 200 V1(0) = 50 h1(0) = 15 v1(0) = 10 andH2(0) = 300 V2(0) = 30 h2(0) = 30 v2(0) = 5 The curves ofthe number of infected humans in patches 1 and 2 respectivelyare blue solid and red dashed (a) when there is no human move-ment R10 = 17571 gt 1 and R20 = 08470 lt 1 indicate the diseasepersists in patch 1 but dies out in patch 2 (b) when human move-ment presents R0 = 13944 gt 1 indicates the disease persists inboth patches and will stabilize in a seasonal pattern

The basic reproduction number for the corresponding time-averaged autonomoussystem (see Wang and Zhao [36] and Zhang et al [38]) is R0 = 12364 By Lemma

3142 DAOZHOU GAO YIJUN LOU AND SHIGUI RUAN

41 or the linearity of R0 in the mosquito biting rates we can multiply the bitingrates by an appropriate constant such that R0 below 1 while R0 above 1 Thus theautonomous malaria model may underestimate disease severity in some transmissionsettings

In order to investigate the effect of possible travel control on the malaria riskwe determine the disease risk when shifting the phases of the movement functionswhile keeping their respective mean values and forcing strengths to be the sameWith the same parameter values and initial data except that the migration ratesm12(t) = 0006 minus 0004 cos( 2πt

365 + φ1) and m21(t) = 0006 + 0004 cos( 2πt365 + φ2)

we evaluate the annual malaria prevalence over the two patches for φ1 φ2 isin [0 2π]Then Figure 2 presents the effect of phase shifting for movement functions on diseaseprevalence The approximate range of malaria prevalence is from 17 to 23 Thisresult indicates that appropriate travel control can definitely reduce the disease riskand therefore an optimal travel control measure can be proposed

φ2

φ 1

0 1 2 3 4 5 60

1

2

3

4

5

6

0175

018

0185

019

0195

02

0205

021

0215

022

0225

Figure 2 Effect of phase shift for movement rates between twopatches on disease risk

In this project we have developed a compartmental model to address seasonalvariations of malaria transmission among different regions by incorporating seasonalhuman movement and periodic changing in mosquito ecology We then derive thebasic reproduction number and establish the global dynamics of the model Math-ematically we generalize the model and results presented in Cosner et al [7] andAuger et al [3] from a constant environment to a periodic environment Biologi-cally our result gives a possible explanation to the fact that malaria incidence showseasonal peaks in most endemic areas (Roca-Feltrer et al [30]) The numericalexamples suggest that a better understanding of seasonal human migration can

PERIODIC ROSS-MACDONALD MODEL 3143

help us to prevent and control malaria transmission in a spatially heterogeneousenvironment

It would be interesting to assess the impacts of climate change on the emergingand reemerging of mosquito-borne diseases especially malaria It is believed thatglobal warming would cause climate change while climatic factors such as precip-itation temperature humidity and wind speed affect the biological activity andgeographic distribution of the pathogen and its vector It sometimes even affectshuman behavior through sea-level rise more extreme weather events etc

The current and potential future impact of climate change on human healthhas attracted considerable attention in recent years Many of the early studies(see Ostfeld [27] and the references cited therein) claimed that recent and futuretrends in climate warming were likely to increase the severity and global distribu-tion of vector-borne diseases while there is still substantial debate about the linkbetween climate change and the spread of infectious diseases (Lafferty [16]) Thecontroversy is partially due to the fact that although there is massive work us-ing empirical-statistical models to explore the relationship between climatic factorsand the distribution and prevalence of vector-borne diseases there are very fewstudies incorporating climatic factors into mathematical models to describe diseasetransmission

The research that comes closest to what we expect to do has been conducted byParham and Michael in [28] where they used a system of delay differential equationsto model malaria transmission under varying climatic and environmental conditionsModel parameters are assumed to be temperature-dependent or rainfall-dependentor both Due to the complexity of their model mathematical analysis was notpossible and only numerical simulations were carried out Furthermore the basicreproduction number is also difficult to project through their model We wouldlike to extend our model to a mathematically tractable climate-based model in thefuture

Acknowledgments Part of this work was done while the first author was a grad-uate student in the Department of Mathematics University of Miami We thankDrs Chris Cosner David Smith and Xiao-Qiang Zhao for helpful discussions onthis project

REFERENCES

[1] S Altizer A Dobson P Hosseini P Hudson M Pascual and P Rohani Seasonality and

the dynamics of infectious diseases Ecol Lett 9 (2006) 467ndash484[2] G Aronsson and R B Kellogg On a differential equation arising from compartmental anal-

ysis Math Biosci 38 (1973) 113ndash122[3] P Auger E Kouokam G Sallet M Tchuente and B Tsanou The Ross-Macdonald model

in a patchy environment Math Biosci 216 (2008) 123ndash131

[4] N Bacaer and S Guernaoui The epidemic threshold of vector-borne diseases with seasonalityJ Math Biol 53 (2006) 421ndash436

[5] N Bacaer Approximation of the basic reproduction number R0 for vector-borne diseases with

a periodic vector population Bull Math Biol 69 (2007) 1067ndash1091[6] Z Bai and Y Zhou Threshold dynamics of a Bacillary Dysentery model with seasonal fluc-

tuation Discrete Contin Dyn Syst Ser B 15 (2011) 1ndash14

[7] C Cosner J C Beier R S Cantrell D Impoinvil L Kapitanski M D Potts A Troyoand S Ruan The effects of human movement on the persistence of vector-borne diseases J

Theoret Biol 258 (2009) 550ndash560

PERIODIC ROSS-MACDONALD MODEL 3135

Table 1 The parameters in model (21) and their descriptions

Symbol Descriptionεi(t) gt 0 Recruitment rate of mosquitoesdi(t) gt 0 Mortality rate of mosquitoesai(t) gt 0 Mosquito biting ratebi gt 0 Transmission probability from infectious mosquitoes

to susceptible humansci gt 0 Transmission probability from infectious humans

to susceptible mosquitoes1ri gt 0 Human infectious periodmij(t) ge 0 Human immigration rate from patch j to patch i for i 6= jminusmii(t) ge 0 Human emigration rate from patch i

dHi(t)

dt=

psumj=1

mij(t)Hj(t) 1 le i le p (21a)

dVi(t)

dt= εi(t)minus di(t)Vi(t) 1 le i le p (21b)

dhi(t)

dt= biai(t)

Hi(t)minus hi(t)Hi(t)

vi(t)minus rihi(t) +

psumj=1

mij(t)hj(t) 1 le i le p (21c)

dvi(t)

dt= ciai(t)

hi(t)

Hi(t)(Vi(t)minus vi(t))minus di(t)vi(t) 1 le i le p (21d)

The model parameters and their descriptions are listed in Table 1 To incor-porate the seasonal mosquito dynamics and human migration all time-dependentparameters in system (21) are continuous and periodic functions with the sameperiod ω = 365 days We assume that there is no death or birth during travel sothe emigration rate of humans in patch i minusmii(t) ge 0 satisfies

psumj=1

mji(t) = 0 for i = 1 p and t isin [0 ω]

For travel rates between two patches we assume there must be some travelersbetween different patches at some specific seasons Mathematically we assume thatthe matrix (

int ω0mij(t)dt)ptimesp which is related to the travel rates between different

patches is irreducible The notation H(t) will mean (H1(t) Hp(t)) with similarnotations for other vectors We will denote both zero value and zero vector by 0but this should cause no confusion The following theorem indicates that model(21) is mathematically and epidemiologically well-posed

Theorem 21 For any initial value z in

Γ = (HV h v) isin R4p+ hi le Hi vi le Vi i = 1 p

system (21) has a unique nonnegative bounded solution through z for all t ge 0

Proof Let G(t z) be the vector field described by (21) with z(t) isin Γ Then G(t z)is continuous and Lipschitzian in z on each compact subset of R1 times Γ ClearlyGk(t z) ge 0 whenever z ge 0 and zk = 0 k = 1 4p It follows from Theorem521 in Smith [34] that there exists a unique nonnegative solution for system (21)

3136 DAOZHOU GAO YIJUN LOU AND SHIGUI RUAN

through z isin Γ in its maximal interval of existence The total numbers for hosts and

vectors Nh(t) =psumi=1

Hi(t) and Nv(t) =psumi=1

Vi(t) satisfy

dNh(t)

dt=

psumi=1

psumj=1

mij(t)Hj(t) =

psumi=1

psumj=1

mji(t)Hi(t) = 0

and

dNv(t)

dt=

psumi=1

(εi(t)minus di(t)Vi(t)) le εminus dNv(t)

respectively where ε = max0letleω

( psumi=1

εi(t))

and d = min0letleω

(min

1leilepdi(t)

)are positive

constants Thus Nh(t) equiv Nh(0) The comparison principle (see Theorem B1 inSmith and Waltman [35]) implies that Nv(t) is ultimately bounded Hence everysolution of (21) exists globally

3 Mathematical analysis In this section we first show the existence of a uniquedisease-free periodic solution and then evaluate the basic reproduction number forthe periodic system By using the theory of monotone dynamical systems (Smith[34] Zhao [40]) and internally chain transitive sets (Hirsch Smith and Zhao [15]Zhao [40]) we establish the global dynamics of the system The following result isanalogous to Lemma 1 in Cosner et al [7] for the autonomous case

Lemma 31 The human migration model (21a) with Hi(0) ge 0 for i = 1 pand Nh(0) gt 0 has a unique positive ω-periodic solution Hlowast(t) equiv (Hlowast1 (t) Hlowastp (t))which is globally asymptotically stable The mosquito growth model (21b) withVi(0) ge 0 for i = 1 p has a unique positive ω-periodic solution V lowast(t) equiv(V lowast1 (t) V lowastp (t)) which is globally asymptotically stable

Proof Note that the travel rate matrix M(t) equiv (mij(t))ptimesp has nonnegative off-diagonal entries and the matrix (

int ω0mij(t)dt)ptimesp is irreducible Let φ(t) be the fun-

damental matrix of system (21a) which satisfies dφ(t)dt = M(t)φ(t) and φ(0) = IpThen the pω period fundamental matrix φ(pω) is a strongly positive operator (Smith[34]) By using the pω period fundamental matrix φ(pω) in the proof of Theorem 1in Aronsson and Kellogg [2] we see that the pω periodic system (21a) has a positivepω periodic solution Hlowast(t) which is globally asymptotically stable with respect toall nonzero solutions By using the global attractivity of the pω periodic solutionwe can further show it is also ω periodic by using a similar argument in Weng and

Zhao [37] Moreover it is easy to see thatpsumi=1

Hlowasti (t) = Nh(0)

The i-th equation of (21b) has a unique positive periodic solution

V lowasti (t) = eminus

tint0

di(s)ds(V lowasti (0)+

tint0

e

sint0

di(τ)dτεi(s)ds

)with V lowasti (0) =

ωint0

e

sint0

di(τ)dτεi(s)ds

e

ωint0

di(s)dsminus 1

which is globally asymptotically stable

Remark 31 We can relax the positivity assumption on εi(t) to [εi] gt 0 where[εi] = 1

ω

int ω0εi(t)dt is the average value of εi(t) over [0 ω] or equivalently εi(t) gt 0

PERIODIC ROSS-MACDONALD MODEL 3137

for some t isin [0 2π) See Lemma 21 in Zhang and Teng [39] for results on a generalnonautonomous system of the form dVi(t)dt = εi(t)minus di(t)Vi(t)

The above result guarantees that system (21) admits a unique disease-free peri-odic solution

E0(t) = (Hlowast1 (t) Hlowastp (t) V lowast1 (t) V lowastp (t) 0 0 0 0)

Biologically both human and mosquito populations in each patch are seasonallyforced due to periodically changing human migration and seasonal birth deathand biting rates of mosquitoes respectively Now we consider the asymptoticallyperiodic system for malaria transmission

dhi(t)

dt= biai(t)

Hlowasti (t)minus hi(t)Hlowasti (t)

vi(t)minus rihi(t) +

psumj=1

mij(t)hj(t) 1 le i le p

dvi(t)

dt= ciai(t)

hi(t)

Hlowasti (t)(V lowasti (t)minus vi(t))minus di(t)vi(t) 1 le i le p

(31)

In what follows we use the definition of Bacaer and Guernaoui [4] (see alsoBacaer [5]) and the general calculation method in Wang and Zhao [36] to evaluatethe basic reproduction number R0 for system (31) Then we analyze the thresholddynamics of system (31) Finally we study global dynamics for the whole system(21) by applying the theory of internally chain transitive sets (Hirsch Smith andZhao [15] and Zhao [40])

Let x = (h1 hp v1 vp) be the vector of all infectious class variables Thelinearization of system (31) at the disease-free equilibrium P0 = (0 0 0 0)is

dx

dt= (F (t)minus V (t))x (32)

where

F (t) =

[0 AB 0

]and V (t) =

[C 00 D

]

Here A = (δijai(t)bi)ptimesp B = (δijai(t)ciVlowasti (t)Hlowasti (t))ptimesp C = (δijri minusmij(t))ptimesp

D = (δijdi(t))ptimesp and δij denotes the Kronecker delta function (ie 1 when i = jand 0 elsewhere)

Let Y (t s) t ge s be the evolution operator of the linear ω-periodic system

dy

dt= minusV (t)y

That is for each s isin R1 the 2ptimes 2p matrix Y (t s) satisfies

dY (t s)

dt= minusV (t)Y (t s) forallt ge s Y (s s) = I2p

where I2p is the 2ptimes 2p identity matrixLet Cω be the ordered Banach space of all ω-periodic functions from R1 to R2p

equipped with the maximum norm We define a linear operator L Cω rarr Cω by

(Lφ)(t) =

int t

minusinfinY (t s)F (s)φ(s)ds

=

int infin0

Y (t tminus a)F (tminus a)φ(tminus a)da forallt isin R1 φ isin Cω(33)

It is easy to verify that conditions (A1)-(A7) in Wang and Zhao [36] are satisfiedTherefore the basic reproduction number of the periodic system (31) is then defined

3138 DAOZHOU GAO YIJUN LOU AND SHIGUI RUAN

as R0 = ρ(L) the spectral radius of L The following lemma shows that the basicreproduction numberR0 is the threshold parameter for local stability of the disease-free steady state

Lemma 32 (Theorem 22 in Wang and Zhao [36]) Let ΦFminusV (t) and ρ(ΦFminusV (ω))be the monodromy matrix of system (32) and the spectral radius of ΦFminusV (ω) re-spectively The following statements are valid

(i) R0 = 1 if and only if ρ(ΦFminusV (ω)) = 1(ii) R0 gt 1 if and only if ρ(ΦFminusV (ω)) gt 1

(iii) R0 lt 1 if and only if ρ(ΦFminusV (ω)) lt 1

Thus the disease-free equilibrium P0 of system (31) is locally asymptotically stableif R0 lt 1 and unstable if R0 gt 1

Lou and Zhao [20] presented a global qualitative analysis for system (31) with asingle patch We will show that the system with multiple patches does not exhibitmore complicated dynamics

Lemma 33 Let Dt = (h v) 0 le h le Hlowast(t) 0 le v le V lowast(t) For each x(0) =

(h(0) v(0)) isin R2p+ system (31) admits a unique solution x(t) = (h(t) v(t)) isin R2p

+

through x(0) for all t ge 0 Moreover x(t) isin Dt forallt ge 0 provided that x(0) isin D0

Proof Let K(t x) denote the vector field described by (31) Since K(t x) is con-tinuous and locally Lipschitizan in x in any bounded set By Theorem 521 inSmith [34] and Theorem 21 system (31) has a unique solution (h(t) v(t)) through

x(0) isin R2p+ which exists globally

To prove the second statement we let u(t) = Hlowast(t)minush(t) and w(t) = V lowast(t)minusv(t)where (h(t) v(t)) is the solution through x(0) isin D0 Then (u(t) w(t)) satisfies thefollowing system with u(0) = Hlowast(0)minus h(0) ge 0 and w(0) = V lowast(0)minus v(0) ge 0

dui(t)

dt= minusai(t)bi

ui(t)

Hlowasti (t)(V lowasti (t)minus wi(t))minus riui(t) + riH

lowasti (t) +

psumj=1

mij(t)uj(t)

dwi(t)

dt= εi(t)minus ai(t)ci

Hlowasti (t)minus ui(t)Hlowasti (t)

wi(t)minus di(t)wi(t)

for i = 1 p It then follows from Theorem 511 in Smith [34] that (u(t) w(t)) ge0 That is the solution for system (31) x(t) = (h(t) v(t)) le (Hlowast(t) V lowast(t)) holdsfor all t ge 0 whenever x(0) isin D0

Theorem 31 System (31) admits a unique positive ω-periodic solution de-noted by P lowast(t) = (hlowast(t) vlowast(t)) = (hlowast1(t) hlowastp(t) v

lowast1(t) vlowastp(t)) which is globally

asymptotically stable with initial values in D00 if R0 gt 1 and the disease-freeequilibrium P0 is globally asymptotically stable in D0 if R0 le 1

