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Dynamics of an epidemic model with host migration
Applied Mathematics and Computation 218 (2011) 4614–4625
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Dynamics of an epidemic model with host migration Zhipeng Qiu ⇑ Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing 210094, PR China
a r t i c l e
i n f o
Keywords: Vector–host Epidemic model Stability Migration The reproduction number
a b s t r a c t Host migration among discrete geographical regions is demonstrated as an important factor that brings about the diffusion and outbreak of many vector–host diseases. In the paper, we develop a mathematical model to explore the effect of host migration between two patches on the spread of a vector–host disease. Analytical results show that the reproduction number R0 provides a threshold condition that determines the uniform persistence and extinction of the disease. If both the patches are identical, it is shown that an endemic equilibrium is locally stable. It is also shown that a unique endemic equilibrium, which exists when the disease cannot induce the death of the host, is globally asymptotically stable. Finally, two examples are given to illustrate the effect of host migration on the spread of the vector–host disease. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Many diseases are spread by vectors, as we can see with plague (spread by direct contamination, but usually by fleas), typhus (spread by lice), malaria, or dengue fever (the latter two spread by mosquitoes), so the emergence and reemergence of the vector–host diseases have promoted much interest in their mathematical models. Recently there has been some effort in the mathematical modeling of the vector–host epidemic transmission dynamics (see, for example, [1–3] etc.). These models have provided much insight into understanding the mechanism of the vector–host epidemic transmission. Nevertheless, all these models have only considered transmission dynamics of the vector–host disease in an isolated patch, ignoring the effect of host migration among patches (discrete geographical regions). However, for some kind of vector–host diseases such as West Nile virus (WNV) and dengue fever, the migration of host is an important factor that leads to the worldwide spread of the vector–host diseases. Wonham et al. [4] have suggested that the WNV model should be extended biologically to consider bird migration. The correlation between migratory birds and the spread of WNV has been studied in many papers [5–7]. In paper [6], Rappole and Hubalek have provided some factors supporting the ‘migrant bird as introductory host’ hypothesis for the spread of WNV, and in paper [7], Owen et al. have demonstrated that migrating passerine birds are potential dispersal vehicles for WNV. All the facts inspire us to consider the effect of host migration among multiple patches on the spread of the vector–host diseases. Modeling the spatial spread of the infectious disease is a complex task. One possible approach is to consider the migration of individuals between discrete geographical regions. In mathematical epidemiology a few models which incorporate discrete geographical regions have been studied (see [8–11] and the references cited therein). However, up till now no one has considered the effect of host migration among multiple patches on the dynamics of vector–host diseases. Therefore, we expect to explore the effect of host migration among multiple patches on the spread of the vector–host diseases in order to answer the interesting questions in mathematical epidemiology: How does the host migration affect the dynamics and outbreak of the vector–host diseases in discrete regions? ⇑ Corresponding author. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.10.045
Z. Qiu / Applied Mathematics and Computation 218 (2011) 4614–4625
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The purpose of the paper is to model the transmission dynamics of a vector–host disease spread between two patches due to host migration, and describe the dynamics of the system. The remaining part of this paper is organized as follows: in Section 2, we mainly formulate our model. In Section 3, we mainly present some preliminary results and derive the reproduction number. In Section 4, the threshold dynamics of the model is analyzed. The stability of endemic equilibrium is studied in Section 5. In the last section we summarize our results, and give two examples to illustrate how the host migration affects the dynamics and outbreak of the vector–host diseases.
2. Model description In this section we mainly formulate an epidemic model to describe the transmission dynamics of a vector–host disease spread between two discrete patches due to host migration. First, let us formulate a model for the spread of a vector–host disease in the ith patch. Suppose there is no host migration among patches, i.e., the patches are isolated. We assume that the total host population Ni(t) in the ith patch is partitioned into two distinct epidemiological subclasses which are susceptible and infectious, with sizes denoted by Si(t) and Ii(t), respectively, and the total vector population Ti(t) in the ith patch is also divided into susceptible and infectious subclasses, with the sizes denoted by Mi(t) and Vi(t), respectively. Our assumptions on the dynamical transfer of the host and vector population in the ith patch are demonstrated in Fig. 1. In Fig. 1, Bi(Ni) and Fi(Ti) are the growth rates of the hosts and the vectors in the ith patch, respectively. The growth rate of the hosts Bi(Ni) and the rate of the vectors Fi(Ti) are generally assumed to satisfy the following basic assumptions for Ni, Ti 2 (0, +1) [9,12]: (a1) Bi ð0þÞ ¼ limNi !0þ Bi ðN i Þ > 0; F i ð0þÞ ¼ limT i !0þ F i ðT i Þ > 0; i ¼ 1; 2; (a2) Bi(Ni) and Fi(Ti) are continuously differentiable with B0i ðN i Þ < 0; F 0i ðT i Þ < 0; i ¼ 1; 2; (a3) li > Bi(+1), i > Fi(+1), i = 1, 2. In this paper, we shall choose the function Bi ðN i Þ ¼ NAii þ ci with Ai > 0, li > ci P 0 as the growth rate of the hosts and the Ai function F i ðT i Þ ¼ GT ii þ hi with Gi > 0, i > hi P 0 as the growth rate of the vectors. Biologically, the function Ni þ ci N i represents a constant immigration rate Ai together with a linear birth term ciNi of the hosts. The function GT i þ hi T i has the same i biological meaning. bi(Ni) denotes the per capita rate of contacts on hosts by vectors, which is continuously differential function of Ni. Generally, bi(Ni) is a monotonic increasing function on Ni. In this paper, we assume that bi(Ni) = biNi. ai and bi are the disease transmission probabilities from infected hosts to uninfected vectors and from infected vectors to uninfected hosts in the ith patch, respectively; i, li are the natural death rates of the vectors and the hosts, respectively; di is the disease-induced death rate of the hosts, and ci is the recovery rate of infected hosts in the ith patch. All the above parameters are positive except ci P 0 and di P 0. Using the transfer diagram and the assumptions that bi ðN i Þ ¼ bi N i ; Bi ðN i Þ ¼ NAii þ ci ; F i ðT i Þ ¼ GT ii þ hi , the system which describes the spread of a vector–host disease in ith patch can be derived:
8 0 S ¼ Ai þ ci Ni li Si bi bi V i Si þ ci Ii ; > > > i > > < I0 ¼ bi bi V i Si ðl þ c þ di ÞIi ; i i i > > M 0i ¼ Gi þ hi T i bi ai Ii M i i M i ; > > > : 0 V i ¼ bi a i I i M i i V i :
Fig. 1. Transfer diagram for the vector–host epidemic model.
ð2:1Þ
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When two patches are connected, we assume that only hosts can migrate among the patches since vectors are usually arthropod whose migration can be ignored. We assume that susceptible and infected hosts of every patch i leave for patch j at a per capita rate ai. Then the dynamics of the hosts and the vectors is governed by the following model:
8 0 Si ¼ Ai þ ci Ni li Si bi bi V i Si þ ci Ii ai Si þ aj Sj ; > > > < I0 ¼ b b V S ðl þ c þ d ÞI a I þ a I ; i i i i i i i i j j i i i > M0i ¼ Gi þ hi T i bi ai Ii M i i M i ; > > : 0 V i ¼ bi ai Ii M i i V i ;
i; j ¼ 1; 2; i – j:
ð2:2Þ
The purpose of the paper is to analyze the dynamics of the model (2.2) and to answer how the host migration affects the dynamics and outbreak of the vector–host disease in discrete regions. 3. The preliminary results and reproduction number In this section, we mainly present the preliminary results and derive the reproduction number for system (2.2). In order to investigate the dynamics of the system (2.2), we begin with stating some results on system (2.1). The system (2.1) has been 0 analyzed in [13]. Straight forward computation yields that system (2.1) admits a boundary equilibria Ei01 N 00 ; 0; T ; 0 and a i i 0 0 0 , where potential unique positive equilibrium Ei0 S0 i ; Ii ; M i ; V i
Ai ; li c i
V 0 i ¼
ðli þ ci þ di ÞðAi ðli ci ÞN0 Þ i ; 0 bi bi ðli þ di ci ÞNi Ai
T 0i ¼
Gi ; i hi
ðli þ di ci ÞN0 i Ai ; di
N00 i ¼
S0 i ¼
I0 i ¼
Ai ðli ci ÞN0 i ; di
Mi ¼ T 0i V 0 i
and 2
N0 i ¼
ðli þ ci þ di Þðbi ai Ai þ di i Þ þ T 0i bi ai bi Ai 2 T 0i bi
ai bi ðli þ di ci Þ þ ðli þ di þ ci Þbi ai ðli ci Þ
:
The reproduction number of system (2.1) is established in paper [13], which can be expressed as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 0 bi ai bi N00 i Ti : R0i ¼ i ðli þ ci þ di Þ By Theorems 3.2 and 5.1 in paper [13], we can obtain the following lemma. Lemma 3.1. If R0i < 1, then Ei01 is globally asymptotically stable; if R0i > 1, then Ei0 is globally asymptotically stable. We now derive the reproduction number for system (2.2). Adding up the third and fourth equations of system (2.2) gives the equations for Ti, i = 1, 2:
T 0i ¼ Gi ðei hi ÞT i : It is easy to see that T i ðtÞ ! T 0i as t ? +1. From the first and second equations of system (2.2) we have
(
N01 6 A1 ðl1 c1 ÞN1 a1 N1 þ a2 N2 ; N02 6 A2 ðl2 c2 ÞN2 þ a1 N1 a2 N2 :
Define an auxiliary system by
(
e 0 ¼ A1 ð l c 1 Þ N e 1 þ a2 N e 2; e 1 a1 N N 1 1 0 e e 2: e e N 2 ¼ A2 ðl2 c2 Þ N 2 þ a1 N 1 a2 N
ð3:1Þ
We can verify that the system (3.1) is cooperative [14] and there exists a unique equilibrium N 01 ; N 02 of system (3.1) in R2þ , where
ðA1 þ A2 Þa2 þ ðl2 c2 ÞA1 ; ðl1 c1 Þðl2 c2 Þ þ ðl1 c1 Þa2 þ ðl2 c2 Þa1 ðA1 þ A2 Þa1 þ ðl1 c1 ÞA2 N02 ¼ : ðl1 c1 Þðl2 c2 Þ þ ðl1 c1 Þa2 þ ðl2 c2 Þa1
N01 ¼
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By the results in [15] we conclude that the equilibrium N 01 ; N 02 is globally asymptotically stable in R2þ . By the comparison principle we have N 1 ðtÞ 6 N 01 ; N 2 ðtÞ 6 N 02 . Let
n
o
C ¼ ðS1 ; I1 ; M1 ; V 1 ; S2 ; I2 ; M2 ; V 2 Þ : 0 6 Mi ; V i ; Mi þ V i 6 T 0i ; 0 6 Si ; Ii ; Si þ Ii 6 N0i ; i ¼ 1; 2; : Then it can be seen that all solutions of the systems (2.2) starting in C remain in C for all t P 0. Thus C is positively invariant and it is sufficient to consider solutions in C. In this region, the usual existence, uniqueness and continuation results hold for the system (2.2). In what follows, we always assume that the initial points lie in C. Straight forward computation yields that the disease free equilibrium (DFE) of system (2.2) is
E0 N01 ; 0; T 01 ; 0; N02 ; 0; T 02 ; 0 : A common method of determining its local stability is finding the reproduction number, which determines the persistence and extinction of a disease. Note that the system (2.2) has four infected populations, namely, I1, I2, V1,V2, it follows that, using the notation of van den Driessche and Watmough [16], the matrix F and V for the new infection terms and the remaining transfer terms, respectively, are given by
F¼
0
F12
F21
0
;
V¼
V11
0
0
V22
;
where
F12 ¼ V11 ¼
!
b1 b1 N01
0
0
b2 b2 N02
F21 ¼
;
b1 a1 T 01
0
0
b2 a2 T 02
l1 þ c1 þ d1 þ a1
a2
a1
l2 þ c2 þ d2 þ a2
;
! ;
V22 ¼
1
0
0
2
:
If R0 :¼ qðFV1 Þ; where qðFV1 Þ represents the spectral radius of the matrix FV1 , it follows from [16] that R0 is the reproduction number of system (2.2). After extensive algebraic calculations, we can obtain
R0 ¼
pffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 e2 e 2 vf 2 ; e 2 ð1 fÞ þ ð R e 2 ð1 vÞ R e 2 ð1 fÞÞ2 þ 4 R e2 R R 01 ð1 vÞ þ R 02 01 02 01 02 2
where
e 0i ¼ R
v¼ f¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 bi ai bi N0i T 0i ; i ðli þ ci þ di Þ
i ¼ 1; 2;
a1 l1 þc1 þd1 ; a1 1 þ l þc þd1 þ l þac2 þd2 1 1 2 2 a2 l2 þc2 þd2 : a1 1 þ l þc þd1 þ l þac2 þd2 1 2 1 2
Especially, if the two patches are isolated, i.e., a1 = a2 = 0, then we have
R0 ¼ maxfR01 ; R02 g: From the proof of Theorem 2 in [16], it follows that
R0 < 1ðR0 ¼ 1; R0 > 1Þ () sðJÞ < 0ðsðJÞ ¼ 0;
sðJÞ > 0Þ;
ð3:2Þ
where
0 B B J ¼FV¼B B @
ðl1 þ c1 þ d1 þ a1 Þ
a2
b1 b1 N01
a1
ðl2 þ c2 þ d2 þ a2 Þ
0
0
1
0 1T1
b1 a
0
0 2T 2
b2 a
0
0
1
C b2 b2 N02 C C C 0 A 2
and s(J) is the maximum real part of the eigenvalues of the matrix J. Since J is irreducible and has non-negative off-diagonal elements. Then it follows from Theorem A.5 in [17] that s(J) is a simple eigenvalue of J with a positive eigenvector.
