An epidemic model in a patchy environment
Mathematical Biosciences 190 (2004) 97–112 www.elsevier.com/locate/mbs
An epidemic model in a patchy environment Wendi Wang
a,1
, Xiao-Qiang Zhao
b,*,2
a
b
Department of Mathematics, Southwest Normal University Chongqing, 400715, People’s Republic of China Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, Canada NF A1C 5S7 Received 6 September 2001; received in revised form 9 August 2002; accepted 18 November 2002 Available online
Abstract An epidemic model is proposed to describe the dynamics of disease spread among patches due to population dispersal. We establish a threshold above which the disease is uniformly persistent and below which disease-free equilibrium is locally attractive, and globally attractive when both susceptible and infective individuals in each patch have the same dispersal rate. Two examples are given to illustrate that the population dispersal plays an important role for the disease spread. The first one shows that the population dispersal can intensify the disease spread if the reproduction number for one patch is large, and can reduce the disease spread if the reproduction numbers for all patches are suitable and the population dispersal rate is strong. The second example indicates that a population dispersal results in the spread of the disease in all patches, even though the disease can not spread in each isolated patch. Ó 2004 Published by Elsevier Inc. Keywords: Epidemic model; Population dispersal; Threshold dynamics
1. Introduction Many epidemic models have been proposed and studied to understand mechanism of disease transmission (see, for example, [1,3,12] and the references cited therein). One of the most
*
Corresponding author. Tel.: +1-709 737 8098; fax: +1-709 737 3010. E-mail addresses: [email protected] (W. Wang), [email protected] (X.-Q. Zhao). 1 Research supported by the Key Teachers Fund from the Minister of Education of China and the National Science Foundation of China. 2 Research supported by the NSERC of Canada. 0025-5564/$ - see front matter Ó 2004 Published by Elsevier Inc. doi:10.1016/j.mbs.2002.11.001
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important subjects in this field is to obtain a threshold that determines the persistence and extinction of a disease. Reproduction numbers for ordinary differential equation models were investigated by Diekmann et al. [6,7], Hyman and Li [15], and van den Driessche and Watmough [19]. Reproduction numbers for SIS epidemiological models with delays were studied by Cooke and van den Driessche [4], Hethcote and van den Driessche [13]. The thresholds for the uniform persistence and extinction of a disease in epidemic models with delays were also considered recently by Thieme [18], Zhao and Zou [23], Wang and Ma [20]. Reproduction number and persistence in an endemic model with many infection stages were proposed and studied by Feng and Thieme [8,9]. Communicable diseases such as influenza and sexual diseases can be easily transmitted from one country (or one region ) to other countries (or other regions). Thus, it is important to consider the effect of population dispersal on spread of a disease. Hethcote [11] proposed an epidemic model with population dispersal between two patches. Brauer and van den Driessche [2] proposed a model with immigration of infectives. In this paper, we consider a disease transmission model with population dispersal among n patches. We assume that demographic structure is described by the following equation: N 0 ¼ BðN ÞN lN ; where N is the number of a population, BðN Þ is the birth rate of the population, and l is its death rate. This type of demographic structure with variable population size was proposed by Cooke et al. [5]. We adopt it here as a basis to develop an epidemic model with population dispersal. We consider SIS type of disease transmission. The population is divided into two classes: susceptible individuals and infectious individuals. Susceptible individuals become infective after contact with infective individuals. Infective individuals return to susceptible class when they are recovered. Gonorrhea and other sexually transmitted diseases or bacterial infections exhibit this phenomenon. We denote the numbers of susceptible individuals at time t by SðtÞ and the numbers of infective individuals at time t by IðtÞ. If there is no population dispersal among patches, i.e., the patches are isolated, we suppose that the population dynamics in ith patch is governed by 0 Si ¼ Bi ðNi ÞNi li Si bi Si Ii þ ci Ii ; ð1:1Þ Ii0 ¼ bi Si Ii ðli þ ci ÞIi ; where Si is the number of susceptible individuals in patch i, Ii the number of infectious individuals in patch i, Ni ¼ Si þ Ii is the number of the population in patch i, Bi ðNi Þ is the birth rate of the population in the ith patch, bi is the disease transmission coefficient, li is the death rate of the population in the ith patch, and ci is the recovery rate of infective individuals in the ith patch. Here, we adopt mass action incidence rate for convenience. Our method also applies to standard incidence rate. Following [5], we assume that Bi ðNi Þ satisfy the following basic assumptions for Ni 2 ð0; 1Þ: (A1) Bi ðNi Þ > 0, i ¼ 1; 2; . . . ; n; (A2) Bi ðNi Þ is continuously differentiable with B0i ðNi Þ < 0, i ¼ 1; 2; . . . ; n; (A3) li > Bi ð1Þ, i ¼ 1; 2; . . . ; n.
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As mentioned in [5], the following three types of birth functions Bi ðNi Þ can be found in the biological literature: (B1) Bi ðNi Þ ¼ bi eai Ni with ai > 0, bi > 0; pi (B2) Bi ðNi Þ ¼ qi þN m with pi , qi , m > 0; i
(B3) Bi ðNi Þ ¼ NAii þ ci with Ai > 0, ci > 0. When the patches are connected, we suppose that the dynamics of those individuals is governed by the following model: 8 n P > > > Si0 ¼ Bi ðNi ÞNi li Si bi Si Ii þ ci Ii þ aij Sj ; 1 6 i 6 n; < j¼1 ð1:2Þ n P > 0 > > : Ii ¼ bi Si Ii ðli þ ci ÞIi þ bij Ij ; 1 6 i 6 n; j¼1
where aii , bii , 1 6 i 6 n, are non-positive constants, aij and bij with i 6¼ j are non-negative constants. aii P 0 represents the emigration rate of susceptible individuals in the ith patch, bii P 0 represents the emigration rate of infective individuals in the ith patch, aij , j 6¼ i, represents the immigration rate of susceptible individuals from jth patch to ith patch, and bij , j 6¼ i, the immigration rate of infective individuals from jth patch to ith patch. For simplicity, we neglect death rates and birth rates of the individuals during the dispersal process. Thus, we have n X j¼1
aji ¼ 0;
n X
bji ¼ 0;
8 1 6 i 6 n:
ð1:3Þ
j¼1
We further assume that the n patches cannot be separated into two groups such that there is no immigration of susceptible and infective individuals from first group to second group. Mathematically, this means that two n n matrices ðaij Þ and ðbij Þ are irreducible (see, e.g., [16, Appendix A]). Note that system (1.2) indicates that the population can have different demographic structures and different infection forces among different patches, aij 6¼ bij implies that we also consider the variation of the dispersal rates of susceptible individuals and infective individuals. The remaining parts of this paper is organized as follows. In the next section, we establish a threshold between the extinction and persistence of the disease. In Section 3, the results are applied to the model with the patch number being 2. Section 4 gives a brief discussion of main results.
2. Threshold dynamics In order to find the disease-free equilibrium of (1.2), we consider Si0 ¼ Bi ðSi ÞSi li Si þ
n X j¼1
aij Sj ;
i ¼ 1; . . . ; n:
ð2:1Þ
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If z is a non-negative constant, we define an auxiliary matrix 2 3 B1 ðzÞ l1 þ a11 a12 a1n 6 7 a21 B2 ðzÞ l2 þ a22 a2n 7: MðzÞ ¼ 6 4 5 ... ... ... ... an2 Bn ðzÞ ln þ ann an1 This matrix is used to prove the existence and the uniqueness of a positive equilibrium in (2.1) and is different from the standard Jacobian matrix. Recall that the stability modulus of an n n matrix M, denoted by sðMÞ, is defined by sðMÞ :¼ maxfRe k : k is an eigenvalue of Mg: Note that MðzÞ is irreducible and has non-negative off-diagonal elements. By [16, Theorem A.5], it then follows that sðMðzÞÞ is a simple eigenvalue of MðzÞ with a (componentwise) positive eigenvector. Let F : Rnþ ! Rn be defined by the right-hand side of (2.1). Clearly, F is cooperative and DF ðSÞ is irreducible for every S 2 Rnþ . For any a 2 ð0; 1Þ and S ¼ ðS1 ; . . . ; Sn Þ 2 intðRnþ Þ, there holds " # n n X X aij asj > a Bi ðSi ÞSi li Si þ aij Sj ; i ¼ 1; 2; . . . ; n: aSi Bi ðaSi Þ li aSi þ j¼1
j¼1
Thus F is strongly sublinear on Rnþ (see, e.g., [22]). It then follows that for any S0 2 Rnþ , the unique solution Sðt; S0 Þ of (2.1) satisfying Sð0; S0 Þ ¼ S0 exists globally on ½0; 1Þ and Sðt; S0 Þ P 0, 8t P 0. We further claim that (2.1) admits a bounded positive solution. Indeed, in view of (A3), we can choose a sufficiently large real number K > 0 such that Bi ðKÞ < li , i ¼ 1; 2; . . . ; n. Let v ¼ ðv1 ; . . . ; vn Þ be a positive eigenvector associated with sðMðKÞÞ. Then V ðtÞ ¼ ðV1 ðtÞ; . . . ; Vn ðtÞÞ ¼ 0 vesðMðKÞÞtPis a positive solution Pn of the linear ordinary differential system V ¼ MðKÞV . Let n sðMðKÞÞt vn . By the first equation in (1.3), it easily follows that RðtÞ ¼ i¼1 Vi ðtÞ ¼ e i¼1 R0 ðtÞ 6 aRðtÞ, 8t P 0, where a ¼ maxfBi ðKÞ li : 1 6 i 6 ng < 0. Thus limt!1 RðtÞ ¼ 0, and hence sðMðKÞÞ < 0. Choose l > 0 large enough such that lvi > K, i ¼ 1; 2; . . . ; n. Set xðtÞ lv. If we rewrite (2.1) as S 0 ¼ F ðSÞ, it is easy to see that x0 ðtÞ ¼ 0 > sðMðKÞÞxðtÞ ¼ MðKÞxðtÞ > F ðxðtÞÞ;
8t P 0;
ð2:2Þ
where (A2) is used. By the standard comparison theorem(see, e.g., [16, Theorem B.1]), it follows that: 0 < Sðt; lvÞ 6 xðtÞ ¼ lv;
8t P 0:
Consequently, Sðt; lvÞ is a bounded positive solution of (2.1). In order for (2.1) to admit a positive equilibrium, we need to assume that ðA4Þ
sðMð0ÞÞ > 0:
By [22, Corollary 3.2], it then follows that (2.1) has a unique positive equilibrium S ¼ ðS1 ; S2 ; . . . ; Sn Þ and S is globally asymptotically stable for S 2 Rnþ n f0g. Thus, E0 ¼ ðS1 ; S2 ; . . . ; Sn ; 0; . . . ; 0Þ is a disease-free equilibrium of (1.2).