Proof To show the global asymptotic stability of P lowast(t) or P0 it suffices to verifythat K(t x) R1

+ times D0 rarr R2p satisfies (A1)-(A3) in Theorem 312 in Zhao [40]For every x = (h v) ge 0 with hi = 0 or vi = 0 t isin R1

+ we have

Ki(t x) = ai(t)bivi +sum

1lejlepj 6=i

mij(t)hj ge 0

or

Kp+i(t x) = ai(t)cihi

Hlowasti (t)V lowasti (t) ge 0

for i = 1 p So (A1) is satisfied

PERIODIC ROSS-MACDONALD MODEL 3139

By Lemma 33 the system (31) is cooperative in Dt and the ω-periodic semi-flow is monotone We can further prove that the pω-periodic semiflow is stronglymonotone in IntD0 the interior of D0 due to the irreducibility of (

int ω0mij(t)dt)ptimesp

For each t ge 0 i = 1 p there hold

Ki(t αx) = ai(t)biHlowasti (t)minus αhi

Hlowasti (t)αvi minus riαhi +

psumj=1

mij(t)αhj

gt α

(ai(t)bi

Hlowasti (t)minus hiHlowasti (t)

vi minus rihi +

psumj=1

mij(t)hj

)= αKi(t x)

and

Kp+i(t αx) = ai(t)ciαhiHlowasti (t)

(V lowasti (t)minus αvi)minus di(t)αvi

gt α

(ai(t)ci

hiHlowasti (t)

(V lowasti (t)minus vi)minus di(t)vi)

= αKp+i(t x)

for each x 0 α isin (0 1) That is K(t middot) is strictly subhomogeneous on R2p+

Moreover K(t 0) equiv 0 and all solutions are ultimately bounded Therefore byTheorem 312 in Zhao [40] or Theorem 234 in Zhao [40] as applied to the pωsolution map associated with (31) the proof is complete

Using the theory of internally chain transitive sets (Hirsch Smith and Zhao [15]or Zhao [40]) as argued in Lou and Zhao [22] we find that the disease either diesout or persists and stabilizes at a periodic state

Theorem 32 For system (21) if R0 gt 1 then there is a unique positive ω-periodic solution denoted by Elowast(t) = (Hlowast(t) V lowast(t) hlowast(t) vlowast(t)) which is globallyasymptotically stable if R0 le 1 then the disease-free periodic solution E0(t) isglobally asymptotically stable

Proof Let P be the Poincare map of the system (21) on R4p+ that is

P (H(0) V (0) h(0) v(0)) = (H(ω) V (ω) h(ω) v(ω))

Then P is a compact map Let Ω =Ω(H(0) V (0) h(0) v(0)) be the omega limitset of Pn(H(0) V (0) h(0) v(0)) It then follows from Lemma 21prime in Hirsch Smithand Zhao [15] (see also Lemma 121prime in Zhao [40]) that Ω is an internally chaintransitive set for P Based on Lemma 31 we have

limtrarrinfin

(H(t) V (t)) = (Hlowast(t) V lowast(t))

for any H(0) gt 0 and V (0) gt 0 Thus Ω = (Hlowast(0) V lowast(0)) times Ω1 for some

Ω1 sub R2p+ and

Pn|Ω(Hlowast(0) V lowast(0) h(0) v(0)) = (Hlowast(0) V lowast(0) Pn1 (h(0) v(0)))

where P1 is the Poincare map associated with the system (31) Since Ω is aninternally chain transitive set for Pn then Ω1 is an internally chain transitive setfor Pn1 Next we will discuss different scenarios when R0 le 1 and R0 gt 1

Case 1 R0 le 1 In this case the zero equilibrium P0 is globally asymptoticallystable for system (31) according to Theorem 31 It then follows from Theorem 32and Remark 46 of Hirsch Smith and Zhao [15] that Ω1 = (0 0) which further

3140 DAOZHOU GAO YIJUN LOU AND SHIGUI RUAN

implies that Ω = (Hlowast(0) V lowast(0) 0 0) Therefore the disease free periodic solution(Hlowast(t) V lowast(t) 0 0) is globally asymptotically stable

Case 2 R0 gt 1 In this case there are two fixed points for the map P1 (0 0)and (hlowast(0) vlowast(0)) We claim that Ω1 6= (0 0) for all h(0) gt 0 or v(0) gt 0Assume that by contradiction Ω1 = (0 0) for some h(0) gt 0 or v(0) gt 0 ThenΩ = (Hlowast(0) V lowast(0) 0 0) and we have

limtrarrinfin

(H(t) V (t) h(t) v(t)) = (Hlowast(t) V lowast(t) 0 0)

Since R0 gt 1 there exists some δ gt 0 such that the following linear system isunstable

dhi(t)

dt= biai(t)(1minus δ)vi(t)minus rihi(t) +

psumj=1

mij(t)hj(t) 1 le i le p

dvi(t)

dt= ciai(t)(1minus δ)

V lowasti (t)

Hlowasti (t)hi(t)minus di(t)vi(t) 1 le i le p

(34)

Moreover there exists some T0 gt 0 such that

|(H(t) V (t) h(t) v(t))minus (Hlowast(t) V lowast(t) 0 0)| lt δ min0letleω

Hlowast(t) V lowast(t) forallt gt T0

Hence we have

dhi(t)

dtge biai(t)(1minus δ)vi(t)minus rihi(t) +

psumj=1

mij(t)hj(t) 1 le i le p

dvi(t)

dtge ciai(t)(1minus δ)

V lowasti (t)

Hlowasti (t)hi(t)minus di(t)vi(t) 1 le i le p

when t gt T0 According to the instability of system (34) and the comparison princi-ple hi(t) and vi(t) will go unbounded as trarrinfin which contradicts the boundednessof solutions (Theorem 21)

Since Ω1 6= (0 0) and (hlowast(0) vlowast(0)) is globally asymptotically stable for Pn1 in

R2p+ (0 0) it follows that Ω1

⋂W s((hlowast(0) vlowast(0))) 6= empty where W s((hlowast(0) vlowast(0)))

is the stable set for (hlowast(0) vlowast(0)) By Theorem 31 and Remark 46 in Hirsch Smithand Zhao [15] we then get Ω1 = (hlowast(0) vlowast(0)) Thus Ω = (Hlowast(0) V lowast(0) hlowast(0)vlowast(0)) and hence the statement for the case when R0 gt 1 is valid

4 Simulations and discussions In this section we provide some numerical sim-ulations for the two-patch case to support our analytical conclusions The followingmatrix and lemma will be used in numerical computation for the basic reproduc-tion number R0 Let W (t λ) be the monodromy matrix (see Hale [14]) of thehomogeneous linear ω-periodic system

dx

dt=(minus V (t) +

F (t)

λ

)x t isin R1

with parameter λ isin (0infin)

Lemma 41 (Theorem 21 in Wang and Zhao [36]) The following statements arevalid

(i) If ρ(W (ω λ)) = 1 has a positive solution λ0 then λ0 is an eigenvalue of Land hence R0 gt 0 where L is the next generation operator defined in (33)

(ii) If R0 gt 0 then λ = R0 is the unique solution of ρ(W (ω λ)) = 1(iii) R0 = 0 if and only if ρ(W (ω λ)) lt 1 for all λ gt 0

PERIODIC ROSS-MACDONALD MODEL 3141

According to Lemma 41(ii) we can show that the basic reproduction number

for (31) is proportional to the biting rate that is R0 = kR0 if βi(t) = kβi(t) fork gt 0 and 1 le i le p (see Lou and Zhao [21] or Liu and Zhao [18])

To implement numerical simulations we choose the baseline parameter values asfollows bi = 02 ri = 002 ci = 03 di(t) = 01 for i = 1 2 For the first patch

ε1(t) = 125minus(5 cos( 2πt365 )+25 sin( 2πt

365 ))minus(5 cos( 4πt365 )minus25 sin( 4πt

365 )) ge 125minus5radic

5 gt 0

a1(t) = 0028ε1(t) gt 0 In the second patch ε2(t) = 15 minus 12 cos( 2πt365 ) a2(t) =

018 minus 012 cos( 2πt365 ) Then using Lemma 41 we can numerically compute that

the respective reproduction numbers of the disease in patches 1 and 2 are R10 =17571 gt 1 and R20 = 08470 lt 1 and the disease persists in the first patch but diesout in the second patch (see Figure 1(a)) When humans migrate between these twopatches with m12(t) = 0006 minus 0004 cos( 2πt

365 ) and m21(t) = 0006 + 0004 cos( 2πt365 )

the basic reproduction number is R0 = 13944 gt 1 Thus the disease becomesendemic in both patches (see Figure 1(b))

0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

time t

h 1amph 2

(a)

h

1

h2

0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

time t

h 1amph 2

(b)

h

1

h2

Figure 1 Numerical solution of system (31) with initial con-ditions H1(0) = 200 V1(0) = 50 h1(0) = 15 v1(0) = 10 andH2(0) = 300 V2(0) = 30 h2(0) = 30 v2(0) = 5 The curves ofthe number of infected humans in patches 1 and 2 respectivelyare blue solid and red dashed (a) when there is no human move-ment R10 = 17571 gt 1 and R20 = 08470 lt 1 indicate the diseasepersists in patch 1 but dies out in patch 2 (b) when human move-ment presents R0 = 13944 gt 1 indicates the disease persists inboth patches and will stabilize in a seasonal pattern

The basic reproduction number for the corresponding time-averaged autonomoussystem (see Wang and Zhao [36] and Zhang et al [38]) is R0 = 12364 By Lemma

3142 DAOZHOU GAO YIJUN LOU AND SHIGUI RUAN

41 or the linearity of R0 in the mosquito biting rates we can multiply the bitingrates by an appropriate constant such that R0 below 1 while R0 above 1 Thus theautonomous malaria model may underestimate disease severity in some transmissionsettings

In order to investigate the effect of possible travel control on the malaria riskwe determine the disease risk when shifting the phases of the movement functionswhile keeping their respective mean values and forcing strengths to be the sameWith the same parameter values and initial data except that the migration ratesm12(t) = 0006 minus 0004 cos( 2πt

365 + φ1) and m21(t) = 0006 + 0004 cos( 2πt365 + φ2)

we evaluate the annual malaria prevalence over the two patches for φ1 φ2 isin [0 2π]Then Figure 2 presents the effect of phase shifting for movement functions on diseaseprevalence The approximate range of malaria prevalence is from 17 to 23 Thisresult indicates that appropriate travel control can definitely reduce the disease riskand therefore an optimal travel control measure can be proposed

φ2

φ 1

0 1 2 3 4 5 60

1

2

3

4

5

6

0175

018

0185

019

0195

02

0205

021

0215

022

0225

Figure 2 Effect of phase shift for movement rates between twopatches on disease risk

In this project we have developed a compartmental model to address seasonalvariations of malaria transmission among different regions by incorporating seasonalhuman movement and periodic changing in mosquito ecology We then derive thebasic reproduction number and establish the global dynamics of the model Math-ematically we generalize the model and results presented in Cosner et al [7] andAuger et al [3] from a constant environment to a periodic environment Biologi-cally our result gives a possible explanation to the fact that malaria incidence showseasonal peaks in most endemic areas (Roca-Feltrer et al [30]) The numericalexamples suggest that a better understanding of seasonal human migration can

PERIODIC ROSS-MACDONALD MODEL 3143

help us to prevent and control malaria transmission in a spatially heterogeneousenvironment

It would be interesting to assess the impacts of climate change on the emergingand reemerging of mosquito-borne diseases especially malaria It is believed thatglobal warming would cause climate change while climatic factors such as precip-itation temperature humidity and wind speed affect the biological activity andgeographic distribution of the pathogen and its vector It sometimes even affectshuman behavior through sea-level rise more extreme weather events etc

The current and potential future impact of climate change on human healthhas attracted considerable attention in recent years Many of the early studies(see Ostfeld [27] and the references cited therein) claimed that recent and futuretrends in climate warming were likely to increase the severity and global distribu-tion of vector-borne diseases while there is still substantial debate about the linkbetween climate change and the spread of infectious diseases (Lafferty [16]) Thecontroversy is partially due to the fact that although there is massive work us-ing empirical-statistical models to explore the relationship between climatic factorsand the distribution and prevalence of vector-borne diseases there are very fewstudies incorporating climatic factors into mathematical models to describe diseasetransmission

The research that comes closest to what we expect to do has been conducted byParham and Michael in [28] where they used a system of delay differential equationsto model malaria transmission under varying climatic and environmental conditionsModel parameters are assumed to be temperature-dependent or rainfall-dependentor both Due to the complexity of their model mathematical analysis was notpossible and only numerical simulations were carried out Furthermore the basicreproduction number is also difficult to project through their model We wouldlike to extend our model to a mathematically tractable climate-based model in thefuture

Acknowledgments Part of this work was done while the first author was a grad-uate student in the Department of Mathematics University of Miami We thankDrs Chris Cosner David Smith and Xiao-Qiang Zhao for helpful discussions onthis project

REFERENCES

[1] S Altizer A Dobson P Hosseini P Hudson M Pascual and P Rohani Seasonality and

the dynamics of infectious diseases Ecol Lett 9 (2006) 467ndash484[2] G Aronsson and R B Kellogg On a differential equation arising from compartmental anal-

ysis Math Biosci 38 (1973) 113ndash122[3] P Auger E Kouokam G Sallet M Tchuente and B Tsanou The Ross-Macdonald model

in a patchy environment Math Biosci 216 (2008) 123ndash131

[4] N Bacaer and S Guernaoui The epidemic threshold of vector-borne diseases with seasonalityJ Math Biol 53 (2006) 421ndash436

[5] N Bacaer Approximation of the basic reproduction number R0 for vector-borne diseases with

a periodic vector population Bull Math Biol 69 (2007) 1067ndash1091[6] Z Bai and Y Zhou Threshold dynamics of a Bacillary Dysentery model with seasonal fluc-

tuation Discrete Contin Dyn Syst Ser B 15 (2011) 1ndash14

[7] C Cosner J C Beier R S Cantrell D Impoinvil L Kapitanski M D Potts A Troyoand S Ruan The effects of human movement on the persistence of vector-borne diseases J

Theoret Biol 258 (2009) 550ndash560

3136 DAOZHOU GAO YIJUN LOU AND SHIGUI RUAN

through z isin Γ in its maximal interval of existence The total numbers for hosts and

vectors Nh(t) =psumi=1

Hi(t) and Nv(t) =psumi=1

Vi(t) satisfy

dNh(t)

dt=

psumi=1

psumj=1

mij(t)Hj(t) =

psumi=1

psumj=1

mji(t)Hi(t) = 0

and

dNv(t)

dt=

psumi=1

(εi(t)minus di(t)Vi(t)) le εminus dNv(t)

respectively where ε = max0letleω

( psumi=1

εi(t))

and d = min0letleω

(min

1leilepdi(t)

)are positive

constants Thus Nh(t) equiv Nh(0) The comparison principle (see Theorem B1 inSmith and Waltman [35]) implies that Nv(t) is ultimately bounded Hence everysolution of (21) exists globally

3 Mathematical analysis In this section we first show the existence of a uniquedisease-free periodic solution and then evaluate the basic reproduction number forthe periodic system By using the theory of monotone dynamical systems (Smith[34] Zhao [40]) and internally chain transitive sets (Hirsch Smith and Zhao [15]Zhao [40]) we establish the global dynamics of the system The following result isanalogous to Lemma 1 in Cosner et al [7] for the autonomous case

Lemma 31 The human migration model (21a) with Hi(0) ge 0 for i = 1 pand Nh(0) gt 0 has a unique positive ω-periodic solution Hlowast(t) equiv (Hlowast1 (t) Hlowastp (t))which is globally asymptotically stable The mosquito growth model (21b) withVi(0) ge 0 for i = 1 p has a unique positive ω-periodic solution V lowast(t) equiv(V lowast1 (t) V lowastp (t)) which is globally asymptotically stable

Proof Note that the travel rate matrix M(t) equiv (mij(t))ptimesp has nonnegative off-diagonal entries and the matrix (

int ω0mij(t)dt)ptimesp is irreducible Let φ(t) be the fun-

damental matrix of system (21a) which satisfies dφ(t)dt = M(t)φ(t) and φ(0) = IpThen the pω period fundamental matrix φ(pω) is a strongly positive operator (Smith[34]) By using the pω period fundamental matrix φ(pω) in the proof of Theorem 1in Aronsson and Kellogg [2] we see that the pω periodic system (21a) has a positivepω periodic solution Hlowast(t) which is globally asymptotically stable with respect toall nonzero solutions By using the global attractivity of the pω periodic solutionwe can further show it is also ω periodic by using a similar argument in Weng and

Zhao [37] Moreover it is easy to see thatpsumi=1

Hlowasti (t) = Nh(0)

