a generalized nonautonomous sirvs model

Research Article

Received 14 January 2011 Published online in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/mma.2586MOS subject classification: 92D30; 37B55

A generalized nonautonomous SIRVS model

Edgar Pereiraa*†, César M. Silvab‡ and Jacques A. L. da Silvac

Communicated by K. P. Hadeler

A nonautonomous SIRVS model with time-dependent parameters is considered. The global dynamics is investigated, andconditions for the permanence, extinction, and disease-free equilibrium are studied. Substitute quantities that replace thebasic reproduction number are presented. Numerical simulations illustrate the dynamic behavior of the proposed model.Copyright © 2012 John Wiley & Sons, Ltd.

Keywords: epidemic model; nonautonomous; global stability

1. Introduction

The classic models in epidemiology make use of the homogeneous mixing hypothesis or the law of mass action, stating that the rateof change in the disease incidence is directly proportional to the product of the number of susceptible and the number of infectiveindividuals [1–3]. Let S and I denote the number of susceptible and infective individuals, respectively. The bilinear incidence rate ˇSI,ˇ > 0, is consistent with the homogeneous mixing, and it is present in many disease transmission models studied in literature [4].The observed periodicity in the incidence of many diseases [5, 6] justifies the interest for models that have periodic outputs. Classicalmodels, based on the bilinear incidence rate, constant rates of department from model compartments (susceptible, exposed, infective,recovered, vaccinated, etc.), no forcing terms, cannot reproduce the oscillations in the epidemiologic data [7]. These models have eithera globally stable trivial equilibrium corresponding to the disease-free state or a globally stable nontrivial equilibrium corresponding tothe endemic state [2].

Epidemic models using nonlinear incidence rates can show a rich variety of nonlinear phenomena such as multiple stable equilibria,periodic solutions, and Hopf bifurcations [8]. Some of the criticisms on the homogeneous mixing hypothesis is made on its lack of somesaturation effect on contacts between infective and susceptible individuals when the number of infected is large. An incidence rate ofthe form

ˇIpS

1C ˛Iq, ˛,ˇ, p, q > 0,

was proposed in [9]. The cases p D q D 1 studied in [10] and p D q D 2 used in [11] are two different examples of saturation on thecontact rates. A nonmonotonic contact rate was proposed in [12], assuming an incidence rate of the general form given previously withp D 1 and q D 2. In this case, the contact rate decreases at high levels of infective individuals, which may be because of the successof the isolation of the infected individuals and protection measures by the susceptible individuals. Incidence rates of the form ˇIpSq,ˇ, p, q > 0, were studied in [9, 13], and a quite general form was studied in [14], considering an incidence rate of the form f .S, I, N/,where N is the total population and the function f is increasing with respect to S and I and is concave with respect to I. Of course, theparticular choice of the functional form of the incidence rate depends on the disease itself and on the particular form of interactionbetween susceptible and infective individuals.

The models in the preceding paragraph belong to the class of the autonomous differential equations. More realistic models arenonautonomous dynamical systems because they allow for time-dependent coefficients or time-dependent recruitment. Nonau-tonomous epidemic models even with the simple bilinear incidence rate can display interesting behavior such as periodic oscillations[15, 16], subharmonic bifurcations [17, 18], Feigenbaum cascades, and chaos [14, 19, 20]. In autonomous epidemic models, a typicalthreshold condition involves the basic reproduction number R0 (the average number of secondary infections produced when one

aDepartamento de Informática, Instituto de Telecomunicações, Universidade da Beira Interior, 6201-001 Covilhã, PortugalbDepartamento de Matemática, Universidade da Beira Interior, 6201-001 Covilhã, PortugalcInstituto de Matemática, Departamento de Matemática Pura e Aplicada, Universidade Federal do Rio Grande do Sul, 91501-970 Porto Alegre, RS, Brazil*Correspondence to: Edgar Pereira, Departamento de Informática, Instituto de Telecomunicações, Universidade da Beira Interior, 6201-001 Covilhã, Portugal.†E-mail: [email protected]‡Partially supported by FCT throught CMUBI (project PEst-OE/MAT/UI0212/2011)

Copyright © 2012 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2012

E. PEREIRA, C. M. SILVA AND J. A. L. da SILVA

infected individual is introduced to a host population where everyone is susceptible [4]). A typical result states that, if R0 < 1, thedisease is extinct (the disease-free equilibrium is stable) and, if R0 > 1, the disease is permanent (endemic equilibrium is stable). Inthe nonautonomous case, it may not be simple to define a quantity such as the basic reproduction number. Despite this intrinsic diffi-culty, some important results related to the permanence and extinction of the disease in nonautonomous epidemic models have beenobtained. An SIRS model [21, 22], an SIR and an SEIR model [23], an SEIRS model [24], and an SIRVS model [25], all of which are nonau-tonomous epidemic models with bilinear incidence rates, were studied, and threshold conditions for the eradication and permanenceof the disease were obtained. Threshold conditions for the infection persistence were also obtained in [26], considering a SIRS modelwith bilinear incidence rate and density-dependent birth rate.