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Define 2
b1 S1 l1 c1 þ b11 6 b21 M1 ¼ 6 4 bn1
b12 b2 S2 l2 c2 þ b22 bn2
3 b1n 7 b2n 7: 5 bn Sn ln cn þ bnn
Clearly, M1 is irreducible and has non-negative off-diagonal elements. Then sðM1 Þ is a simple eigenvalue of M1 with a positive eigenvector (see, e.g., [16, Theorem A.5]). According to the concepts of next generation matrix and reproduction number presented in [6,19], we define 3 2 b1 S1 0 0 6 0 0 7 b2 S2 7 F :¼ 6 4 5 0 0 bn Sn and 2
l1 c1 þ b11 6 b21 V :¼ 6 4 bn1
b12 l2 c2 þ b22 bn2
3 b1n 7 b2n 7: 5 ln cn þ bnn
Set R0 :¼ qðFV1 Þ, where q represents the spectral radius of the a matrix. Then R0 is called the reproduction number for (1.2). Note that M1 ¼ F V. Thus, the following observation is implied by the proof of [19, Theorem 2] with J1 ¼ M1 . Lemma 2.1. There hold two equivalences: R0 > 1 () sðM1 Þ > 0;
R0 < I () sðM1 Þ < 0:
ð2:3Þ
By [19, Theorem 2], the disease-free equilibrium E0 is locally asymptotically stable if R0 < 1 and is unstable if R0 > 1. We will show that R0 is a threshold parameter for the uniform persistence and extinction of the disease, which is more general. To investigate the global dynamics of (1.2), we first show that (1.2) admits a compact, positively invariant set which absorbs all forward orbits in R2n þ , and hence (1.2) has a global compact (see, e.g., [10, Theorem 3.4.8]). For convenience, we denote the positive solution attractor on R2n þ ðS1 ðtÞ; . . . ; Sn ðtÞ; I1 ðtÞ; . . . ; In ðtÞÞ of (1.2) by ðSðtÞ; IðtÞÞ. Lemma 2.2. Let (A1)–(A4) hold. Then there isPan N > 0 such that every forward orbit in R2n þ of n : ðS þ I Þ 6 N g, and G is positively invariant for (1.2) eventually enters into G :¼ fðS; IÞ 2 R2n i i þ i¼1 (1.2).
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P Proof. Let N ¼ ni¼1 Ni , Ni ¼ Si þ Ii . By (1.2) and (1.3), we have n X 0 ðBi ðNi Þ li ÞNi : N ¼
ð2:4Þ
i¼1
If Bi ð0þÞ < li , i ¼ 1; 2; . . . ; n, then there exists an a > 0 such that N 0 ðtÞ 6 aN ðtÞ, 8t P 0, and hence, Lemma 2.2 holds for any positive number N . Otherwise, we partition f1; 2; . . . ; ng into two sets P1 and P2 such that Bi ð0þÞ > li ;
8i 2 P1 ;
Bi ð0þÞ 6 li ;
8i 2 P2 :
Without loss of generality, we suppose that P1 ¼ f1; . . . ; mg and P2 ¼ fm þ 1; . . . ; ng. For i 2 P1 , since Bi ð0þÞ > li and Bi ð1Þ < li , (A2) implies that there is a unique ki > 0 such that Bi ðki Þ li ¼ 0. It follows from (A3) that there is an N0 > 0 such that m X kj Bj ð0þÞ 1; 8N P N0 ; i ¼ 1; 2; . . . ; n: ðBi ðN Þ li ÞN < j¼1
Let N ¼ nN0 . By the definition of N , it is easy to see that N P N implies Ni0 > N0 for some 1 6 i0 6 n. It then follows from (2.4) that m X 0 Bj ð0þÞkj þ ðBi0 ðNi0 Þ li0 ÞNi0 < 1; if NðtÞ P N ; N ðtÞ 6 j¼1
which implies that G is positively invariant and every forward orbit enters into G after a certain time. h In the case where the susceptible and infective individuals in each patch have the same dispersal rate, we have the following result on the global attractivity of E0 . Theorem 2.1. Let (A1)–(A4) hold and R0 < 1. If aij ¼ bij for i ¼ 1; . . . ; n, j ¼ 1; . . . ; n, then E0 is globally attractive for ðS0 ; I0 Þ 2 ðRnþ n f0gÞ Rnþ . Proof. Let us consider a non-negative solution ðSðtÞ; IðtÞÞ of (1.2). We wish to show that lim IðtÞ ¼ 0: t!1
ð2:5Þ
By (1.2), we have Ni0 ¼ Bi ðNi ÞNi li Ni þ
n X
aij Nj ;
i ¼ 1; . . . ; n:
ð2:6Þ
j¼1
By the afore-mentioned conclusion for (2.1), (2.6) admits a unique positive equilibrium S which is globally asymptotically stable for N 2 Rnþ n f0g. It then follows that for any > 0, there holds NðtÞ ¼ SðtÞ þ IðtÞ < S þ , where ¼ ð; . . . ; Þ > 0, when t is sufficiently large. Thus, if t is sufficiently large, we have n X bij Ij ; 81 6 i 6 n: ð2:7Þ Ii0 < bi ðSi þ ÞIi ðli þ ci ÞIi þ j¼1
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It then suffices to show that positive solutions of the following auxiliary system: n X bij Ij ; 1 6 i 6 n Ii0 ¼ bi ðSi þ ÞIi ðli þ ci ÞIi þ
ð2:8Þ
j¼1
tend to zero as t goes to infinity. Let M2 be the matrix defined by M2 ¼ diagðb1 ; b2 ; . . . ; bn Þ: Since sðM1 Þ < 0 and sðM1 þ M2 Þ is continuous for small , we can fix an > 0 small enough such that sðM1 þ M2 Þ < 0. As a consequence, solutions of (2.8) tend to zero as t goes to infinity, which implies (2.5). For any ðS0 ; I0 Þ 2 Rnþ Rnþ with S0 6¼ 0, we have N0 ¼ S0 þ I0 2 Rnþ n f0g. By the global attractivity of S for system (2.6), it then follows that limt!1 SðtÞ ¼ limt!1 ðN ðtÞ IðtÞÞ ¼ S 0 ¼ S. h If the susceptible and infective individuals have different dispersal rates, by using the arguments in obtaining the unique positive equilibrium of (2.1) or by [19, Theorem 2], we see that (Al)–(A4) and R0 < 1 imply that the disease-free equilibrium is locally asymptotically stable. This means that a positive solution (SðtÞ, IðtÞ of (1.2) satisfies IðtÞ ! 0 as t ! 1 if its initial position is near to the disease-free equilibrium. Here, we give a little more general result where it is only needed that the values of infective individuals are small. Theorem 2.2. Let (A1)–(A4) hold and R0 < 1. Then there exists d > 0 such that for every ðSð0Þ; Ið0ÞÞ 2 G with Ii ð0Þ < d, i ¼ 1; 2; . . . ; n, the solution ðSðtÞ; IðtÞÞ of (1.2) satisfies limt!1 ðS 0 ðtÞ; IðtÞÞ ¼ ðS ; 0Þ. Proof. Let us consider an auxiliary system Si0 ¼ Bi ðSi ÞSi li Si þ ðBi ð0þÞ þ ci Þ þ
n X
aij Sj ;
i ¼ 1; . . . ; n;
ð2:9Þ
j¼1
where > 0 is a small constant to be determined. By (A4) and the previous analysis of system (2.1), we can restrict small enough such that (2.9) admits a unique positive equilibrium S ðÞ which is globally asymptotically stable. Let Sðt; N Þ be the solution of (2.9) through ðN ; . . . ; N Þ at t ¼ 0. Select T ðÞ > 0 such that Sðt; N Þ < S ðÞ þ ; 8t P T ðÞ; where ¼ ð; . . . ; Þ. Define a matrix M1 ðÞ by 2 b12 6 b1 S1 ðÞ l1 c1 þ b11 6 b21 b2 S2 ðÞ l2 c2 þ b22 6 6 .. . .. 4 . bn1 bn2
.. . .. . .. . .. .
b1n b2n .. .
3 7 7 7: 7 5
bn Sn ðÞ ln cn þ bnn
Since M1 ð0Þ ¼ M1 and sðM1 ðÞ þ M2 Þ is continuous for small , we can now restrict small enough such that sðM1 ðÞ þ M2 Þ < 0. Let v ¼ ðv1 ; . . . ; vn Þ be a positive eigenvector associated with sðM1 ðÞ þ M2 Þ. Choose k > 0 small enough such that kv < .