The i-th equation of (21b) has a unique positive periodic solution

V lowasti (t) = eminus

tint0

di(s)ds(V lowasti (0)+

tint0

e

sint0

di(τ)dτεi(s)ds

)with V lowasti (0) =

ωint0

e

sint0

di(τ)dτεi(s)ds

e

ωint0

di(s)dsminus 1

which is globally asymptotically stable

Remark 31 We can relax the positivity assumption on εi(t) to [εi] gt 0 where[εi] = 1

ω

int ω0εi(t)dt is the average value of εi(t) over [0 ω] or equivalently εi(t) gt 0

PERIODIC ROSS-MACDONALD MODEL 3137

for some t isin [0 2π) See Lemma 21 in Zhang and Teng [39] for results on a generalnonautonomous system of the form dVi(t)dt = εi(t)minus di(t)Vi(t)

The above result guarantees that system (21) admits a unique disease-free peri-odic solution

E0(t) = (Hlowast1 (t) Hlowastp (t) V lowast1 (t) V lowastp (t) 0 0 0 0)

Biologically both human and mosquito populations in each patch are seasonallyforced due to periodically changing human migration and seasonal birth deathand biting rates of mosquitoes respectively Now we consider the asymptoticallyperiodic system for malaria transmission

dhi(t)

dt= biai(t)

Hlowasti (t)minus hi(t)Hlowasti (t)

vi(t)minus rihi(t) +

psumj=1

mij(t)hj(t) 1 le i le p

dvi(t)

dt= ciai(t)

hi(t)

Hlowasti (t)(V lowasti (t)minus vi(t))minus di(t)vi(t) 1 le i le p

(31)

In what follows we use the definition of Bacaer and Guernaoui [4] (see alsoBacaer [5]) and the general calculation method in Wang and Zhao [36] to evaluatethe basic reproduction number R0 for system (31) Then we analyze the thresholddynamics of system (31) Finally we study global dynamics for the whole system(21) by applying the theory of internally chain transitive sets (Hirsch Smith andZhao [15] and Zhao [40])

Let x = (h1 hp v1 vp) be the vector of all infectious class variables Thelinearization of system (31) at the disease-free equilibrium P0 = (0 0 0 0)is

dx

dt= (F (t)minus V (t))x (32)

where

F (t) =

[0 AB 0

]and V (t) =

[C 00 D

]

Here A = (δijai(t)bi)ptimesp B = (δijai(t)ciVlowasti (t)Hlowasti (t))ptimesp C = (δijri minusmij(t))ptimesp

D = (δijdi(t))ptimesp and δij denotes the Kronecker delta function (ie 1 when i = jand 0 elsewhere)

Let Y (t s) t ge s be the evolution operator of the linear ω-periodic system

dy

dt= minusV (t)y

That is for each s isin R1 the 2ptimes 2p matrix Y (t s) satisfies

dY (t s)

dt= minusV (t)Y (t s) forallt ge s Y (s s) = I2p

where I2p is the 2ptimes 2p identity matrixLet Cω be the ordered Banach space of all ω-periodic functions from R1 to R2p

equipped with the maximum norm We define a linear operator L Cω rarr Cω by

(Lφ)(t) =

int t

minusinfinY (t s)F (s)φ(s)ds

=

int infin0

Y (t tminus a)F (tminus a)φ(tminus a)da forallt isin R1 φ isin Cω(33)

It is easy to verify that conditions (A1)-(A7) in Wang and Zhao [36] are satisfiedTherefore the basic reproduction number of the periodic system (31) is then defined

3138 DAOZHOU GAO YIJUN LOU AND SHIGUI RUAN

as R0 = ρ(L) the spectral radius of L The following lemma shows that the basicreproduction numberR0 is the threshold parameter for local stability of the disease-free steady state

Lemma 32 (Theorem 22 in Wang and Zhao [36]) Let ΦFminusV (t) and ρ(ΦFminusV (ω))be the monodromy matrix of system (32) and the spectral radius of ΦFminusV (ω) re-spectively The following statements are valid

(i) R0 = 1 if and only if ρ(ΦFminusV (ω)) = 1(ii) R0 gt 1 if and only if ρ(ΦFminusV (ω)) gt 1

(iii) R0 lt 1 if and only if ρ(ΦFminusV (ω)) lt 1

Thus the disease-free equilibrium P0 of system (31) is locally asymptotically stableif R0 lt 1 and unstable if R0 gt 1

Lou and Zhao [20] presented a global qualitative analysis for system (31) with asingle patch We will show that the system with multiple patches does not exhibitmore complicated dynamics

Lemma 33 Let Dt = (h v) 0 le h le Hlowast(t) 0 le v le V lowast(t) For each x(0) =

(h(0) v(0)) isin R2p+ system (31) admits a unique solution x(t) = (h(t) v(t)) isin R2p

+

through x(0) for all t ge 0 Moreover x(t) isin Dt forallt ge 0 provided that x(0) isin D0

Proof Let K(t x) denote the vector field described by (31) Since K(t x) is con-tinuous and locally Lipschitizan in x in any bounded set By Theorem 521 inSmith [34] and Theorem 21 system (31) has a unique solution (h(t) v(t)) through

x(0) isin R2p+ which exists globally

To prove the second statement we let u(t) = Hlowast(t)minush(t) and w(t) = V lowast(t)minusv(t)where (h(t) v(t)) is the solution through x(0) isin D0 Then (u(t) w(t)) satisfies thefollowing system with u(0) = Hlowast(0)minus h(0) ge 0 and w(0) = V lowast(0)minus v(0) ge 0

dui(t)

dt= minusai(t)bi

ui(t)

Hlowasti (t)(V lowasti (t)minus wi(t))minus riui(t) + riH

lowasti (t) +

psumj=1

mij(t)uj(t)

dwi(t)

dt= εi(t)minus ai(t)ci

Hlowasti (t)minus ui(t)Hlowasti (t)

wi(t)minus di(t)wi(t)

for i = 1 p It then follows from Theorem 511 in Smith [34] that (u(t) w(t)) ge0 That is the solution for system (31) x(t) = (h(t) v(t)) le (Hlowast(t) V lowast(t)) holdsfor all t ge 0 whenever x(0) isin D0

Theorem 31 System (31) admits a unique positive ω-periodic solution de-noted by P lowast(t) = (hlowast(t) vlowast(t)) = (hlowast1(t) hlowastp(t) v

lowast1(t) vlowastp(t)) which is globally

asymptotically stable with initial values in D00 if R0 gt 1 and the disease-freeequilibrium P0 is globally asymptotically stable in D0 if R0 le 1

Proof To show the global asymptotic stability of P lowast(t) or P0 it suffices to verifythat K(t x) R1

+ times D0 rarr R2p satisfies (A1)-(A3) in Theorem 312 in Zhao [40]For every x = (h v) ge 0 with hi = 0 or vi = 0 t isin R1

+ we have

Ki(t x) = ai(t)bivi +sum

1lejlepj 6=i

mij(t)hj ge 0

or

Kp+i(t x) = ai(t)cihi

Hlowasti (t)V lowasti (t) ge 0

for i = 1 p So (A1) is satisfied

PERIODIC ROSS-MACDONALD MODEL 3139

By Lemma 33 the system (31) is cooperative in Dt and the ω-periodic semi-flow is monotone We can further prove that the pω-periodic semiflow is stronglymonotone in IntD0 the interior of D0 due to the irreducibility of (

int ω0mij(t)dt)ptimesp

For each t ge 0 i = 1 p there hold

Ki(t αx) = ai(t)biHlowasti (t)minus αhi

Hlowasti (t)αvi minus riαhi +

psumj=1

mij(t)αhj

gt α

(ai(t)bi

Hlowasti (t)minus hiHlowasti (t)

vi minus rihi +

psumj=1

mij(t)hj

)= αKi(t x)

and

Kp+i(t αx) = ai(t)ciαhiHlowasti (t)

(V lowasti (t)minus αvi)minus di(t)αvi

gt α

(ai(t)ci

hiHlowasti (t)

(V lowasti (t)minus vi)minus di(t)vi)

= αKp+i(t x)

for each x 0 α isin (0 1) That is K(t middot) is strictly subhomogeneous on R2p+

Moreover K(t 0) equiv 0 and all solutions are ultimately bounded Therefore byTheorem 312 in Zhao [40] or Theorem 234 in Zhao [40] as applied to the pωsolution map associated with (31) the proof is complete

Using the theory of internally chain transitive sets (Hirsch Smith and Zhao [15]or Zhao [40]) as argued in Lou and Zhao [22] we find that the disease either diesout or persists and stabilizes at a periodic state

Theorem 32 For system (21) if R0 gt 1 then there is a unique positive ω-periodic solution denoted by Elowast(t) = (Hlowast(t) V lowast(t) hlowast(t) vlowast(t)) which is globallyasymptotically stable if R0 le 1 then the disease-free periodic solution E0(t) isglobally asymptotically stable

Proof Let P be the Poincare map of the system (21) on R4p+ that is

P (H(0) V (0) h(0) v(0)) = (H(ω) V (ω) h(ω) v(ω))

Then P is a compact map Let Ω =Ω(H(0) V (0) h(0) v(0)) be the omega limitset of Pn(H(0) V (0) h(0) v(0)) It then follows from Lemma 21prime in Hirsch Smithand Zhao [15] (see also Lemma 121prime in Zhao [40]) that Ω is an internally chaintransitive set for P Based on Lemma 31 we have

limtrarrinfin

(H(t) V (t)) = (Hlowast(t) V lowast(t))

for any H(0) gt 0 and V (0) gt 0 Thus Ω = (Hlowast(0) V lowast(0)) times Ω1 for some

Ω1 sub R2p+ and

Pn|Ω(Hlowast(0) V lowast(0) h(0) v(0)) = (Hlowast(0) V lowast(0) Pn1 (h(0) v(0)))

where P1 is the Poincare map associated with the system (31) Since Ω is aninternally chain transitive set for Pn then Ω1 is an internally chain transitive setfor Pn1 Next we will discuss different scenarios when R0 le 1 and R0 gt 1

Case 1 R0 le 1 In this case the zero equilibrium P0 is globally asymptoticallystable for system (31) according to Theorem 31 It then follows from Theorem 32and Remark 46 of Hirsch Smith and Zhao [15] that Ω1 = (0 0) which further

3140 DAOZHOU GAO YIJUN LOU AND SHIGUI RUAN

implies that Ω = (Hlowast(0) V lowast(0) 0 0) Therefore the disease free periodic solution(Hlowast(t) V lowast(t) 0 0) is globally asymptotically stable

Case 2 R0 gt 1 In this case there are two fixed points for the map P1 (0 0)and (hlowast(0) vlowast(0)) We claim that Ω1 6= (0 0) for all h(0) gt 0 or v(0) gt 0Assume that by contradiction Ω1 = (0 0) for some h(0) gt 0 or v(0) gt 0 ThenΩ = (Hlowast(0) V lowast(0) 0 0) and we have

limtrarrinfin

(H(t) V (t) h(t) v(t)) = (Hlowast(t) V lowast(t) 0 0)

Since R0 gt 1 there exists some δ gt 0 such that the following linear system isunstable

dhi(t)

dt= biai(t)(1minus δ)vi(t)minus rihi(t) +

psumj=1

mij(t)hj(t) 1 le i le p

dvi(t)

dt= ciai(t)(1minus δ)

V lowasti (t)

Hlowasti (t)hi(t)minus di(t)vi(t) 1 le i le p

(34)

Moreover there exists some T0 gt 0 such that

|(H(t) V (t) h(t) v(t))minus (Hlowast(t) V lowast(t) 0 0)| lt δ min0letleω

Hlowast(t) V lowast(t) forallt gt T0

Hence we have

dhi(t)

dtge biai(t)(1minus δ)vi(t)minus rihi(t) +

psumj=1

mij(t)hj(t) 1 le i le p

dvi(t)

dtge ciai(t)(1minus δ)

V lowasti (t)

Hlowasti (t)hi(t)minus di(t)vi(t) 1 le i le p

when t gt T0 According to the instability of system (34) and the comparison princi-ple hi(t) and vi(t) will go unbounded as trarrinfin which contradicts the boundednessof solutions (Theorem 21)

Since Ω1 6= (0 0) and (hlowast(0) vlowast(0)) is globally asymptotically stable for Pn1 in

R2p+ (0 0) it follows that Ω1

⋂W s((hlowast(0) vlowast(0))) 6= empty where W s((hlowast(0) vlowast(0)))

is the stable set for (hlowast(0) vlowast(0)) By Theorem 31 and Remark 46 in Hirsch Smithand Zhao [15] we then get Ω1 = (hlowast(0) vlowast(0)) Thus Ω = (Hlowast(0) V lowast(0) hlowast(0)vlowast(0)) and hence the statement for the case when R0 gt 1 is valid

4 Simulations and discussions In this section we provide some numerical sim-ulations for the two-patch case to support our analytical conclusions The followingmatrix and lemma will be used in numerical computation for the basic reproduc-tion number R0 Let W (t λ) be the monodromy matrix (see Hale [14]) of thehomogeneous linear ω-periodic system

dx

dt=(minus V (t) +

F (t)

λ

)x t isin R1

with parameter λ isin (0infin)

Lemma 41 (Theorem 21 in Wang and Zhao [36]) The following statements arevalid

(i) If ρ(W (ω λ)) = 1 has a positive solution λ0 then λ0 is an eigenvalue of Land hence R0 gt 0 where L is the next generation operator defined in (33)

(ii) If R0 gt 0 then λ = R0 is the unique solution of ρ(W (ω λ)) = 1(iii) R0 = 0 if and only if ρ(W (ω λ)) lt 1 for all λ gt 0

PERIODIC ROSS-MACDONALD MODEL 3141

According to Lemma 41(ii) we can show that the basic reproduction number

for (31) is proportional to the biting rate that is R0 = kR0 if βi(t) = kβi(t) fork gt 0 and 1 le i le p (see Lou and Zhao [21] or Liu and Zhao [18])

To implement numerical simulations we choose the baseline parameter values asfollows bi = 02 ri = 002 ci = 03 di(t) = 01 for i = 1 2 For the first patch

ε1(t) = 125minus(5 cos( 2πt365 )+25 sin( 2πt

365 ))minus(5 cos( 4πt365 )minus25 sin( 4πt

365 )) ge 125minus5radic

5 gt 0

a1(t) = 0028ε1(t) gt 0 In the second patch ε2(t) = 15 minus 12 cos( 2πt365 ) a2(t) =

018 minus 012 cos( 2πt365 ) Then using Lemma 41 we can numerically compute that

the respective reproduction numbers of the disease in patches 1 and 2 are R10 =17571 gt 1 and R20 = 08470 lt 1 and the disease persists in the first patch but diesout in the second patch (see Figure 1(a)) When humans migrate between these twopatches with m12(t) = 0006 minus 0004 cos( 2πt

365 ) and m21(t) = 0006 + 0004 cos( 2πt365 )

the basic reproduction number is R0 = 13944 gt 1 Thus the disease becomesendemic in both patches (see Figure 1(b))

0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

time t

h 1amph 2

(a)

h

1

h2

0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

time t

h 1amph 2

(b)

h

1

h2

Figure 1 Numerical solution of system (31) with initial con-ditions H1(0) = 200 V1(0) = 50 h1(0) = 15 v1(0) = 10 andH2(0) = 300 V2(0) = 30 h2(0) = 30 v2(0) = 5 The curves ofthe number of infected humans in patches 1 and 2 respectivelyare blue solid and red dashed (a) when there is no human move-ment R10 = 17571 gt 1 and R20 = 08470 lt 1 indicate the diseasepersists in patch 1 but dies out in patch 2 (b) when human move-ment presents R0 = 13944 gt 1 indicates the disease persists inboth patches and will stabilize in a seasonal pattern

The basic reproduction number for the corresponding time-averaged autonomoussystem (see Wang and Zhao [36] and Zhang et al [38]) is R0 = 12364 By Lemma

3142 DAOZHOU GAO YIJUN LOU AND SHIGUI RUAN

41 or the linearity of R0 in the mosquito biting rates we can multiply the bitingrates by an appropriate constant such that R0 below 1 while R0 above 1 Thus theautonomous malaria model may underestimate disease severity in some transmissionsettings

In order to investigate the effect of possible travel control on the malaria riskwe determine the disease risk when shifting the phases of the movement functionswhile keeping their respective mean values and forcing strengths to be the sameWith the same parameter values and initial data except that the migration ratesm12(t) = 0006 minus 0004 cos( 2πt

365 + φ1) and m21(t) = 0006 + 0004 cos( 2πt365 + φ2)

we evaluate the annual malaria prevalence over the two patches for φ1 φ2 isin [0 2π]Then Figure 2 presents the effect of phase shifting for movement functions on diseaseprevalence The approximate range of malaria prevalence is from 17 to 23 Thisresult indicates that appropriate travel control can definitely reduce the disease riskand therefore an optimal travel control measure can be proposed

φ2

φ 1

0 1 2 3 4 5 60

1

2

3

4

5

6

0175

018

0185

019

0195

02

0205

021

0215

022

0225

Figure 2 Effect of phase shift for movement rates between twopatches on disease risk