In this paper, we explore the ideas used in [24, 25] to study the extinction and persistence of the disease in a nonautonomous SIRVSmodel with a nonlinear incidence rate. We assume an incidence rate of the form '.t, S/I, where ' is continuous with bounded derivativewith respect to the second variable. We obtain threshold conditions for the eradication and permanence of the disease, depending oncertain time averages of the parameter functions; this conditions serve as substitutes for the basic reproduction number. Our resultsreduce to the ones in [25] if we consider that the bilinear incidence rates have the special forms '.t, S/D ˇ.t/S and .t, V/D �.t/V . Wealso perform numerical simulations, illustrating various scenarios that reveal the interesting possibility of having sustained oscillationsof the model state variables as in field observations.

2. Notation and preliminaries

We consider the following model

8̂̂<̂ˆ̂̂:

S0 Dƒ.t/� '.t, S/I� .�.t/C p.t//SC �.t/V ,

I0 D Œ'.t, S/C .t, V/��.t/� ˛.t/� �.t/� I,

R0 D �.t/I��.t/R,

V 0 D p.t/S� .�.t/C �.t//V � .t, V/I,

(1)

where, besides the compartments or classes S, I, R, and V , the following parameter functions are considered:

� ƒ.t/ denotes the inflow of newborns in the susceptible class;� '.t, S/ is the incidence (into the infective class) from the susceptible individuals;� .t, V/ is the incidence from the vaccinated individuals;� �.t/ are the natural deaths;� p.t/ represents the vaccination of susceptibles;� �.t/ represents the loss of immunity and consequence influx in the susceptible class;� ˛.t/ are the deaths occurring in the infective class; and� �.t/ is the recovery.

Furthermore, we assume thatƒ,�, p, �,˛, and � are continuous bounded and nonnegative functions on RC0 and that

(H1) the functions ' : RC0 �RC0 !R and : RC0 �R

C0 !R are continuous and nonnegative on RC0 �R

C0 ; and

(H2) for each t 2 RC0 , the functions 't and t given respectively by 't.x/ D '.t, x/ and t.x/ D .t, x/ belong to C1.RC0 /,'t.0/D t.0/D 0, and there exists K > 0 such that

sup.t,x/2RC0 �R

C0

�ˇ̌̌ˇ@'@x

.t, x/

ˇ̌̌ˇ ,

ˇ̌̌ˇ@ @x

.t, x/

ˇ̌̌ˇ�< K .

By (H2) and the mean value theorem, we get

j'.t, x/� '.t, y/j � sup�2Œ0,1�

ˇ̌̌ˇ@'@x

.t, �xC .1� �/y/

ˇ̌̌ˇ jx � yj � Kjx � yj

and also, by the same reasoning,

j .t, x/� .t, y/j � Kjx � yj.

In particular, because '.t, 0/D .t, 0/D 0, we have j'.t, x/j � Kjxj and j .t, x/j � Kjxj.Furthermore, we also need to assume that there are positive constants w�, wƒ, wp, and w� such that

(P1) lim inft!C1

R tCw�t �.s/ds> 0;

(P2) lim inft!C1

R tCwpt p.s/ds > 0 and lim inf

t!C1

R tCwƒt ƒ.s/ds > 0; and

(P3) lim inft!C1

R tCw�t �.s/ds > 0.



Now, we need to make some definitions. Let .S.t/, I.t/, R.t/, V.t// be a solution of (1). We say that

(i) the infectives I are permanent if there exist constants 0< m < M such that

m < lim inft!1

I.t/� lim supt!1

I.t/ < M;

(ii) the infectives I go to extinction if limt!1 I.t/D 0.

Similar definitions can be made for the other compartments. For instance, if there exist constants 0< m < M such that

m < lim inft!1

S.t/� lim supt!1

S.t/ < M,

we say that the susceptibles are permanent.

3. Statement of the results

We need to consider the auxiliary system

�x0 Dƒ.t/� Œ�.t/C p.t/�xC �.t/y,y0 D p.t/x � Œ�.t/C �.t/�y,

(2)

and the function

b.t, x, y/D '.t, x/C .t, y/��.t/� ˛.t/� �.t/.

Note that the auxiliary system describes the behavior of the system in the absence of infection. Ifƒ.t/,�.t/, p.t/, �.t/, p.t/,�.t/, and�.t/ are constant functions, then (2) becomes an autonomous linear system that can be solved explicitly and that corresponds to thelinearization of the equations for S0 and V 0 in the classical (autonomous) SIRVS model.

For each solution �.t/D .x.t/, y.t// of (2) with x.t/ > 0 and y.t/ > 0, we define the numbers

Rp.� ,/D lim inft!1

Z tC�

tb.s, x.s/, y.s//ds,

Re,1.� ,/D lim supt!1

Z tC�

tb.s, x.s/, y.s//ds,

Re,2.�/D lim supt!1

1

t

Z t

0b.s, x.s/, y.s//ds.

Contrarily to what one could expect, the next lemma shows that the aforementioned numbers do not depend on the particular solution�.t/D .x.t/, y.t// of (2) with x.0/ > 0 and y.0/ > 0. We thus write

Rp./, Re,1./, and Re,2

instead of Rp.� ,/, Re,1.� ,/, and Re,2.�/.We prove the next lemma in Section 4.