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Now, we define another auxiliary system Ii0 ¼ bi N Ii ðli þ ci ÞIi þ
n X
i ¼ 1; . . . ; n:
bij Ij ;
ð2:10Þ
j¼1
Let Iðt; dÞ be the solution of (2.10) through ðd; . . . ; dÞ at t ¼ 0. We restrict d > 0 small enough such that Iðt; dÞ < kv;
8t 2 ½0; T ðÞ:
ð2:11Þ
Let ðSðtÞ; IðtÞÞ be a non-negative solution of (1.2) with ðSð0Þ; Ið0ÞÞ 2 G and Ii ð0Þ < d, i ¼ 1; 2; . . . ; n. We claim that IðtÞ 6 kv, 8t P 0. Suppose not. By the comparison principle and (2.11), there exist a q, 1 6 q 6 n, and a T1 > T ðÞ such that IðtÞ 6 kv Iq ðT1 Þ ¼ kvq ;
Iq ðtÞ > kvq
for 0 6 t 6 T1 ; for 0 < t T1 1:
ð2:12Þ
Notice that for 0 6 t 6 T1 , we have Si0 < Bi ðSi ÞSi li Si þ ðBð0þÞ þ ci Þ þ
n X
aij Sj ;
i ¼ 1; . . . ; n:
ð2:13Þ
j¼1
It follows from the comparison principle that SðT1 Þ < S ðÞ þ . Hence, for 0 6 t T1 1, we have n X bij Ij ; i ¼ 1; . . . ; n: Ii0 < bi ðSi ðÞ þ ÞIi ðli þ ci ÞIi þ j¼1
Since IðT1 Þ 6 kv, the comparison principle implies that IðtÞ < kvesðM1 ðÞþM2 ÞðtT1 Þ
for 0 6 t T1 1
and hence, Iq ðtÞ < kvq esðM1 ðÞþM2 ÞðtT1 Þ < kvq
for 0 < t T1 1;
which contradicts to (2.12). This proves the claim. Now, (2.13) holds for all t P 0 due to the claim, and hence, the comparison principle implies that SðtÞ < S ðÞ þ , 8t P T ðÞ. By a similar argument as above, it then follows that IðtÞ < kvesðM1 ðÞþM2 ÞðtT ðÞ ;
8t > T ðÞ:
Consequently, IðtÞ ! 0 as t ! 1. Let UðtÞ : Rnþ ! Rnþ be the solution semiflow of (1.2), that is, UðtÞðS0 ; I0 Þ ¼ ðSðtÞ; IðtÞÞ is the solution of (1.2) with ðSð0Þ; Ið0ÞÞ ¼ ðS0 ; I0 Þ. Given ðS0 ; I0 Þ 2 G with S0 6¼ 0 and I0i < d, i ¼ 1; . . . ; n, it easily follows that SðtÞ 2 intðRnþ Þ, 8t > 0. Let x ¼ xðS0 ; I0 Þ be the omega limit set of UðtÞðS0 ; I0 Þ. f0g. We claim that x 6¼ f0g. Assume that, by Since IðtÞ ! 0 as t ! 1, there holds x ¼ x ¼ f0g. Then limt!1 ðSðtÞ; IðtÞÞ ¼ ð0; 0Þ. By assumption (A4), we can choose a contradiction, x small g > 0 such that sðMð0Þ gEÞ > 0, where E ¼ diagð1; . . . ; 1Þ. It follows that there exists a t > 0 such that Bi ðNi ðtÞÞ bi Ii ðtÞ P Bi ð0þÞ g for 8t P t, i ¼ 1; . . . ; n. Then SðtÞ ¼ ðS1 ðtÞ; . . . ; Sn ðtÞÞ satisfies
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Si0 ðtÞ > ðBi ð0þÞ gÞSi li Si þ
n X
aij Sj ;
8t P t;
i ¼ 1; . . . ; n:
105
ð2:14Þ
j¼1
Let w ¼ ðw1 ; . . . ; wn Þ be a positive eigenvector of Mð0Þ gE associated with the eigenvalue sðMð0Þ gEÞ. Choose a small number a a > 0 such that SðtÞ > aw. Then the comparison theorem implies that SðtÞ P awesðMð0ÞgEÞðttÞ ;
8t P t
and hence limt!1 Si ðtÞ ¼ 1, i ¼ 1; 2; . . . ; n, a contradiction. It is easy to see that UðtÞjx ðS; 0Þ ¼ ðU1 ðtÞS; 0Þ; where U1 ðtÞ is the solution semiflow of system (2.1). By [14, Lemma 2.10 ], x is an internal chain 6¼ f0g and is an internal chain transitive set for U1 ðtÞ. Since x transitive set for UðtÞ, and hence, x \ W s ðS Þ 6¼ ;. By [14, Theorem S is globally asymptotically stable for (2.1) in Rnþ n f0g, we have x ¼ S . This proves x ¼ fðS ; 0Þg. Consequently, 3.1 and Remark 4.6], we then get x ðSðtÞ; IðtÞÞ ! ðS ; 0Þ as t ! 1. h It is easy to see that the eigenvalues of M1 are also the eigenvalues of the Jacobian matrix of (1.2) at E0 . It follows that E0 is unstable if R0 > 1. The following result shows that R0 > 1 actually implies that model (1.2) admits at least one endemic equilibrium and the disease is uniformly persistent. Theorem 2.3. Let (A1)–(A4) hold and R0 > 1. Then (1.2) admits at least one (componentwise) positive equilibrium, and there is a positive constant such that every solution ðSðtÞ; IðtÞÞ of (1.2) with ðSð0Þ; Ið0ÞÞ 2 Rnþ intðRnþ Þ satisfies lim inf Ii ðtÞ P ; t!1
i ¼ 1; 2; . . . ; n:
Proof. Define X ¼ fðS1 ; . . . ; Sn ; I1 ; . . . ; In Þ : Si P 0; Ii P 0; i ¼ 1; 2; . . . ; ng; X0 ¼ fðS1 ; . . . ; Sn ; I1 ; . . . ; In Þ 2 X : Ii > 0; i ¼ 1; 2; . . . ; ng; oX0 ¼ X n X0 : It then suffices to show that (1.2) is uniformly persistent with respect to ðX0 ; oX0 Þ. First, by the form of (1.2), it is easy to see that both X and X0 are positively invariant. Clearly, oX0 is relatively closed in X . Furthermore, system (1.2) is point dissipative(see Lemma 2.2). Set Mo ¼ fðSð0Þ; Ið0ÞÞ : ðSðtÞ; IðtÞÞ satisfiesð1:2Þ and ðSðtÞ; IðtÞÞ 2 oX0 ; 8t P 0g: We now show that Mo ¼ fðS; 0Þ : S P 0g:
ð2:15Þ
Assume ðSð0Þ; Ið0ÞÞ 2 Mo . It suffices to show that IðtÞ ¼ 0, 8t P 0. Suppose not. Then there exist an i0 , 1 6 i0 6 n, and a t0 P 0 such that Ii0 ðt0 Þ > 0. We partition f1; 2; . . . ; ng into two sets Q1 and Q2 such that
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Ii ðt0 Þ ¼ 0; Ii ðt0 Þ > 0;
8i 2 Q1 ; 8i 2 Q2 :
Q1 is non-empty due to the definition of Mo . Q2 is non-empty since Ii0 ðt0 Þ > 0. For any j 2 Q1 , we have Ii0 ðt0 Þ P bji0 Ii0 ðt0 Þ > 0: It follows that there is an 0 > 0 such that Ij ðtÞ > 0, j 2 Q1 for t0 < t < t0 þ 0 . Clearly, we can restrict 0 > 0 small enough such that Ii ðtÞ > 0, i 2 Q2 for t0 < t < t0 þ 0 . This means that ðSðtÞ; IðtÞÞ 62 oX0 for t0 < t < t0 þ 0 , which contradicts the assumption that ðSð0Þ; Ið0ÞÞ 2 Mo . This proves (2.15). It is clear that there are two equilibria (0; 0) and (S ; 0) in Mo . Choose g > 0 small enough such that sðM1 gM2 Þ > 0. Let us consider a perturbed system of (2.1) n X Si0 ¼ Bi ðSi þ 1 ÞSi ðli þ bi 1 ÞSi þ aij Sj ; i ¼ 1; . . . ; n: ð2:16Þ j¼1
First, as in our previous analysis of system (2.1), we can restrict 1 > 0 small enough such that (2.16) admits a unique positive equilibrium S ð1 Þ which is globally asymptotically stable. By the implicit function theorem, it follows that S ð1 Þ is continuous in 1 . Thus, we can further restrict 1 small enough such that S ð1 Þ > S g. Let us consider an arbitrary positive solution ðSðtÞ; IðtÞÞ of (1.2). We now claim that lim sup maxfIi ðtÞg > 1 : t!1
i
Suppose, for the sake of contradiction, that there is a T > 0 such that Ii ðtÞ 6 1 , i ¼ 1; 2; . . . ; n, for all t P T . Then for t P T , we have N X 0 aij Sj ; i ¼ 1; . . . ; n: ð2:17Þ Si P Bi ðSi þ 1 ÞSi ðli þ Bi 1 ÞSi þ j¼1
Since the equilibrium S ð1 Þ of (2.16) is globally asymptotically stable and S ð1 Þ > S g, there is a T1 > 0 such that SðtÞ P S g for t P T þ T1 . As a consequence, for t > T þ T1 , there holds n X Ii0 P bi ðSi gÞIi ðli þ ci ÞIi þ bij Ij ; i ¼ 1; . . . ; n: ð2:18Þ j¼1
Since the matrix M1 gM2 has a positive eigenvalue sðM1 gM2 Þ with a positive eigenvector, it is easy to see that Ii ðtÞ ! 1 as t ! 1, i ¼ 1; 2; . . . ; n, which leads to a contradiction. Note that S is globally asymptotically stable in Rnþ n f0g for system (2.1). By the afore-mentioned claim, it then follows that ð0; 0Þ and E0 are isolated invariant sets in X , W s ðð0; 0ÞÞ \ X0 ¼ ;, and W s ðE0 Þ \ X0 ¼ ;. Clearly, every orbit in Mo converges to either ð0; 0Þ or E0 , and ð0; 0Þ and E0 are acyclic in Mo . By [17, Theorem 4.6] (see also [14, Theorem 4.3 and Remark 4.3], for a stronger repelling property of oX0 ), we conclude that system (1.2) is uniformly persistent with respect to IÞ 2 X0 . Then S 2 Rn and ðX0 ; oX0 Þ. By [21, Theorem 2.4], system (1.2) has an equilibrium ðS; þ I 2 intðRn Þ. We further claim that S 2 Rn n f0g. Suppose that S ¼ 0. By the second equation in þ þ P (1.3), we then get 0 ¼ ni¼1 ðli þ ci ÞIi , and hence Ii ¼ 0, i ¼ 1; 2; . . . ; n, a contradiction. By the first equation in (1.2) and the irreducibility of the cooperative matrix ðaij Þ, it follows that IÞ 2 intðRn Þ, 8t > 0. Then ðS; IÞ is a componentwise positive equilibrium of (1.2). h S ¼ Sðt; S; þ
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3. Application to two patches In the case where the patch number n is 2, the assumption (1.3) is equivalent to that a12 ¼ a22 , a21 ¼ a11 , b12 ¼ b22 , b21 ¼ b11 , and hence, (1.2) reduces to S10 ¼ B1 ðN1 ÞN1 ðl1 a11 ÞS1 b1 S1 I1 þ c1 I1 a22 S2 ; I10 ¼ b1 S1 I1 ðl1 þ c1 b11 ÞI1 b22 I2 ; S20 ¼ B2 ðN2 ÞN2 ðl2 a22 ÞS2 b2 S2 I2 þ c2 I2 a11 S1 ;
ð3:1Þ
I20 ¼ b2 S2 I2 ðl2 þ c2 b22 ÞI2 b11 I1 ; where aii and bii , 1 6 i 6 2, are negative constants. Assume that ðA5Þ
Bi ð0þÞ > li ;
81 6 i 6 2:
It then follows that sðMð0ÞÞ > 0. Let ðS1 ; S2 Þ be the positive solution of the following system: B1 ðS1 ÞS1 ðl1 a11 ÞS1 a22 S2 ¼ 0; B2 ðS2 ÞS2 ðl2 a22 ÞS2 a11 S1 ¼ 0: Now M1 becomes b1 S1 þ b11 l1 c1 b11
ð3:2Þ
b22 : b2 S2 þ b22 l2 c2
Its characteristic equation is k2 h1 k þ h2 ¼ 0 where h1 ¼ b1 S1 þ b2 S2 þ b11 þ b22 c1 c2 l1 l2 ; h2 ¼ b1 S1 b2 S2 b11 b22 b1 S1 ðl2 þ c2 b22 Þ
ð3:3Þ
b2 S2 ðl1 þ c1 b11 Þ þ ðl1 þ c1 b11 Þðl2 þ c2 b22 Þ: It is easy to see that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h1 þ h21 4h2 : sðM1 Þ ¼ 2 Then R0 > 1 if and only if h1 P 0 or 8 < h1 < 0; b1 S1 b2 S2 b11 b22 b1 S1 b2 S2 :1 þ þ : < ðl1 þ c1 b11 Þðl2 þ c2 b22 Þ l1 þ c1 b11 l2 þ c2 b22
ð3:4Þ
ð3:5Þ