In this project we have developed a compartmental model to address seasonalvariations of malaria transmission among different regions by incorporating seasonalhuman movement and periodic changing in mosquito ecology We then derive thebasic reproduction number and establish the global dynamics of the model Math-ematically we generalize the model and results presented in Cosner et al [7] andAuger et al [3] from a constant environment to a periodic environment Biologi-cally our result gives a possible explanation to the fact that malaria incidence showseasonal peaks in most endemic areas (Roca-Feltrer et al [30]) The numericalexamples suggest that a better understanding of seasonal human migration can

PERIODIC ROSS-MACDONALD MODEL 3143

help us to prevent and control malaria transmission in a spatially heterogeneousenvironment

It would be interesting to assess the impacts of climate change on the emergingand reemerging of mosquito-borne diseases especially malaria It is believed thatglobal warming would cause climate change while climatic factors such as precip-itation temperature humidity and wind speed affect the biological activity andgeographic distribution of the pathogen and its vector It sometimes even affectshuman behavior through sea-level rise more extreme weather events etc

The current and potential future impact of climate change on human healthhas attracted considerable attention in recent years Many of the early studies(see Ostfeld [27] and the references cited therein) claimed that recent and futuretrends in climate warming were likely to increase the severity and global distribu-tion of vector-borne diseases while there is still substantial debate about the linkbetween climate change and the spread of infectious diseases (Lafferty [16]) Thecontroversy is partially due to the fact that although there is massive work us-ing empirical-statistical models to explore the relationship between climatic factorsand the distribution and prevalence of vector-borne diseases there are very fewstudies incorporating climatic factors into mathematical models to describe diseasetransmission

The research that comes closest to what we expect to do has been conducted byParham and Michael in [28] where they used a system of delay differential equationsto model malaria transmission under varying climatic and environmental conditionsModel parameters are assumed to be temperature-dependent or rainfall-dependentor both Due to the complexity of their model mathematical analysis was notpossible and only numerical simulations were carried out Furthermore the basicreproduction number is also difficult to project through their model We wouldlike to extend our model to a mathematically tractable climate-based model in thefuture

Acknowledgments Part of this work was done while the first author was a grad-uate student in the Department of Mathematics University of Miami We thankDrs Chris Cosner David Smith and Xiao-Qiang Zhao for helpful discussions onthis project

REFERENCES

[1] S Altizer A Dobson P Hosseini P Hudson M Pascual and P Rohani Seasonality and

the dynamics of infectious diseases Ecol Lett 9 (2006) 467ndash484[2] G Aronsson and R B Kellogg On a differential equation arising from compartmental anal-

ysis Math Biosci 38 (1973) 113ndash122[3] P Auger E Kouokam G Sallet M Tchuente and B Tsanou The Ross-Macdonald model

in a patchy environment Math Biosci 216 (2008) 123ndash131

[4] N Bacaer and S Guernaoui The epidemic threshold of vector-borne diseases with seasonalityJ Math Biol 53 (2006) 421ndash436

[5] N Bacaer Approximation of the basic reproduction number R0 for vector-borne diseases with

a periodic vector population Bull Math Biol 69 (2007) 1067ndash1091[6] Z Bai and Y Zhou Threshold dynamics of a Bacillary Dysentery model with seasonal fluc-

tuation Discrete Contin Dyn Syst Ser B 15 (2011) 1ndash14

[7] C Cosner J C Beier R S Cantrell D Impoinvil L Kapitanski M D Potts A Troyoand S Ruan The effects of human movement on the persistence of vector-borne diseases J

Theoret Biol 258 (2009) 550ndash560

PERIODIC ROSS-MACDONALD MODEL 3137

for some t isin [0 2π) See Lemma 21 in Zhang and Teng [39] for results on a generalnonautonomous system of the form dVi(t)dt = εi(t)minus di(t)Vi(t)

The above result guarantees that system (21) admits a unique disease-free peri-odic solution

E0(t) = (Hlowast1 (t) Hlowastp (t) V lowast1 (t) V lowastp (t) 0 0 0 0)

Biologically both human and mosquito populations in each patch are seasonallyforced due to periodically changing human migration and seasonal birth deathand biting rates of mosquitoes respectively Now we consider the asymptoticallyperiodic system for malaria transmission

dhi(t)

dt= biai(t)

Hlowasti (t)minus hi(t)Hlowasti (t)

vi(t)minus rihi(t) +

psumj=1

mij(t)hj(t) 1 le i le p

dvi(t)

dt= ciai(t)

hi(t)

Hlowasti (t)(V lowasti (t)minus vi(t))minus di(t)vi(t) 1 le i le p

(31)

In what follows we use the definition of Bacaer and Guernaoui [4] (see alsoBacaer [5]) and the general calculation method in Wang and Zhao [36] to evaluatethe basic reproduction number R0 for system (31) Then we analyze the thresholddynamics of system (31) Finally we study global dynamics for the whole system(21) by applying the theory of internally chain transitive sets (Hirsch Smith andZhao [15] and Zhao [40])

Let x = (h1 hp v1 vp) be the vector of all infectious class variables Thelinearization of system (31) at the disease-free equilibrium P0 = (0 0 0 0)is

dx

dt= (F (t)minus V (t))x (32)

where

F (t) =

[0 AB 0

]and V (t) =

[C 00 D

]

Here A = (δijai(t)bi)ptimesp B = (δijai(t)ciVlowasti (t)Hlowasti (t))ptimesp C = (δijri minusmij(t))ptimesp

D = (δijdi(t))ptimesp and δij denotes the Kronecker delta function (ie 1 when i = jand 0 elsewhere)

Let Y (t s) t ge s be the evolution operator of the linear ω-periodic system

dy

dt= minusV (t)y

That is for each s isin R1 the 2ptimes 2p matrix Y (t s) satisfies

dY (t s)

dt= minusV (t)Y (t s) forallt ge s Y (s s) = I2p

where I2p is the 2ptimes 2p identity matrixLet Cω be the ordered Banach space of all ω-periodic functions from R1 to R2p

equipped with the maximum norm We define a linear operator L Cω rarr Cω by

(Lφ)(t) =

int t

minusinfinY (t s)F (s)φ(s)ds

=

int infin0

Y (t tminus a)F (tminus a)φ(tminus a)da forallt isin R1 φ isin Cω(33)

It is easy to verify that conditions (A1)-(A7) in Wang and Zhao [36] are satisfiedTherefore the basic reproduction number of the periodic system (31) is then defined

3138 DAOZHOU GAO YIJUN LOU AND SHIGUI RUAN

as R0 = ρ(L) the spectral radius of L The following lemma shows that the basicreproduction numberR0 is the threshold parameter for local stability of the disease-free steady state

Lemma 32 (Theorem 22 in Wang and Zhao [36]) Let ΦFminusV (t) and ρ(ΦFminusV (ω))be the monodromy matrix of system (32) and the spectral radius of ΦFminusV (ω) re-spectively The following statements are valid

(i) R0 = 1 if and only if ρ(ΦFminusV (ω)) = 1(ii) R0 gt 1 if and only if ρ(ΦFminusV (ω)) gt 1

(iii) R0 lt 1 if and only if ρ(ΦFminusV (ω)) lt 1

Thus the disease-free equilibrium P0 of system (31) is locally asymptotically stableif R0 lt 1 and unstable if R0 gt 1

Lou and Zhao [20] presented a global qualitative analysis for system (31) with asingle patch We will show that the system with multiple patches does not exhibitmore complicated dynamics

Lemma 33 Let Dt = (h v) 0 le h le Hlowast(t) 0 le v le V lowast(t) For each x(0) =

(h(0) v(0)) isin R2p+ system (31) admits a unique solution x(t) = (h(t) v(t)) isin R2p

+

through x(0) for all t ge 0 Moreover x(t) isin Dt forallt ge 0 provided that x(0) isin D0

Proof Let K(t x) denote the vector field described by (31) Since K(t x) is con-tinuous and locally Lipschitizan in x in any bounded set By Theorem 521 inSmith [34] and Theorem 21 system (31) has a unique solution (h(t) v(t)) through

x(0) isin R2p+ which exists globally

To prove the second statement we let u(t) = Hlowast(t)minush(t) and w(t) = V lowast(t)minusv(t)where (h(t) v(t)) is the solution through x(0) isin D0 Then (u(t) w(t)) satisfies thefollowing system with u(0) = Hlowast(0)minus h(0) ge 0 and w(0) = V lowast(0)minus v(0) ge 0

dui(t)

dt= minusai(t)bi

ui(t)

Hlowasti (t)(V lowasti (t)minus wi(t))minus riui(t) + riH

lowasti (t) +

psumj=1

mij(t)uj(t)

dwi(t)

dt= εi(t)minus ai(t)ci

Hlowasti (t)minus ui(t)Hlowasti (t)

wi(t)minus di(t)wi(t)

for i = 1 p It then follows from Theorem 511 in Smith [34] that (u(t) w(t)) ge0 That is the solution for system (31) x(t) = (h(t) v(t)) le (Hlowast(t) V lowast(t)) holdsfor all t ge 0 whenever x(0) isin D0

Theorem 31 System (31) admits a unique positive ω-periodic solution de-noted by P lowast(t) = (hlowast(t) vlowast(t)) = (hlowast1(t) hlowastp(t) v

lowast1(t) vlowastp(t)) which is globally

asymptotically stable with initial values in D00 if R0 gt 1 and the disease-freeequilibrium P0 is globally asymptotically stable in D0 if R0 le 1

Proof To show the global asymptotic stability of P lowast(t) or P0 it suffices to verifythat K(t x) R1

+ times D0 rarr R2p satisfies (A1)-(A3) in Theorem 312 in Zhao [40]For every x = (h v) ge 0 with hi = 0 or vi = 0 t isin R1

+ we have

Ki(t x) = ai(t)bivi +sum

1lejlepj 6=i

mij(t)hj ge 0

or

Kp+i(t x) = ai(t)cihi

Hlowasti (t)V lowasti (t) ge 0

for i = 1 p So (A1) is satisfied

PERIODIC ROSS-MACDONALD MODEL 3139

By Lemma 33 the system (31) is cooperative in Dt and the ω-periodic semi-flow is monotone We can further prove that the pω-periodic semiflow is stronglymonotone in IntD0 the interior of D0 due to the irreducibility of (

int ω0mij(t)dt)ptimesp

For each t ge 0 i = 1 p there hold

Ki(t αx) = ai(t)biHlowasti (t)minus αhi

Hlowasti (t)αvi minus riαhi +

psumj=1

mij(t)αhj

gt α

(ai(t)bi

Hlowasti (t)minus hiHlowasti (t)

vi minus rihi +

psumj=1

mij(t)hj

)= αKi(t x)

and

Kp+i(t αx) = ai(t)ciαhiHlowasti (t)

(V lowasti (t)minus αvi)minus di(t)αvi

gt α

(ai(t)ci

hiHlowasti (t)

(V lowasti (t)minus vi)minus di(t)vi)

= αKp+i(t x)

for each x 0 α isin (0 1) That is K(t middot) is strictly subhomogeneous on R2p+

Moreover K(t 0) equiv 0 and all solutions are ultimately bounded Therefore byTheorem 312 in Zhao [40] or Theorem 234 in Zhao [40] as applied to the pωsolution map associated with (31) the proof is complete

Using the theory of internally chain transitive sets (Hirsch Smith and Zhao [15]or Zhao [40]) as argued in Lou and Zhao [22] we find that the disease either diesout or persists and stabilizes at a periodic state

Theorem 32 For system (21) if R0 gt 1 then there is a unique positive ω-periodic solution denoted by Elowast(t) = (Hlowast(t) V lowast(t) hlowast(t) vlowast(t)) which is globallyasymptotically stable if R0 le 1 then the disease-free periodic solution E0(t) isglobally asymptotically stable

Proof Let P be the Poincare map of the system (21) on R4p+ that is

P (H(0) V (0) h(0) v(0)) = (H(ω) V (ω) h(ω) v(ω))

Then P is a compact map Let Ω =Ω(H(0) V (0) h(0) v(0)) be the omega limitset of Pn(H(0) V (0) h(0) v(0)) It then follows from Lemma 21prime in Hirsch Smithand Zhao [15] (see also Lemma 121prime in Zhao [40]) that Ω is an internally chaintransitive set for P Based on Lemma 31 we have

limtrarrinfin

(H(t) V (t)) = (Hlowast(t) V lowast(t))

for any H(0) gt 0 and V (0) gt 0 Thus Ω = (Hlowast(0) V lowast(0)) times Ω1 for some

Ω1 sub R2p+ and

Pn|Ω(Hlowast(0) V lowast(0) h(0) v(0)) = (Hlowast(0) V lowast(0) Pn1 (h(0) v(0)))

where P1 is the Poincare map associated with the system (31) Since Ω is aninternally chain transitive set for Pn then Ω1 is an internally chain transitive setfor Pn1 Next we will discuss different scenarios when R0 le 1 and R0 gt 1

Case 1 R0 le 1 In this case the zero equilibrium P0 is globally asymptoticallystable for system (31) according to Theorem 31 It then follows from Theorem 32and Remark 46 of Hirsch Smith and Zhao [15] that Ω1 = (0 0) which further

3140 DAOZHOU GAO YIJUN LOU AND SHIGUI RUAN

implies that Ω = (Hlowast(0) V lowast(0) 0 0) Therefore the disease free periodic solution(Hlowast(t) V lowast(t) 0 0) is globally asymptotically stable

Case 2 R0 gt 1 In this case there are two fixed points for the map P1 (0 0)and (hlowast(0) vlowast(0)) We claim that Ω1 6= (0 0) for all h(0) gt 0 or v(0) gt 0Assume that by contradiction Ω1 = (0 0) for some h(0) gt 0 or v(0) gt 0 ThenΩ = (Hlowast(0) V lowast(0) 0 0) and we have

limtrarrinfin

(H(t) V (t) h(t) v(t)) = (Hlowast(t) V lowast(t) 0 0)

Since R0 gt 1 there exists some δ gt 0 such that the following linear system isunstable

dhi(t)

dt= biai(t)(1minus δ)vi(t)minus rihi(t) +

psumj=1

mij(t)hj(t) 1 le i le p

dvi(t)

dt= ciai(t)(1minus δ)

V lowasti (t)

Hlowasti (t)hi(t)minus di(t)vi(t) 1 le i le p

(34)

Moreover there exists some T0 gt 0 such that

|(H(t) V (t) h(t) v(t))minus (Hlowast(t) V lowast(t) 0 0)| lt δ min0letleω

Hlowast(t) V lowast(t) forallt gt T0

Hence we have

dhi(t)

dtge biai(t)(1minus δ)vi(t)minus rihi(t) +

psumj=1

mij(t)hj(t) 1 le i le p

dvi(t)

dtge ciai(t)(1minus δ)

V lowasti (t)

Hlowasti (t)hi(t)minus di(t)vi(t) 1 le i le p

when t gt T0 According to the instability of system (34) and the comparison princi-ple hi(t) and vi(t) will go unbounded as trarrinfin which contradicts the boundednessof solutions (Theorem 21)

Since Ω1 6= (0 0) and (hlowast(0) vlowast(0)) is globally asymptotically stable for Pn1 in

R2p+ (0 0) it follows that Ω1

⋂W s((hlowast(0) vlowast(0))) 6= empty where W s((hlowast(0) vlowast(0)))

is the stable set for (hlowast(0) vlowast(0)) By Theorem 31 and Remark 46 in Hirsch Smithand Zhao [15] we then get Ω1 = (hlowast(0) vlowast(0)) Thus Ω = (Hlowast(0) V lowast(0) hlowast(0)vlowast(0)) and hence the statement for the case when R0 gt 1 is valid

4 Simulations and discussions In this section we provide some numerical sim-ulations for the two-patch case to support our analytical conclusions The followingmatrix and lemma will be used in numerical computation for the basic reproduc-tion number R0 Let W (t λ) be the monodromy matrix (see Hale [14]) of thehomogeneous linear ω-periodic system

dx

dt=(minus V (t) +

F (t)

λ

)x t isin R1

with parameter λ isin (0infin)

Lemma 41 (Theorem 21 in Wang and Zhao [36]) The following statements arevalid

(i) If ρ(W (ω λ)) = 1 has a positive solution λ0 then λ0 is an eigenvalue of Land hence R0 gt 0 where L is the next generation operator defined in (33)

(ii) If R0 gt 0 then λ = R0 is the unique solution of ρ(W (ω λ)) = 1(iii) R0 = 0 if and only if ρ(W (ω λ)) lt 1 for all λ gt 0

PERIODIC ROSS-MACDONALD MODEL 3141

According to Lemma 41(ii) we can show that the basic reproduction number

for (31) is proportional to the biting rate that is R0 = kR0 if βi(t) = kβi(t) fork gt 0 and 1 le i le p (see Lou and Zhao [21] or Liu and Zhao [18])

To implement numerical simulations we choose the baseline parameter values asfollows bi = 02 ri = 002 ci = 03 di(t) = 01 for i = 1 2 For the first patch