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Furthermore, since the diagonal elements of J are positive and its off-diagonal elements are non-positive, it follows from Mmatrix [18] that
8 J 1 ¼ ðl1 þ c1 þ d1 þ a1 Þ < 0; > > > > > J 2 ¼ ðl1 þ c1 þ d1 þ a1 Þðl2 þ c2 þ d2 þ a2 Þ a1 a2 > 0; > > > > > a1 a2 > > < J 3 ¼ ðl1 þ c1 þ d1 Þðl2 þ c2 þ d2 Þ1 1 þ l1 þc1 þd1 þ l2 þc2 þd2 sðJÞ < 0 () e 2 ð1 vÞ 1 < 0; R > 01 > > > > > > > J 4 ¼ ðl1 þ c1 þ d1 Þðl2 þ c2 þ d2 Þ1 2 1 þ l þac1 þd1 þ l þac2 þd2 > > 1 1 2 2 > > : e 2 ð1 vÞ R e2 e 2 ð1 fÞ þ R e2 R ð1 R 01 02 01 02 ð1 v fÞÞ > 0; where Ji, j = 1, 2, 3, 4, are the leading principal minors of J with i rows. Consequently, we have
(
R0 < 1 ()
e 2 ð1 fÞ < 1Þ; e 2 ð1 vÞ < 1 ðor R R 01 02 2 2 e 2 ð1 v fÞ < 1: e ð1 fÞ R e2 R e ð1 vÞ þ R R 01
02
01
ð3:3Þ
02
Using Theorem 2 in [16], we can easily obtain the following stability result. Theorem 3.2. If R0 < 1, then the DFE (E0) is locally asymptotically stable; if R0 > 1, then the DFE (E0) is unstable. 4. Threshold dynamics The purpose of this section is to discuss the global extinction and persistence of the disease described by system (2.2). We will show that R0 is a threshold between the extinction and uniform persistence of the disease. Theorem 4.1. If R0 < 1, then the disease-free equilibrium of system (2.2) is globally asymptotically stable. Proof. If R0 < 1, then it follows from Theorem 3.2 that the disease-free equilibrium E0 is locally asymptotically stable. In the following we only need to prove that E0 is a global attractor. Let us consider a positive solution (S1(t), I1(t), M1(t), V1(t), S2(t), I2(t), M2(t), V2(t)) of system (2.2). From the second and fourth equations of (2.2), it follows that
8 0 I 6 b1 b1 V 1 N01 ðl1 þ c1 þ d1 ÞI1 a1 I1 þ a2 I2 ; > > > 1 > < V0 6 b a I T0 V ; 1 1 1 1 1 1 1 > I0 6 b2 b2 V 2 N0 ðl þ c þ d2 ÞI2 a2 I2 þ a1 I1 ; > 2 2 2 2 > > : 0 V 2 6 b2 a2 I2 T 02 2 V 2 :
ð4:1Þ
We consider the following differential equations
8 0 I1 ¼ b1 b1 V 1 N01 ðl1 þ c1 þ d1 ÞI1 a1 I1 þ a2 I2 ; > > > > < V0 ¼ b a I T0 V ; 1 1 1 1 1 1 1 0 > > I02 ¼ b2 b2 V 2 N2 ðl2 þ c2 þ d2 ÞI2 a2 I2 þ a1 I1 ; > > : 0 V 2 ¼ b2 a2 I2 T 02 2 V 2 :
ð4:2Þ
Since the system (4.2) is a linear system, the global stability of the origin of system (4.2) is determined from the stability of the matrix J. From (3.2) it can be seen that if R0 < 1 then all the eigenvalues of the matrix J have negative real parts. It then follows that each solution of system (4.2) satisfies
lim I1 ðtÞ ¼ 0;
t!þ1
lim I2 ðtÞ ¼ 0;
t!þ1
lim V 1 ðtÞ ¼ 0;
t!þ1
lim V 2 ðtÞ ¼ 0:
t!þ1
By the comparison principle we have that each positive solution of (4.1) satisfies Ii(t) ? 0, Vi(t) ? 0, i = 1, 2 as t ? +1. Then the limiting system of (2.2) is
8 0 S1 ¼ A1 þ c1 S1 l1 S1 a1 S1 þ a2 S2 ; > > > < S0 ¼ A þ c S l S þ a S a S ; 2 2 2 1 1 2 2 2 2 2 0 > M ¼ G þ h M M ; 1 1 1 1 1 > 1 > : 0 M2 ¼ G2 þ h2 M2 2 M2 :
ð4:3Þ
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We can verify that system (4.3) is a cooperative system [14] and there exists a unique equilibrium N 01 ; N 02 ; T 01 ; T 02 . By the results in [15] we conclude that the equilibrium N 01 ; N 02 ; T 01 ; T 02 is globally asymptotically stable. Since system (4.3) is the limiting system of (2.2), it follows from Theorem 2.3 in paper [19] that the unique positive equilibrium E0 is a global attractor of system (2.2). This completes the proof of Theorem 4.1. h Theorem 4.2. Let ai > 0, i = 1, 2. If R0 > 1, then the system (2.2) is uniformly persistent, i.e., there is a positive constant d such that every positive solution (S1(t), I1(t), M1(t), V1(t), S2(t), I2(t), M2(t), V2(t)) of (2.2) satisfies
lim inf Si ðtÞ P d; lim inf Ii ðtÞ P d; lim inf Mi ðtÞ P d; lim inf V i ðtÞ P d; t!þ1
t!þ1
t!þ1
t!þ1
i ¼ 1; 2:
Proof. Define
X ¼ fðS1 ; I1 ; M 1 ; V 1 ; S2 ; I2 ; M2 ; V 2 Þ : Si P 0; Ii P 0; Mi P 0; V i P 0; i ¼ 1; 2g; X 0 ¼ fðS1 ; I1 ; M 1 ; V 1 ; S2 ; I2 ; M 2 ; V 2 Þ : Si > 0; Ii > 0; M i > 0; V i > 0; i ¼ 1; 2g; @X 0 ¼ X n X 0 : In order to prove that the disease is uniformly persistent, it suffices to show that @X0 repels uniformly the solutions of X0. First, by the form of (2.2), it follows that both X and X0 are positively invariant. Clearly, @X0 is relatively closed in X and system (2.2) is point dissipative. Set
M@ ¼ fðS1 ð0Þ; I1 ð0Þ; M 1 ð0Þ; V 1 ð0Þ; S2 ð0Þ; I2 ð0Þ; M 2 ð0Þ; V 2 ð0ÞÞ : ðS1 ðtÞ; I1 ðtÞ; M1 ðtÞ; V 1 ðtÞ; S2 ðtÞ; I2 ðtÞ; M2 ðtÞ; V 2 ðtÞÞ satisfies ð2:2Þ and ðS1 ðtÞ; I1 ðtÞ; M 1 ðtÞ; V 1 ðtÞ; S2 ðtÞ; I2 ðtÞ; M 2 ðtÞ; V 2 ðtÞÞ 2 @X 0 ; 8t P 0g: We now show that
n o M@ ¼ ðS1 ; I1 ; M 1 ; V 1 ; S2 ; I2 ; M 2 ; V 2 Þ 2 X : I21 þ V 21 þ I22 þ V 22 ¼ 0 ;
ð4:4Þ
if ai > 0, i = 1, 2. Assume (S1(0), I1(0), M1(0), V1(0), S2(0), I2(0), M2(0), V2(0)) 2 M@ . It suffices to show that I21 ðtÞ þ V 21 ðtÞþ I22 ðtÞ þ V 22 ðtÞ ¼ 0 for all t P 0. Suppose not, then there exists a t0 P 0 such that I21 ðt 0 Þ þ V 21 ðt 0 Þ þ I22 ðt 0 Þ þ V 22 ðt0 Þ > 0. Here we only consider the case I1(t0) > 0, I2(t0) = 0, Si(t0) = 0, Mi(t0) = 0, Vi(t0) = 0, i = 1, 2. The other cases can be deduced in the same way. Since
S0i ðt0 Þ P Ai þ ci Ni ðt 0 Þ li Si ðt 0 Þ bi bi V i ðt 0 ÞSi ðt 0 Þ þ ci Ii ðt 0 Þ P Ai > 0; M 0i ðt 0 Þ P Gi þ hi T i ðt 0 Þ bi ai Ii ðt0 ÞM i ðt 0 Þ i M i ðt0 Þ P Gi > 0; I02 ðt0 Þ ¼ b2 b2 V 2 ðt 0 ÞS2 ðt 0 Þ ðl2 þ c2 þ d2 ÞI2 ðt 0 Þ a2 I2 ðt0 Þ þ a1 I1 ðt 0 Þ ¼ a1 I1 ðt 0 Þ > 0 and
I01 ðtÞ P ðli þ ci þ di þ a1 ÞI1 ðtÞ; it follows that there is an e0 small enough such that Si(t) > 0, Ii(t) > 0, Mi(t) > 0, i = 1, 2, for all t0 < t < t0 + e0. If V i ðt 0 þ e20 Þ > 0; i ¼ 1; 2, then we have
V 0i ðtÞ P i V i ðtÞ: This means that Vi(t) > 0, i = 1, 2 for all t P t0 þ e20 ; if V i ðt 0 þ e20 Þ ¼ 0, it then follows from the fourth equation of system (2.2) that
V 0i ðt0 þ
e0 e0 Þ ¼ bi ai Ii t 0 þ Mi ðt0 þ Þ > 0: 2 2 2
e0
It then follows that there exists e1 < e20 such that Vi(t) > 0, i = 1, 2 for all t 2 t0 þ e20 ; t0 þ e20 þ e1 . Thus for all e0 e0 t 2 t0 þ 2 ; t 0 þ 2 þ e1 we have Si(t) > 0, Ii(t) > 0, Mi(t) > 0, Vi(t) > 0, i = 1, 2. This contradicts the assumption that (S1(0), I1(0), M1(0), V1(0), S2(0), I2(0), M2(0), V2(0)) 2 M@ . This proves (4.4). It is clear that E0 is the unique equilibrium in M@ . We now show that E0 repels the solution in X0. By (3.3) we know that e 2 ð1 vÞ > 1 or R0 > 1 if and only if R 01
(
e 2 ð1 vÞ 6 1; R 01 e2 e 2 ð1 fÞ R e2 R e 2 ð1 vÞ þ R R 01 02 01 02 ð1 v fÞ > 1:
ð4:5Þ
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e 2 ð1 vÞ > 1, i.e., If R 01 2
b1 a1 b1 N01 T 01 ðl2 þ c2 þ d2 þ a2 Þ þ 1 a1 a2 > 1 ðl1 þ c1 þ d1 þ a1 Þðl2 þ c2 þ d2 þ a2 Þ; we can choose h1 > 0 small enough such that
2 b1 a1 b1 N01 h1 T 01 h1 ðl2 þ c2 þ d2 þ a2 Þ þ 1 a1 a2 > 1 ðl1 þ c1 þ d1 þ a1 Þðl2 þ c2 þ d2 þ a2 Þ:
ð4:6Þ
If (4.5) holds, i.e., det (J) < 0, we can also choose h1 small enough such that
ðl þ c þ d þ a Þ a2 b1 b1 N01 h1 0 1 1 1 1 ðl2 þ c2 þ d2 þ a2 Þ 0 b2 b2 N02 h1 a1 < 0: b1 a1 T 01 h1 0 0 1 0 2 0 b2 a2 T 02 h1
ð4:7Þ
By direct calculations, we can choose d1 > 0 small enough such that
8 ðA1 þA2 Þa2 þðl2 þb2 b2 d1 c2 ÞA1 > > N01 h1 ; > > ðl1 þb1 b1 d1 c1 Þðl2 þb2 b2 d1 c2 Þþðl1 þb1 b1 d1 c1 Þa2 þðl2 þb2 b2 d1 c2 Þa1 > > > Þa1 þðl1 þb1 b1 d1 c1 ÞA2 > < ðl þb b d c Þðl þbðA1bþAd 2c > N02 h1 ; 1 1 1 1 2 2 1 2 Þþðl þb1 b1 d1 c 1 Þa2 þðl þb2 b2 d1 c 2 Þa1 1