h1 P 0 means that the total infection force of two patches overpasses the sum of recovery rates, b S1 can be viewed as a death rates and emigration rates of infectives in the two patches. l1 þc11 b 11 b1 S1 reproduction number at the first patch, and l1 þc1 b11 a reproduction number at the second patch. b1 S1 b2 S2 b11 b22 represents the quantity that new infectives minus moving infectives during the ðl1 þc1 b11 Þðl2 þc2 b22 Þ interval of product of infection periods in two patches. Thus, the second inequality of (3.5) can be interpreted as the sum of reproduction numbers in the two patches is stronger than 1 plus the net
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value that the numbers of new infectives minus moving infectives during the interval of product of infection periods in two patches. The conditions clearly show the effect of population dispersal on the spread of the disease. In order to be specific, we choose the birth rates Bi of the population as ri Bi ðNi Þ ¼ þ ci ; ci < li ; 1 6 i 6 2: Ni Then (3.1) reduces to S10 ¼ r1 þ c1 I1 ðl1 c1 a11 ÞS1 b1 S1 I1 þ c1 I1 a22 S2 ; I10 ¼ b1 S1 I1 ðl1 þ c1 b11 ÞI1 b22 I2 ; S20 ¼ r2 þ c2 I2 ðl2 c2 a22 ÞS2 b2 S2 I2 þ c2 I2 a11 S1 ;
ð3:6Þ
I20 ¼ b2 S2 I2 ðl2 þ c2 b22 ÞI2 b11 I1 : In this case ðS1 ; S2 Þ can be given explicitly as r2 a22 l2 r1 þ c2 r1 þ r1 a22 ; S1 ¼ l2 l1 þ l2 c1 þ l2 a11 þ c2 l1 c2 c1 c2 a11 þ l1 a22 c1 a22 l1 r2 þ c1 r2 þ a11 r2 þ a11 r1 S2 ¼ : l2 l1 þ l2 c1 þ l2 a11 þ c2 l1 c2 c1 c2 a11 þ l1 a22 c1 a22
ð3:7Þ
In the absence of population dispersal between two patches, i.e., a11 ¼ a22 ¼ b11 ¼ b22 ¼ 0, (3.6) becomes S10 ¼ r1 þ c1 I1 ðl1 c1 ÞS1 b1 S1 I1 þ c1 I1 ; I10 ¼ b1 S1 I1 ðl1 þ c1 ÞI1
ð3:8Þ
and S20 ¼ r2 þ c2 I2 ðl2 c2 ÞS2 b2 S2 I2 þ c2 I2 ; I20 ¼ b2 S2 I2 ðl2 þ c2 ÞI2 :
ð3:9Þ
Set R01 :¼
b1 r1 ; ðl1 c1 Þðl1 þ c1 Þ
ð3:10Þ
R02 :¼
b2 r2 : ðl2 c2 Þðl2 þ c2 Þ
ð3:11Þ
It is well known that R01 is a reproduction number of the disease in the first patch and R02 is a reproduction number of the disease in the second patch. Let ðf1 ; f2 Þ be the vector field defined by system (3.8). Then, for the Dulac function DðS1 ; I1 Þ :¼ S11I1 , there holds oðDf1 Þ oðDf2 Þ r1 c1 þ c1 þ ¼ 2 < 0: oS1 oI1 S1 I1 S12 Thus, (3.8) does not have a limit cycle. Then it is easy to see that the disease will disappear in the first patch if R01 < 1 and there is an endemic equilibrium in (3.8) which is globally asymptotically stable if R02 > 1. Similarly, the disease will disappear in the second patch if R02 < 1 and there is an
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endemic equilibrium in (3.9) which is globally asymptotically stable if R02 > 1. Based upon these results, we can present two examples that illustrate the effect of population dispersal on the spread of disease. Example 3.1. Suppose r1 ¼ r2 ¼ r, c1 ¼ c2 ¼ c, l1 ¼ l2 ¼ l, c1 ¼ c2 ¼ c, a11 ¼ a22 ¼ b11 ¼ b22 ¼ h in (3.6). This means that we assume that two patches have the same demographic structure and the same recovery rate for the disease. In this way, it is easier to find the effect of the variance of the contact rates in two patches and the population dispersal rate on the disease spread. Then, r . It follows from (3.4) that it is easy to obtain S1 ¼ S2 ¼ lc sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 rðb1 þ b2 Þ r2 ðb2 b1 Þ þ 4h2 : 2h 2ðl þ cÞ þ ð3:12Þ 2sðM1 Þ ¼ 2 lc ðl cÞ If the two patches are isolated, the discussions above show that the disease will be persistent in the first patch if rb1 >1 R01 ¼ ðl cÞðl þ cÞ and the disease will disappear in the first patch if R01 < 1. Further, the disease will be persistent in the second patch if rb2 >1 R02 ¼ ðl cÞðl þ cÞ and the disease will disappear in the second patch if R02 < 1. Assume that the disease spreads in each isolated patch, i.e., R0i > 1, i ¼ 1; 2. Notice that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 ðb2 b1 Þ2 þ 4h2 > 2h; if b1 6¼ b2 : 2 ðl cÞ By (3.12), we have sðM1 Þ > 0, and hence R0 > 1. It follows that the disease also spreads in both patches when population dispersal occurs. Now, we suppose that the disease dies out in each isolated patch, i.e., R0i < 1, i ¼ 1; 2. Note that 2 3 ðR01 R02 Þ2 6 7 2sðM1 Þ ¼ ðl þ cÞ4R01 þ R02 2 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 0 (i.e., R0 > 1) for all h > 0 if b1 P 5 (see Fig. 1), Thus, the disease spreads in both patches when population dispersal occurs. Since R01 P 5=3 and R02 ¼ 1=3, we see that the population dispersal intensifies the disease spread. If 3 < b1 < 5, We find that there is a h0 > 0 such that sðM1 Þ > 0 (i.e., R0 > 1) for all 0 < h < h0 and sðM1 Þ < 0 (i.e., R0 < 1) for all h > h0 (see Fig. 2). Hence, the disease will disappear in both patches if the population dispersal rate is large. Since 1 < R01 < 5=3 and R02 ¼ 1=3, we see that the population dispersal reduces the disease spread and is beneficial to disease control.
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Fig. 1. The curve of 2sðM1 Þ when b1 ¼ 6.
Fig. 2. The curve of 2sðM1 Þ when b1 ¼ 4.
Example 3.2. We consider the case where susceptible individuals and infective individuals in each patch have the same dispersal rate, but the population has different birth rates and different contact rates in different patches. Fix r1 ¼ 1, r2 ¼ 5, c1 ¼ c2 ¼ 1, l1 ¼ l2 ¼ 2, c1 ¼ c2 ¼ 0, a11 ¼ b11 ¼ 0:3, a22 ¼ b22 ¼ 0:3k, b1 ¼ 1:5, b2 ¼ 0:1 in (3.6), where k is a positive constant. In the absence of population dispersal, it is easy to see that the reproduction numbers in the two patches are R01 ¼ 0:75, R02 ¼ 0:25. Thus, the disease dies out in each patch when they are isolated. If the population dispersal occurs, by calculating the hi defined in (3.3) we have 14:9 þ 20:1k ðk þ 8:486078811Þðk þ 2:513921189Þ 0:9 ; 13:0 þ 3:0k 13:0 þ 3:0k 5:0 þ 9:0k 6:8 h2 ¼ 3:0 2:3 0:3k 2 0:09k: 13:0 þ 3:0k 13:0 þ 3:0k h1 ¼
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111
By numerical calculations, we obtain 8 > < h1 < 0 for all k > 0; h2 > 0 if k < 0:7138728627; > : h2 < 0 if k > 0:7138728627: It then follows that R0 > 1 when k > 0:7138728627. Thus, the disease will spread in the two patches if k > 0:7138728627, although the disease cannot spread in any patch when they are isolated.
4. Discussions In this paper, we have proposed an epidemic model in order to simulate the dynamics of disease transmission under the influence of a population dispersal among patches. The population dispersal among patches can be interpreted as the movement that people travel or migrate from one city to another city or from one country to another country. In order to be more realistic, We have incorporated more general demographic structure, proposed in [5], into the model and incorporated both the difference of demographic structure and disease transmission among different patches and the difference between the dispersal rates of susceptible individual and the dispersal rates of infective individuals, which simulates the process of disease control. We establish a threshold above which the disease is uniformly persistent and below which disease-free equilibrium is locally attractive, and globally attractive when both susceptible and infective individuals in each patch have the same dispersal rate. We have also applied our result to a specific demographic structure. For this special case, we have constructed two examples to show that population dispersal can both intensify and reduce the spread of disease in patches. In the first example, we consider a case where the disease spreads in one patch and cannot spread in the other patch when they are isolated. We have shown that the population dispersal leads to the disease spread in both patches if the reproduction number for one patch is large. We have also shown that the disease dies out in the two patches if the reproduction numbers for two patches are suitable and the population dispersal rate is strong. In the second example, we suppose that susceptible individuals and infective individuals in each patch have the same dispersal rate, but the population has different birth rates and different contact rates in different patches. We have shown that a population dispersal results in the spread of the disease in all patches, even though the disease cannot spread in each isolated patch. It will be interesting to consider the global stability of an endemic equilibrium of the model under the influence of population dispersal among patches. This seems to be a difficult problem since the dimension of the model is higher. We leave this for future investigations.
Acknowledgements We are very grateful to two referees for their valuable comments and suggestions which led to an improvement of our original manuscript.