Lemma 1If '1.t/D .x1.t/, y1.t// and '2.t/D .x2.t/, y2.t// are two solutions of (2) with xi.0/ > 0 and yi.0/ > 0, iD 1, 2, then

Rp.'1,/D Rp.'2,/, Re,1.'1,/D Re,1.'2,/, and Re,2.'1/D Re,2.'2/.

Our main results are the following.

Theorem 1 (Permanence of the disease)Assume that conditions (H1), (H2), and (P1) hold. Then, we have the following:

(a) if there is a constant > 0 such that Rp./ > 0, then the infectives I are permanent in system (1);(b) if the infectives I are permanent in system (1) and condition (P2) holds, then the susceptible and the vaccinated are permanent in

system (1); and(c) if the infectives I are permanent in system (1) and condition (P3) holds, then the removed are permanent in system (1).

Theorem 2 (Extinction of the disease)Assume that conditions (H1), (H2), and (P1) hold. Then, we have the following:

(a) if there is a constant > 0 such that Re,1./ < 0, then the infectives I go to extinction; and(b) if Re,2 < 0, then the infectives I go to extinction.



Because R.t/ in system (1) does not appear in the other equations, we consider the three-dimensional system8<:

S0 Dƒ.t/� '.t, S/I� .�.t/C p.t//SC �.t/V ,I0 D Œ'.t, S/C .t, V/��.t/� ˛.t/� �.t/�I,V 0 D p.t/S� .�.t/C �.t//V � .t, V/I.

(3)

Theorem 3 (Disease-free solution)Assume that conditions (H1), (H2), and (P1) hold and that .S.t/, V.t// is a solution of (2) with initial conditions S.0/ � 0 and V.0/ � 0. Ifthere is a constant such that Re,1./ < 0 or if Re,2 < 0, then the disease-free solution .S.t/, 0, V.t// of (3) is globally attractive.

We now introduce some constants that play the role of the basic reproduction number in this case. For each > 0 and each solution�.t/D .Nx.t/, Ny.t// of system (2) with Nx.0/ > 0 and Ny.0/ > 0, we define

Ra., �/Dlim inft!C1

R tC�t '.s, Nx.s//C .s, Ny.s//ds

lim supt!C1

R tC�t �.s/C ˛.s/C �.s/ds

,

Rb.�/D

lim inft,s!C1

1t

R t0 '.rC s, Nx.rC s//C .rC s, Ny.rC s//dr

lim supt,s!C1

1t

R t0 �.rC s/C ˛.rC s/C �.rC s/dr

,

Rc., �/D

lim supt!C1

R tC�t '.s, Nx.s//C .s, Ny.s//ds

lim inft!C1

R tC�t �.s/C ˛.s/C �.s/ds

,

Rd.�/D

lim supt!C1

1t

R t0 '.s, Nx.s//C .s, Ny.s//ds

lim inft!C1

1t

R t0 �.s/C ˛.s/C �.s/ds

.

Using the same arguments as in the proof of Lemma 1, we can show that the aforementioned numbers do not depend on theparticular solution �.t/ of system (2). Therefore, we write

Ra./, Rb, Rc./, and Rd

instead of Ra., �/, Rb.�/, Rc., �/, and Rd.�/.The next result relates the aforementioned constants with the asymptotic behavior of the infectives in system (2).

Theorem 4Assume that conditions (H1), (H2), and (P1) hold. Then, we have the following:

(a) if there is a constant > 0 such that Ra./ > 1, then the infectives I are permanent in system (1);(b) if Rb > 1, then the infectives I are permanent in system (1);(c) if there is a constant > 0 such that Rc./ < 1, then the infectives I go to extinction in system (1); and(d) if Rd < 1, then the infectives I go to extinction in system (1).

Before proving the theorems, we show that the classical permanence/extinction results of autonomous SIRVS models can be easilyobtained from the main results stated previously. For these, we set ƒ.t/ D ƒ, '.t, S/ D ˇS, .t, V/ D �V , �.t/ D �, p.t/ D p, �.t/ D �,˛.t/D ˛, and �.t/D � ; thus, the basic reproduction number R0 is given by

R0 Dƒ.ˇ.�C �/C �p/

�.�C pC �/.�C ˛C �/.

In this case, the auxiliary system (2) becomes an autonomous system that can be explicitly solved. We get the general solution8<:

x.t/D C1e�.�C�Cp/t C C2e��t C .�C�/ƒ�.�CpC�/ ,

y.t/D C1e�.�C�Cp/t C C2p� e��t C

pƒ�.�CpC�/ .

Taking C1 D C2 D 0, we obtain a particular solution .xp, yp/ for which we have

b.t, xp, yp/D .�C ˛C �/.R0 � 1/.

Therefore, in the autonomous setting, we can easily compute the numbers that play the role of the basic reproduction number. Namely,we can easily see that

Ra./D Rb D Rc D Rd./D R0,



Rp./D Re,1./D .˛C � C�/.R0 � 1/,

and

Re,2 D .˛C � C�/.R0 � 1/.

Therefore, there exists a > 0 such that Rp./ > 0 if and only if R0 > 1, there exists a > 0 such that Re,1./ < 0 if and only if R0 < 1,and Re,2.ƒ/ < 0 if and only if R0 < 1. Thus, in the classical autonomous setting, we conclude that our conditions yield exactly the sameamount of information given by the basic reproduction number.