2
1
2
> G1 > > T 01 h1 ; > b1 a1 d1 þ1 h1 > > > > : G2 > T 02 h1 : b2 a2 d1 þ2 h2
ð4:8Þ
Suppose (S1(t), I1(t), M1(t), V1(t), S2(t), I2(t), M2(t), V2(t)) is a solution of system (2.2) with (S1(0), I1(0), M1(0), V1(0), S2(0), I2(0), M2(0), V2(0)) 2 X0. We now claim that
lim sup maxfI1 ðtÞ; I2 ðtÞ; V 1 ðtÞ; V 2 ðtÞg > d1 :
ð4:9Þ
t!þ1
Suppose, for the sake of contradiction, that there is a T > 0 such that Ii(t) 6 d1, Vi(t) 6 d1, i = 1, 2, for all t P T. Then by the first and third equations of system (2.2), we have
8 0 > S1 ðtÞ P A1 þ c1 S1 ðtÞ ðl1 þ b1 b1 d1 ÞS1 ðtÞ a1 S1 ðtÞ þ a2 S2 ðtÞ; > > > > < S0 ðtÞ P A2 þ c2 S2 ðtÞ ðl þ b2 b d1 ÞS2 ðtÞ þ a1 S1 ðtÞ a2 S2 ðtÞ; 2 2 2 0 > ðtÞ P G þ h M ðtÞ ð þ b M > 1 1 1 1 1 1 a1 d1 ÞM 1 ðtÞ; > > > : 0 M2 ðtÞ P G2 þ h2 M 2 ðtÞ ð2 þ b2 a2 d1 ÞM 2 ðtÞ for t > T. Consider the following system:
8 e e0 e e e > > > S 1 ðtÞ ¼ A1 þ c1 S 1 ðtÞ ðl1 þ b1 b1 d1 Þ S 1 ðtÞ a1 S 1 ðtÞ þ a2 S 2 ðtÞ; > > > 0 e e e > M 1 ðtÞ ¼ G1 þ h1 M 1 ðtÞ ð1 þ b1 a1 d1 Þ M 1 ðtÞ; > > > > : e0 e 2 ðtÞ ð2 þ b2 a2 d1 Þ M e 2 ðtÞ: M 2 ðtÞ ¼ G2 þ h2 M
ð4:10Þ
As in our previous analysis of system (4.3), we can restrict d1 to be small enough such that (4.10) admits a positive equilib e 0; M e 0 , where rium e S 01 ; e S 02 ; M 1 2
e S 01 ¼
ðA1 þ A2 Þa2 þ ðl2 þ b2 b2 d1 c2 ÞA1 ; ðl1 þ b1 b1 d1 c1 Þðl2 þ b2 b2 d1 c2 Þ þ ðl1 þ b1 b1 d1 c1 Þa2 þ ðl2 þ b2 b2 d1 c2 Þa1
e S 02 ¼
ðA1 þ A2 Þa1 þ ðl1 þ b1 b1 d1 c1 ÞA2 ; ðl1 þ b1 b1 d1 c1 Þðl2 þ b2 b2 d1 c2 Þ þ ðl1 þ b1 b1 d1 c1 Þa2 þ ðl2 þ b2 b2 d1 c2 Þa1
e0 ¼ M 1
G1 G2 e0 ¼ ;M : 2 b1 a1 d1 þ 1 h1 b2 a2 d1 þ 2 h2
Z. Qiu / Applied Mathematics and Computation 218 (2011) 4614–4625
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e 0; M e 0 is globally asymptotically stable for system (4.10). By (4.8) Furthermore, the unique positive equilibrium e S 01 ; e S 02 ; M 1 2 and the standard comparison principle, there is a s > 0 such that S1 ðtÞ P N 01 h1 ; S2 ðtÞ P N 02 h1 ; M 1 ðtÞ P T 01 h1 ; M2 ðtÞ P T 02 h1 for t > T + s. Consequently, for t P T + s, we have
8 > I01 P b1 b1 V 1 N01 h1 ðl1 þ c1 þ d1 ÞI1 a1 I1 þ a2 I2 ; > > > > > > > < I02 P b2 b2 V 2 N02 h1 ðl2 þ c2 þ d2 ÞI2 a2 I2 þ a1 I1 ; 0 0 > > > V 1 P b1 a1 I1 T 1 h1 1 V 1 ; > > > > > : V 0 P b2 a2 I2 T 0 h1 2 V 2 : 2 2
ð4:11Þ
Consider an auxiliary system
8 0 b bI 0 ¼ b1 b V b b b > > 1 1 N 1 h1 ðl1 þ c1 þ d1 Þ I 1 a1 I 1 þ a2 I 2 ; 1 > > > > > > b 2 N0 h1 ðl þ c þ d2 ÞbI 2 a2bI 2 þ a1bI 1 ; < bI 02 ¼ b2 b2 V 2 2 2 0 > 0 b b b > V 1 ¼ b1 a1 I 1 T 1 h1 1 V 1 ; > > > > > > b 2: b 0 ¼ b2 a2bI 2 T 0 h1 2 V :V 2 2
ð4:12Þ
The coefficient matrix bJ of the right hand of (4.12) is defined by
0 B B bJ ¼ B B @
ðl1 þ c1 þ d1 þ a1 Þ
a2
b1 b1 ðN01 h1 Þ
a1
ðl2 þ c2 þ d2 þ a2 Þ
0
b1 a1 ðT 01 h1 Þ
0
1
0
0 2 ðT 2
b2 a
h1 Þ
0
0
1
C b2 b2 ðN02 h1 Þ C C: C 0 A 2
Since bJ admits positive off-diagonal elements, Perron–Frobenius Theorem implies that there is a positive eigenvector vm for the maximin eigenvalue km of bJ. After extensive computations, we have km > 0 since (4.6) and (4.7) hold. By the theory of linear system, it is easy to see that any positive solution of (4.12) tends to infinity as t tends to infinity. Then by the stand comparison principle, we have Ii(t) ? +1, Vi(t) ? +1, i = 1, 2 as t ? +1. This contradicts Ii(t) 6 d1, Vi(t) 6 d1, i = 1, 2 for all t P T. This proves (4.9). Hence Ws(E0) \ X0 = ;. Clearly, every forward orbit in M@ converges to E0. By Theorem 4.6 of [20] we are able to conclude that the system (2.2) is uniformly persistent with respect to (X0, oX0). This completes the proof of Theorem 4.2. h
5. Stability of the endemic equilibrium In this section, we consider the stability of endemic equilibrium of system (2.2). In order to be tractable in mathematics, we only consider two special cases. First, we suppose that the both region are identical, i.e., demographic parameters are the same for each region. This means that Ai = A, ci = c, li = l, bi = b, ai = a, di = d, ci = c, Gi = G, hi = h, i = , ai = a, i = 1, 2. Then (2.2) becomes
8 0 Si ¼ A þ cN i lSi bbV i Si þ cIi aSi þ aSj ; > > > < I0 ¼ bbV S ðl þ c þ dÞI aI þ aI ; i i i i j i i; j ¼ 1; 2; i – j: > M 0i ¼ G þ hT i baIi Mi M i ; > > : 0 V i ¼ baIi M i V i ;
ð5:1Þ
G In this case, the disease free equilibrium of system (5.1) is E0(N0, 0, T0, 0, N0, 0, T0, 0), where N 0 ¼ lAc ; T 0 ¼ h . Therefore, the reproduction number for the model (5.1) can be expressed as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b abN0 T 0 R0 ¼ : ðl þ c þ dÞ After extensive algebraic calculations, we have that system (5.1) has no positive equilibrium if R0 < 1 and if R0 > 1 system (5.1) has an endemic equilibrium E⁄(S⁄, I⁄, M⁄, V⁄, S⁄, I⁄, M⁄, V⁄), where S⁄, I⁄, M⁄, V⁄ can be expressed as
ðl þ d cÞN A A ðl cÞN ;I ¼ ; d d ðl þ d þ cÞðA ðl cÞN Þ V ¼ ; M ¼ T 0 V bbððl þ d cÞN AÞ
S ¼
4622
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and 2
N ¼
2
ðl þ c þ dÞðb aA þ dÞ þ T 0 b abA 0 2
ðl þ d þ cÞðl cÞ þ T b abðl þ d cÞ
:
Now let us consider the local stability of the positive equilibrium E⁄ when R0 > 1. At the positive equilibrium E (S⁄, I⁄, M⁄, V⁄, S⁄, I⁄, M⁄, V⁄) of the system (5.1), the Jacobian matrix is given by ⁄
JðE Þ ¼
A
B
B
A
;
where
0 B B A¼B @
ðl þ a þ bbV cÞ
cþc
bbV
ðl þ a þ c þ dÞ
0
baM
0
baM
bbS
0
0 h baI
baI
1
bbS C C C h A
and
1 a 0 0 0 B0 a 0 0C C B B¼B C: @0 0 0 0A 0
0 0 0 0 Calculating the characteristic polynomial, we have
kI A B ¼ jkI ðA þ BÞjjkI ðA BÞj: jkI Jj ¼ B kI A It follows that the stability of the equilibrium E⁄ is determined by the eigenvalues of the matrices A + B and A B since the eigenvalues of the matrix J are made up of the eigenvalues of the matrices A + B and A B. After extensive algebraic calculations, the eigenvalues of A + B are h and the roots of
k3 þ A1 k2 þ A2 k þ A3 ¼ 0; where
A1 ¼ baI þ þ 2l þ bbV þ c þ d c > 0; A2 ¼ ðbaI þ ÞðbbV þ l cÞ þ bbV ðl þ d cÞ þ ðl þ d þ cÞðl c þ baI Þ > 0; A3 ¼ ðl cÞðl þ c þ dÞbaI þ bbV ð þ baI Þðl c þ dÞ > 0: Direct calculation yields that
A1 A2 A3 ¼ ðbaI þ ÞððbaI þ ÞðbbV þ l cÞ þ ðl þ d þ cÞðl c þ baI ÞÞ þ ðbbV þ l þ d þ cÞððbaI þ ÞðbbV þ l cÞ þ bbV ðl þ d cÞ þ ðl þ d þ cÞðl c þ baI ÞÞ þ ðl cÞððbaI þ ÞðbbV þ l cÞ þ bbV ðl þ d cÞ þ ðl þ d þ cÞðl cÞÞ > 0: Similarly, the eigenvalues of A B are h < 0 and the roots of
e 1 k2 þ A e 2k þ A e 3 ¼ 0; k3 þ A where
e 1 ¼ baI þ þ 2l þ 4a þ bbV þ c þ d c > 0; A e 2 ¼ ðbaI þ ÞðbbV þ l þ 2a cÞ þ bbV ðl þ 2a þ d cÞ þ ðl þ 2a þ d þ cÞðl þ 2a c þ baI Þ þ 2a > 0; A e 3 ¼ ðl þ 2a cÞðl þ c þ dÞbaI þ bbV ð þ baI Þðl þ 2a c þ dÞ þ 2aðl þ 2a cÞð þ baI Þ > 0 A and
e 3 ¼ ðbaI þ ÞððbaI þ ÞðbbV þ l þ 2a cÞ þ ðl þ 2a þ d þ cÞðl þ 2a c þ baI ÞÞ e1 A e2 A A þ ðbbV þ l þ 2a þ d þ cÞððbaI þ ÞðbbV þ l þ 2a cÞ þ bbV ðl þ 2a þ d cÞ þ ðl þ 2a þ d þ cÞðl þ 2a c þ baI ÞÞ þ ðl cÞððbaI þ ÞðbbV þ l þ 2a cÞ þ bbV ðl þ d cÞ þ ðl þ 2a þ d þ cÞðl þ 2a cÞÞ þ 2aððbaI þ ÞbbV þ bbV ðl þ 2a þ d cÞ þ ðl þ 2a þ d þ cÞðl þ 2a cÞÞ > 0:
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By Routh–Hurwitz criteria, it follows that all the eigenvalues of A B and A + B have negative real parts. Therefore, we have the following theorem. 2
G Theorem 5.1. If b ab lAc h > ðl þ c þ dÞ, then system (5.1) has an endemic equilibrium E⁄(S⁄, I⁄, M⁄, V⁄, S⁄, I⁄, M⁄, V⁄), which is locally asymptotically stable.