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References [1] R.M. Anderson, R.M. May, Infectious Diseases of Humans, Dynamics and Control, Oxford University, Oxford, 1991. [2] F. Brauer, P. van den Driessche, Models for transmission of disease with immigration of infectives, Math. Biosci. 171 (2001) 143. [3] V. Capasso, in: Mathematical Structures of Epidemic Systems, Lecture Notes in Biomath., vol. 97, Springer, Heidelberg, 1993. [4] K. Cooke, P. van den Driessche, Analysis of an SEIRS epidemic model with two delays, J. Math. Biol. 35 (1996) 240. [5] 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. [6] O. Diekmann, J.A.P. Heesterbeek, J.A.J. Metz, On the definition and the computation of the basic reproduction ratio R0 in the models for infectious disease in heterogeneous populations, J. Math. Biol. 28 (1990) 365. [7] O. Diekmann, J.A.P. Heesterbeek, The basic reproduction ratio for sexually transmitted disease: I. theoretical considerations, Math. Biosci. 107 (1991) 325. [8] Z. Feng, H.R. Thieme, Endemic models with arbitrarily distributed periods of infection I: fundamental properties of the models, SIAM Appl. Math. 61 (2000) 803. [9] Z. Feng, H.R. Thieme, Endemic models with arbitrarily distributed periods of infection II: fast disease dynamics and permanent recovery, SIAM Appl. Math. 61 (2000) 983. [10] J.K. Hale, in: Asymptotic Behavior of Dissipative Systems, Math. Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. [11] H.W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci. 28 (1976) 335. [12] H.W. Hethcote, The mathematics of infectious diseases, SIAM Rev. 42 (2000) 599. [13] H.W. Hethcote, P. van den Driessche, Two SIS epidemiologic models with delays, J. Math. Biol. 40 (2000) 3. [14] W.M. Hirsch, H.L. Smith, X.-Q. Zhao, Chain transitivity attractivity and strong repellors for semidynamical systems, J. Dynam. Differ. Equat. 13 (2001) 107. [15] J.M. Hyman, J. Li, An intuitive formulation for the reproductive number for the spread of diseases in heterogeneous populations, Math. Biosci. 167 (2000) 65. [16] H.L. Smith, P. Waltman, The Theory of the Chemostat, Cambridge University, 1995. [17] H.R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal. 24 (1993) 407. [18] H.R. Thieme, Uniform persistence and permanence for non-autonomous semi-flows in population biology, Math. Biosci. 166 (2000) 173. [19] P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002) 29. [20] W. Wang, Z. Ma, Global dynamics of an epidemic model with time delay, Nonlinear Anal.: Real World Appl. 3 (2002) 365. [21] X.-Q. Zhao, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflows with applications, Can. Appl. Math. Quart. 3 (1995) 473. [22] X.-Q. Zhao, Z.-J. Jing, Global asymptotic behavior in some cooperative systems of functional differential equations, Can. Appl. Math. Quart. 4 (1996) 421. [23] X.-Q. Zhao, X. Zou, Threshold dynamics in a delayed SIS epidemic model, J. Math. Anal. Appl. 257 (2001) 282.
An epidemic model in a patchy environment Wendi Wang
a,1
, Xiao-Qiang Zhao
b,*,2
a
b
Department of Mathematics, Southwest Normal University Chongqing, 400715, People’s Republic of China Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, Canada NF A1C 5S7 Received 6 September 2001; received in revised form 9 August 2002; accepted 18 November 2002 Available online
Abstract An epidemic model is proposed to describe the dynamics of disease spread among patches due to population dispersal. We establish a threshold above which the disease is uniformly persistent and below which disease-free equilibrium is locally attractive, and globally attractive when both susceptible and infective individuals in each patch have the same dispersal rate. Two examples are given to illustrate that the population dispersal plays an important role for the disease spread. The first one shows that the population dispersal can intensify the disease spread if the reproduction number for one patch is large, and can reduce the disease spread if the reproduction numbers for all patches are suitable and the population dispersal rate is strong. The second example indicates that a population dispersal results in the spread of the disease in all patches, even though the disease can not spread in each isolated patch. Ó 2004 Published by Elsevier Inc. Keywords: Epidemic model; Population dispersal; Threshold dynamics
1. Introduction Many epidemic models have been proposed and studied to understand mechanism of disease transmission (see, for example, [1,3,12] and the references cited therein). One of the most
*
Corresponding author. Tel.: +1-709 737 8098; fax: +1-709 737 3010. E-mail addresses: [email protected] (W. Wang), [email protected] (X.-Q. Zhao). 1 Research supported by the Key Teachers Fund from the Minister of Education of China and the National Science Foundation of China. 2 Research supported by the NSERC of Canada. 0025-5564/$ - see front matter Ó 2004 Published by Elsevier Inc. doi:10.1016/j.mbs.2002.11.001
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important subjects in this field is to obtain a threshold that determines the persistence and extinction of a disease. Reproduction numbers for ordinary differential equation models were investigated by Diekmann et al. [6,7], Hyman and Li [15], and van den Driessche and Watmough [19]. Reproduction numbers for SIS epidemiological models with delays were studied by Cooke and van den Driessche [4], Hethcote and van den Driessche [13]. The thresholds for the uniform persistence and extinction of a disease in epidemic models with delays were also considered recently by Thieme [18], Zhao and Zou [23], Wang and Ma [20]. Reproduction number and persistence in an endemic model with many infection stages were proposed and studied by Feng and Thieme [8,9]. Communicable diseases such as influenza and sexual diseases can be easily transmitted from one country (or one region ) to other countries (or other regions). Thus, it is important to consider the effect of population dispersal on spread of a disease. Hethcote [11] proposed an epidemic model with population dispersal between two patches. Brauer and van den Driessche [2] proposed a model with immigration of infectives. In this paper, we consider a disease transmission model with population dispersal among n patches. We assume that demographic structure is described by the following equation: N 0 ¼ BðN ÞN lN ; where N is the number of a population, BðN Þ is the birth rate of the population, and l is its death rate. This type of demographic structure with variable population size was proposed by Cooke et al. [5]. We adopt it here as a basis to develop an epidemic model with population dispersal. We consider SIS type of disease transmission. The population is divided into two classes: susceptible individuals and infectious individuals. Susceptible individuals become infective after contact with infective individuals. Infective individuals return to susceptible class when they are recovered. Gonorrhea and other sexually transmitted diseases or bacterial infections exhibit this phenomenon. We denote the numbers of susceptible individuals at time t by SðtÞ and the numbers of infective individuals at time t by IðtÞ. If there is no population dispersal among patches, i.e., the patches are isolated, we suppose that the population dynamics in ith patch is governed by 0 Si ¼ Bi ðNi ÞNi li Si bi Si Ii þ ci Ii ; ð1:1Þ Ii0 ¼ bi Si Ii ðli þ ci ÞIi ; where Si is the number of susceptible individuals in patch i, Ii the number of infectious individuals in patch i, Ni ¼ Si þ Ii is the number of the population in patch i, Bi ðNi Þ is the birth rate of the population in the ith patch, bi is the disease transmission coefficient, li is the death rate of the population in the ith patch, and ci is the recovery rate of infective individuals in the ith patch. Here, we adopt mass action incidence rate for convenience. Our method also applies to standard incidence rate. Following [5], we assume that Bi ðNi Þ satisfy the following basic assumptions for Ni 2 ð0; 1Þ: (A1) Bi ðNi Þ > 0, i ¼ 1; 2; . . . ; n; (A2) Bi ðNi Þ is continuously differentiable with B0i ðNi Þ < 0, i ¼ 1; 2; . . . ; n; (A3) li > Bi ð1Þ, i ¼ 1; 2; . . . ; n.
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As mentioned in [5], the following three types of birth functions Bi ðNi Þ can be found in the biological literature: (B1) Bi ðNi Þ ¼ bi eai Ni with ai > 0, bi > 0; pi (B2) Bi ðNi Þ ¼ qi þN m with pi , qi , m > 0; i
(B3) Bi ðNi Þ ¼ NAii þ ci with Ai > 0, ci > 0. When the patches are connected, we suppose that the dynamics of those individuals is governed by the following model: 8 n P > > > Si0 ¼ Bi ðNi ÞNi li Si bi Si Ii þ ci Ii þ aij Sj ; 1 6 i 6 n; < j¼1 ð1:2Þ n P > 0 > > : Ii ¼ bi Si Ii ðli þ ci ÞIi þ bij Ij ; 1 6 i 6 n; j¼1
where aii , bii , 1 6 i 6 n, are non-positive constants, aij and bij with i 6¼ j are non-negative constants. aii P 0 represents the emigration rate of susceptible individuals in the ith patch, bii P 0 represents the emigration rate of infective individuals in the ith patch, aij , j 6¼ i, represents the immigration rate of susceptible individuals from jth patch to ith patch, and bij , j 6¼ i, the immigration rate of infective individuals from jth patch to ith patch. For simplicity, we neglect death rates and birth rates of the individuals during the dispersal process. Thus, we have n X j¼1
aji ¼ 0;
n X
bji ¼ 0;
8 1 6 i 6 n:
ð1:3Þ
j¼1
We further assume that the n patches cannot be separated into two groups such that there is no immigration of susceptible and infective individuals from first group to second group. Mathematically, this means that two n n matrices ðaij Þ and ðbij Þ are irreducible (see, e.g., [16, Appendix A]). Note that system (1.2) indicates that the population can have different demographic structures and different infection forces among different patches, aij 6¼ bij implies that we also consider the variation of the dispersal rates of susceptible individuals and infective individuals. The remaining parts of this paper is organized as follows. In the next section, we establish a threshold between the extinction and persistence of the disease. In Section 3, the results are applied to the model with the patch number being 2. Section 4 gives a brief discussion of main results.
2. Threshold dynamics In order to find the disease-free equilibrium of (1.2), we consider Si0 ¼ Bi ðSi ÞSi li Si þ
n X j¼1
aij Sj ;
i ¼ 1; . . . ; n:
ð2:1Þ
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If z is a non-negative constant, we define an auxiliary matrix 2 3 B1 ðzÞ l1 þ a11 a12 a1n 6 7 a21 B2 ðzÞ l2 þ a22 a2n 7: MðzÞ ¼ 6 4 5 ... ... ... ... an2 Bn ðzÞ ln þ ann an1 This matrix is used to prove the existence and the uniqueness of a positive equilibrium in (2.1) and is different from the standard Jacobian matrix. Recall that the stability modulus of an n n matrix M, denoted by sðMÞ, is defined by sðMÞ :¼ maxfRe k : k is an eigenvalue of Mg: Note that MðzÞ is irreducible and has non-negative off-diagonal elements. By [16, Theorem A.5], it then follows that sðMðzÞÞ is a simple eigenvalue of MðzÞ with a (componentwise) positive eigenvector. Let F : Rnþ ! Rn be defined by the right-hand side of (2.1). Clearly, F is cooperative and DF ðSÞ is irreducible for every S 2 Rnþ . For any a 2 ð0; 1Þ and S ¼ ðS1 ; . . . ; Sn Þ 2 intðRnþ Þ, there holds " # n n X X aij asj > a Bi ðSi ÞSi li Si þ aij Sj ; i ¼ 1; 2; . . . ; n: aSi Bi ðaSi Þ li aSi þ j¼1
j¼1
Thus F is strongly sublinear on Rnþ (see, e.g., [22]). It then follows that for any S0 2 Rnþ , the unique solution Sðt; S0 Þ of (2.1) satisfying Sð0; S0 Þ ¼ S0 exists globally on ½0; 1Þ and Sðt; S0 Þ P 0, 8t P 0. We further claim that (2.1) admits a bounded positive solution. Indeed, in view of (A3), we can choose a sufficiently large real number K > 0 such that Bi ðKÞ < li , i ¼ 1; 2; . . . ; n. Let v ¼ ðv1 ; . . . ; vn Þ be a positive eigenvector associated with sðMðKÞÞ. Then V ðtÞ ¼ ðV1 ðtÞ; . . . ; Vn ðtÞÞ ¼ 0 vesðMðKÞÞtPis a positive solution Pn of the linear ordinary differential system V ¼ MðKÞV . Let n sðMðKÞÞt vn . By the first equation in (1.3), it easily follows that RðtÞ ¼ i¼1 Vi ðtÞ ¼ e i¼1 R0 ðtÞ 6 aRðtÞ, 8t P 0, where a ¼ maxfBi ðKÞ li : 1 6 i 6 ng < 0. Thus limt!1 RðtÞ ¼ 0, and hence sðMðKÞÞ < 0. Choose l > 0 large enough such that lvi > K, i ¼ 1; 2; . . . ; n. Set xðtÞ lv. If we rewrite (2.1) as S 0 ¼ F ðSÞ, it is easy to see that x0 ðtÞ ¼ 0 > sðMðKÞÞxðtÞ ¼ MðKÞxðtÞ > F ðxðtÞÞ;
8t P 0;
ð2:2Þ
where (A2) is used. By the standard comparison theorem(see, e.g., [16, Theorem B.1]), it follows that: 0 < Sðt; lvÞ 6 xðtÞ ¼ lv;
8t P 0:
Consequently, Sðt; lvÞ is a bounded positive solution of (2.1). In order for (2.1) to admit a positive equilibrium, we need to assume that ðA4Þ
sðMð0ÞÞ > 0:
By [22, Corollary 3.2], it then follows that (2.1) has a unique positive equilibrium S ¼ ðS1 ; S2 ; . . . ; Sn Þ and S is globally asymptotically stable for S 2 Rnþ n f0g. Thus, E0 ¼ ðS1 ; S2 ; . . . ; Sn ; 0; . . . ; 0Þ is a disease-free equilibrium of (1.2).