4. Proof of the theorems

We will divide the proof in several steps. Namely, we first establish some simple results about the solutions of (1), then analyze thenonhomogeneous linear system (2), and finally prove the theorems.

4.1. Boundedness of solutions

Assume that conditions (H1) and (H2) hold. Then, we have the following:

(i) all solutions .S.t/, I.t/, R.t/, V.t// of (1) with nonnegative initial conditions are nonnegative for all t � 0;(ii) all solutions .S.t/, I.t/, R.t/, V.t// of (1) with positive initial conditions are positive for all t � 0;

(iii) there is a constant M > 0 such that, if .S.t/, I.t/, R.t/, V.t// is a solution of (1) with nonnegative initial conditions, then

lim supt!C1

S.t/ < M, lim supt!C1

I.t/ < M, lim supt!C1

R.t/ < M, and lim supt!C1

V.t/ < M.

Properties (i) and (ii) are easy to prove. In fact, because t 7! ƒ.t/ and t 7! �.t/ are bounded, setting U.t/ D S.t/C I.t/C R.t/C V.t/and adding the equations in (1), we obtain, for nonnegative initial conditions,

U0 D��.t/UCƒ.t/� ˛I ��UCƒ,

whereƒD supCt2R0ƒ.t/ and�D infCt2R0

�.t/. Moreover, because the general solution of equation u0C�uDƒ is u.t/D Ce��tCƒ=�,we obtain, by comparison, property (iii).

4.2. Auxiliary system

Assume that conditions (H1), (H2), and (P1) hold. Then, it is straightforward to see that

(i) all solutions .x.t/, y.t// of system (2) with initial condition x.0/� 0 and y.0/� 0 are nonnegative for all t � 0;(ii) each fixed solution .x.t/, y.t// of (2) is bounded and globally uniformly attractive on Œ0,C1Œ;

(iii) if .x.t/, y.t// is a solution of (2) and .Qx.t/, Qy.t// is a solution of the system�

x0 Dƒ.t/� Œ�.t/C p.t/�xC �.t/yC f .t/,y0 D p.t/x � Œ�.t/C �.t/�yC g.t/,

with .Qx.0/, Qy.0//D .x.0/, y.0//, then there is a constant L > 0, only depending on �.t/, satisfying

supt�0fjQx.t/� x.t/j C jQy.t/� y.t/jg � L sup

t�0.jf .t/j C jg.t/j/;

(iv) if (P2) holds, then there exists constants m, M > 0, such that, for each solution of (2), we have

m� lim inft!1

x.t/� lim supt!1

x.t/�M,

m� lim inft!1

y.t/� lim supt!1

y.t/�M.

4.3. Proof of Lemma 1

We will show that, in fact, Rp./ does not depend on the particular solution .x.t/, y.t// of (2) with x.0/ > 0 and y.0/ > 0.By (ii) in Section 4.2, for any " > 0 and any solution .x1.t/, y1.t// of (2) with x1.0/ > 0 and y1.0/ > 0, there is a T > 0 such that

x.t/� "� x1.t/� x.t/C " and y.t/� "� y1.t/� y.t/C " (4)

for every t � T . Moreover, for each 't , t 2 C1.R/, " > 0, and t � T , we have

'.t, x.t/˙ "/D '.t, x.t//˙@'

@x.t, x.t/C �"/"

� '.t, x.t//C "K



for some � 2 Œ0, 1� and also, with the use of (4),

'.t, x.t/˙ "/D '.t, x.t/� x1.t/C x1.t/˙ "/

D '.t, x1.t//C@'

@x.t, x1.t/C �Œx.t/� x1.t/˙ "�/Œx.t/� x1.t/˙ "�

� '.t, x1.t//C KŒ"C jx.t/� x1.t/j�

� '.t, x1.t//C 2"K .

Analogously, '.t, x.t/˙ "/� '.t, x.t//� "K and '.t, x.t/˙ "/� '.t, x1.t//� 2"K . Therefore,

'.t, x.t/˙ "/� "K � '.t, x.t//� '.t, x.t/˙ "/C "K

and

'.t, x.t/˙ "/� 2"K � '.t, x1.t//� '.t, x.t/˙ "/C 2"K ,

and proceeding in a similar way, we get

.t, x.t/˙ "/� "K � .t, x.t//� .t, x.t/˙ "/C "K

and

.t, x.t/˙ "/� 2"K � .t, x1.t//� .t, x.t/˙ "/C 2"K .

Thus, for each t � T , we have

b.t, x.t/C ", y.t/C "/� 4"K � b.t, x1.t/, y1.t//� b.t, x.t/� ", y.t/� "/C 4"K .

Therefore,

lim inft!C1

Z tC�

tb.t, x.t/C ", y.t/C "/ds� 4"K� lim inf

t!C1

Z tC�

tb.t, x1.t/, y1.t//ds� lim inf

t!C1

Z tC�

tb.t, x.t/C ", y.t/C "/dsC 4"K,

lim supt!C1

Z tC�

tb.t, x.t/C ", y.t/C "/ds� 4"K� lim sup

t!C1

Z tC�

tb.t, x1.t/, y1.t//ds� lim sup

t!C1

Z tC�

tb.t, x.t/C ", y.t/C "/dsC 4"K,

and also

lim supt!C1

1

t

Z t

0b.t, x.t/C ", y.t/C "/ds� 4"K� lim sup

t!C1

1

t

Z t

0b.t, x1.t/, y1.t//ds� lim sup

t!C1

1

t

Z t

0b.t, x.t/C ", y.t/C "/dsC 4"K.