Second, we suppose that there is no disease-induced death rate of hosts, i.e., di = 0, i = 1, 2. Then the system (2.2) becomes
8 0 Si ¼ Ai þ ci Ni li Si bi bi V i Si þ ci Ii ai Si þ aj Sj ; > > > 0 < Ii ¼ bi bi V i Si ðli þ ci ÞIi ai Ii þ aj Ij ; 0 > > M i ¼ Gi þ hi T i bi ai Ii M i i M i ; > : 0 V i ¼ bi a i I i M i i V i ;
i; j ¼ 1; 2; i – j:
ð5:2Þ
Theorem 5.2. Let ai > 0, i = 1, 2. If R0 > 1, then system (5.2) has only a unique positive equilibrium E⁄, which is globally asymptotically stable in IntC.
Proof. Adding up the first and second equations of system (5.2) gives
N0i ¼ Ai ðli ci ÞN i ai Ni þ aj Nj ; It is easy to see that N i ðtÞ ! be read as
N 0i
i; j ¼ 1; 2; i – j:
as t ? +1. Similarly, we have M i þ V i ! T 0i as t ? +1. Then the limiting system of (5.2) can
8 > I01 ¼ b1 b1 V 1 N01 I01 ðl1 þ c1 ÞI1 a1 I1 þ a2 I2 ; > > > > > > > < I02 ¼ b2 b2 V 2 N02 I02 ðl2 þ c2 ÞI2 a2 I2 þ a1 I1 ; 0 0 > > V ¼ b a I T V 1 V 1 ; > 1 1 1 1 1 1 > > > > > 0 0 : V ¼ b2 a2 I2 T V 2 2 V 2 ; 2 2
ð5:3Þ
From the Jacobian of system (5.3) we can verify that system (5.2) is a cooperative irreducible system in R4þ [14]. By Theorem 3.2 it follows that if R0 < 1 the origin of system (5.3) is local asymptotically stable and if R0 > 1 the origin of system (5.3) is unstable. Following Smith [21], system (5.3) is strongly concave. Then it follows that if R0 > 1 system (5.3) has an equilibrium E I1 ; I2 ; V 1 ; V 2 , which is globally asymptotically stable. Since system (5.3) is the limiting system of (5.2), it follows from Theorem 2.3 in paper [19] that the unique positive equilibrium E N 01 I1 ; I1 ; T 01 V 1 ; V 1 ; N 02 I2 ; I2 ; T 02 V 2 ; V 2 is a globally asymptotically stable equilibrium of system (5.2). This completes the proof of Theorem 5.2 h
6. Discussions In this section, we mainly summarize our results and provide two numerical examples to illustrate the effect of host migration on the spread of the vector–host diseases. In this paper, we mainly proposed an epidemic model to investigate the impact of host dispersal on the diffusion and outbreak of a vector–host disease. The host population dispersal among patches can be interpreted as the movement that the host, such as birds, people, migrate from one region to another. Due to the spread of the vector–host disease between and within two populations, we incorporated two species into our model, which is different from the classical compartmental epidemiological models [8,9,22]. We adopted the function Bi ðN i Þ ¼ NAi þ ci for the growth rates of the hosts and the function i F i ðN i Þ ¼ GT ii þ hi for the growth rate of the vectors, which is widely used in literature. Our model is suited to describe the dynamical transmission of many vector–host diseases between two patches in which the vital and epidemiological parameters may depend on the patches. Systemic analysis of the model is then provided. By analyzing the local stability of disease free equilibrium E0, we derived the reproduction number, which can be expressed as
R0 ¼
pffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 e2 e 2 vf 2 ; e 2 ð1 fÞ þ ð R e 2 ð1 vÞ R e 2 ð1 fÞÞ2 þ 4 R e2 R R 01 ð1 vÞ þ R 02 01 02 01 02 2
e 0i ¼ where R
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b2i ai bi N0i T 0i i ðli þci þdi Þ;
i ¼ 1; 2;
a1 l1 þc1 þd1 a1 a2 l1 þc1 þd1 þl2 þc2 þd2
v ¼ 1þ
; f¼
a2 l2 þc2 þd2 a1 a 1þl þc þd þl þc2 þd 1 1 1 2 2 2
. It is proved that the reproduction number R0 is
the threshold condition that determines the uniform persistence and extinction of the disease. When R0 < 1, by Theorem 4.1 the disease free equilibrium is globally asymptotically stable and the disease will die out. For R0 > 1 it follows from
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Z. Qiu / Applied Mathematics and Computation 218 (2011) 4614–4625
Theorem 4.2 that the disease will be uniformly persistent. Especially, if the two patches are isolated, the results in [13] show that the disease will be persistent in the first patch if
R01
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 0 b1 a1 b1 N00 1 T1 >1 ¼ 1 ðl1 þ c1 þ d1 Þ
and the disease will disappear in the first patch if R01 < 1. Further, the disease will be persistent in the second patch if
R02
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 0 b2 a2 b2 N00 2 T2 ¼ >1 2 ðl2 þ c2 þ d2 Þ
and the disease will disappear in the second patch if R02 < 1. Thus, the quantities R01 ; R02 ; R0 can provide us more important information for analyzing the effect of host migration on the spread of the vector–host disease. In the following, we give two examples to illustrate it. All the values of the parameters selected in the two examples are for illustration purpose only. Example 6.1. We use the parameters given below:
l1 ¼ 0:6; c1 ¼ 0:1; 1 ¼ 0:6; h1 ¼ 0:1; b1 ¼ 1; a1 ¼ 0:01; c1 ¼ 0:2; d1 ¼ 0:2; A2 ¼ 20; G2 ¼ 1; l2 ¼ 0:6; c2 ¼ 0:5; 2 ¼ 0:6; h2 ¼ 0:5; b1 ¼ 1; a2 ¼ 0:01; b2 ¼ 0:01; c2 ¼ 0:2; d2 ¼ 0:2: A1 ¼ 10;
G1 ¼ 80;
b1 ¼ 0:01;
If the two patches are isolated, numerical calculations give
R01 ¼ 0:7303;
R02 ¼ 0:5774:
It follows from Lemma 3.1 that the disease will die out in both two patches when they are isolated. By means of Matlab software, we can easily see that R0 < 1 for small a1, a2 (see Fig. 2), but if the host migration rates a1, a2 are large then we have R0 > 1. This suggests that the host migration will cause the spread of the vector–host disease in the two patches. Example 6.2. We use the parameters given in Example 6.1 except that
A1 ¼ 10;
G1 ¼ 10;
A2 ¼ 10;
G2 ¼ 10:
If the two patches are isolated, numerical calculations give
R01 ¼ 0:2582;
R02 ¼ 1:2910:
It follows from Lemma 3.1 that the disease will be uniformly persistent in the second patch and will die out in the first patch. By means of Matlab software, we can easily see that R0 > 1 for small a1, a2 (see Fig. 3). This means that the host dispersal intensifies the vector-disease spread. However, if the host migration rate a1, a2 is large, it also follows from Fig. 3 that R0 < 1. This implies that host migration can reduces the vector-disease spread and is beneficial to disease control. Finally, we are interested in the global stability of the system (2.2). This seems to be a difficult problem since the dimension of the model is higher. Numerical simulations suggest that if R0 > 1 then system (2.2) has a unique endemic equilibrium, which is globally asymptotically stable. Additionally, the model presented here can be extended to describe the
Fig. 2. Surface plot of R0 as a function of a1 and a2 for the parameters given in Example 6.1.
Z. Qiu / Applied Mathematics and Computation 218 (2011) 4614–4625
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Fig. 3. Surface plot of R0 as a function of a1 and a2 for the parameters given in Example 6.2.
dynamical transmission of the vector–host disease with host (and vector) migration among more than 2 patches. It can also be extended to incorporate the other ingredients, such as the different incidences and different compartmental structures. It will also be interesting to study the effect of the worldwide spread of the vector–host diseases caused by patches structure or/and the migration of the vectors and the hosts. We leave these for future investigation. Acknowledgements The author greatly appreciates the anonymous referees for their valuable comments which help to improve the paper. The research of this paper is supported by the NSF of China grants 10801074, 60874007, NUST Research Funding No. 2011YBXM31, the Zijin Star Project of Excellence Plan of NJUST and Qinglan Project of Jiangsu Province. References [1] Z.L. Feng, J.X. Velasco-Hernández, Competitive exclusion in a vector–host model for the dengue fever, J. Math. Biol. 35 (1997) 523–544. [2] G. Cruz-Pachecoa, L. Estevab, J.A. Montano-Hirosec, C. Vargasd, Modelling the dynamics of West Nile Virus, Bull. Math. Biol. 67 (2005) 1157–1172. [3] C. Bowman, A.B. Gumel, J. Wu, P. van den Driessche, H. Zhu, A mathematical model for assessing control strategies against West Nile virus, Bull. Math. Biol. 67 (2005) 1107–1133. [4] M.J. Wonham, T. de-Camino-Beck, M. Lewis, An epidemiological model for West Nile virus: invasion analysis and control applications, Proc. Roy. Soc. Lond. Ser. B 1538 (2004) 501–507. [5] J.H. Rappole, S.R. Derrickson, Z. Hubalek, Migratory birds and spread of West Nile virus in the Western Hemisphere, Emer. Infect. Dis. 6 (2000) 319–328. [6] J.H. Rappole, Z. Hubalek, Migratory birds and West Nile virus, J. Appl. Microbiol. 94 (2003) 47–58. [7] J. Owen, F. Moore, N. Panella, et al, Migrating birds as dispersal vehicles for West Nile virus, EcoHealth 3 (2006) 79–85. [8] W.D. Wang, G. Mulone, Threshold of disease transmission in a patch environment, J. Math. Anal. Appl. 285 (2003) 321–335. [9] W.D. Wang, X.Q. Zhao, An epidemic model in a patchy environment, Math. Biosci. 190 (2004) 97–112. [10] J.A. Cui, Y. Takeuchi, Y. Saito, Spreading disease with transport-related infection, J. Theor. Biol. 239 (2006) 376–390. [11] J. Arino, R. Jordan, P. van den Driessche, Quarantine in a multi-species epidemic model with spatial dynamics, Math. Biosci. 206 (2007) 46–60. [12] K. Cooke, P. van den Driessche, X. Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol. 39 (1999) 332–352. [13] Z.P. Qiu, Dynamical behavior of a vector–host epidemic model with demographic structure, Comput. Math. Appl. 56 (2008) 3118–3129. [14] H.L. Smith, Monotone Dynamical Systems: An Introduction to Theory of Competitive and Cooperative Systems, Math. Surveys Monogr., 41, AMS, Providence, RI, 1995. [15] J.F. Jiang, On the global stability of cooperative systems, Bull. Lond. Math. Soc. 26 (1994) 455–458. [16] P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002) 29–48. [17] H.L. Smith, P. Waltman, The Theory of the Chemostat, Cambridge University, 1995. [18] J. Franklin, Matrix Theory, Prentice-Hall, Englewood Cliffs, NJ, 1968. [19] C. Castillo-Chavez, H.R. Thieme, Asymptotically autonomous epidemic models, in: O. Arino, M. Kimmel (Eds.), Mathematical Population Dynamics: Analysis of Heterogeneity, Theory of Epidemics, Wuetz 1, 1995, pp. 33–50. [20] H.R. Thieme, Persistence under relaxed point-dissipativity with an application to an epidemic model, SIAM J. Math. Anal. 24 (1993) 407–435. [21] H.L. Smith, Cooperative systems of differential equations with concave nonlinearities, Nonlinear Anal.: T.M.A. 10 (1986) 1037–1052. [22] J. Arino, P. van den Driessche, The basic reproduction number in a multi-city compartmental epidemic model, Lect. Notes Contr. Inf. Sci. 294 (2003) 135–142.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Dynamics of an epidemic model with host migration Zhipeng Qiu ⇑ Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing 210094, PR China
a r t i c l e
i n f o
Keywords: Vector–host Epidemic model Stability Migration The reproduction number
a b s t r a c t Host migration among discrete geographical regions is demonstrated as an important factor that brings about the diffusion and outbreak of many vector–host diseases. In the paper, we develop a mathematical model to explore the effect of host migration between two patches on the spread of a vector–host disease. Analytical results show that the reproduction number R0 provides a threshold condition that determines the uniform persistence and extinction of the disease. If both the patches are identical, it is shown that an endemic equilibrium is locally stable. It is also shown that a unique endemic equilibrium, which exists when the disease cannot induce the death of the host, is globally asymptotically stable. Finally, two examples are given to illustrate the effect of host migration on the spread of the vector–host disease. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Many diseases are spread by vectors, as we can see with plague (spread by direct contamination, but usually by fleas), typhus (spread by lice), malaria, or dengue fever (the latter two spread by mosquitoes), so the emergence and reemergence of the vector–host diseases have promoted much interest in their mathematical models. Recently there has been some effort in the mathematical modeling of the vector–host epidemic transmission dynamics (see, for example, [1–3] etc.). These models have provided much insight into understanding the mechanism of the vector–host epidemic transmission. Nevertheless, all these models have only considered transmission dynamics of the vector–host disease in an isolated patch, ignoring the effect of host migration among patches (discrete geographical regions). However, for some kind of vector–host diseases such as West Nile virus (WNV) and dengue fever, the migration of host is an important factor that leads to the worldwide spread of the vector–host diseases. Wonham et al. [4] have suggested that the WNV model should be extended biologically to consider bird migration. The correlation between migratory birds and the spread of WNV has been studied in many papers [5–7]. In paper [6], Rappole and Hubalek have provided some factors supporting the ‘migrant bird as introductory host’ hypothesis for the spread of WNV, and in paper [7], Owen et al. have demonstrated that migrating passerine birds are potential dispersal vehicles for WNV. All the facts inspire us to consider the effect of host migration among multiple patches on the spread of the vector–host diseases. Modeling the spatial spread of the infectious disease is a complex task. One possible approach is to consider the migration of individuals between discrete geographical regions. In mathematical epidemiology a few models which incorporate discrete geographical regions have been studied (see [8–11] and the references cited therein). However, up till now no one has considered the effect of host migration among multiple patches on the dynamics of vector–host diseases. Therefore, we expect to explore the effect of host migration among multiple patches on the spread of the vector–host diseases in order to answer the interesting questions in mathematical epidemiology: How does the host migration affect the dynamics and outbreak of the vector–host diseases in discrete regions? ⇑ Corresponding author. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.10.045
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The purpose of the paper is to model the transmission dynamics of a vector–host disease spread between two patches due to host migration, and describe the dynamics of the system. The remaining part of this paper is organized as follows: in Section 2, we mainly formulate our model. In Section 3, we mainly present some preliminary results and derive the reproduction number. In Section 4, the threshold dynamics of the model is analyzed. The stability of endemic equilibrium is studied in Section 5. In the last section we summarize our results, and give two examples to illustrate how the host migration affects the dynamics and outbreak of the vector–host diseases.