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101
Define 2
b1 S1 l1 c1 þ b11 6 b21 M1 ¼ 6 4 bn1
b12 b2 S2 l2 c2 þ b22 bn2
3 b1n 7 b2n 7: 5 bn Sn ln cn þ bnn
Clearly, M1 is irreducible and has non-negative off-diagonal elements. Then sðM1 Þ is a simple eigenvalue of M1 with a positive eigenvector (see, e.g., [16, Theorem A.5]). According to the concepts of next generation matrix and reproduction number presented in [6,19], we define 3 2 b1 S1 0 0 6 0 0 7 b2 S2 7 F :¼ 6 4 5 0 0 bn Sn and 2
l1 c1 þ b11 6 b21 V :¼ 6 4 bn1
b12 l2 c2 þ b22 bn2
3 b1n 7 b2n 7: 5 ln cn þ bnn
Set R0 :¼ qðFV1 Þ, where q represents the spectral radius of the a matrix. Then R0 is called the reproduction number for (1.2). Note that M1 ¼ F V. Thus, the following observation is implied by the proof of [19, Theorem 2] with J1 ¼ M1 . Lemma 2.1. There hold two equivalences: R0 > 1 () sðM1 Þ > 0;
R0 < I () sðM1 Þ < 0:
ð2:3Þ
By [19, Theorem 2], the disease-free equilibrium E0 is locally asymptotically stable if R0 < 1 and is unstable if R0 > 1. We will show that R0 is a threshold parameter for the uniform persistence and extinction of the disease, which is more general. To investigate the global dynamics of (1.2), we first show that (1.2) admits a compact, positively invariant set which absorbs all forward orbits in R2n þ , and hence (1.2) has a global compact (see, e.g., [10, Theorem 3.4.8]). For convenience, we denote the positive solution attractor on R2n þ ðS1 ðtÞ; . . . ; Sn ðtÞ; I1 ðtÞ; . . . ; In ðtÞÞ of (1.2) by ðSðtÞ; IðtÞÞ. Lemma 2.2. Let (A1)–(A4) hold. Then there isPan N > 0 such that every forward orbit in R2n þ of n : ðS þ I Þ 6 N g, and G is positively invariant for (1.2) eventually enters into G :¼ fðS; IÞ 2 R2n i i þ i¼1 (1.2).
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P Proof. Let N ¼ ni¼1 Ni , Ni ¼ Si þ Ii . By (1.2) and (1.3), we have n X 0 ðBi ðNi Þ li ÞNi : N ¼
ð2:4Þ
i¼1
If Bi ð0þÞ < li , i ¼ 1; 2; . . . ; n, then there exists an a > 0 such that N 0 ðtÞ 6 aN ðtÞ, 8t P 0, and hence, Lemma 2.2 holds for any positive number N . Otherwise, we partition f1; 2; . . . ; ng into two sets P1 and P2 such that Bi ð0þÞ > li ;
8i 2 P1 ;
Bi ð0þÞ 6 li ;
8i 2 P2 :
Without loss of generality, we suppose that P1 ¼ f1; . . . ; mg and P2 ¼ fm þ 1; . . . ; ng. For i 2 P1 , since Bi ð0þÞ > li and Bi ð1Þ < li , (A2) implies that there is a unique ki > 0 such that Bi ðki Þ li ¼ 0. It follows from (A3) that there is an N0 > 0 such that m X kj Bj ð0þÞ 1; 8N P N0 ; i ¼ 1; 2; . . . ; n: ðBi ðN Þ li ÞN < j¼1
Let N ¼ nN0 . By the definition of N , it is easy to see that N P N implies Ni0 > N0 for some 1 6 i0 6 n. It then follows from (2.4) that m X 0 Bj ð0þÞkj þ ðBi0 ðNi0 Þ li0 ÞNi0 < 1; if NðtÞ P N ; N ðtÞ 6 j¼1
which implies that G is positively invariant and every forward orbit enters into G after a certain time. h In the case where the susceptible and infective individuals in each patch have the same dispersal rate, we have the following result on the global attractivity of E0 . Theorem 2.1. Let (A1)–(A4) hold and R0 < 1. If aij ¼ bij for i ¼ 1; . . . ; n, j ¼ 1; . . . ; n, then E0 is globally attractive for ðS0 ; I0 Þ 2 ðRnþ n f0gÞ Rnþ . Proof. Let us consider a non-negative solution ðSðtÞ; IðtÞÞ of (1.2). We wish to show that lim IðtÞ ¼ 0: t!1
ð2:5Þ
By (1.2), we have Ni0 ¼ Bi ðNi ÞNi li Ni þ
n X
aij Nj ;
i ¼ 1; . . . ; n:
ð2:6Þ
j¼1
By the afore-mentioned conclusion for (2.1), (2.6) admits a unique positive equilibrium S which is globally asymptotically stable for N 2 Rnþ n f0g. It then follows that for any > 0, there holds NðtÞ ¼ SðtÞ þ IðtÞ < S þ , where ¼ ð; . . . ; Þ > 0, when t is sufficiently large. Thus, if t is sufficiently large, we have n X bij Ij ; 81 6 i 6 n: ð2:7Þ Ii0 < bi ðSi þ ÞIi ðli þ ci ÞIi þ j¼1
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103
It then suffices to show that positive solutions of the following auxiliary system: n X bij Ij ; 1 6 i 6 n Ii0 ¼ bi ðSi þ ÞIi ðli þ ci ÞIi þ
ð2:8Þ
j¼1
tend to zero as t goes to infinity. Let M2 be the matrix defined by M2 ¼ diagðb1 ; b2 ; . . . ; bn Þ: Since sðM1 Þ < 0 and sðM1 þ M2 Þ is continuous for small , we can fix an > 0 small enough such that sðM1 þ M2 Þ < 0. As a consequence, solutions of (2.8) tend to zero as t goes to infinity, which implies (2.5). For any ðS0 ; I0 Þ 2 Rnþ Rnþ with S0 6¼ 0, we have N0 ¼ S0 þ I0 2 Rnþ n f0g. By the global attractivity of S for system (2.6), it then follows that limt!1 SðtÞ ¼ limt!1 ðN ðtÞ IðtÞÞ ¼ S 0 ¼ S. h If the susceptible and infective individuals have different dispersal rates, by using the arguments in obtaining the unique positive equilibrium of (2.1) or by [19, Theorem 2], we see that (Al)–(A4) and R0 < 1 imply that the disease-free equilibrium is locally asymptotically stable. This means that a positive solution (SðtÞ, IðtÞ of (1.2) satisfies IðtÞ ! 0 as t ! 1 if its initial position is near to the disease-free equilibrium. Here, we give a little more general result where it is only needed that the values of infective individuals are small. Theorem 2.2. Let (A1)–(A4) hold and R0 < 1. Then there exists d > 0 such that for every ðSð0Þ; Ið0ÞÞ 2 G with Ii ð0Þ < d, i ¼ 1; 2; . . . ; n, the solution ðSðtÞ; IðtÞÞ of (1.2) satisfies limt!1 ðS 0 ðtÞ; IðtÞÞ ¼ ðS ; 0Þ. Proof. Let us consider an auxiliary system Si0 ¼ Bi ðSi ÞSi li Si þ ðBi ð0þÞ þ ci Þ þ
n X
aij Sj ;
i ¼ 1; . . . ; n;
ð2:9Þ
j¼1
where > 0 is a small constant to be determined. By (A4) and the previous analysis of system (2.1), we can restrict small enough such that (2.9) admits a unique positive equilibrium S ðÞ which is globally asymptotically stable. Let Sðt; N Þ be the solution of (2.9) through ðN ; . . . ; N Þ at t ¼ 0. Select T ðÞ > 0 such that Sðt; N Þ < S ðÞ þ ; 8t P T ðÞ; where ¼ ð; . . . ; Þ. Define a matrix M1 ðÞ by 2 b12 6 b1 S1 ðÞ l1 c1 þ b11 6 b21 b2 S2 ðÞ l2 c2 þ b22 6 6 .. . .. 4 . bn1 bn2
.. . .. . .. . .. .
b1n b2n .. .
3 7 7 7: 7 5
bn Sn ðÞ ln cn þ bnn
Since M1 ð0Þ ¼ M1 and sðM1 ðÞ þ M2 Þ is continuous for small , we can now restrict small enough such that sðM1 ðÞ þ M2 Þ < 0. Let v ¼ ðv1 ; . . . ; vn Þ be a positive eigenvector associated with sðM1 ðÞ þ M2 Þ. Choose k > 0 small enough such that kv < .