ε1(t) = 125minus(5 cos( 2πt365 )+25 sin( 2πt

365 ))minus(5 cos( 4πt365 )minus25 sin( 4πt

365 )) ge 125minus5radic

5 gt 0

a1(t) = 0028ε1(t) gt 0 In the second patch ε2(t) = 15 minus 12 cos( 2πt365 ) a2(t) =

018 minus 012 cos( 2πt365 ) Then using Lemma 41 we can numerically compute that

the respective reproduction numbers of the disease in patches 1 and 2 are R10 =17571 gt 1 and R20 = 08470 lt 1 and the disease persists in the first patch but diesout in the second patch (see Figure 1(a)) When humans migrate between these twopatches with m12(t) = 0006 minus 0004 cos( 2πt

365 ) and m21(t) = 0006 + 0004 cos( 2πt365 )

the basic reproduction number is R0 = 13944 gt 1 Thus the disease becomesendemic in both patches (see Figure 1(b))

0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

time t

h 1amph 2

(a)

h

1

h2

0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

time t

h 1amph 2

(b)

h

1

h2

Figure 1 Numerical solution of system (31) with initial con-ditions H1(0) = 200 V1(0) = 50 h1(0) = 15 v1(0) = 10 andH2(0) = 300 V2(0) = 30 h2(0) = 30 v2(0) = 5 The curves ofthe number of infected humans in patches 1 and 2 respectivelyare blue solid and red dashed (a) when there is no human move-ment R10 = 17571 gt 1 and R20 = 08470 lt 1 indicate the diseasepersists in patch 1 but dies out in patch 2 (b) when human move-ment presents R0 = 13944 gt 1 indicates the disease persists inboth patches and will stabilize in a seasonal pattern

The basic reproduction number for the corresponding time-averaged autonomoussystem (see Wang and Zhao [36] and Zhang et al [38]) is R0 = 12364 By Lemma

3142 DAOZHOU GAO YIJUN LOU AND SHIGUI RUAN

41 or the linearity of R0 in the mosquito biting rates we can multiply the bitingrates by an appropriate constant such that R0 below 1 while R0 above 1 Thus theautonomous malaria model may underestimate disease severity in some transmissionsettings

In order to investigate the effect of possible travel control on the malaria riskwe determine the disease risk when shifting the phases of the movement functionswhile keeping their respective mean values and forcing strengths to be the sameWith the same parameter values and initial data except that the migration ratesm12(t) = 0006 minus 0004 cos( 2πt

365 + φ1) and m21(t) = 0006 + 0004 cos( 2πt365 + φ2)

we evaluate the annual malaria prevalence over the two patches for φ1 φ2 isin [0 2π]Then Figure 2 presents the effect of phase shifting for movement functions on diseaseprevalence The approximate range of malaria prevalence is from 17 to 23 Thisresult indicates that appropriate travel control can definitely reduce the disease riskand therefore an optimal travel control measure can be proposed

φ2

φ 1

0 1 2 3 4 5 60

1

2

3

4

5

6

0175

018

0185

019

0195

02

0205

021

0215

022

0225

Figure 2 Effect of phase shift for movement rates between twopatches on disease risk

In this project we have developed a compartmental model to address seasonalvariations of malaria transmission among different regions by incorporating seasonalhuman movement and periodic changing in mosquito ecology We then derive thebasic reproduction number and establish the global dynamics of the model Math-ematically we generalize the model and results presented in Cosner et al [7] andAuger et al [3] from a constant environment to a periodic environment Biologi-cally our result gives a possible explanation to the fact that malaria incidence showseasonal peaks in most endemic areas (Roca-Feltrer et al [30]) The numericalexamples suggest that a better understanding of seasonal human migration can

PERIODIC ROSS-MACDONALD MODEL 3143

help us to prevent and control malaria transmission in a spatially heterogeneousenvironment

It would be interesting to assess the impacts of climate change on the emergingand reemerging of mosquito-borne diseases especially malaria It is believed thatglobal warming would cause climate change while climatic factors such as precip-itation temperature humidity and wind speed affect the biological activity andgeographic distribution of the pathogen and its vector It sometimes even affectshuman behavior through sea-level rise more extreme weather events etc

The current and potential future impact of climate change on human healthhas attracted considerable attention in recent years Many of the early studies(see Ostfeld [27] and the references cited therein) claimed that recent and futuretrends in climate warming were likely to increase the severity and global distribu-tion of vector-borne diseases while there is still substantial debate about the linkbetween climate change and the spread of infectious diseases (Lafferty [16]) Thecontroversy is partially due to the fact that although there is massive work us-ing empirical-statistical models to explore the relationship between climatic factorsand the distribution and prevalence of vector-borne diseases there are very fewstudies incorporating climatic factors into mathematical models to describe diseasetransmission

The research that comes closest to what we expect to do has been conducted byParham and Michael in [28] where they used a system of delay differential equationsto model malaria transmission under varying climatic and environmental conditionsModel parameters are assumed to be temperature-dependent or rainfall-dependentor both Due to the complexity of their model mathematical analysis was notpossible and only numerical simulations were carried out Furthermore the basicreproduction number is also difficult to project through their model We wouldlike to extend our model to a mathematically tractable climate-based model in thefuture

Acknowledgments Part of this work was done while the first author was a grad-uate student in the Department of Mathematics University of Miami We thankDrs Chris Cosner David Smith and Xiao-Qiang Zhao for helpful discussions onthis project

REFERENCES

[1] S Altizer A Dobson P Hosseini P Hudson M Pascual and P Rohani Seasonality and

the dynamics of infectious diseases Ecol Lett 9 (2006) 467ndash484[2] G Aronsson and R B Kellogg On a differential equation arising from compartmental anal-

ysis Math Biosci 38 (1973) 113ndash122[3] P Auger E Kouokam G Sallet M Tchuente and B Tsanou The Ross-Macdonald model

in a patchy environment Math Biosci 216 (2008) 123ndash131

[4] N Bacaer and S Guernaoui The epidemic threshold of vector-borne diseases with seasonalityJ Math Biol 53 (2006) 421ndash436

[5] N Bacaer Approximation of the basic reproduction number R0 for vector-borne diseases with

a periodic vector population Bull Math Biol 69 (2007) 1067ndash1091[6] Z Bai and Y Zhou Threshold dynamics of a Bacillary Dysentery model with seasonal fluc-

tuation Discrete Contin Dyn Syst Ser B 15 (2011) 1ndash14

[7] C Cosner J C Beier R S Cantrell D Impoinvil L Kapitanski M D Potts A Troyoand S Ruan The effects of human movement on the persistence of vector-borne diseases J

Theoret Biol 258 (2009) 550ndash560

3138 DAOZHOU GAO YIJUN LOU AND SHIGUI RUAN

as R0 = ρ(L) the spectral radius of L The following lemma shows that the basicreproduction numberR0 is the threshold parameter for local stability of the disease-free steady state

Lemma 32 (Theorem 22 in Wang and Zhao [36]) Let ΦFminusV (t) and ρ(ΦFminusV (ω))be the monodromy matrix of system (32) and the spectral radius of ΦFminusV (ω) re-spectively The following statements are valid

(i) R0 = 1 if and only if ρ(ΦFminusV (ω)) = 1(ii) R0 gt 1 if and only if ρ(ΦFminusV (ω)) gt 1

(iii) R0 lt 1 if and only if ρ(ΦFminusV (ω)) lt 1

Thus the disease-free equilibrium P0 of system (31) is locally asymptotically stableif R0 lt 1 and unstable if R0 gt 1

Lou and Zhao [20] presented a global qualitative analysis for system (31) with asingle patch We will show that the system with multiple patches does not exhibitmore complicated dynamics

Lemma 33 Let Dt = (h v) 0 le h le Hlowast(t) 0 le v le V lowast(t) For each x(0) =

(h(0) v(0)) isin R2p+ system (31) admits a unique solution x(t) = (h(t) v(t)) isin R2p

+

through x(0) for all t ge 0 Moreover x(t) isin Dt forallt ge 0 provided that x(0) isin D0

Proof Let K(t x) denote the vector field described by (31) Since K(t x) is con-tinuous and locally Lipschitizan in x in any bounded set By Theorem 521 inSmith [34] and Theorem 21 system (31) has a unique solution (h(t) v(t)) through

x(0) isin R2p+ which exists globally

To prove the second statement we let u(t) = Hlowast(t)minush(t) and w(t) = V lowast(t)minusv(t)where (h(t) v(t)) is the solution through x(0) isin D0 Then (u(t) w(t)) satisfies thefollowing system with u(0) = Hlowast(0)minus h(0) ge 0 and w(0) = V lowast(0)minus v(0) ge 0

dui(t)

dt= minusai(t)bi

ui(t)

Hlowasti (t)(V lowasti (t)minus wi(t))minus riui(t) + riH

lowasti (t) +

psumj=1

mij(t)uj(t)

dwi(t)

dt= εi(t)minus ai(t)ci

Hlowasti (t)minus ui(t)Hlowasti (t)

wi(t)minus di(t)wi(t)

for i = 1 p It then follows from Theorem 511 in Smith [34] that (u(t) w(t)) ge0 That is the solution for system (31) x(t) = (h(t) v(t)) le (Hlowast(t) V lowast(t)) holdsfor all t ge 0 whenever x(0) isin D0

Theorem 31 System (31) admits a unique positive ω-periodic solution de-noted by P lowast(t) = (hlowast(t) vlowast(t)) = (hlowast1(t) hlowastp(t) v

lowast1(t) vlowastp(t)) which is globally

asymptotically stable with initial values in D00 if R0 gt 1 and the disease-freeequilibrium P0 is globally asymptotically stable in D0 if R0 le 1

Proof To show the global asymptotic stability of P lowast(t) or P0 it suffices to verifythat K(t x) R1

+ times D0 rarr R2p satisfies (A1)-(A3) in Theorem 312 in Zhao [40]For every x = (h v) ge 0 with hi = 0 or vi = 0 t isin R1

+ we have

Ki(t x) = ai(t)bivi +sum

1lejlepj 6=i

mij(t)hj ge 0

or

Kp+i(t x) = ai(t)cihi

Hlowasti (t)V lowasti (t) ge 0

for i = 1 p So (A1) is satisfied

PERIODIC ROSS-MACDONALD MODEL 3139

By Lemma 33 the system (31) is cooperative in Dt and the ω-periodic semi-flow is monotone We can further prove that the pω-periodic semiflow is stronglymonotone in IntD0 the interior of D0 due to the irreducibility of (

int ω0mij(t)dt)ptimesp

For each t ge 0 i = 1 p there hold

Ki(t αx) = ai(t)biHlowasti (t)minus αhi

Hlowasti (t)αvi minus riαhi +

psumj=1

mij(t)αhj

gt α

(ai(t)bi

Hlowasti (t)minus hiHlowasti (t)

vi minus rihi +

psumj=1

mij(t)hj

)= αKi(t x)

and

Kp+i(t αx) = ai(t)ciαhiHlowasti (t)

(V lowasti (t)minus αvi)minus di(t)αvi

gt α

(ai(t)ci

hiHlowasti (t)

(V lowasti (t)minus vi)minus di(t)vi)

= αKp+i(t x)

for each x 0 α isin (0 1) That is K(t middot) is strictly subhomogeneous on R2p+

Moreover K(t 0) equiv 0 and all solutions are ultimately bounded Therefore byTheorem 312 in Zhao [40] or Theorem 234 in Zhao [40] as applied to the pωsolution map associated with (31) the proof is complete

Using the theory of internally chain transitive sets (Hirsch Smith and Zhao [15]or Zhao [40]) as argued in Lou and Zhao [22] we find that the disease either diesout or persists and stabilizes at a periodic state

Theorem 32 For system (21) if R0 gt 1 then there is a unique positive ω-periodic solution denoted by Elowast(t) = (Hlowast(t) V lowast(t) hlowast(t) vlowast(t)) which is globallyasymptotically stable if R0 le 1 then the disease-free periodic solution E0(t) isglobally asymptotically stable

Proof Let P be the Poincare map of the system (21) on R4p+ that is

P (H(0) V (0) h(0) v(0)) = (H(ω) V (ω) h(ω) v(ω))

Then P is a compact map Let Ω =Ω(H(0) V (0) h(0) v(0)) be the omega limitset of Pn(H(0) V (0) h(0) v(0)) It then follows from Lemma 21prime in Hirsch Smithand Zhao [15] (see also Lemma 121prime in Zhao [40]) that Ω is an internally chaintransitive set for P Based on Lemma 31 we have

limtrarrinfin

(H(t) V (t)) = (Hlowast(t) V lowast(t))

for any H(0) gt 0 and V (0) gt 0 Thus Ω = (Hlowast(0) V lowast(0)) times Ω1 for some

Ω1 sub R2p+ and

Pn|Ω(Hlowast(0) V lowast(0) h(0) v(0)) = (Hlowast(0) V lowast(0) Pn1 (h(0) v(0)))

where P1 is the Poincare map associated with the system (31) Since Ω is aninternally chain transitive set for Pn then Ω1 is an internally chain transitive setfor Pn1 Next we will discuss different scenarios when R0 le 1 and R0 gt 1

Case 1 R0 le 1 In this case the zero equilibrium P0 is globally asymptoticallystable for system (31) according to Theorem 31 It then follows from Theorem 32and Remark 46 of Hirsch Smith and Zhao [15] that Ω1 = (0 0) which further

3140 DAOZHOU GAO YIJUN LOU AND SHIGUI RUAN

implies that Ω = (Hlowast(0) V lowast(0) 0 0) Therefore the disease free periodic solution(Hlowast(t) V lowast(t) 0 0) is globally asymptotically stable

Case 2 R0 gt 1 In this case there are two fixed points for the map P1 (0 0)and (hlowast(0) vlowast(0)) We claim that Ω1 6= (0 0) for all h(0) gt 0 or v(0) gt 0Assume that by contradiction Ω1 = (0 0) for some h(0) gt 0 or v(0) gt 0 ThenΩ = (Hlowast(0) V lowast(0) 0 0) and we have

limtrarrinfin

(H(t) V (t) h(t) v(t)) = (Hlowast(t) V lowast(t) 0 0)

Since R0 gt 1 there exists some δ gt 0 such that the following linear system isunstable

dhi(t)

dt= biai(t)(1minus δ)vi(t)minus rihi(t) +

psumj=1

mij(t)hj(t) 1 le i le p

dvi(t)

dt= ciai(t)(1minus δ)

V lowasti (t)

Hlowasti (t)hi(t)minus di(t)vi(t) 1 le i le p

(34)

Moreover there exists some T0 gt 0 such that

|(H(t) V (t) h(t) v(t))minus (Hlowast(t) V lowast(t) 0 0)| lt δ min0letleω

Hlowast(t) V lowast(t) forallt gt T0

Hence we have

dhi(t)

dtge biai(t)(1minus δ)vi(t)minus rihi(t) +

psumj=1

mij(t)hj(t) 1 le i le p

dvi(t)

dtge ciai(t)(1minus δ)

V lowasti (t)

Hlowasti (t)hi(t)minus di(t)vi(t) 1 le i le p

when t gt T0 According to the instability of system (34) and the comparison princi-ple hi(t) and vi(t) will go unbounded as trarrinfin which contradicts the boundednessof solutions (Theorem 21)

Since Ω1 6= (0 0) and (hlowast(0) vlowast(0)) is globally asymptotically stable for Pn1 in

R2p+ (0 0) it follows that Ω1

⋂W s((hlowast(0) vlowast(0))) 6= empty where W s((hlowast(0) vlowast(0)))

is the stable set for (hlowast(0) vlowast(0)) By Theorem 31 and Remark 46 in Hirsch Smithand Zhao [15] we then get Ω1 = (hlowast(0) vlowast(0)) Thus Ω = (Hlowast(0) V lowast(0) hlowast(0)vlowast(0)) and hence the statement for the case when R0 gt 1 is valid

4 Simulations and discussions In this section we provide some numerical sim-ulations for the two-patch case to support our analytical conclusions The followingmatrix and lemma will be used in numerical computation for the basic reproduc-tion number R0 Let W (t λ) be the monodromy matrix (see Hale [14]) of thehomogeneous linear ω-periodic system

dx

dt=(minus V (t) +

F (t)

λ

)x t isin R1

with parameter λ isin (0infin)

Lemma 41 (Theorem 21 in Wang and Zhao [36]) The following statements arevalid

(i) If ρ(W (ω λ)) = 1 has a positive solution λ0 then λ0 is an eigenvalue of Land hence R0 gt 0 where L is the next generation operator defined in (33)

(ii) If R0 gt 0 then λ = R0 is the unique solution of ρ(W (ω λ)) = 1(iii) R0 = 0 if and only if ρ(W (ω λ)) lt 1 for all λ gt 0

PERIODIC ROSS-MACDONALD MODEL 3141

According to Lemma 41(ii) we can show that the basic reproduction number

for (31) is proportional to the biting rate that is R0 = kR0 if βi(t) = kβi(t) fork gt 0 and 1 le i le p (see Lou and Zhao [21] or Liu and Zhao [18])