From the arbitrariness of " > 0, we conclude that Rp.'1,/ D Rp.',/, Re,1.'1,/ D Re,1.',/, and Re,2.'1/ D Re,2.'/, and this provesthe lemma.

4.4. Proof of Theorem 1

Because R.t/ in system (1) does not appear in the other equations, we only need to consider system (3).For "1 > 0, we consider the auxiliary system

�x0 Dƒ.t/� Œ�.t/C p.t/�xC �.t/y � KM"1,y0 D p.t/x � Œ�.t/C �.t/�y � KM"1.

(5)

Given t0, x0, y0 2 RC, let .Nx.t/, Ny.t// be the solution of (2) with Nx.t0/ D x0 and Ny.t0/ D y0 and let .x.t/, y.t// be the solution of (5) alsowith x.t0/D x0 and y.t0/D y0. Also, let .x�.t/, y�.t// be some fixed solution of (2) with x�.t0/ > 0 and y�.t0/ > 0. By (iii) in Section 4.2,there exists L> 0, only depending on � such that, for all t � t0, we have

jx.t/� Nx.t/j C jy.t/� Ny.t/j � LKM"1.

By hypothesis, there are constants ı > 0 and T0 > 0 such that

Z tC�

tb.s, x�.s/� � , y�.s/� �/ds> ı (6)

for all t � T0 and � > 0 sufficiently small, say, for 0< � < N". We put

"0 <min

�ı

4M, N"

�. (7)



Letting "1 > 0 be sufficiently small so that "1 < "0=2LMK , we obtain, for all t � t0,

jx.t/� Nx.t/j C jy.t/� Ny.t/j �"0

2. (8)

By (ii) in Section 4.2, .x�.t/, y�.t// is globally uniformly attractive on RC0 . Thus, there exists T1 > 0, independent of t0, such that, for allt � t0C T1, we have

jNx.t/� x�.t/j C jNy.t/� y�.t/j �"0

2. (9)

Let .S.t/, I.t/, V.t// be any solution of (3) with S.t0/ > 0, I.t0/ > 0, and V.t0/ > 0 and let Rp./ > 0. We claim that

lim supt!C1

I.t/� "1. (10)

In fact, suppose that (10) is not true. Then there exists T2 � t0 satisfying I.t/ < "1 for all t � T2. From (3) and (iii) in Section 4.1, we have�

S0 �ƒ.t/� Œ�.t/C p.t/�SC �.t/V � KM"1,V 0 � p.t/S� Œ�.t/C �.t/�V � KM"1,

(11)

for all t � T2. Let .x.t/, y.t// be the solution of (5) with initial values x.T2/ D S.T2/ and y.T2/ D V.T2/. Comparing (5) and (11), weconclude that S.t/� x.t/ and V.t/� y.t/ for all t � T2. For t0 D T2, x0 D S.T2/, and y0 D V.T2/, we get, by (8),


2,

for all t � T2. Write T3 DmaxfT0, T1C T2g. By (9), we have, for all t � T3,

S.t/� x.t/ > Nx.t/�1

2"0 > x�.t/� "0 and S.t/� y.t/ > Ny.t/�

1

2"0 > y�.t/� "0.

Proceeding as in Section 4.3, we get, for all t � T3,

b.t, S.t/, V.t//� b.t, x�.t/� "0, y�.t/� "0/� 4"0M,

and, by (6), we get

Z t

T3

b. , x�./� "0, y�./� "0/d �

�1

.t� T3/� 1

�ı.

Thus, integrating the second equation of (3) from T3 to t, we obtain

I.t/D I.T3/eR t

T3b.� ,S.�/,V.�// d�

� I.T3/eR t

T3b.� ,x�.�/�"0,y�.�/�"0/d��4"M.t�T3/

� I.T3/e. ı��4"0M/.t�T3/�ı .

We conclude by (7) that I.t/!C1. This contradicts the assumption that I.t/ < "1 for all t � T2. From this, we conclude that

lim supt!C1

I.t/� "1. (12)

Next, we will prove that, for some constant ` > 0, we have, in fact,

lim inft!C1

I.t/ > `.

By hypothesis, there is P > 0 such that

Z tC

tb.s, x�.s/� "0, y�.s/� "0/ds > ı (13)

for all � � P and t � T0.We proceed by contradiction. Assume that lim inft!C1 I.t/ < ` for all ` > 0. Then there exists a sequence of initial values .xn/n2N,

with xn D .sn, In, Vn/, sn > 0, In > 0, and Vn > 0 such that

lim inft!C1

I.t, xn/ <"1

n2

By (12), given n 2N, there are two sequences .tn,k/k2N and .sn,k/k2N with

T0 < sn,1 < tn,1 < sn,2 < tn,2 < � � �< sn,k < tn,k < � � �



and lim infk!C1 sn,k DC1, such that

I.sk,n, xn/D"1

n, I.tk,n, xn/D

"1

n2,

and

"1

n2< I.t, xn/ <

"1

n, for all t 2�sn,k , tn,kŒ. (14)

By (iii) in Section 4.1, we can choose Kn such that

S.t, xn/ < M, I.t, xn/ < M, V.t, xn/ < M for k > Kn, t 2�sk,n, tk,nŒ.