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Now, we define another auxiliary system Ii0 ¼ bi N Ii ðli þ ci ÞIi þ
n X
i ¼ 1; . . . ; n:
bij Ij ;
ð2:10Þ
j¼1
Let Iðt; dÞ be the solution of (2.10) through ðd; . . . ; dÞ at t ¼ 0. We restrict d > 0 small enough such that Iðt; dÞ < kv;
8t 2 ½0; T ðÞ:
ð2:11Þ
Let ðSðtÞ; IðtÞÞ be a non-negative solution of (1.2) with ðSð0Þ; Ið0ÞÞ 2 G and Ii ð0Þ < d, i ¼ 1; 2; . . . ; n. We claim that IðtÞ 6 kv, 8t P 0. Suppose not. By the comparison principle and (2.11), there exist a q, 1 6 q 6 n, and a T1 > T ðÞ such that IðtÞ 6 kv Iq ðT1 Þ ¼ kvq ;
Iq ðtÞ > kvq
for 0 6 t 6 T1 ; for 0 < t T1 1:
ð2:12Þ
Notice that for 0 6 t 6 T1 , we have Si0 < Bi ðSi ÞSi li Si þ ðBð0þÞ þ ci Þ þ
n X
aij Sj ;
i ¼ 1; . . . ; n:
ð2:13Þ
j¼1
It follows from the comparison principle that SðT1 Þ < S ðÞ þ . Hence, for 0 6 t T1 1, we have n X bij Ij ; i ¼ 1; . . . ; n: Ii0 < bi ðSi ðÞ þ ÞIi ðli þ ci ÞIi þ j¼1
Since IðT1 Þ 6 kv, the comparison principle implies that IðtÞ < kvesðM1 ðÞþM2 ÞðtT1 Þ
for 0 6 t T1 1
and hence, Iq ðtÞ < kvq esðM1 ðÞþM2 ÞðtT1 Þ < kvq
for 0 < t T1 1;
which contradicts to (2.12). This proves the claim. Now, (2.13) holds for all t P 0 due to the claim, and hence, the comparison principle implies that SðtÞ < S ðÞ þ , 8t P T ðÞ. By a similar argument as above, it then follows that IðtÞ < kvesðM1 ðÞþM2 ÞðtT ðÞ ;
8t > T ðÞ:
Consequently, IðtÞ ! 0 as t ! 1. Let UðtÞ : Rnþ ! Rnþ be the solution semiflow of (1.2), that is, UðtÞðS0 ; I0 Þ ¼ ðSðtÞ; IðtÞÞ is the solution of (1.2) with ðSð0Þ; Ið0ÞÞ ¼ ðS0 ; I0 Þ. Given ðS0 ; I0 Þ 2 G with S0 6¼ 0 and I0i < d, i ¼ 1; . . . ; n, it easily follows that SðtÞ 2 intðRnþ Þ, 8t > 0. Let x ¼ xðS0 ; I0 Þ be the omega limit set of UðtÞðS0 ; I0 Þ. f0g. We claim that x 6¼ f0g. Assume that, by Since IðtÞ ! 0 as t ! 1, there holds x ¼ x ¼ f0g. Then limt!1 ðSðtÞ; IðtÞÞ ¼ ð0; 0Þ. By assumption (A4), we can choose a contradiction, x small g > 0 such that sðMð0Þ gEÞ > 0, where E ¼ diagð1; . . . ; 1Þ. It follows that there exists a t > 0 such that Bi ðNi ðtÞÞ bi Ii ðtÞ P Bi ð0þÞ g for 8t P t, i ¼ 1; . . . ; n. Then SðtÞ ¼ ðS1 ðtÞ; . . . ; Sn ðtÞÞ satisfies
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Si0 ðtÞ > ðBi ð0þÞ gÞSi li Si þ
n X
aij Sj ;
8t P t;
i ¼ 1; . . . ; n:
105
ð2:14Þ
j¼1
Let w ¼ ðw1 ; . . . ; wn Þ be a positive eigenvector of Mð0Þ gE associated with the eigenvalue sðMð0Þ gEÞ. Choose a small number a a > 0 such that SðtÞ > aw. Then the comparison theorem implies that SðtÞ P awesðMð0ÞgEÞðttÞ ;
8t P t
and hence limt!1 Si ðtÞ ¼ 1, i ¼ 1; 2; . . . ; n, a contradiction. It is easy to see that UðtÞjx ðS; 0Þ ¼ ðU1 ðtÞS; 0Þ; where U1 ðtÞ is the solution semiflow of system (2.1). By [14, Lemma 2.10 ], x is an internal chain 6¼ f0g and is an internal chain transitive set for U1 ðtÞ. Since x transitive set for UðtÞ, and hence, x \ W s ðS Þ 6¼ ;. By [14, Theorem S is globally asymptotically stable for (2.1) in Rnþ n f0g, we have x ¼ S . This proves x ¼ fðS ; 0Þg. Consequently, 3.1 and Remark 4.6], we then get x ðSðtÞ; IðtÞÞ ! ðS ; 0Þ as t ! 1. h It is easy to see that the eigenvalues of M1 are also the eigenvalues of the Jacobian matrix of (1.2) at E0 . It follows that E0 is unstable if R0 > 1. The following result shows that R0 > 1 actually implies that model (1.2) admits at least one endemic equilibrium and the disease is uniformly persistent. Theorem 2.3. Let (A1)–(A4) hold and R0 > 1. Then (1.2) admits at least one (componentwise) positive equilibrium, and there is a positive constant such that every solution ðSðtÞ; IðtÞÞ of (1.2) with ðSð0Þ; Ið0ÞÞ 2 Rnþ intðRnþ Þ satisfies lim inf Ii ðtÞ P ; t!1
i ¼ 1; 2; . . . ; n:
Proof. Define X ¼ fðS1 ; . . . ; Sn ; I1 ; . . . ; In Þ : Si P 0; Ii P 0; i ¼ 1; 2; . . . ; ng; X0 ¼ fðS1 ; . . . ; Sn ; I1 ; . . . ; In Þ 2 X : Ii > 0; i ¼ 1; 2; . . . ; ng; oX0 ¼ X n X0 : It then suffices to show that (1.2) is uniformly persistent with respect to ðX0 ; oX0 Þ. First, by the form of (1.2), it is easy to see that both X and X0 are positively invariant. Clearly, oX0 is relatively closed in X . Furthermore, system (1.2) is point dissipative(see Lemma 2.2). Set Mo ¼ fðSð0Þ; Ið0ÞÞ : ðSðtÞ; IðtÞÞ satisfiesð1:2Þ and ðSðtÞ; IðtÞÞ 2 oX0 ; 8t P 0g: We now show that Mo ¼ fðS; 0Þ : S P 0g:
ð2:15Þ
Assume ðSð0Þ; Ið0ÞÞ 2 Mo . It suffices to show that IðtÞ ¼ 0, 8t P 0. Suppose not. Then there exist an i0 , 1 6 i0 6 n, and a t0 P 0 such that Ii0 ðt0 Þ > 0. We partition f1; 2; . . . ; ng into two sets Q1 and Q2 such that
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Ii ðt0 Þ ¼ 0; Ii ðt0 Þ > 0;
8i 2 Q1 ; 8i 2 Q2 :
Q1 is non-empty due to the definition of Mo . Q2 is non-empty since Ii0 ðt0 Þ > 0. For any j 2 Q1 , we have Ii0 ðt0 Þ P bji0 Ii0 ðt0 Þ > 0: It follows that there is an 0 > 0 such that Ij ðtÞ > 0, j 2 Q1 for t0 < t < t0 þ 0 . Clearly, we can restrict 0 > 0 small enough such that Ii ðtÞ > 0, i 2 Q2 for t0 < t < t0 þ 0 . This means that ðSðtÞ; IðtÞÞ 62 oX0 for t0 < t < t0 þ 0 , which contradicts the assumption that ðSð0Þ; Ið0ÞÞ 2 Mo . This proves (2.15). It is clear that there are two equilibria (0; 0) and (S ; 0) in Mo . Choose g > 0 small enough such that sðM1 gM2 Þ > 0. Let us consider a perturbed system of (2.1) n X Si0 ¼ Bi ðSi þ 1 ÞSi ðli þ bi 1 ÞSi þ aij Sj ; i ¼ 1; . . . ; n: ð2:16Þ j¼1
First, as in our previous analysis of system (2.1), we can restrict 1 > 0 small enough such that (2.16) admits a unique positive equilibrium S ð1 Þ which is globally asymptotically stable. By the implicit function theorem, it follows that S ð1 Þ is continuous in 1 . Thus, we can further restrict 1 small enough such that S ð1 Þ > S g. Let us consider an arbitrary positive solution ðSðtÞ; IðtÞÞ of (1.2). We now claim that lim sup maxfIi ðtÞg > 1 : t!1
i
Suppose, for the sake of contradiction, that there is a T > 0 such that Ii ðtÞ 6 1 , i ¼ 1; 2; . . . ; n, for all t P T . Then for t P T , we have N X 0 aij Sj ; i ¼ 1; . . . ; n: ð2:17Þ Si P Bi ðSi þ 1 ÞSi ðli þ Bi 1 ÞSi þ j¼1
Since the equilibrium S ð1 Þ of (2.16) is globally asymptotically stable and S ð1 Þ > S g, there is a T1 > 0 such that SðtÞ P S g for t P T þ T1 . As a consequence, for t > T þ T1 , there holds n X Ii0 P bi ðSi gÞIi ðli þ ci ÞIi þ bij Ij ; i ¼ 1; . . . ; n: ð2:18Þ j¼1
Since the matrix M1 gM2 has a positive eigenvalue sðM1 gM2 Þ with a positive eigenvector, it is easy to see that Ii ðtÞ ! 1 as t ! 1, i ¼ 1; 2; . . . ; n, which leads to a contradiction. Note that S is globally asymptotically stable in Rnþ n f0g for system (2.1). By the afore-mentioned claim, it then follows that ð0; 0Þ and E0 are isolated invariant sets in X , W s ðð0; 0ÞÞ \ X0 ¼ ;, and W s ðE0 Þ \ X0 ¼ ;. Clearly, every orbit in Mo converges to either ð0; 0Þ or E0 , and ð0; 0Þ and E0 are acyclic in Mo . By [17, Theorem 4.6] (see also [14, Theorem 4.3 and Remark 4.3], for a stronger repelling property of oX0 ), we conclude that system (1.2) is uniformly persistent with respect to IÞ 2 X0 . Then S 2 Rn and ðX0 ; oX0 Þ. By [21, Theorem 2.4], system (1.2) has an equilibrium ðS; þ I 2 intðRn Þ. We further claim that S 2 Rn n f0g. Suppose that S ¼ 0. By the second equation in þ þ P (1.3), we then get 0 ¼ ni¼1 ðli þ ci ÞIi , and hence Ii ¼ 0, i ¼ 1; 2; . . . ; n, a contradiction. By the first equation in (1.2) and the irreducibility of the cooperative matrix ðaij Þ, it follows that IÞ 2 intðRn Þ, 8t > 0. Then ðS; IÞ is a componentwise positive equilibrium of (1.2). h S ¼ Sðt; S; þ
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107
3. Application to two patches In the case where the patch number n is 2, the assumption (1.3) is equivalent to that a12 ¼ a22 , a21 ¼ a11 , b12 ¼ b22 , b21 ¼ b11 , and hence, (1.2) reduces to S10 ¼ B1 ðN1 ÞN1 ðl1 a11 ÞS1 b1 S1 I1 þ c1 I1 a22 S2 ; I10 ¼ b1 S1 I1 ðl1 þ c1 b11 ÞI1 b22 I2 ; S20 ¼ B2 ðN2 ÞN2 ðl2 a22 ÞS2 b2 S2 I2 þ c2 I2 a11 S1 ;
ð3:1Þ
I20 ¼ b2 S2 I2 ðl2 þ c2 b22 ÞI2 b11 I1 ; where aii and bii , 1 6 i 6 2, are negative constants. Assume that ðA5Þ
Bi ð0þÞ > li ;
81 6 i 6 2:
It then follows that sðMð0ÞÞ > 0. Let ðS1 ; S2 Þ be the positive solution of the following system: B1 ðS1 ÞS1 ðl1 a11 ÞS1 a22 S2 ¼ 0; B2 ðS2 ÞS2 ðl2 a22 ÞS2 a11 S1 ¼ 0: Now M1 becomes b1 S1 þ b11 l1 c1 b11
ð3:2Þ
b22 : b2 S2 þ b22 l2 c2
Its characteristic equation is k2 h1 k þ h2 ¼ 0 where h1 ¼ b1 S1 þ b2 S2 þ b11 þ b22 c1 c2 l1 l2 ; h2 ¼ b1 S1 b2 S2 b11 b22 b1 S1 ðl2 þ c2 b22 Þ
ð3:3Þ
b2 S2 ðl1 þ c1 b11 Þ þ ðl1 þ c1 b11 Þðl2 þ c2 b22 Þ: It is easy to see that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h1 þ h21 4h2 : sðM1 Þ ¼ 2 Then R0 > 1 if and only if h1 P 0 or 8 < h1 < 0; b1 S1 b2 S2 b11 b22 b1 S1 b2 S2 :1 þ þ : < ðl1 þ c1 b11 Þðl2 þ c2 b22 Þ l1 þ c1 b11 l2 þ c2 b22
ð3:4Þ
ð3:5Þ