To implement numerical simulations we choose the baseline parameter values asfollows bi = 02 ri = 002 ci = 03 di(t) = 01 for i = 1 2 For the first patch

ε1(t) = 125minus(5 cos( 2πt365 )+25 sin( 2πt

365 ))minus(5 cos( 4πt365 )minus25 sin( 4πt

365 )) ge 125minus5radic

5 gt 0

a1(t) = 0028ε1(t) gt 0 In the second patch ε2(t) = 15 minus 12 cos( 2πt365 ) a2(t) =

018 minus 012 cos( 2πt365 ) Then using Lemma 41 we can numerically compute that

the respective reproduction numbers of the disease in patches 1 and 2 are R10 =17571 gt 1 and R20 = 08470 lt 1 and the disease persists in the first patch but diesout in the second patch (see Figure 1(a)) When humans migrate between these twopatches with m12(t) = 0006 minus 0004 cos( 2πt

365 ) and m21(t) = 0006 + 0004 cos( 2πt365 )

the basic reproduction number is R0 = 13944 gt 1 Thus the disease becomesendemic in both patches (see Figure 1(b))

0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

time t

h 1amph 2

(a)

h

1

h2

0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

time t

h 1amph 2

(b)

h

1

h2

Figure 1 Numerical solution of system (31) with initial con-ditions H1(0) = 200 V1(0) = 50 h1(0) = 15 v1(0) = 10 andH2(0) = 300 V2(0) = 30 h2(0) = 30 v2(0) = 5 The curves ofthe number of infected humans in patches 1 and 2 respectivelyare blue solid and red dashed (a) when there is no human move-ment R10 = 17571 gt 1 and R20 = 08470 lt 1 indicate the diseasepersists in patch 1 but dies out in patch 2 (b) when human move-ment presents R0 = 13944 gt 1 indicates the disease persists inboth patches and will stabilize in a seasonal pattern

The basic reproduction number for the corresponding time-averaged autonomoussystem (see Wang and Zhao [36] and Zhang et al [38]) is R0 = 12364 By Lemma

3142 DAOZHOU GAO YIJUN LOU AND SHIGUI RUAN

41 or the linearity of R0 in the mosquito biting rates we can multiply the bitingrates by an appropriate constant such that R0 below 1 while R0 above 1 Thus theautonomous malaria model may underestimate disease severity in some transmissionsettings

In order to investigate the effect of possible travel control on the malaria riskwe determine the disease risk when shifting the phases of the movement functionswhile keeping their respective mean values and forcing strengths to be the sameWith the same parameter values and initial data except that the migration ratesm12(t) = 0006 minus 0004 cos( 2πt

365 + φ1) and m21(t) = 0006 + 0004 cos( 2πt365 + φ2)

we evaluate the annual malaria prevalence over the two patches for φ1 φ2 isin [0 2π]Then Figure 2 presents the effect of phase shifting for movement functions on diseaseprevalence The approximate range of malaria prevalence is from 17 to 23 Thisresult indicates that appropriate travel control can definitely reduce the disease riskand therefore an optimal travel control measure can be proposed

φ2

φ 1

0 1 2 3 4 5 60

1

2

3

4

5

6

0175

018

0185

019

0195

02

0205

021

0215

022

0225

Figure 2 Effect of phase shift for movement rates between twopatches on disease risk

In this project we have developed a compartmental model to address seasonalvariations of malaria transmission among different regions by incorporating seasonalhuman movement and periodic changing in mosquito ecology We then derive thebasic reproduction number and establish the global dynamics of the model Math-ematically we generalize the model and results presented in Cosner et al [7] andAuger et al [3] from a constant environment to a periodic environment Biologi-cally our result gives a possible explanation to the fact that malaria incidence showseasonal peaks in most endemic areas (Roca-Feltrer et al [30]) The numericalexamples suggest that a better understanding of seasonal human migration can

PERIODIC ROSS-MACDONALD MODEL 3143

help us to prevent and control malaria transmission in a spatially heterogeneousenvironment

It would be interesting to assess the impacts of climate change on the emergingand reemerging of mosquito-borne diseases especially malaria It is believed thatglobal warming would cause climate change while climatic factors such as precip-itation temperature humidity and wind speed affect the biological activity andgeographic distribution of the pathogen and its vector It sometimes even affectshuman behavior through sea-level rise more extreme weather events etc

The current and potential future impact of climate change on human healthhas attracted considerable attention in recent years Many of the early studies(see Ostfeld [27] and the references cited therein) claimed that recent and futuretrends in climate warming were likely to increase the severity and global distribu-tion of vector-borne diseases while there is still substantial debate about the linkbetween climate change and the spread of infectious diseases (Lafferty [16]) Thecontroversy is partially due to the fact that although there is massive work us-ing empirical-statistical models to explore the relationship between climatic factorsand the distribution and prevalence of vector-borne diseases there are very fewstudies incorporating climatic factors into mathematical models to describe diseasetransmission

The research that comes closest to what we expect to do has been conducted byParham and Michael in [28] where they used a system of delay differential equationsto model malaria transmission under varying climatic and environmental conditionsModel parameters are assumed to be temperature-dependent or rainfall-dependentor both Due to the complexity of their model mathematical analysis was notpossible and only numerical simulations were carried out Furthermore the basicreproduction number is also difficult to project through their model We wouldlike to extend our model to a mathematically tractable climate-based model in thefuture

Acknowledgments Part of this work was done while the first author was a grad-uate student in the Department of Mathematics University of Miami We thankDrs Chris Cosner David Smith and Xiao-Qiang Zhao for helpful discussions onthis project

REFERENCES

[1] S Altizer A Dobson P Hosseini P Hudson M Pascual and P Rohani Seasonality and

the dynamics of infectious diseases Ecol Lett 9 (2006) 467ndash484[2] G Aronsson and R B Kellogg On a differential equation arising from compartmental anal-

ysis Math Biosci 38 (1973) 113ndash122[3] P Auger E Kouokam G Sallet M Tchuente and B Tsanou The Ross-Macdonald model

in a patchy environment Math Biosci 216 (2008) 123ndash131

[4] N Bacaer and S Guernaoui The epidemic threshold of vector-borne diseases with seasonalityJ Math Biol 53 (2006) 421ndash436

[5] N Bacaer Approximation of the basic reproduction number R0 for vector-borne diseases with

a periodic vector population Bull Math Biol 69 (2007) 1067ndash1091[6] Z Bai and Y Zhou Threshold dynamics of a Bacillary Dysentery model with seasonal fluc-

tuation Discrete Contin Dyn Syst Ser B 15 (2011) 1ndash14

[7] C Cosner J C Beier R S Cantrell D Impoinvil L Kapitanski M D Potts A Troyoand S Ruan The effects of human movement on the persistence of vector-borne diseases J

Theoret Biol 258 (2009) 550ndash560

PERIODIC ROSS-MACDONALD MODEL 3139

By Lemma 33 the system (31) is cooperative in Dt and the ω-periodic semi-flow is monotone We can further prove that the pω-periodic semiflow is stronglymonotone in IntD0 the interior of D0 due to the irreducibility of (

int ω0mij(t)dt)ptimesp

For each t ge 0 i = 1 p there hold

Ki(t αx) = ai(t)biHlowasti (t)minus αhi

Hlowasti (t)αvi minus riαhi +

psumj=1

mij(t)αhj

gt α

(ai(t)bi

Hlowasti (t)minus hiHlowasti (t)

vi minus rihi +

psumj=1

mij(t)hj

)= αKi(t x)

and

Kp+i(t αx) = ai(t)ciαhiHlowasti (t)

(V lowasti (t)minus αvi)minus di(t)αvi

gt α

(ai(t)ci

hiHlowasti (t)

(V lowasti (t)minus vi)minus di(t)vi)

= αKp+i(t x)

for each x 0 α isin (0 1) That is K(t middot) is strictly subhomogeneous on R2p+

Moreover K(t 0) equiv 0 and all solutions are ultimately bounded Therefore byTheorem 312 in Zhao [40] or Theorem 234 in Zhao [40] as applied to the pωsolution map associated with (31) the proof is complete

Using the theory of internally chain transitive sets (Hirsch Smith and Zhao [15]or Zhao [40]) as argued in Lou and Zhao [22] we find that the disease either diesout or persists and stabilizes at a periodic state

Theorem 32 For system (21) if R0 gt 1 then there is a unique positive ω-periodic solution denoted by Elowast(t) = (Hlowast(t) V lowast(t) hlowast(t) vlowast(t)) which is globallyasymptotically stable if R0 le 1 then the disease-free periodic solution E0(t) isglobally asymptotically stable

Proof Let P be the Poincare map of the system (21) on R4p+ that is

P (H(0) V (0) h(0) v(0)) = (H(ω) V (ω) h(ω) v(ω))

Then P is a compact map Let Ω =Ω(H(0) V (0) h(0) v(0)) be the omega limitset of Pn(H(0) V (0) h(0) v(0)) It then follows from Lemma 21prime in Hirsch Smithand Zhao [15] (see also Lemma 121prime in Zhao [40]) that Ω is an internally chaintransitive set for P Based on Lemma 31 we have

limtrarrinfin

(H(t) V (t)) = (Hlowast(t) V lowast(t))

for any H(0) gt 0 and V (0) gt 0 Thus Ω = (Hlowast(0) V lowast(0)) times Ω1 for some

Ω1 sub R2p+ and

Pn|Ω(Hlowast(0) V lowast(0) h(0) v(0)) = (Hlowast(0) V lowast(0) Pn1 (h(0) v(0)))

where P1 is the Poincare map associated with the system (31) Since Ω is aninternally chain transitive set for Pn then Ω1 is an internally chain transitive setfor Pn1 Next we will discuss different scenarios when R0 le 1 and R0 gt 1

Case 1 R0 le 1 In this case the zero equilibrium P0 is globally asymptoticallystable for system (31) according to Theorem 31 It then follows from Theorem 32and Remark 46 of Hirsch Smith and Zhao [15] that Ω1 = (0 0) which further

3140 DAOZHOU GAO YIJUN LOU AND SHIGUI RUAN

implies that Ω = (Hlowast(0) V lowast(0) 0 0) Therefore the disease free periodic solution(Hlowast(t) V lowast(t) 0 0) is globally asymptotically stable

Case 2 R0 gt 1 In this case there are two fixed points for the map P1 (0 0)and (hlowast(0) vlowast(0)) We claim that Ω1 6= (0 0) for all h(0) gt 0 or v(0) gt 0Assume that by contradiction Ω1 = (0 0) for some h(0) gt 0 or v(0) gt 0 ThenΩ = (Hlowast(0) V lowast(0) 0 0) and we have

limtrarrinfin

(H(t) V (t) h(t) v(t)) = (Hlowast(t) V lowast(t) 0 0)

Since R0 gt 1 there exists some δ gt 0 such that the following linear system isunstable

dhi(t)

dt= biai(t)(1minus δ)vi(t)minus rihi(t) +

psumj=1

mij(t)hj(t) 1 le i le p

dvi(t)

dt= ciai(t)(1minus δ)

V lowasti (t)

Hlowasti (t)hi(t)minus di(t)vi(t) 1 le i le p

(34)

Moreover there exists some T0 gt 0 such that

|(H(t) V (t) h(t) v(t))minus (Hlowast(t) V lowast(t) 0 0)| lt δ min0letleω

Hlowast(t) V lowast(t) forallt gt T0

Hence we have

dhi(t)

dtge biai(t)(1minus δ)vi(t)minus rihi(t) +

psumj=1

mij(t)hj(t) 1 le i le p

dvi(t)

dtge ciai(t)(1minus δ)

V lowasti (t)

Hlowasti (t)hi(t)minus di(t)vi(t) 1 le i le p

when t gt T0 According to the instability of system (34) and the comparison princi-ple hi(t) and vi(t) will go unbounded as trarrinfin which contradicts the boundednessof solutions (Theorem 21)

Since Ω1 6= (0 0) and (hlowast(0) vlowast(0)) is globally asymptotically stable for Pn1 in

R2p+ (0 0) it follows that Ω1

⋂W s((hlowast(0) vlowast(0))) 6= empty where W s((hlowast(0) vlowast(0)))

is the stable set for (hlowast(0) vlowast(0)) By Theorem 31 and Remark 46 in Hirsch Smithand Zhao [15] we then get Ω1 = (hlowast(0) vlowast(0)) Thus Ω = (Hlowast(0) V lowast(0) hlowast(0)vlowast(0)) and hence the statement for the case when R0 gt 1 is valid

4 Simulations and discussions In this section we provide some numerical sim-ulations for the two-patch case to support our analytical conclusions The followingmatrix and lemma will be used in numerical computation for the basic reproduc-tion number R0 Let W (t λ) be the monodromy matrix (see Hale [14]) of thehomogeneous linear ω-periodic system

dx

dt=(minus V (t) +

F (t)

λ

)x t isin R1

with parameter λ isin (0infin)

Lemma 41 (Theorem 21 in Wang and Zhao [36]) The following statements arevalid

(i) If ρ(W (ω λ)) = 1 has a positive solution λ0 then λ0 is an eigenvalue of Land hence R0 gt 0 where L is the next generation operator defined in (33)

(ii) If R0 gt 0 then λ = R0 is the unique solution of ρ(W (ω λ)) = 1(iii) R0 = 0 if and only if ρ(W (ω λ)) lt 1 for all λ gt 0

PERIODIC ROSS-MACDONALD MODEL 3141

According to Lemma 41(ii) we can show that the basic reproduction number

for (31) is proportional to the biting rate that is R0 = kR0 if βi(t) = kβi(t) fork gt 0 and 1 le i le p (see Lou and Zhao [21] or Liu and Zhao [18])

To implement numerical simulations we choose the baseline parameter values asfollows bi = 02 ri = 002 ci = 03 di(t) = 01 for i = 1 2 For the first patch

ε1(t) = 125minus(5 cos( 2πt365 )+25 sin( 2πt

365 ))minus(5 cos( 4πt365 )minus25 sin( 4πt

365 )) ge 125minus5radic

5 gt 0

a1(t) = 0028ε1(t) gt 0 In the second patch ε2(t) = 15 minus 12 cos( 2πt365 ) a2(t) =

018 minus 012 cos( 2πt365 ) Then using Lemma 41 we can numerically compute that

the respective reproduction numbers of the disease in patches 1 and 2 are R10 =17571 gt 1 and R20 = 08470 lt 1 and the disease persists in the first patch but diesout in the second patch (see Figure 1(a)) When humans migrate between these twopatches with m12(t) = 0006 minus 0004 cos( 2πt

365 ) and m21(t) = 0006 + 0004 cos( 2πt365 )

the basic reproduction number is R0 = 13944 gt 1 Thus the disease becomesendemic in both patches (see Figure 1(b))

0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

time t

h 1amph 2

(a)

h

1

h2

0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

time t

h 1amph 2

(b)

h

1

h2

Figure 1 Numerical solution of system (31) with initial con-ditions H1(0) = 200 V1(0) = 50 h1(0) = 15 v1(0) = 10 andH2(0) = 300 V2(0) = 30 h2(0) = 30 v2(0) = 5 The curves ofthe number of infected humans in patches 1 and 2 respectivelyare blue solid and red dashed (a) when there is no human move-ment R10 = 17571 gt 1 and R20 = 08470 lt 1 indicate the diseasepersists in patch 1 but dies out in patch 2 (b) when human move-ment presents R0 = 13944 gt 1 indicates the disease persists inboth patches and will stabilize in a seasonal pattern

The basic reproduction number for the corresponding time-averaged autonomoussystem (see Wang and Zhao [36] and Zhang et al [38]) is R0 = 12364 By Lemma

3142 DAOZHOU GAO YIJUN LOU AND SHIGUI RUAN

41 or the linearity of R0 in the mosquito biting rates we can multiply the bitingrates by an appropriate constant such that R0 below 1 while R0 above 1 Thus theautonomous malaria model may underestimate disease severity in some transmissionsettings

In order to investigate the effect of possible travel control on the malaria riskwe determine the disease risk when shifting the phases of the movement functionswhile keeping their respective mean values and forcing strengths to be the sameWith the same parameter values and initial data except that the migration ratesm12(t) = 0006 minus 0004 cos( 2πt

365 + φ1) and m21(t) = 0006 + 0004 cos( 2πt365 + φ2)

we evaluate the annual malaria prevalence over the two patches for φ1 φ2 isin [0 2π]Then Figure 2 presents the effect of phase shifting for movement functions on diseaseprevalence The approximate range of malaria prevalence is from 17 to 23 Thisresult indicates that appropriate travel control can definitely reduce the disease riskand therefore an optimal travel control measure can be proposed

φ2

φ 1

0 1 2 3 4 5 60

1

2

3

4

5

6

0175

018

0185

019

0195

02

0205

021

0215

022

0225

Figure 2 Effect of phase shift for movement rates between twopatches on disease risk