2. Model description In this section we mainly formulate an epidemic model to describe the transmission dynamics of a vector–host disease spread between two discrete patches due to host migration. First, let us formulate a model for the spread of a vector–host disease in the ith patch. Suppose there is no host migration among patches, i.e., the patches are isolated. We assume that the total host population Ni(t) in the ith patch is partitioned into two distinct epidemiological subclasses which are susceptible and infectious, with sizes denoted by Si(t) and Ii(t), respectively, and the total vector population Ti(t) in the ith patch is also divided into susceptible and infectious subclasses, with the sizes denoted by Mi(t) and Vi(t), respectively. Our assumptions on the dynamical transfer of the host and vector population in the ith patch are demonstrated in Fig. 1. In Fig. 1, Bi(Ni) and Fi(Ti) are the growth rates of the hosts and the vectors in the ith patch, respectively. The growth rate of the hosts Bi(Ni) and the rate of the vectors Fi(Ti) are generally assumed to satisfy the following basic assumptions for Ni, Ti 2 (0, +1) [9,12]: (a1) Bi ð0þÞ ¼ limNi !0þ Bi ðN i Þ > 0; F i ð0þÞ ¼ limT i !0þ F i ðT i Þ > 0; i ¼ 1; 2; (a2) Bi(Ni) and Fi(Ti) are continuously differentiable with B0i ðN i Þ < 0; F 0i ðT i Þ < 0; i ¼ 1; 2; (a3) li > Bi(+1), i > Fi(+1), i = 1, 2. In this paper, we shall choose the function Bi ðN i Þ ¼ NAii þ ci with Ai > 0, li > ci P 0 as the growth rate of the hosts and the Ai function F i ðT i Þ ¼ GT ii þ hi with Gi > 0, i > hi P 0 as the growth rate of the vectors. Biologically, the function Ni þ ci N i represents a constant immigration rate Ai together with a linear birth term ciNi of the hosts. The function GT i þ hi T i has the same i biological meaning. bi(Ni) denotes the per capita rate of contacts on hosts by vectors, which is continuously differential function of Ni. Generally, bi(Ni) is a monotonic increasing function on Ni. In this paper, we assume that bi(Ni) = biNi. ai and bi are the disease transmission probabilities from infected hosts to uninfected vectors and from infected vectors to uninfected hosts in the ith patch, respectively; i, li are the natural death rates of the vectors and the hosts, respectively; di is the disease-induced death rate of the hosts, and ci is the recovery rate of infected hosts in the ith patch. All the above parameters are positive except ci P 0 and di P 0. Using the transfer diagram and the assumptions that bi ðN i Þ ¼ bi N i ; Bi ðN i Þ ¼ NAii þ ci ; F i ðT i Þ ¼ GT ii þ hi , the system which describes the spread of a vector–host disease in ith patch can be derived:
8 0 S ¼ Ai þ ci Ni li Si bi bi V i Si þ ci Ii ; > > > i > > < I0 ¼ bi bi V i Si ðl þ c þ di ÞIi ; i i i > > M 0i ¼ Gi þ hi T i bi ai Ii M i i M i ; > > > : 0 V i ¼ bi a i I i M i i V i :
Fig. 1. Transfer diagram for the vector–host epidemic model.
ð2:1Þ
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When two patches are connected, we assume that only hosts can migrate among the patches since vectors are usually arthropod whose migration can be ignored. We assume that susceptible and infected hosts of every patch i leave for patch j at a per capita rate ai. Then the dynamics of the hosts and the vectors is governed by the following model:
8 0 Si ¼ Ai þ ci Ni li Si bi bi V i Si þ ci Ii ai Si þ aj Sj ; > > > < I0 ¼ b b V S ðl þ c þ d ÞI a I þ a I ; i i i i i i i i j j i i i > M0i ¼ Gi þ hi T i bi ai Ii M i i M i ; > > : 0 V i ¼ bi ai Ii M i i V i ;
i; j ¼ 1; 2; i – j:
ð2:2Þ
The purpose of the paper is to analyze the dynamics of the model (2.2) and to answer how the host migration affects the dynamics and outbreak of the vector–host disease in discrete regions. 3. The preliminary results and reproduction number In this section, we mainly present the preliminary results and derive the reproduction number for system (2.2). In order to investigate the dynamics of the system (2.2), we begin with stating some results on system (2.1). The system (2.1) has been 0 analyzed in [13]. Straight forward computation yields that system (2.1) admits a boundary equilibria Ei01 N 00 ; 0; T ; 0 and a i i 0 0 0 , where potential unique positive equilibrium Ei0 S0 i ; Ii ; M i ; V i
Ai ; li c i
V 0 i ¼
ðli þ ci þ di ÞðAi ðli ci ÞN0 Þ i ; 0 bi bi ðli þ di ci ÞNi Ai
T 0i ¼
Gi ; i hi
ðli þ di ci ÞN0 i Ai ; di
N00 i ¼
S0 i ¼
I0 i ¼
Ai ðli ci ÞN0 i ; di
Mi ¼ T 0i V 0 i
and 2
N0 i ¼
ðli þ ci þ di Þðbi ai Ai þ di i Þ þ T 0i bi ai bi Ai 2 T 0i bi
ai bi ðli þ di ci Þ þ ðli þ di þ ci Þbi ai ðli ci Þ
:
The reproduction number of system (2.1) is established in paper [13], which can be expressed as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 0 bi ai bi N00 i Ti : R0i ¼ i ðli þ ci þ di Þ By Theorems 3.2 and 5.1 in paper [13], we can obtain the following lemma. Lemma 3.1. If R0i < 1, then Ei01 is globally asymptotically stable; if R0i > 1, then Ei0 is globally asymptotically stable. We now derive the reproduction number for system (2.2). Adding up the third and fourth equations of system (2.2) gives the equations for Ti, i = 1, 2:
T 0i ¼ Gi ðei hi ÞT i : It is easy to see that T i ðtÞ ! T 0i as t ? +1. From the first and second equations of system (2.2) we have
(
N01 6 A1 ðl1 c1 ÞN1 a1 N1 þ a2 N2 ; N02 6 A2 ðl2 c2 ÞN2 þ a1 N1 a2 N2 :
Define an auxiliary system by
(
e 0 ¼ A1 ð l c 1 Þ N e 1 þ a2 N e 2; e 1 a1 N N 1 1 0 e e 2: e e N 2 ¼ A2 ðl2 c2 Þ N 2 þ a1 N 1 a2 N
ð3:1Þ
We can verify that the system (3.1) is cooperative [14] and there exists a unique equilibrium N 01 ; N 02 of system (3.1) in R2þ , where
ðA1 þ A2 Þa2 þ ðl2 c2 ÞA1 ; ðl1 c1 Þðl2 c2 Þ þ ðl1 c1 Þa2 þ ðl2 c2 Þa1 ðA1 þ A2 Þa1 þ ðl1 c1 ÞA2 N02 ¼ : ðl1 c1 Þðl2 c2 Þ þ ðl1 c1 Þa2 þ ðl2 c2 Þa1
N01 ¼
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By the results in [15] we conclude that the equilibrium N 01 ; N 02 is globally asymptotically stable in R2þ . By the comparison principle we have N 1 ðtÞ 6 N 01 ; N 2 ðtÞ 6 N 02 . Let
n
o
C ¼ ðS1 ; I1 ; M1 ; V 1 ; S2 ; I2 ; M2 ; V 2 Þ : 0 6 Mi ; V i ; Mi þ V i 6 T 0i ; 0 6 Si ; Ii ; Si þ Ii 6 N0i ; i ¼ 1; 2; : Then it can be seen that all solutions of the systems (2.2) starting in C remain in C for all t P 0. Thus C is positively invariant and it is sufficient to consider solutions in C. In this region, the usual existence, uniqueness and continuation results hold for the system (2.2). In what follows, we always assume that the initial points lie in C. Straight forward computation yields that the disease free equilibrium (DFE) of system (2.2) is
E0 N01 ; 0; T 01 ; 0; N02 ; 0; T 02 ; 0 : A common method of determining its local stability is finding the reproduction number, which determines the persistence and extinction of a disease. Note that the system (2.2) has four infected populations, namely, I1, I2, V1,V2, it follows that, using the notation of van den Driessche and Watmough [16], the matrix F and V for the new infection terms and the remaining transfer terms, respectively, are given by
F¼
0
F12
F21
0
;
V¼
V11
0
0
V22
;
where
F12 ¼ V11 ¼
!
b1 b1 N01
0
0
b2 b2 N02
F21 ¼
;
b1 a1 T 01
0
0
b2 a2 T 02
l1 þ c1 þ d1 þ a1
a2
a1
l2 þ c2 þ d2 þ a2
;
! ;
V22 ¼
1
0
0
2
:
If R0 :¼ qðFV1 Þ; where qðFV1 Þ represents the spectral radius of the matrix FV1 , it follows from [16] that R0 is the reproduction number of system (2.2). After extensive algebraic calculations, we can obtain
R0 ¼
pffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 e2 e 2 vf 2 ; e 2 ð1 fÞ þ ð R e 2 ð1 vÞ R e 2 ð1 fÞÞ2 þ 4 R e2 R R 01 ð1 vÞ þ R 02 01 02 01 02 2
where
e 0i ¼ R
v¼ f¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 bi ai bi N0i T 0i ; i ðli þ ci þ di Þ
i ¼ 1; 2;
a1 l1 þc1 þd1 ; a1 1 þ l þc þd1 þ l þac2 þd2 1 1 2 2 a2 l2 þc2 þd2 : a1 1 þ l þc þd1 þ l þac2 þd2 1 2 1 2
Especially, if the two patches are isolated, i.e., a1 = a2 = 0, then we have
R0 ¼ maxfR01 ; R02 g: From the proof of Theorem 2 in [16], it follows that
R0 < 1ðR0 ¼ 1; R0 > 1Þ () sðJÞ < 0ðsðJÞ ¼ 0;
sðJÞ > 0Þ;
ð3:2Þ
where
0 B B J ¼FV¼B B @
ðl1 þ c1 þ d1 þ a1 Þ
a2
b1 b1 N01
a1
ðl2 þ c2 þ d2 þ a2 Þ
0
0
1
0 1T1
b1 a
0
0 2T 2
b2 a
0
0
1
C b2 b2 N02 C C C 0 A 2
and s(J) is the maximum real part of the eigenvalues of the matrix J. Since J is irreducible and has non-negative off-diagonal elements. Then it follows from Theorem A.5 in [17] that s(J) is a simple eigenvalue of J with a positive eigenvector.