Let k � Kn; then, for any t 2�sk,n, tk,nŒ, we have

I0.t, xn/D I.t, xn/b.t, S.t, xn/, V.t, xn//

��.�.t/C ˛.t/C �.t//I.t, xn/

��0I.t, xn/,

where �0 D supt�0f�.t/C ˛.t/C �.t/g. Therefore, we obtain

Z tk,n

sk,n

I0. , xn/

I. , xn/d ��0.tk,n � sk,n/

and thus I.tk,n, xn/� I.sk,n, xn/e��0.tk,n�sk,n/. By (14), we get 1n � e��0.tk,n�sk,n/ and therefore

tk,n � sk,n �log n

�0!C1 (15)

as n!C1, k > Kn. Because, for all t 2�sk,n, tk,nŒ, we have I.t, xn/ < "1=n < "1, we conclude that

'.t, S/I � KM"1 and .t, S/I � KM"1

for all t � T0, k � Kn; thus, by (3), we obtain

�S0 �ƒ.t/� .�.t/C p.t//SC �.t/V � KM"1,V 0 � p.t/S� .�.t/C �.t//V � KM"1,

for all t � T0, k � Kn.By comparison, we have S.t, xn/� x.t/ and V.t, xn/� y.t/ for all t 2 Œsk,n, tk,n� and k � Kn. Using (8), we obtain, for all t 2 Œsk,n, tk,n�,


2, (16)

where .Nx.t/, Ny.t// is a solution of (2) with Nx.sk,n/ D S.sk,n, xn/ and Ny.sk,n/ D S.sk,n, xn/. Because .x�.t/, y�.t// is globally uniformlyattractive, there exists T� > 0, independent of n and k, such that

jNx.t/� x�.t/j C jNy.t/� y�.t/j �"0

2, (17)

for all t � sk,n C T�. By (15), we can choose K > 0 such that tn,k � sn,k > PC T� for all n� K and k � Kn. Given n� B and k > Kn, by (13),(16), and (17), and by the second equation in (3), we get

"1

n2D I.tk,n, xn/D I.sk,nC T�, xn/e

R tk,nsk,nCT�

b.� ,S.� ,xn/,V.� ,xn//d�

�"1

n2e

R tk,nsk,nCT�

b.� ,x�.�/�"0,y�.�/�"0/d�>"1

n2.

This leads to a contradiction and establishes (a) of Theorem 1.Statements (b) and (c) in the theorem can be proved in a similar way.




Assuming that we have (H1) and (H2), we can choose " > 0, ı > 0, and T0 > 0 such that

Z tC�

tb.s, x�.s/C ", y�.s/C "/ds < �ı (18)

or

1

t

Z t

0b.s, x�.s/C ", y�.s/C "/ds < �ı (19)

for all t � T0.Given a solution of (3) with S.t0/ > 0, I.t0/ > 0, and V.t0/ > 0 for some t0 � 0, we have

�S0.t/�ƒ.t/� .�.t/C p.t//SC �.t/V ,V 0.t/� p.t/S� .�.t/C �.t//V ,

for all t � t0. For any solution .x.t/, y.t// of (2) with .x.t0/, y.t0//D .S.t0/, V.t0//, we have, by comparison, S.t/� x.t/ and V.t/� y.t/ forall t � t0. Because .x�.t/, y�.t// is globally uniformly attractive, given " > 0, there exists T1 > T such that, for all t > T1, we have

jx.t/� x�.t/j C jy.t/� y�.t/j< ".

Therefore, for all t � T1, we have S.t/� x�.t/C " and V.t/� y�.t/C ".We thus conclude by the second equation in (3) that

logI.t/

I.T1/D

Z t

t1

b. , S.//, V.//d

and therefore

I.t/D I.T1/eR t

t1b.� ,S.�//,V.�// d�

� I.T1/eR t

t1b.� ,x�.�/C",y�.�/C"/d� .

We finally conclude by (18) or (19) that limt!C1 I.t/D 0, and this proves Theorem 2.


Let .S0.t/, V0.t// be a solution of (2) with initial conditions S0.0/ � 0 and V0.0/ � 0 and let .S.t/, I.t/, V.t// be a solution of (3) withnonnegative initial conditions. By (iii) in Section 4.1, we have lim supt!C1 S.t/ < M and lim supt!C1 V.t/ < M. By Theorem 2, wehave lim supt!C1 I.t/D 0. Therefore, given " > 0, there exists T1 � 0 such that I.t/ < ", S.t/ < M, and V.t/ < M for t � T1. Thus,