In this project we have developed a compartmental model to address seasonalvariations of malaria transmission among different regions by incorporating seasonalhuman movement and periodic changing in mosquito ecology We then derive thebasic reproduction number and establish the global dynamics of the model Math-ematically we generalize the model and results presented in Cosner et al [7] andAuger et al [3] from a constant environment to a periodic environment Biologi-cally our result gives a possible explanation to the fact that malaria incidence showseasonal peaks in most endemic areas (Roca-Feltrer et al [30]) The numericalexamples suggest that a better understanding of seasonal human migration can

PERIODIC ROSS-MACDONALD MODEL 3143

help us to prevent and control malaria transmission in a spatially heterogeneousenvironment

It would be interesting to assess the impacts of climate change on the emergingand reemerging of mosquito-borne diseases especially malaria It is believed thatglobal warming would cause climate change while climatic factors such as precip-itation temperature humidity and wind speed affect the biological activity andgeographic distribution of the pathogen and its vector It sometimes even affectshuman behavior through sea-level rise more extreme weather events etc

The current and potential future impact of climate change on human healthhas attracted considerable attention in recent years Many of the early studies(see Ostfeld [27] and the references cited therein) claimed that recent and futuretrends in climate warming were likely to increase the severity and global distribu-tion of vector-borne diseases while there is still substantial debate about the linkbetween climate change and the spread of infectious diseases (Lafferty [16]) Thecontroversy is partially due to the fact that although there is massive work us-ing empirical-statistical models to explore the relationship between climatic factorsand the distribution and prevalence of vector-borne diseases there are very fewstudies incorporating climatic factors into mathematical models to describe diseasetransmission

The research that comes closest to what we expect to do has been conducted byParham and Michael in [28] where they used a system of delay differential equationsto model malaria transmission under varying climatic and environmental conditionsModel parameters are assumed to be temperature-dependent or rainfall-dependentor both Due to the complexity of their model mathematical analysis was notpossible and only numerical simulations were carried out Furthermore the basicreproduction number is also difficult to project through their model We wouldlike to extend our model to a mathematically tractable climate-based model in thefuture

Acknowledgments Part of this work was done while the first author was a grad-uate student in the Department of Mathematics University of Miami We thankDrs Chris Cosner David Smith and Xiao-Qiang Zhao for helpful discussions onthis project

REFERENCES

[1] S Altizer A Dobson P Hosseini P Hudson M Pascual and P Rohani Seasonality and

the dynamics of infectious diseases Ecol Lett 9 (2006) 467ndash484[2] G Aronsson and R B Kellogg On a differential equation arising from compartmental anal-

ysis Math Biosci 38 (1973) 113ndash122[3] P Auger E Kouokam G Sallet M Tchuente and B Tsanou The Ross-Macdonald model

in a patchy environment Math Biosci 216 (2008) 123ndash131

[4] N Bacaer and S Guernaoui The epidemic threshold of vector-borne diseases with seasonalityJ Math Biol 53 (2006) 421ndash436

[5] N Bacaer Approximation of the basic reproduction number R0 for vector-borne diseases with

a periodic vector population Bull Math Biol 69 (2007) 1067ndash1091[6] Z Bai and Y Zhou Threshold dynamics of a Bacillary Dysentery model with seasonal fluc-

tuation Discrete Contin Dyn Syst Ser B 15 (2011) 1ndash14

[7] C Cosner J C Beier R S Cantrell D Impoinvil L Kapitanski M D Potts A Troyoand S Ruan The effects of human movement on the persistence of vector-borne diseases J

Theoret Biol 258 (2009) 550ndash560

3140 DAOZHOU GAO YIJUN LOU AND SHIGUI RUAN

implies that Ω = (Hlowast(0) V lowast(0) 0 0) Therefore the disease free periodic solution(Hlowast(t) V lowast(t) 0 0) is globally asymptotically stable

Case 2 R0 gt 1 In this case there are two fixed points for the map P1 (0 0)and (hlowast(0) vlowast(0)) We claim that Ω1 6= (0 0) for all h(0) gt 0 or v(0) gt 0Assume that by contradiction Ω1 = (0 0) for some h(0) gt 0 or v(0) gt 0 ThenΩ = (Hlowast(0) V lowast(0) 0 0) and we have

limtrarrinfin

(H(t) V (t) h(t) v(t)) = (Hlowast(t) V lowast(t) 0 0)

Since R0 gt 1 there exists some δ gt 0 such that the following linear system isunstable

dhi(t)

dt= biai(t)(1minus δ)vi(t)minus rihi(t) +

psumj=1

mij(t)hj(t) 1 le i le p

dvi(t)

dt= ciai(t)(1minus δ)

V lowasti (t)

Hlowasti (t)hi(t)minus di(t)vi(t) 1 le i le p

(34)

Moreover there exists some T0 gt 0 such that

|(H(t) V (t) h(t) v(t))minus (Hlowast(t) V lowast(t) 0 0)| lt δ min0letleω

Hlowast(t) V lowast(t) forallt gt T0

Hence we have

dhi(t)

dtge biai(t)(1minus δ)vi(t)minus rihi(t) +

psumj=1

mij(t)hj(t) 1 le i le p

dvi(t)

dtge ciai(t)(1minus δ)

V lowasti (t)

Hlowasti (t)hi(t)minus di(t)vi(t) 1 le i le p

when t gt T0 According to the instability of system (34) and the comparison princi-ple hi(t) and vi(t) will go unbounded as trarrinfin which contradicts the boundednessof solutions (Theorem 21)

Since Ω1 6= (0 0) and (hlowast(0) vlowast(0)) is globally asymptotically stable for Pn1 in

R2p+ (0 0) it follows that Ω1

⋂W s((hlowast(0) vlowast(0))) 6= empty where W s((hlowast(0) vlowast(0)))

is the stable set for (hlowast(0) vlowast(0)) By Theorem 31 and Remark 46 in Hirsch Smithand Zhao [15] we then get Ω1 = (hlowast(0) vlowast(0)) Thus Ω = (Hlowast(0) V lowast(0) hlowast(0)vlowast(0)) and hence the statement for the case when R0 gt 1 is valid

4 Simulations and discussions In this section we provide some numerical sim-ulations for the two-patch case to support our analytical conclusions The followingmatrix and lemma will be used in numerical computation for the basic reproduc-tion number R0 Let W (t λ) be the monodromy matrix (see Hale [14]) of thehomogeneous linear ω-periodic system

dx

dt=(minus V (t) +

F (t)

λ

)x t isin R1

with parameter λ isin (0infin)

Lemma 41 (Theorem 21 in Wang and Zhao [36]) The following statements arevalid

(i) If ρ(W (ω λ)) = 1 has a positive solution λ0 then λ0 is an eigenvalue of Land hence R0 gt 0 where L is the next generation operator defined in (33)

(ii) If R0 gt 0 then λ = R0 is the unique solution of ρ(W (ω λ)) = 1(iii) R0 = 0 if and only if ρ(W (ω λ)) lt 1 for all λ gt 0

PERIODIC ROSS-MACDONALD MODEL 3141

According to Lemma 41(ii) we can show that the basic reproduction number

for (31) is proportional to the biting rate that is R0 = kR0 if βi(t) = kβi(t) fork gt 0 and 1 le i le p (see Lou and Zhao [21] or Liu and Zhao [18])

To implement numerical simulations we choose the baseline parameter values asfollows bi = 02 ri = 002 ci = 03 di(t) = 01 for i = 1 2 For the first patch

ε1(t) = 125minus(5 cos( 2πt365 )+25 sin( 2πt

365 ))minus(5 cos( 4πt365 )minus25 sin( 4πt

365 )) ge 125minus5radic

5 gt 0

a1(t) = 0028ε1(t) gt 0 In the second patch ε2(t) = 15 minus 12 cos( 2πt365 ) a2(t) =

018 minus 012 cos( 2πt365 ) Then using Lemma 41 we can numerically compute that

the respective reproduction numbers of the disease in patches 1 and 2 are R10 =17571 gt 1 and R20 = 08470 lt 1 and the disease persists in the first patch but diesout in the second patch (see Figure 1(a)) When humans migrate between these twopatches with m12(t) = 0006 minus 0004 cos( 2πt

365 ) and m21(t) = 0006 + 0004 cos( 2πt365 )

the basic reproduction number is R0 = 13944 gt 1 Thus the disease becomesendemic in both patches (see Figure 1(b))

0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

time t

h 1amph 2

(a)

h

1

h2

0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

time t

h 1amph 2

(b)

h

1

h2

Figure 1 Numerical solution of system (31) with initial con-ditions H1(0) = 200 V1(0) = 50 h1(0) = 15 v1(0) = 10 andH2(0) = 300 V2(0) = 30 h2(0) = 30 v2(0) = 5 The curves ofthe number of infected humans in patches 1 and 2 respectivelyare blue solid and red dashed (a) when there is no human move-ment R10 = 17571 gt 1 and R20 = 08470 lt 1 indicate the diseasepersists in patch 1 but dies out in patch 2 (b) when human move-ment presents R0 = 13944 gt 1 indicates the disease persists inboth patches and will stabilize in a seasonal pattern

The basic reproduction number for the corresponding time-averaged autonomoussystem (see Wang and Zhao [36] and Zhang et al [38]) is R0 = 12364 By Lemma

3142 DAOZHOU GAO YIJUN LOU AND SHIGUI RUAN

41 or the linearity of R0 in the mosquito biting rates we can multiply the bitingrates by an appropriate constant such that R0 below 1 while R0 above 1 Thus theautonomous malaria model may underestimate disease severity in some transmissionsettings

In order to investigate the effect of possible travel control on the malaria riskwe determine the disease risk when shifting the phases of the movement functionswhile keeping their respective mean values and forcing strengths to be the sameWith the same parameter values and initial data except that the migration ratesm12(t) = 0006 minus 0004 cos( 2πt

365 + φ1) and m21(t) = 0006 + 0004 cos( 2πt365 + φ2)

we evaluate the annual malaria prevalence over the two patches for φ1 φ2 isin [0 2π]Then Figure 2 presents the effect of phase shifting for movement functions on diseaseprevalence The approximate range of malaria prevalence is from 17 to 23 Thisresult indicates that appropriate travel control can definitely reduce the disease riskand therefore an optimal travel control measure can be proposed

φ2

φ 1

0 1 2 3 4 5 60

1

2

3

4

5

6

0175

018

0185

019

0195

02

0205

021

0215

022

0225

Figure 2 Effect of phase shift for movement rates between twopatches on disease risk

In this project we have developed a compartmental model to address seasonalvariations of malaria transmission among different regions by incorporating seasonalhuman movement and periodic changing in mosquito ecology We then derive thebasic reproduction number and establish the global dynamics of the model Math-ematically we generalize the model and results presented in Cosner et al [7] andAuger et al [3] from a constant environment to a periodic environment Biologi-cally our result gives a possible explanation to the fact that malaria incidence showseasonal peaks in most endemic areas (Roca-Feltrer et al [30]) The numericalexamples suggest that a better understanding of seasonal human migration can

PERIODIC ROSS-MACDONALD MODEL 3143

help us to prevent and control malaria transmission in a spatially heterogeneousenvironment

It would be interesting to assess the impacts of climate change on the emergingand reemerging of mosquito-borne diseases especially malaria It is believed thatglobal warming would cause climate change while climatic factors such as precip-itation temperature humidity and wind speed affect the biological activity andgeographic distribution of the pathogen and its vector It sometimes even affectshuman behavior through sea-level rise more extreme weather events etc

The current and potential future impact of climate change on human healthhas attracted considerable attention in recent years Many of the early studies(see Ostfeld [27] and the references cited therein) claimed that recent and futuretrends in climate warming were likely to increase the severity and global distribu-tion of vector-borne diseases while there is still substantial debate about the linkbetween climate change and the spread of infectious diseases (Lafferty [16]) Thecontroversy is partially due to the fact that although there is massive work us-ing empirical-statistical models to explore the relationship between climatic factorsand the distribution and prevalence of vector-borne diseases there are very fewstudies incorporating climatic factors into mathematical models to describe diseasetransmission

The research that comes closest to what we expect to do has been conducted byParham and Michael in [28] where they used a system of delay differential equationsto model malaria transmission under varying climatic and environmental conditionsModel parameters are assumed to be temperature-dependent or rainfall-dependentor both Due to the complexity of their model mathematical analysis was notpossible and only numerical simulations were carried out Furthermore the basicreproduction number is also difficult to project through their model We wouldlike to extend our model to a mathematically tractable climate-based model in thefuture

Acknowledgments Part of this work was done while the first author was a grad-uate student in the Department of Mathematics University of Miami We thankDrs Chris Cosner David Smith and Xiao-Qiang Zhao for helpful discussions onthis project

REFERENCES

[1] S Altizer A Dobson P Hosseini P Hudson M Pascual and P Rohani Seasonality and

the dynamics of infectious diseases Ecol Lett 9 (2006) 467ndash484[2] G Aronsson and R B Kellogg On a differential equation arising from compartmental anal-

ysis Math Biosci 38 (1973) 113ndash122[3] P Auger E Kouokam G Sallet M Tchuente and B Tsanou The Ross-Macdonald model

in a patchy environment Math Biosci 216 (2008) 123ndash131

[4] N Bacaer and S Guernaoui The epidemic threshold of vector-borne diseases with seasonalityJ Math Biol 53 (2006) 421ndash436

[5] N Bacaer Approximation of the basic reproduction number R0 for vector-borne diseases with

a periodic vector population Bull Math Biol 69 (2007) 1067ndash1091[6] Z Bai and Y Zhou Threshold dynamics of a Bacillary Dysentery model with seasonal fluc-

tuation Discrete Contin Dyn Syst Ser B 15 (2011) 1ndash14

[7] C Cosner J C Beier R S Cantrell D Impoinvil L Kapitanski M D Potts A Troyoand S Ruan The effects of human movement on the persistence of vector-borne diseases J

Theoret Biol 258 (2009) 550ndash560

PERIODIC ROSS-MACDONALD MODEL 3141

According to Lemma 41(ii) we can show that the basic reproduction number

for (31) is proportional to the biting rate that is R0 = kR0 if βi(t) = kβi(t) fork gt 0 and 1 le i le p (see Lou and Zhao [21] or Liu and Zhao [18])

To implement numerical simulations we choose the baseline parameter values asfollows bi = 02 ri = 002 ci = 03 di(t) = 01 for i = 1 2 For the first patch

ε1(t) = 125minus(5 cos( 2πt365 )+25 sin( 2πt

365 ))minus(5 cos( 4πt365 )minus25 sin( 4πt

365 )) ge 125minus5radic

5 gt 0

a1(t) = 0028ε1(t) gt 0 In the second patch ε2(t) = 15 minus 12 cos( 2πt365 ) a2(t) =

018 minus 012 cos( 2πt365 ) Then using Lemma 41 we can numerically compute that

the respective reproduction numbers of the disease in patches 1 and 2 are R10 =17571 gt 1 and R20 = 08470 lt 1 and the disease persists in the first patch but diesout in the second patch (see Figure 1(a)) When humans migrate between these twopatches with m12(t) = 0006 minus 0004 cos( 2πt

365 ) and m21(t) = 0006 + 0004 cos( 2πt365 )

the basic reproduction number is R0 = 13944 gt 1 Thus the disease becomesendemic in both patches (see Figure 1(b))

0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

time t

h 1amph 2

(a)

h

1

h2

0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

time t

h 1amph 2

(b)

h

1

h2

Figure 1 Numerical solution of system (31) with initial con-ditions H1(0) = 200 V1(0) = 50 h1(0) = 15 v1(0) = 10 andH2(0) = 300 V2(0) = 30 h2(0) = 30 v2(0) = 5 The curves ofthe number of infected humans in patches 1 and 2 respectivelyare blue solid and red dashed (a) when there is no human move-ment R10 = 17571 gt 1 and R20 = 08470 lt 1 indicate the diseasepersists in patch 1 but dies out in patch 2 (b) when human move-ment presents R0 = 13944 gt 1 indicates the disease persists inboth patches and will stabilize in a seasonal pattern

The basic reproduction number for the corresponding time-averaged autonomoussystem (see Wang and Zhao [36] and Zhang et al [38]) is R0 = 12364 By Lemma

3142 DAOZHOU GAO YIJUN LOU AND SHIGUI RUAN

41 or the linearity of R0 in the mosquito biting rates we can multiply the bitingrates by an appropriate constant such that R0 below 1 while R0 above 1 Thus theautonomous malaria model may underestimate disease severity in some transmissionsettings

In order to investigate the effect of possible travel control on the malaria riskwe determine the disease risk when shifting the phases of the movement functionswhile keeping their respective mean values and forcing strengths to be the sameWith the same parameter values and initial data except that the migration ratesm12(t) = 0006 minus 0004 cos( 2πt

365 + φ1) and m21(t) = 0006 + 0004 cos( 2πt365 + φ2)

we evaluate the annual malaria prevalence over the two patches for φ1 φ2 isin [0 2π]Then Figure 2 presents the effect of phase shifting for movement functions on diseaseprevalence The approximate range of malaria prevalence is from 17 to 23 Thisresult indicates that appropriate travel control can definitely reduce the disease riskand therefore an optimal travel control measure can be proposed