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Furthermore, since the diagonal elements of J are positive and its off-diagonal elements are non-positive, it follows from Mmatrix [18] that
8 J 1 ¼ ðl1 þ c1 þ d1 þ a1 Þ < 0; > > > > > J 2 ¼ ðl1 þ c1 þ d1 þ a1 Þðl2 þ c2 þ d2 þ a2 Þ a1 a2 > 0; > > > > > a1 a2 > > < J 3 ¼ ðl1 þ c1 þ d1 Þðl2 þ c2 þ d2 Þ1 1 þ l1 þc1 þd1 þ l2 þc2 þd2 sðJÞ < 0 () e 2 ð1 vÞ 1 < 0; R > 01 > > > > > > > J 4 ¼ ðl1 þ c1 þ d1 Þðl2 þ c2 þ d2 Þ1 2 1 þ l þac1 þd1 þ l þac2 þd2 > > 1 1 2 2 > > : e 2 ð1 vÞ R e2 e 2 ð1 fÞ þ R e2 R ð1 R 01 02 01 02 ð1 v fÞÞ > 0; where Ji, j = 1, 2, 3, 4, are the leading principal minors of J with i rows. Consequently, we have
(
R0 < 1 ()
e 2 ð1 fÞ < 1Þ; e 2 ð1 vÞ < 1 ðor R R 01 02 2 2 e 2 ð1 v fÞ < 1: e ð1 fÞ R e2 R e ð1 vÞ þ R R 01
02
01
ð3:3Þ
02
Using Theorem 2 in [16], we can easily obtain the following stability result. Theorem 3.2. If R0 < 1, then the DFE (E0) is locally asymptotically stable; if R0 > 1, then the DFE (E0) is unstable. 4. Threshold dynamics The purpose of this section is to discuss the global extinction and persistence of the disease described by system (2.2). We will show that R0 is a threshold between the extinction and uniform persistence of the disease. Theorem 4.1. If R0 < 1, then the disease-free equilibrium of system (2.2) is globally asymptotically stable. Proof. If R0 < 1, then it follows from Theorem 3.2 that the disease-free equilibrium E0 is locally asymptotically stable. In the following we only need to prove that E0 is a global attractor. Let us consider a positive solution (S1(t), I1(t), M1(t), V1(t), S2(t), I2(t), M2(t), V2(t)) of system (2.2). From the second and fourth equations of (2.2), it follows that
8 0 I 6 b1 b1 V 1 N01 ðl1 þ c1 þ d1 ÞI1 a1 I1 þ a2 I2 ; > > > 1 > < V0 6 b a I T0 V ; 1 1 1 1 1 1 1 > I0 6 b2 b2 V 2 N0 ðl þ c þ d2 ÞI2 a2 I2 þ a1 I1 ; > 2 2 2 2 > > : 0 V 2 6 b2 a2 I2 T 02 2 V 2 :
ð4:1Þ
We consider the following differential equations
8 0 I1 ¼ b1 b1 V 1 N01 ðl1 þ c1 þ d1 ÞI1 a1 I1 þ a2 I2 ; > > > > < V0 ¼ b a I T0 V ; 1 1 1 1 1 1 1 0 > > I02 ¼ b2 b2 V 2 N2 ðl2 þ c2 þ d2 ÞI2 a2 I2 þ a1 I1 ; > > : 0 V 2 ¼ b2 a2 I2 T 02 2 V 2 :
ð4:2Þ
Since the system (4.2) is a linear system, the global stability of the origin of system (4.2) is determined from the stability of the matrix J. From (3.2) it can be seen that if R0 < 1 then all the eigenvalues of the matrix J have negative real parts. It then follows that each solution of system (4.2) satisfies
lim I1 ðtÞ ¼ 0;
t!þ1
lim I2 ðtÞ ¼ 0;
t!þ1
lim V 1 ðtÞ ¼ 0;
t!þ1
lim V 2 ðtÞ ¼ 0:
t!þ1
By the comparison principle we have that each positive solution of (4.1) satisfies Ii(t) ? 0, Vi(t) ? 0, i = 1, 2 as t ? +1. Then the limiting system of (2.2) is
8 0 S1 ¼ A1 þ c1 S1 l1 S1 a1 S1 þ a2 S2 ; > > > < S0 ¼ A þ c S l S þ a S a S ; 2 2 2 1 1 2 2 2 2 2 0 > M ¼ G þ h M M ; 1 1 1 1 1 > 1 > : 0 M2 ¼ G2 þ h2 M2 2 M2 :
ð4:3Þ
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We can verify that system (4.3) is a cooperative system [14] and there exists a unique equilibrium N 01 ; N 02 ; T 01 ; T 02 . By the results in [15] we conclude that the equilibrium N 01 ; N 02 ; T 01 ; T 02 is globally asymptotically stable. Since system (4.3) is the limiting system of (2.2), it follows from Theorem 2.3 in paper [19] that the unique positive equilibrium E0 is a global attractor of system (2.2). This completes the proof of Theorem 4.1. h Theorem 4.2. Let ai > 0, i = 1, 2. If R0 > 1, then the system (2.2) is uniformly persistent, i.e., there is a positive constant d such that every positive solution (S1(t), I1(t), M1(t), V1(t), S2(t), I2(t), M2(t), V2(t)) of (2.2) satisfies
lim inf Si ðtÞ P d; lim inf Ii ðtÞ P d; lim inf Mi ðtÞ P d; lim inf V i ðtÞ P d; t!þ1
t!þ1
t!þ1
t!þ1
i ¼ 1; 2:
Proof. Define
X ¼ fðS1 ; I1 ; M 1 ; V 1 ; S2 ; I2 ; M2 ; V 2 Þ : Si P 0; Ii P 0; Mi P 0; V i P 0; i ¼ 1; 2g; X 0 ¼ fðS1 ; I1 ; M 1 ; V 1 ; S2 ; I2 ; M 2 ; V 2 Þ : Si > 0; Ii > 0; M i > 0; V i > 0; i ¼ 1; 2g; @X 0 ¼ X n X 0 : In order to prove that the disease is uniformly persistent, it suffices to show that @X0 repels uniformly the solutions of X0. First, by the form of (2.2), it follows that both X and X0 are positively invariant. Clearly, @X0 is relatively closed in X and system (2.2) is point dissipative. Set
M@ ¼ fðS1 ð0Þ; I1 ð0Þ; M 1 ð0Þ; V 1 ð0Þ; S2 ð0Þ; I2 ð0Þ; M 2 ð0Þ; V 2 ð0ÞÞ : ðS1 ðtÞ; I1 ðtÞ; M1 ðtÞ; V 1 ðtÞ; S2 ðtÞ; I2 ðtÞ; M2 ðtÞ; V 2 ðtÞÞ satisfies ð2:2Þ and ðS1 ðtÞ; I1 ðtÞ; M 1 ðtÞ; V 1 ðtÞ; S2 ðtÞ; I2 ðtÞ; M 2 ðtÞ; V 2 ðtÞÞ 2 @X 0 ; 8t P 0g: We now show that
n o M@ ¼ ðS1 ; I1 ; M 1 ; V 1 ; S2 ; I2 ; M 2 ; V 2 Þ 2 X : I21 þ V 21 þ I22 þ V 22 ¼ 0 ;
ð4:4Þ
if ai > 0, i = 1, 2. Assume (S1(0), I1(0), M1(0), V1(0), S2(0), I2(0), M2(0), V2(0)) 2 M@ . It suffices to show that I21 ðtÞ þ V 21 ðtÞþ I22 ðtÞ þ V 22 ðtÞ ¼ 0 for all t P 0. Suppose not, then there exists a t0 P 0 such that I21 ðt 0 Þ þ V 21 ðt 0 Þ þ I22 ðt 0 Þ þ V 22 ðt0 Þ > 0. Here we only consider the case I1(t0) > 0, I2(t0) = 0, Si(t0) = 0, Mi(t0) = 0, Vi(t0) = 0, i = 1, 2. The other cases can be deduced in the same way. Since
S0i ðt0 Þ P Ai þ ci Ni ðt 0 Þ li Si ðt 0 Þ bi bi V i ðt 0 ÞSi ðt 0 Þ þ ci Ii ðt 0 Þ P Ai > 0; M 0i ðt 0 Þ P Gi þ hi T i ðt 0 Þ bi ai Ii ðt0 ÞM i ðt 0 Þ i M i ðt0 Þ P Gi > 0; I02 ðt0 Þ ¼ b2 b2 V 2 ðt 0 ÞS2 ðt 0 Þ ðl2 þ c2 þ d2 ÞI2 ðt 0 Þ a2 I2 ðt0 Þ þ a1 I1 ðt 0 Þ ¼ a1 I1 ðt 0 Þ > 0 and
I01 ðtÞ P ðli þ ci þ di þ a1 ÞI1 ðtÞ; it follows that there is an e0 small enough such that Si(t) > 0, Ii(t) > 0, Mi(t) > 0, i = 1, 2, for all t0 < t < t0 + e0. If V i ðt 0 þ e20 Þ > 0; i ¼ 1; 2, then we have
V 0i ðtÞ P i V i ðtÞ: This means that Vi(t) > 0, i = 1, 2 for all t P t0 þ e20 ; if V i ðt 0 þ e20 Þ ¼ 0, it then follows from the fourth equation of system (2.2) that
V 0i ðt0 þ
e0 e0 Þ ¼ bi ai Ii t 0 þ Mi ðt0 þ Þ > 0: 2 2 2
e0
It then follows that there exists e1 < e20 such that Vi(t) > 0, i = 1, 2 for all t 2 t0 þ e20 ; t0 þ e20 þ e1 . Thus for all e0 e0 t 2 t0 þ 2 ; t 0 þ 2 þ e1 we have Si(t) > 0, Ii(t) > 0, Mi(t) > 0, Vi(t) > 0, i = 1, 2. This contradicts the assumption that (S1(0), I1(0), M1(0), V1(0), S2(0), I2(0), M2(0), V2(0)) 2 M@ . This proves (4.4). It is clear that E0 is the unique equilibrium in M@ . We now show that E0 repels the solution in X0. By (3.3) we know that e 2 ð1 vÞ > 1 or R0 > 1 if and only if R 01
(
e 2 ð1 vÞ 6 1; R 01 e2 e 2 ð1 fÞ R e2 R e 2 ð1 vÞ þ R R 01 02 01 02 ð1 v fÞ > 1:
ð4:5Þ
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e 2 ð1 vÞ > 1, i.e., If R 01 2
b1 a1 b1 N01 T 01 ðl2 þ c2 þ d2 þ a2 Þ þ 1 a1 a2 > 1 ðl1 þ c1 þ d1 þ a1 Þðl2 þ c2 þ d2 þ a2 Þ; we can choose h1 > 0 small enough such that
2 b1 a1 b1 N01 h1 T 01 h1 ðl2 þ c2 þ d2 þ a2 Þ þ 1 a1 a2 > 1 ðl1 þ c1 þ d1 þ a1 Þðl2 þ c2 þ d2 þ a2 Þ:
ð4:6Þ
If (4.5) holds, i.e., det (J) < 0, we can also choose h1 small enough such that
ðl þ c þ d þ a Þ a2 b1 b1 N01 h1 0 1 1 1 1 ðl2 þ c2 þ d2 þ a2 Þ 0 b2 b2 N02 h1 a1 < 0: b1 a1 T 01 h1 0 0 1 0 2 0 b2 a2 T 02 h1
ð4:7Þ
By direct calculations, we can choose d1 > 0 small enough such that
8 ðA1 þA2 Þa2 þðl2 þb2 b2 d1 c2 ÞA1 > > N01 h1 ; > > ðl1 þb1 b1 d1 c1 Þðl2 þb2 b2 d1 c2 Þþðl1 þb1 b1 d1 c1 Þa2 þðl2 þb2 b2 d1 c2 Þa1 > > > Þa1 þðl1 þb1 b1 d1 c1 ÞA2 > < ðl þb b d c Þðl þbðA1bþAd 2c > N02 h1 ; 1 1 1 1 2 2 1 2 Þþðl þb1 b1 d1 c 1 Þa2 þðl þb2 b2 d1 c 2 Þa1 1