�S0.t/�ƒ.t/� .�.t/C p.t//SC �.t/V � KM",V 0.t/� p.t/S� .�.t/C �.t//V � KM",

for all t � T1. By comparison, we get S.t/ � Nx.t/ and V.t/ � Ny.t/, where .Nx.t/, Ny.t// is the solution of (2) with "1 D ", Nx.T1/ D S.T1/, andNy.T1/D V.T1/. By (iii) in Section 4.2, we obtain

jNx.t/� x.t/j C jNy.t/� y.t/j � LKM"

for all t � T1, where .x.t/, y.t// is the solution of (2) with initial conditions x.T1/ D Nx.T1/ and y.T1/ D Ny.T1/. Because .S0.t/, V0.t// isglobally uniformly attractive, there exists T2 � T1 such that

jx.t/� S0.t/j C jy.t/� V0.t/j � "

for all t � T1C T2. We conclude that

S.t/� S0.t/� ".1C LMK/ and V.t/� V0.t/� ".1C LMK/, (20)

for all t � T1C T2. Like in the proof of Theorem 2, we can see that

S.t/� S0.t/C " and V.t/� V0.t/C " (21)

for all t large enough. By (20) and (21), we obtain

limt!C1

S.t/D limt!C1

S0.t/ and limt!C1

V.t/D limt!C1

V0.t/,

and it is proved that the disease-free solution .S0.t/, 0, V0.t// is globally attractive.




Assume that there is a constant > 0 such that Ra./ > 1. Then, we have

lim inft!C1

Z tC�

t'.s, Nx.s//C .s, Ny.s//ds > lim sup

t!C1

Z tC�

t�.s/C ˛.s/C �.s/ds

and thus

lim inft!1

Z tC�

tb.s, Nx.s/, Ny.s//ds > lim inf

t!C1

Z tC�

t'.s, Nx.s//C .s, Ny.s//ds� lim sup

t!C1

Z tC�

t�.s/C ˛.s/C �.s/ds > 0.

Therefore, by Theorem 1, we obtain (a).Assume that there is a constant > 0 such that Rb./ > 1. Then there is " > 0 such that

lim inft,s!C1

1

t

Z t

0'.rC s, Nx.rC s//C .rC s, Ny.rC s//dr � lim sup

t,s!C1

1

t

Z t

0�.rC s/C ˛.rC s/C �.rC s/dr � 2" > 0. (22)

For T > 0 sufficiently large, we have

1

t

Z tCs

s'.r, Nx.r//C .r, Ny.r//dr > lim inf

t,s!C1

1

t

Z t

0'.rC s, Nx.rC s//C .rC s, Ny.rC s//dr � "

and also

1

t

Z tCs

s�.r/C ˛.r/C �.r/dr < lim sup

t,s!C1

1

t

Z t

0�.rC s/C ˛.rC s/C �.rC s/drC ",

for all s, t � T . Hence,

1

T

Z TCs

s'.r, Nx.r//C .r, Ny.r//� .�.r/C ˛.r/C �.r//dr > lim inf

t,s!C1

1

t

Z t

0�.rC s/C ˛.rC s/C �.rC s/dr

� lim supt,s!C1

1

t

Z t

0�.rC s/C ˛.rC s/C �.rC s/dr � 2",

for all s� T . By (22), multiplying by T and letting s!C1, we obtain

lim infs!C1

Z TCs

s'.r, Nx.r//C .r, Ny.r//� .�.r/C ˛.r/C �.r//dr > 0;

thus, by Theorem 1, we obtain (b).Assume that there is a constant > 0 such that Rc./ < 1. Then, we have

lim supt!C1

Z tC�

t'.s, Nx.s//C .s, Ny.s// ds < lim inf

t!C1

Z tC�

t�.s/C ˛.s/C �.s/ds

and thus

lim supt!1

Z tC�

tb.s, Nx.s/, Ny.s//ds < lim sup

t!C1

Z tC�

t'.s, Nx.s//C .s, Ny.s//ds� lim inf

t!C1

Z tC�

t�.s/C ˛.s/C �.s/ds < 0.

Therefore, by Theorem 2, we obtain (c).Assume that there is a constant > 0 such that Rd./ < 1. Then, we have

lim supt!C1

1

t

Z t

0'.s, Nx.s//C .s, Ny.s//ds < lim inf

t!C1

1

t

Z t

0�.s/C ˛.s/C �.s/ds

and thus

lim supt!1

1

t

Z t

0b.s, Nx.s/, Ny.s//ds < lim sup

t!C1

1

t

Z t

0'.s, Nx.s//C .s, Ny.s//ds� lim inf

t!C1

1

t

Z t

0�.s/C ˛.s/C �.s/ds < 0.

Therefore, by Theorem 2, we obtain (d).



5. Simulation

We carried out the experiments by using the following functions: sin.at/ and cos.bt/, a, b > 0, which are periodic functions thatcan reflect phenomena that occur seasonally; ect , c < 0, which is a function that converges monotonically to zero and can describecharacteristics that tend to disappear with time; and .tc/ect , c < 0, which is used to picture an outbreak of an epidemic.

For model (1), first we consider the functions:

ƒ.t/D kƒ�1C e�t C qƒ cos.bƒt/

�,

'.t, S/D�

k'.1C ts'e�s' t/C q'.1� sin.a' t//� S.t/

S.t/C r',

.t, V/D k �1C e�c t� V.t/

V.t/C r ,

�.t/D k�.1C q� sin.a�t//,

p.t/D kp.1C e�cpt C bp sin.apt//,

�.t/D k� C q�t

tC 1,

˛.t/D k˛.1C e�a˛ t/,

�.t/D k� .1C e�a� t/.