h1 P 0 means that the total infection force of two patches overpasses the sum of recovery rates, b S1 can be viewed as a death rates and emigration rates of infectives in the two patches. l1 þc11 b 11 b1 S1 reproduction number at the first patch, and l1 þc1 b11 a reproduction number at the second patch. b1 S1 b2 S2 b11 b22 represents the quantity that new infectives minus moving infectives during the ðl1 þc1 b11 Þðl2 þc2 b22 Þ interval of product of infection periods in two patches. Thus, the second inequality of (3.5) can be interpreted as the sum of reproduction numbers in the two patches is stronger than 1 plus the net
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value that the numbers of new infectives minus moving infectives during the interval of product of infection periods in two patches. The conditions clearly show the effect of population dispersal on the spread of the disease. In order to be specific, we choose the birth rates Bi of the population as ri Bi ðNi Þ ¼ þ ci ; ci < li ; 1 6 i 6 2: Ni Then (3.1) reduces to S10 ¼ r1 þ c1 I1 ðl1 c1 a11 ÞS1 b1 S1 I1 þ c1 I1 a22 S2 ; I10 ¼ b1 S1 I1 ðl1 þ c1 b11 ÞI1 b22 I2 ; S20 ¼ r2 þ c2 I2 ðl2 c2 a22 ÞS2 b2 S2 I2 þ c2 I2 a11 S1 ;
ð3:6Þ
I20 ¼ b2 S2 I2 ðl2 þ c2 b22 ÞI2 b11 I1 : In this case ðS1 ; S2 Þ can be given explicitly as r2 a22 l2 r1 þ c2 r1 þ r1 a22 ; S1 ¼ l2 l1 þ l2 c1 þ l2 a11 þ c2 l1 c2 c1 c2 a11 þ l1 a22 c1 a22 l1 r2 þ c1 r2 þ a11 r2 þ a11 r1 S2 ¼ : l2 l1 þ l2 c1 þ l2 a11 þ c2 l1 c2 c1 c2 a11 þ l1 a22 c1 a22
ð3:7Þ
In the absence of population dispersal between two patches, i.e., a11 ¼ a22 ¼ b11 ¼ b22 ¼ 0, (3.6) becomes S10 ¼ r1 þ c1 I1 ðl1 c1 ÞS1 b1 S1 I1 þ c1 I1 ; I10 ¼ b1 S1 I1 ðl1 þ c1 ÞI1
ð3:8Þ
and S20 ¼ r2 þ c2 I2 ðl2 c2 ÞS2 b2 S2 I2 þ c2 I2 ; I20 ¼ b2 S2 I2 ðl2 þ c2 ÞI2 :
ð3:9Þ
Set R01 :¼
b1 r1 ; ðl1 c1 Þðl1 þ c1 Þ
ð3:10Þ
R02 :¼
b2 r2 : ðl2 c2 Þðl2 þ c2 Þ
ð3:11Þ
It is well known that R01 is a reproduction number of the disease in the first patch and R02 is a reproduction number of the disease in the second patch. Let ðf1 ; f2 Þ be the vector field defined by system (3.8). Then, for the Dulac function DðS1 ; I1 Þ :¼ S11I1 , there holds oðDf1 Þ oðDf2 Þ r1 c1 þ c1 þ ¼ 2 < 0: oS1 oI1 S1 I1 S12 Thus, (3.8) does not have a limit cycle. Then it is easy to see that the disease will disappear in the first patch if R01 < 1 and there is an endemic equilibrium in (3.8) which is globally asymptotically stable if R02 > 1. Similarly, the disease will disappear in the second patch if R02 < 1 and there is an
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endemic equilibrium in (3.9) which is globally asymptotically stable if R02 > 1. Based upon these results, we can present two examples that illustrate the effect of population dispersal on the spread of disease. Example 3.1. Suppose r1 ¼ r2 ¼ r, c1 ¼ c2 ¼ c, l1 ¼ l2 ¼ l, c1 ¼ c2 ¼ c, a11 ¼ a22 ¼ b11 ¼ b22 ¼ h in (3.6). This means that we assume that two patches have the same demographic structure and the same recovery rate for the disease. In this way, it is easier to find the effect of the variance of the contact rates in two patches and the population dispersal rate on the disease spread. Then, r . It follows from (3.4) that it is easy to obtain S1 ¼ S2 ¼ lc sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 rðb1 þ b2 Þ r2 ðb2 b1 Þ þ 4h2 : 2h 2ðl þ cÞ þ ð3:12Þ 2sðM1 Þ ¼ 2 lc ðl cÞ If the two patches are isolated, the discussions above show that the disease will be persistent in the first patch if rb1 >1 R01 ¼ ðl cÞðl þ cÞ and the disease will disappear in the first patch if R01 < 1. Further, the disease will be persistent in the second patch if rb2 >1 R02 ¼ ðl cÞðl þ cÞ and the disease will disappear in the second patch if R02 < 1. Assume that the disease spreads in each isolated patch, i.e., R0i > 1, i ¼ 1; 2. Notice that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 ðb2 b1 Þ2 þ 4h2 > 2h; if b1 6¼ b2 : 2 ðl cÞ By (3.12), we have sðM1 Þ > 0, and hence R0 > 1. It follows that the disease also spreads in both patches when population dispersal occurs. Now, we suppose that the disease dies out in each isolated patch, i.e., R0i < 1, i ¼ 1; 2. Note that 2 3 ðR01 R02 Þ2 6 7 2sðM1 Þ ¼ ðl þ cÞ4R01 þ R02 2 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 0 (i.e., R0 > 1) for all h > 0 if b1 P 5 (see Fig. 1), Thus, the disease spreads in both patches when population dispersal occurs. Since R01 P 5=3 and R02 ¼ 1=3, we see that the population dispersal intensifies the disease spread. If 3 < b1 < 5, We find that there is a h0 > 0 such that sðM1 Þ > 0 (i.e., R0 > 1) for all 0 < h < h0 and sðM1 Þ < 0 (i.e., R0 < 1) for all h > h0 (see Fig. 2). Hence, the disease will disappear in both patches if the population dispersal rate is large. Since 1 < R01 < 5=3 and R02 ¼ 1=3, we see that the population dispersal reduces the disease spread and is beneficial to disease control.
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Fig. 1. The curve of 2sðM1 Þ when b1 ¼ 6.
Fig. 2. The curve of 2sðM1 Þ when b1 ¼ 4.
Example 3.2. We consider the case where susceptible individuals and infective individuals in each patch have the same dispersal rate, but the population has different birth rates and different contact rates in different patches. Fix r1 ¼ 1, r2 ¼ 5, c1 ¼ c2 ¼ 1, l1 ¼ l2 ¼ 2, c1 ¼ c2 ¼ 0, a11 ¼ b11 ¼ 0:3, a22 ¼ b22 ¼ 0:3k, b1 ¼ 1:5, b2 ¼ 0:1 in (3.6), where k is a positive constant. In the absence of population dispersal, it is easy to see that the reproduction numbers in the two patches are R01 ¼ 0:75, R02 ¼ 0:25. Thus, the disease dies out in each patch when they are isolated. If the population dispersal occurs, by calculating the hi defined in (3.3) we have 14:9 þ 20:1k ðk þ 8:486078811Þðk þ 2:513921189Þ 0:9 ; 13:0 þ 3:0k 13:0 þ 3:0k 5:0 þ 9:0k 6:8 h2 ¼ 3:0 2:3 0:3k 2 0:09k: 13:0 þ 3:0k 13:0 þ 3:0k h1 ¼
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By numerical calculations, we obtain 8 > < h1 < 0 for all k > 0; h2 > 0 if k < 0:7138728627; > : h2 < 0 if k > 0:7138728627: It then follows that R0 > 1 when k > 0:7138728627. Thus, the disease will spread in the two patches if k > 0:7138728627, although the disease cannot spread in any patch when they are isolated.
4. Discussions In this paper, we have proposed an epidemic model in order to simulate the dynamics of disease transmission under the influence of a population dispersal among patches. The population dispersal among patches can be interpreted as the movement that people travel or migrate from one city to another city or from one country to another country. In order to be more realistic, We have incorporated more general demographic structure, proposed in [5], into the model and incorporated both the difference of demographic structure and disease transmission among different patches and the difference between the dispersal rates of susceptible individual and the dispersal rates of infective individuals, which simulates the process of disease control. We establish a threshold above which the disease is uniformly persistent and below which disease-free equilibrium is locally attractive, and globally attractive when both susceptible and infective individuals in each patch have the same dispersal rate. We have also applied our result to a specific demographic structure. For this special case, we have constructed two examples to show that population dispersal can both intensify and reduce the spread of disease in patches. In the first example, we consider a case where the disease spreads in one patch and cannot spread in the other patch when they are isolated. We have shown that the population dispersal leads to the disease spread in both patches if the reproduction number for one patch is large. We have also shown that the disease dies out in the two patches if the reproduction numbers for two patches are suitable and the population dispersal rate is strong. In the second example, we suppose that susceptible individuals and infective individuals in each patch have the same dispersal rate, but the population has different birth rates and different contact rates in different patches. We have shown that a population dispersal results in the spread of the disease in all patches, even though the disease cannot spread in each isolated patch. It will be interesting to consider the global stability of an endemic equilibrium of the model under the influence of population dispersal among patches. This seems to be a difficult problem since the dimension of the model is higher. We leave this for future investigations.
Acknowledgements We are very grateful to two referees for their valuable comments and suggestions which led to an improvement of our original manuscript.
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