2
1
2
> G1 > > T 01 h1 ; > b1 a1 d1 þ1 h1 > > > > : G2 > T 02 h1 : b2 a2 d1 þ2 h2
ð4:8Þ
Suppose (S1(t), I1(t), M1(t), V1(t), S2(t), I2(t), M2(t), V2(t)) is a solution of system (2.2) with (S1(0), I1(0), M1(0), V1(0), S2(0), I2(0), M2(0), V2(0)) 2 X0. We now claim that
lim sup maxfI1 ðtÞ; I2 ðtÞ; V 1 ðtÞ; V 2 ðtÞg > d1 :
ð4:9Þ
t!þ1
Suppose, for the sake of contradiction, that there is a T > 0 such that Ii(t) 6 d1, Vi(t) 6 d1, i = 1, 2, for all t P T. Then by the first and third equations of system (2.2), we have
8 0 > S1 ðtÞ P A1 þ c1 S1 ðtÞ ðl1 þ b1 b1 d1 ÞS1 ðtÞ a1 S1 ðtÞ þ a2 S2 ðtÞ; > > > > < S0 ðtÞ P A2 þ c2 S2 ðtÞ ðl þ b2 b d1 ÞS2 ðtÞ þ a1 S1 ðtÞ a2 S2 ðtÞ; 2 2 2 0 > ðtÞ P G þ h M ðtÞ ð þ b M > 1 1 1 1 1 1 a1 d1 ÞM 1 ðtÞ; > > > : 0 M2 ðtÞ P G2 þ h2 M 2 ðtÞ ð2 þ b2 a2 d1 ÞM 2 ðtÞ for t > T. Consider the following system:
8 e e0 e e e > > > S 1 ðtÞ ¼ A1 þ c1 S 1 ðtÞ ðl1 þ b1 b1 d1 Þ S 1 ðtÞ a1 S 1 ðtÞ þ a2 S 2 ðtÞ; > > >
ð4:10Þ
As in our previous analysis of system (4.3), we can restrict d1 to be small enough such that (4.10) admits a positive equilib e 0; M e 0 , where rium e S 01 ; e S 02 ; M 1 2
e S 01 ¼
ðA1 þ A2 Þa2 þ ðl2 þ b2 b2 d1 c2 ÞA1 ; ðl1 þ b1 b1 d1 c1 Þðl2 þ b2 b2 d1 c2 Þ þ ðl1 þ b1 b1 d1 c1 Þa2 þ ðl2 þ b2 b2 d1 c2 Þa1
e S 02 ¼
ðA1 þ A2 Þa1 þ ðl1 þ b1 b1 d1 c1 ÞA2 ; ðl1 þ b1 b1 d1 c1 Þðl2 þ b2 b2 d1 c2 Þ þ ðl1 þ b1 b1 d1 c1 Þa2 þ ðl2 þ b2 b2 d1 c2 Þa1
e0 ¼ M 1
G1 G2 e0 ¼ ;M : 2 b1 a1 d1 þ 1 h1 b2 a2 d1 þ 2 h2
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e 0; M e 0 is globally asymptotically stable for system (4.10). By (4.8) Furthermore, the unique positive equilibrium e S 01 ; e S 02 ; M 1 2 and the standard comparison principle, there is a s > 0 such that S1 ðtÞ P N 01 h1 ; S2 ðtÞ P N 02 h1 ; M 1 ðtÞ P T 01 h1 ; M2 ðtÞ P T 02 h1 for t > T + s. Consequently, for t P T + s, we have
8 > I01 P b1 b1 V 1 N01 h1 ðl1 þ c1 þ d1 ÞI1 a1 I1 þ a2 I2 ; > > > > > > > < I02 P b2 b2 V 2 N02 h1 ðl2 þ c2 þ d2 ÞI2 a2 I2 þ a1 I1 ; 0 0 > > > V 1 P b1 a1 I1 T 1 h1 1 V 1 ; > > > > > : V 0 P b2 a2 I2 T 0 h1 2 V 2 : 2 2
ð4:11Þ
Consider an auxiliary system
8 0 b bI 0 ¼ b1 b V b b b > > 1 1 N 1 h1 ðl1 þ c1 þ d1 Þ I 1 a1 I 1 þ a2 I 2 ; 1 > > > > > > b 2 N0 h1 ðl þ c þ d2 ÞbI 2 a2bI 2 þ a1bI 1 ; < bI 02 ¼ b2 b2 V 2 2 2 0 > 0 b b b > V 1 ¼ b1 a1 I 1 T 1 h1 1 V 1 ; > > > > > > b 2: b 0 ¼ b2 a2bI 2 T 0 h1 2 V :V 2 2
ð4:12Þ
The coefficient matrix bJ of the right hand of (4.12) is defined by
0 B B bJ ¼ B B @
ðl1 þ c1 þ d1 þ a1 Þ
a2
b1 b1 ðN01 h1 Þ
a1
ðl2 þ c2 þ d2 þ a2 Þ
0
b1 a1 ðT 01 h1 Þ
0
1
0
0 2 ðT 2
b2 a
h1 Þ
0
0
1
C b2 b2 ðN02 h1 Þ C C: C 0 A 2
Since bJ admits positive off-diagonal elements, Perron–Frobenius Theorem implies that there is a positive eigenvector vm for the maximin eigenvalue km of bJ. After extensive computations, we have km > 0 since (4.6) and (4.7) hold. By the theory of linear system, it is easy to see that any positive solution of (4.12) tends to infinity as t tends to infinity. Then by the stand comparison principle, we have Ii(t) ? +1, Vi(t) ? +1, i = 1, 2 as t ? +1. This contradicts Ii(t) 6 d1, Vi(t) 6 d1, i = 1, 2 for all t P T. This proves (4.9). Hence Ws(E0) \ X0 = ;. Clearly, every forward orbit in M@ converges to E0. By Theorem 4.6 of [20] we are able to conclude that the system (2.2) is uniformly persistent with respect to (X0, oX0). This completes the proof of Theorem 4.2. h
5. Stability of the endemic equilibrium In this section, we consider the stability of endemic equilibrium of system (2.2). In order to be tractable in mathematics, we only consider two special cases. First, we suppose that the both region are identical, i.e., demographic parameters are the same for each region. This means that Ai = A, ci = c, li = l, bi = b, ai = a, di = d, ci = c, Gi = G, hi = h, i = , ai = a, i = 1, 2. Then (2.2) becomes
8 0 Si ¼ A þ cN i lSi bbV i Si þ cIi aSi þ aSj ; > > > < I0 ¼ bbV S ðl þ c þ dÞI aI þ aI ; i i i i j i i; j ¼ 1; 2; i – j: > M 0i ¼ G þ hT i baIi Mi M i ; > > : 0 V i ¼ baIi M i V i ;
ð5:1Þ
G In this case, the disease free equilibrium of system (5.1) is E0(N0, 0, T0, 0, N0, 0, T0, 0), where N 0 ¼ lAc ; T 0 ¼ h . Therefore, the reproduction number for the model (5.1) can be expressed as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b abN0 T 0 R0 ¼ : ðl þ c þ dÞ After extensive algebraic calculations, we have that system (5.1) has no positive equilibrium if R0 < 1 and if R0 > 1 system (5.1) has an endemic equilibrium E⁄(S⁄, I⁄, M⁄, V⁄, S⁄, I⁄, M⁄, V⁄), where S⁄, I⁄, M⁄, V⁄ can be expressed as
ðl þ d cÞN A A ðl cÞN ;I ¼ ; d d ðl þ d þ cÞðA ðl cÞN Þ V ¼ ; M ¼ T 0 V bbððl þ d cÞN AÞ
S ¼
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and 2
N ¼
2
ðl þ c þ dÞðb aA þ dÞ þ T 0 b abA 0 2
ðl þ d þ cÞðl cÞ þ T b abðl þ d cÞ
:
Now let us consider the local stability of the positive equilibrium E⁄ when R0 > 1. At the positive equilibrium E (S⁄, I⁄, M⁄, V⁄, S⁄, I⁄, M⁄, V⁄) of the system (5.1), the Jacobian matrix is given by ⁄
JðE Þ ¼
A
B
B
A
;
where
0 B B A¼B @
ðl þ a þ bbV cÞ
cþc
bbV
ðl þ a þ c þ dÞ
0
baM
0
baM
bbS
0
0 h baI
baI
1
bbS C C C h A
and
1 a 0 0 0 B0 a 0 0C C B B¼B C: @0 0 0 0A 0
0 0 0 0 Calculating the characteristic polynomial, we have
kI A B ¼ jkI ðA þ BÞjjkI ðA BÞj: jkI Jj ¼ B kI A It follows that the stability of the equilibrium E⁄ is determined by the eigenvalues of the matrices A + B and A B since the eigenvalues of the matrix J are made up of the eigenvalues of the matrices A + B and A B. After extensive algebraic calculations, the eigenvalues of A + B are h and the roots of
k3 þ A1 k2 þ A2 k þ A3 ¼ 0; where
A1 ¼ baI þ þ 2l þ bbV þ c þ d c > 0; A2 ¼ ðbaI þ ÞðbbV þ l cÞ þ bbV ðl þ d cÞ þ ðl þ d þ cÞðl c þ baI Þ > 0; A3 ¼ ðl cÞðl þ c þ dÞbaI þ bbV ð þ baI Þðl c þ dÞ > 0: Direct calculation yields that
A1 A2 A3 ¼ ðbaI þ ÞððbaI þ ÞðbbV þ l cÞ þ ðl þ d þ cÞðl c þ baI ÞÞ þ ðbbV þ l þ d þ cÞððbaI þ ÞðbbV þ l cÞ þ bbV ðl þ d cÞ þ ðl þ d þ cÞðl c þ baI ÞÞ þ ðl cÞððbaI þ ÞðbbV þ l cÞ þ bbV ðl þ d cÞ þ ðl þ d þ cÞðl cÞÞ > 0: Similarly, the eigenvalues of A B are h < 0 and the roots of
e 1 k2 þ A e 2k þ A e 3 ¼ 0; k3 þ A where
e 1 ¼ baI þ þ 2l þ 4a þ bbV þ c þ d c > 0; A e 2 ¼ ðbaI þ ÞðbbV þ l þ 2a cÞ þ bbV ðl þ 2a þ d cÞ þ ðl þ 2a þ d þ cÞðl þ 2a c þ baI Þ þ 2a > 0; A e 3 ¼ ðl þ 2a cÞðl þ c þ dÞbaI þ bbV ð þ baI Þðl þ 2a c þ dÞ þ 2aðl þ 2a cÞð þ baI Þ > 0 A and
e 3 ¼ ðbaI þ ÞððbaI þ ÞðbbV þ l þ 2a cÞ þ ðl þ 2a þ d þ cÞðl þ 2a c þ baI ÞÞ e1 A e2 A A þ ðbbV þ l þ 2a þ d þ cÞððbaI þ ÞðbbV þ l þ 2a cÞ þ bbV ðl þ 2a þ d cÞ þ ðl þ 2a þ d þ cÞðl þ 2a c þ baI ÞÞ þ ðl cÞððbaI þ ÞðbbV þ l þ 2a cÞ þ bbV ðl þ d cÞ þ ðl þ 2a þ d þ cÞðl þ 2a cÞÞ þ 2aððbaI þ ÞbbV þ bbV ðl þ 2a þ d cÞ þ ðl þ 2a þ d þ cÞðl þ 2a cÞÞ > 0:
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By Routh–Hurwitz criteria, it follows that all the eigenvalues of A B and A + B have negative real parts. Therefore, we have the following theorem. 2
G Theorem 5.1. If b ab lAc h > ðl þ c þ dÞ, then system (5.1) has an endemic equilibrium E⁄(S⁄, I⁄, M⁄, V⁄, S⁄, I⁄, M⁄, V⁄), which is locally asymptotically stable.