φ2

φ 1

0 1 2 3 4 5 60

1

2

3

4

5

6

0175

018

0185

019

0195

02

0205

021

0215

022

0225

Figure 2 Effect of phase shift for movement rates between twopatches on disease risk

In this project we have developed a compartmental model to address seasonalvariations of malaria transmission among different regions by incorporating seasonalhuman movement and periodic changing in mosquito ecology We then derive thebasic reproduction number and establish the global dynamics of the model Math-ematically we generalize the model and results presented in Cosner et al [7] andAuger et al [3] from a constant environment to a periodic environment Biologi-cally our result gives a possible explanation to the fact that malaria incidence showseasonal peaks in most endemic areas (Roca-Feltrer et al [30]) The numericalexamples suggest that a better understanding of seasonal human migration can

PERIODIC ROSS-MACDONALD MODEL 3143

help us to prevent and control malaria transmission in a spatially heterogeneousenvironment

It would be interesting to assess the impacts of climate change on the emergingand reemerging of mosquito-borne diseases especially malaria It is believed thatglobal warming would cause climate change while climatic factors such as precip-itation temperature humidity and wind speed affect the biological activity andgeographic distribution of the pathogen and its vector It sometimes even affectshuman behavior through sea-level rise more extreme weather events etc

The current and potential future impact of climate change on human healthhas attracted considerable attention in recent years Many of the early studies(see Ostfeld [27] and the references cited therein) claimed that recent and futuretrends in climate warming were likely to increase the severity and global distribu-tion of vector-borne diseases while there is still substantial debate about the linkbetween climate change and the spread of infectious diseases (Lafferty [16]) Thecontroversy is partially due to the fact that although there is massive work us-ing empirical-statistical models to explore the relationship between climatic factorsand the distribution and prevalence of vector-borne diseases there are very fewstudies incorporating climatic factors into mathematical models to describe diseasetransmission

The research that comes closest to what we expect to do has been conducted byParham and Michael in [28] where they used a system of delay differential equationsto model malaria transmission under varying climatic and environmental conditionsModel parameters are assumed to be temperature-dependent or rainfall-dependentor both Due to the complexity of their model mathematical analysis was notpossible and only numerical simulations were carried out Furthermore the basicreproduction number is also difficult to project through their model We wouldlike to extend our model to a mathematically tractable climate-based model in thefuture

Acknowledgments Part of this work was done while the first author was a grad-uate student in the Department of Mathematics University of Miami We thankDrs Chris Cosner David Smith and Xiao-Qiang Zhao for helpful discussions onthis project

REFERENCES

[1] S Altizer A Dobson P Hosseini P Hudson M Pascual and P Rohani Seasonality and

the dynamics of infectious diseases Ecol Lett 9 (2006) 467ndash484[2] G Aronsson and R B Kellogg On a differential equation arising from compartmental anal-

ysis Math Biosci 38 (1973) 113ndash122[3] P Auger E Kouokam G Sallet M Tchuente and B Tsanou The Ross-Macdonald model

in a patchy environment Math Biosci 216 (2008) 123ndash131

[4] N Bacaer and S Guernaoui The epidemic threshold of vector-borne diseases with seasonalityJ Math Biol 53 (2006) 421ndash436

[5] N Bacaer Approximation of the basic reproduction number R0 for vector-borne diseases with

a periodic vector population Bull Math Biol 69 (2007) 1067ndash1091[6] Z Bai and Y Zhou Threshold dynamics of a Bacillary Dysentery model with seasonal fluc-

tuation Discrete Contin Dyn Syst Ser B 15 (2011) 1ndash14

[7] C Cosner J C Beier R S Cantrell D Impoinvil L Kapitanski M D Potts A Troyoand S Ruan The effects of human movement on the persistence of vector-borne diseases J

Theoret Biol 258 (2009) 550ndash560

3142 DAOZHOU GAO YIJUN LOU AND SHIGUI RUAN

41 or the linearity of R0 in the mosquito biting rates we can multiply the bitingrates by an appropriate constant such that R0 below 1 while R0 above 1 Thus theautonomous malaria model may underestimate disease severity in some transmissionsettings

In order to investigate the effect of possible travel control on the malaria riskwe determine the disease risk when shifting the phases of the movement functionswhile keeping their respective mean values and forcing strengths to be the sameWith the same parameter values and initial data except that the migration ratesm12(t) = 0006 minus 0004 cos( 2πt

365 + φ1) and m21(t) = 0006 + 0004 cos( 2πt365 + φ2)

we evaluate the annual malaria prevalence over the two patches for φ1 φ2 isin [0 2π]Then Figure 2 presents the effect of phase shifting for movement functions on diseaseprevalence The approximate range of malaria prevalence is from 17 to 23 Thisresult indicates that appropriate travel control can definitely reduce the disease riskand therefore an optimal travel control measure can be proposed

φ2

φ 1

0 1 2 3 4 5 60

1

2

3

4

5

6

0175

018

0185

019

0195

02

0205

021

0215

022

0225

Figure 2 Effect of phase shift for movement rates between twopatches on disease risk

In this project we have developed a compartmental model to address seasonalvariations of malaria transmission among different regions by incorporating seasonalhuman movement and periodic changing in mosquito ecology We then derive thebasic reproduction number and establish the global dynamics of the model Math-ematically we generalize the model and results presented in Cosner et al [7] andAuger et al [3] from a constant environment to a periodic environment Biologi-cally our result gives a possible explanation to the fact that malaria incidence showseasonal peaks in most endemic areas (Roca-Feltrer et al [30]) The numericalexamples suggest that a better understanding of seasonal human migration can

PERIODIC ROSS-MACDONALD MODEL 3143

help us to prevent and control malaria transmission in a spatially heterogeneousenvironment

It would be interesting to assess the impacts of climate change on the emergingand reemerging of mosquito-borne diseases especially malaria It is believed thatglobal warming would cause climate change while climatic factors such as precip-itation temperature humidity and wind speed affect the biological activity andgeographic distribution of the pathogen and its vector It sometimes even affectshuman behavior through sea-level rise more extreme weather events etc

The current and potential future impact of climate change on human healthhas attracted considerable attention in recent years Many of the early studies(see Ostfeld [27] and the references cited therein) claimed that recent and futuretrends in climate warming were likely to increase the severity and global distribu-tion of vector-borne diseases while there is still substantial debate about the linkbetween climate change and the spread of infectious diseases (Lafferty [16]) Thecontroversy is partially due to the fact that although there is massive work us-ing empirical-statistical models to explore the relationship between climatic factorsand the distribution and prevalence of vector-borne diseases there are very fewstudies incorporating climatic factors into mathematical models to describe diseasetransmission

The research that comes closest to what we expect to do has been conducted byParham and Michael in [28] where they used a system of delay differential equationsto model malaria transmission under varying climatic and environmental conditionsModel parameters are assumed to be temperature-dependent or rainfall-dependentor both Due to the complexity of their model mathematical analysis was notpossible and only numerical simulations were carried out Furthermore the basicreproduction number is also difficult to project through their model We wouldlike to extend our model to a mathematically tractable climate-based model in thefuture

Acknowledgments Part of this work was done while the first author was a grad-uate student in the Department of Mathematics University of Miami We thankDrs Chris Cosner David Smith and Xiao-Qiang Zhao for helpful discussions onthis project

REFERENCES

[1] S Altizer A Dobson P Hosseini P Hudson M Pascual and P Rohani Seasonality and

the dynamics of infectious diseases Ecol Lett 9 (2006) 467ndash484[2] G Aronsson and R B Kellogg On a differential equation arising from compartmental anal-

ysis Math Biosci 38 (1973) 113ndash122[3] P Auger E Kouokam G Sallet M Tchuente and B Tsanou The Ross-Macdonald model

in a patchy environment Math Biosci 216 (2008) 123ndash131

[4] N Bacaer and S Guernaoui The epidemic threshold of vector-borne diseases with seasonalityJ Math Biol 53 (2006) 421ndash436

[5] N Bacaer Approximation of the basic reproduction number R0 for vector-borne diseases with

a periodic vector population Bull Math Biol 69 (2007) 1067ndash1091[6] Z Bai and Y Zhou Threshold dynamics of a Bacillary Dysentery model with seasonal fluc-

tuation Discrete Contin Dyn Syst Ser B 15 (2011) 1ndash14

[7] C Cosner J C Beier R S Cantrell D Impoinvil L Kapitanski M D Potts A Troyoand S Ruan The effects of human movement on the persistence of vector-borne diseases J

Theoret Biol 258 (2009) 550ndash560

PERIODIC ROSS-MACDONALD MODEL 3143

help us to prevent and control malaria transmission in a spatially heterogeneousenvironment

It would be interesting to assess the impacts of climate change on the emergingand reemerging of mosquito-borne diseases especially malaria It is believed thatglobal warming would cause climate change while climatic factors such as precip-itation temperature humidity and wind speed affect the biological activity andgeographic distribution of the pathogen and its vector It sometimes even affectshuman behavior through sea-level rise more extreme weather events etc

The current and potential future impact of climate change on human healthhas attracted considerable attention in recent years Many of the early studies(see Ostfeld [27] and the references cited therein) claimed that recent and futuretrends in climate warming were likely to increase the severity and global distribu-tion of vector-borne diseases while there is still substantial debate about the linkbetween climate change and the spread of infectious diseases (Lafferty [16]) Thecontroversy is partially due to the fact that although there is massive work us-ing empirical-statistical models to explore the relationship between climatic factorsand the distribution and prevalence of vector-borne diseases there are very fewstudies incorporating climatic factors into mathematical models to describe diseasetransmission

The research that comes closest to what we expect to do has been conducted byParham and Michael in [28] where they used a system of delay differential equationsto model malaria transmission under varying climatic and environmental conditionsModel parameters are assumed to be temperature-dependent or rainfall-dependentor both Due to the complexity of their model mathematical analysis was notpossible and only numerical simulations were carried out Furthermore the basicreproduction number is also difficult to project through their model We wouldlike to extend our model to a mathematically tractable climate-based model in thefuture

Acknowledgments Part of this work was done while the first author was a grad-uate student in the Department of Mathematics University of Miami We thankDrs Chris Cosner David Smith and Xiao-Qiang Zhao for helpful discussions onthis project

REFERENCES

[1] S Altizer A Dobson P Hosseini P Hudson M Pascual and P Rohani Seasonality and

the dynamics of infectious diseases Ecol Lett 9 (2006) 467ndash484[2] G Aronsson and R B Kellogg On a differential equation arising from compartmental anal-

ysis Math Biosci 38 (1973) 113ndash122[3] P Auger E Kouokam G Sallet M Tchuente and B Tsanou The Ross-Macdonald model

in a patchy environment Math Biosci 216 (2008) 123ndash131

[4] N Bacaer and S Guernaoui The epidemic threshold of vector-borne diseases with seasonalityJ Math Biol 53 (2006) 421ndash436

[5] N Bacaer Approximation of the basic reproduction number R0 for vector-borne diseases with

a periodic vector population Bull Math Biol 69 (2007) 1067ndash1091[6] Z Bai and Y Zhou Threshold dynamics of a Bacillary Dysentery model with seasonal fluc-

tuation Discrete Contin Dyn Syst Ser B 15 (2011) 1ndash14

[7] C Cosner J C Beier R S Cantrell D Impoinvil L Kapitanski M D Potts A Troyoand S Ruan The effects of human movement on the persistence of vector-borne diseases J

Theoret Biol 258 (2009) 550ndash560

3144 DAOZHOU GAO YIJUN LOU AND SHIGUI RUAN

[8] C Costantini S G Li A D Torre N Sagnon M Coluzzi and C E Taylor Density survivaland dispersal of Anopheles gambiae complex mosquitoes in a West African Sudan savanna

village Med Vet Entomol 10 (1996) 203ndash219

[9] M H Craig I Kleinschmidt J B Nawn D Le Sueur and B L Sharp Exploring 30 yearsof malaria case data in KwaZulu-Natal South Africa Part I The impact of climatic factors

Trop Med Int Health 9 (2004) 1247ndash1257[10] B Dembele A Friedman and A-A Yakubu Malaria model with periodic mosquito birth

and death rates J Biol Dynam 3 (2009) 430ndash445

[11] D Gao and S Ruan A multi-patch malaria model with logistic growth populations SIAMJ Appl Math 72 (2012) 819ndash841

[12] H Gao L Wang S Liang Y Liu S Tong J Wang Y Li X Wang H Yang J Ma L

Fang and W Cao Change in rainfall drives malaria re-emergence in Anhui province ChinaPLoS ONE 7 (2012) e43686

[13] N C Grassly and C Fraser Seasonal infectious disease epidemiology Proc R Soc B 273

(2006) 2541ndash2550[14] J K Hale Ordinary Differential Equations Wiley-Interscience New York 1980

[15] M W Hirsch H L Smith and X-Q Zhao Chain transitivity attractivity and strong repel-

lors for semidynamical systems J Dyn Differ Equ 13 (2001) 107ndash131[16] K D Lafferty The ecology of climate change and infectious diseases Ecology 90 (2009)

888ndash900[17] L Liu X-Q Zhao and Y Zhou A tuberculosis model with seasonality Bull Math Biol

72 (2010) 931ndash952

[18] X Liu and X-Q Zhao A periodic epidemic model with age structure in a patchy environmentSIAM J Appl Math 71 (2011) 1896ndash1917

[19] J Lou Y Lou and J Wu Threshold virus dynamics with impulsive antiretroviral drug effects

J Math Biol 65 (2012) 623ndash652[20] Y Lou and X-Q Zhao The periodic Ross-Macdonald model with diffusion and advection

Appl Anal 89 (2010) 1067ndash1089

[21] Y Lou and X-Q Zhao A climate-based malaria transmission model with structured vectorpopulation SIAM J Appl Math 70 (2010) 2023ndash2044

[22] Y Lou and X-Q Zhao Modelling malaria control by introduction of larvivorous fish Bull

Math Biol 73 (2011) 2384ndash2407[23] G Macdonald The analysis of sporozoite rate Trop Dis Bull 49 (1952) 569ndash585

[24] G Macdonald Epidemiological basis of malaria control Bull World Health Organ 15(1956) 613ndash626

[25] G Macdonald The Epidemiology and Control of Malaria Oxford University Press London

1957[26] J T Midega C M Mbogo H Mwambi M D Wilson G Ojwang J M Mwangangi J G

Nzovu J I Githure G Yan and J C Beier Estimating dispersal and survival of Anophelesgambiae and Anopheles funestus along the Kenyan coast by using mark-release-recapturemethods J Med Entomol 44 (2007) 923ndash929

[27] R S Ostfeld Climate change and the distribution and intensity of infectious diseases Ecology

90 (2009) 903ndash905[28] P E Parham and E Michael Modeling the effects of weather and climate change on malaria

transmission Environ Health Perspect 118 (2010) 620ndash626[29] D Rain Eaters of the Dry Season Circular Labor Migration in the West African Sahel

Westview Press Boulder CO 1999

[30] A Roca-Feltrer J R Armstrong Schellenberg L Smith and I Carneiro A simple method

for defining malaria seasonality Malaria J 8 (2009) 276[31] R Ross The Prevention of Malaria 2nd edn Murray London 1911

[32] S Ruan D Xiao and J C Beier On the delayed Ross-Macdonald model for malaria trans-mission Bull Math Biol 70 (2008) 1098ndash1114

[33] D L Smith J Dushoff and F E McKenzie The risk of a mosquito-borne infection in a

heterogeneous environment PLoS Biology 2 (2004) 1957ndash1964[34] H L Smith Monotone Dynamical Systems An Introduction to the Theory of Competitive

and Cooperative Systems Mathematical Surveys and Monographs vol 41 AMS Provi-

dence RI 1995[35] H L Smith and P Waltman The Theory of the Chemostat Cambridge University Press

Cambridge 1995

PERIODIC ROSS-MACDONALD MODEL 3145

[36] W Wang and X-Q Zhao Threshold dynamics for compartmental epidemic models in periodicenvironments J Dyn Differ Equ 20 (2008) 699ndash717

[37] P Weng and X-Q Zhao Spatial dynamics of a nonlocal and delayed population model in a

periodic habitat Discrete Contin Dyn Syst Ser A 29 (2011) 343ndash366[38] J Zhang Z Jin G Sun X Sun and S Ruan Modeling seasonal Rabies epidemics in China

Bull Math Biol 74 (2012) 1226ndash1251[39] T Zhang and Z Teng On a nonautonomous SEIRS model in epidemiology Bull Math Biol

69 (2007) 2537ndash2559

[40] X-Q Zhao Dynamical Systems in Population Biology Springer-Verlag New York 2003

Received July 2013 1st revision January 2014 2nd revision January 2014

E-mail address daozhougaoucsfedu

E-mail address yijunloupolyueduhk

E-mail address ruanmathmiamiedu