0 50 100 150 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 1. S.t/, I.t/, R.t/, V.t/: typical behavior.

0 500 1000 1500 20000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 2. S.t/, I.t/, R.t/, V.t/: stable behavior.



These functions are bounded continuous and nonnegative. Then, we take the following basic set of parameters: kƒ D 0.01, qƒ D 0.005,bƒ D 0.1, k' D 0.08, q' D 0.02, a' D 0.1, r' D 0.08, s' D 0, k D 0.01, c D 0.4, r D 1, k� D 0.01, q� D 0.5, a� D 1, kp D 0.5, cp D 5,bp D 0.96, ap D 0.001, k� D 0.08, q� D 0.02, k˛ D 0.01, a˛ D 10, k� D 0.02, and a� D 5.

We do not have a basic reproduction number, but Theorem 4 provides the numbers to measure the force of infection. The problemhere is that it is very complicated to compute a solution of the function �.t/ D .Nx.t/, Ny.t// for the auxiliary system with the functionspreviously given. Considering that, we estimate lower bounds lx � Nx.t/ and ly � Ny.t/, for t sufficiently large, such that '.t, lx/� '.t, Nx.t//and .t, ly/ � .t, Ny.t//; thus, we have an 0 � R0 � Rb.�/. So, we obtain Nx.t/ � 0.16 and Ny.t/ � 0.26, giving R0 D 1.69; hence, by (b) ofTheorem 4, the infection is permanent.

For the numerical experiments with model (1), first, we use this basic set, and after, we change only some specific parameters.The initial conditions are set to S.0/D 1 and I.0/D 0.01, and the others are null. Furthermore, we use the following colors: green (S),

red (I), blue (R), and magenta (V).We observe in Figure 1 a typical behavior for 0 < t < 200 and in Figure 2 a first periodic tendency, with 0 < t < 2000. In Figure 3(a),

we see that the system has a stable and periodic behavior as t goes to infinity, with the respective zooms in Figures 3(b,c) showing theperiodicity in more details.

Changing slightly the incidence parameters in function ', we can observe in Figures 4 and 5 that the curve I.t/moves to the right aswe decrease the parameter k' for two different q' .

In Figures 6 and 7, for 0 < t < 400 and 0 < t < 2000, respectively, we observe the effects of a peak function in the classes I and V ,with the curves moving to the left and showing an epidemic outbreak, as we include a nonzero value for s'. Finally, in Figures 8 and9, we depict a simulation in a phase portrait in three dimensions, where we consider three populations: SC V (which can be infected),I, and R.

The SIRVS epidemic model with time-varying coefficients can display an interesting dynamic behavior, especially the sustained oscil-lations throughout time. Numerical simulations are important to reveal the influence of some key parameters such as the frequency ofoscillations in the vaccination rate, in the incidence rate, and in the inflow of susceptible individuals in the population. We performed

1.04 1.06 1.08 1.1 1.12 1.14 1.160

0.01

0.02

0.03

0.04

0.05

0.06

1.15 1.2 1.25 1.3 1.35

0.18

0.2

0.22

0.24

0.26

0.28

x 104

x 104

(b)

(c)

0 0.5 1 1.5 2

x 104

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a)

Figure 3. S.t/, I.t/, R.t/, V.t/: (a) periodic behavior and (b,c) zooms of Figure 3(a).



0 50 100 150 200 250 300 350 4000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Figure 4. I.t/: k' D 0.08, k' D 0.07, and k' D 0.06, for q' D 0.02.

0 50 100 150 200 250 300 350 4000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure 5. I.t/: k' D 0.08, k' D 0.07, and k' D 0.06, for q' D 0.01.

0 50 100 150 200 250 300 350 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 6. I.t/, V.t/, for s' D 0 and s' D 0.001.

several tests, and we could observe that the interplay of the various frequencies are displayed in the sustained oscillations of the infec-tive class. It is interesting to notice that an increase in the frequency of vaccination, besides causing an increase in oscillatory behaviorof the infective class, can cause an unexpected larger epidemic peak. This is rather surprising and certainly deserves further investiga-tion. The effects of a reduction in the incidence rate and an increase in the recovery rate were not a surprise. They cause a reduction inthe number of cases, thereby delaying the peak of epidemic as in the autonomous SIR models.

There are many possibilities for simulation scenario such as the approach used in Earn [27] and Earn et al. [28], giving possibleexplanations for the 2-year cycle in measles epidemics. We decided to follow a more theoretical approach, presenting a quite general



0 500 1000 1500 20000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 7. I.t/, V.t/, for s' D 0 and s' D 0.001.

0.5

1

1.5

0.2

0.4

0.6

0.80

0.2

0.4

0.6

0.8

1

00

Figure 8. For ap D 0.1, SC V , I, and R phase portrait.

Figure 9. The respective zoom of Figure 8.

epidemic model with a nonlinear incidence. We give emphasis to the theoretical results on permanence/extinction of the disease withthis rather general nonlinear incidence. Future investigation plans include a more general nonlinear incidence rate and spatial scalefeatures, allowing us to perform case studies with spatial epidemic data.



Acknowledgements

We thank the anonymous referee for his comments that led to corrections and improvements of the original manuscript.

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a generalized nonautonomous sirvs model

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