Second, we suppose that there is no disease-induced death rate of hosts, i.e., di = 0, i = 1, 2. Then the system (2.2) becomes
8 0 Si ¼ Ai þ ci Ni li Si bi bi V i Si þ ci Ii ai Si þ aj Sj ; > > > 0 < Ii ¼ bi bi V i Si ðli þ ci ÞIi ai Ii þ aj Ij ; 0 > > M i ¼ Gi þ hi T i bi ai Ii M i i M i ; > : 0 V i ¼ bi a i I i M i i V i ;
i; j ¼ 1; 2; i – j:
ð5:2Þ
Theorem 5.2. Let ai > 0, i = 1, 2. If R0 > 1, then system (5.2) has only a unique positive equilibrium E⁄, which is globally asymptotically stable in IntC.
Proof. Adding up the first and second equations of system (5.2) gives
N0i ¼ Ai ðli ci ÞN i ai Ni þ aj Nj ; It is easy to see that N i ðtÞ ! be read as
N 0i
i; j ¼ 1; 2; i – j:
as t ? +1. Similarly, we have M i þ V i ! T 0i as t ? +1. Then the limiting system of (5.2) can
8 > I01 ¼ b1 b1 V 1 N01 I01 ðl1 þ c1 ÞI1 a1 I1 þ a2 I2 ; > > > > > > > < I02 ¼ b2 b2 V 2 N02 I02 ðl2 þ c2 ÞI2 a2 I2 þ a1 I1 ; 0 0 > > V ¼ b a I T V 1 V 1 ; > 1 1 1 1 1 1 > > > > > 0 0 : V ¼ b2 a2 I2 T V 2 2 V 2 ; 2 2
ð5:3Þ
From the Jacobian of system (5.3) we can verify that system (5.2) is a cooperative irreducible system in R4þ [14]. By Theorem 3.2 it follows that if R0 < 1 the origin of system (5.3) is local asymptotically stable and if R0 > 1 the origin of system (5.3) is unstable. Following Smith [21], system (5.3) is strongly concave. Then it follows that if R0 > 1 system (5.3) has an equilibrium E I1 ; I2 ; V 1 ; V 2 , which is globally asymptotically stable. Since system (5.3) is the limiting system of (5.2), it follows from Theorem 2.3 in paper [19] that the unique positive equilibrium E N 01 I1 ; I1 ; T 01 V 1 ; V 1 ; N 02 I2 ; I2 ; T 02 V 2 ; V 2 is a globally asymptotically stable equilibrium of system (5.2). This completes the proof of Theorem 5.2 h
6. Discussions In this section, we mainly summarize our results and provide two numerical examples to illustrate the effect of host migration on the spread of the vector–host diseases. In this paper, we mainly proposed an epidemic model to investigate the impact of host dispersal on the diffusion and outbreak of a vector–host disease. The host population dispersal among patches can be interpreted as the movement that the host, such as birds, people, migrate from one region to another. Due to the spread of the vector–host disease between and within two populations, we incorporated two species into our model, which is different from the classical compartmental epidemiological models [8,9,22]. We adopted the function Bi ðN i Þ ¼ NAi þ ci for the growth rates of the hosts and the function i F i ðN i Þ ¼ GT ii þ hi for the growth rate of the vectors, which is widely used in literature. Our model is suited to describe the dynamical transmission of many vector–host diseases between two patches in which the vital and epidemiological parameters may depend on the patches. Systemic analysis of the model is then provided. By analyzing the local stability of disease free equilibrium E0, we derived the reproduction number, which can be expressed as
R0 ¼
pffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 e2 e 2 vf 2 ; e 2 ð1 fÞ þ ð R e 2 ð1 vÞ R e 2 ð1 fÞÞ2 þ 4 R e2 R R 01 ð1 vÞ þ R 02 01 02 01 02 2
e 0i ¼ where R
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b2i ai bi N0i T 0i i ðli þci þdi Þ;
i ¼ 1; 2;
a1 l1 þc1 þd1 a1 a2 l1 þc1 þd1 þl2 þc2 þd2
v ¼ 1þ
; f¼
a2 l2 þc2 þd2 a1 a 1þl þc þd þl þc2 þd 1 1 1 2 2 2
. It is proved that the reproduction number R0 is
the threshold condition that determines the uniform persistence and extinction of the disease. When R0 < 1, by Theorem 4.1 the disease free equilibrium is globally asymptotically stable and the disease will die out. For R0 > 1 it follows from
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Theorem 4.2 that the disease will be uniformly persistent. Especially, if the two patches are isolated, the results in [13] show that the disease will be persistent in the first patch if
R01
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 0 b1 a1 b1 N00 1 T1 >1 ¼ 1 ðl1 þ c1 þ d1 Þ
and the disease will disappear in the first patch if R01 < 1. Further, the disease will be persistent in the second patch if
R02
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 0 b2 a2 b2 N00 2 T2 ¼ >1 2 ðl2 þ c2 þ d2 Þ
and the disease will disappear in the second patch if R02 < 1. Thus, the quantities R01 ; R02 ; R0 can provide us more important information for analyzing the effect of host migration on the spread of the vector–host disease. In the following, we give two examples to illustrate it. All the values of the parameters selected in the two examples are for illustration purpose only. Example 6.1. We use the parameters given below:
l1 ¼ 0:6; c1 ¼ 0:1; 1 ¼ 0:6; h1 ¼ 0:1; b1 ¼ 1; a1 ¼ 0:01; c1 ¼ 0:2; d1 ¼ 0:2; A2 ¼ 20; G2 ¼ 1; l2 ¼ 0:6; c2 ¼ 0:5; 2 ¼ 0:6; h2 ¼ 0:5; b1 ¼ 1; a2 ¼ 0:01; b2 ¼ 0:01; c2 ¼ 0:2; d2 ¼ 0:2: A1 ¼ 10;
G1 ¼ 80;
b1 ¼ 0:01;
If the two patches are isolated, numerical calculations give
R01 ¼ 0:7303;
R02 ¼ 0:5774:
It follows from Lemma 3.1 that the disease will die out in both two patches when they are isolated. By means of Matlab software, we can easily see that R0 < 1 for small a1, a2 (see Fig. 2), but if the host migration rates a1, a2 are large then we have R0 > 1. This suggests that the host migration will cause the spread of the vector–host disease in the two patches. Example 6.2. We use the parameters given in Example 6.1 except that
A1 ¼ 10;
G1 ¼ 10;
A2 ¼ 10;
G2 ¼ 10:
If the two patches are isolated, numerical calculations give
R01 ¼ 0:2582;
R02 ¼ 1:2910:
It follows from Lemma 3.1 that the disease will be uniformly persistent in the second patch and will die out in the first patch. By means of Matlab software, we can easily see that R0 > 1 for small a1, a2 (see Fig. 3). This means that the host dispersal intensifies the vector-disease spread. However, if the host migration rate a1, a2 is large, it also follows from Fig. 3 that R0 < 1. This implies that host migration can reduces the vector-disease spread and is beneficial to disease control. Finally, we are interested in the global stability of the system (2.2). This seems to be a difficult problem since the dimension of the model is higher. Numerical simulations suggest that if R0 > 1 then system (2.2) has a unique endemic equilibrium, which is globally asymptotically stable. Additionally, the model presented here can be extended to describe the
Fig. 2. Surface plot of R0 as a function of a1 and a2 for the parameters given in Example 6.1.
Z. Qiu / Applied Mathematics and Computation 218 (2011) 4614–4625
4625
Fig. 3. Surface plot of R0 as a function of a1 and a2 for the parameters given in Example 6.2.
dynamical transmission of the vector–host disease with host (and vector) migration among more than 2 patches. It can also be extended to incorporate the other ingredients, such as the different incidences and different compartmental structures. It will also be interesting to study the effect of the worldwide spread of the vector–host diseases caused by patches structure or/and the migration of the vectors and the hosts. We leave these for future investigation. Acknowledgements The author greatly appreciates the anonymous referees for their valuable comments which help to improve the paper. The research of this paper is supported by the NSF of China grants 10801074, 60874007, NUST Research Funding No. 2011YBXM31, the Zijin Star Project of Excellence Plan of NJUST and Qinglan Project of Jiangsu Province. References [1] Z.L. Feng, J.X. Velasco-Hernández, Competitive exclusion in a vector–host model for the dengue fever, J. Math. Biol. 35 (1997) 523–544. [2] G. Cruz-Pachecoa, L. Estevab, J.A. Montano-Hirosec, C. Vargasd, Modelling the dynamics of West Nile Virus, Bull. Math. Biol. 67 (2005) 1157–1172. [3] C. Bowman, A.B. Gumel, J. Wu, P. van den Driessche, H. Zhu, A mathematical model for assessing control strategies against West Nile virus, Bull. Math. Biol. 67 (2005) 1107–1133. [4] M.J. Wonham, T. de-Camino-Beck, M. Lewis, An epidemiological model for West Nile virus: invasion analysis and control applications, Proc. Roy. Soc. Lond. Ser. B 1538 (2004) 501–507. [5] J.H. Rappole, S.R. Derrickson, Z. Hubalek, Migratory birds and spread of West Nile virus in the Western Hemisphere, Emer. Infect. Dis. 6 (2000) 319–328. [6] J.H. Rappole, Z. Hubalek, Migratory birds and West Nile virus, J. Appl. Microbiol. 94 (2003) 47–58. [7] J. Owen, F. Moore, N. Panella, et al, Migrating birds as dispersal vehicles for West Nile virus, EcoHealth 3 (2006) 79–85. [8] W.D. Wang, G. Mulone, Threshold of disease transmission in a patch environment, J. Math. Anal. Appl. 285 (2003) 321–335. [9] W.D. Wang, X.Q. Zhao, An epidemic model in a patchy environment, Math. Biosci. 190 (2004) 97–112. [10] J.A. Cui, Y. Takeuchi, Y. Saito, Spreading disease with transport-related infection, J. Theor. Biol. 239 (2006) 376–390. [11] J. Arino, R. Jordan, P. van den Driessche, Quarantine in a multi-species epidemic model with spatial dynamics, Math. Biosci. 206 (2007) 46–60. [12] K. Cooke, P. van den Driessche, X. Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol. 39 (1999) 332–352. [13] Z.P. Qiu, Dynamical behavior of a vector–host epidemic model with demographic structure, Comput. Math. Appl. 56 (2008) 3118–3129. [14] H.L. Smith, Monotone Dynamical Systems: An Introduction to Theory of Competitive and Cooperative Systems, Math. Surveys Monogr., 41, AMS, Providence, RI, 1995. [15] J.F. Jiang, On the global stability of cooperative systems, Bull. Lond. Math. Soc. 26 (1994) 455–458. [16] P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002) 29–48. [17] H.L. Smith, P. Waltman, The Theory of the Chemostat, Cambridge University, 1995. [18] J. Franklin, Matrix Theory, Prentice-Hall, Englewood Cliffs, NJ, 1968. [19] C. Castillo-Chavez, H.R. Thieme, Asymptotically autonomous epidemic models, in: O. Arino, M. Kimmel (Eds.), Mathematical Population Dynamics: Analysis of Heterogeneity, Theory of Epidemics, Wuetz 1, 1995, pp. 33–50. [20] H.R. Thieme, Persistence under relaxed point-dissipativity with an application to an epidemic model, SIAM J. Math. Anal. 24 (1993) 407–435. [21] H.L. Smith, Cooperative systems of differential equations with concave nonlinearities, Nonlinear Anal.: T.M.A. 10 (1986) 1037–1052. [22] J. Arino, P. van den Driessche, The basic reproduction number in a multi-city compartmental epidemic model, Lect. Notes Contr. Inf. Sci. 294 (2003) 135–142.