Research Article | Open Access
The Global Behavior of a Periodic Epidemic Model with Travel between Patches
We establish an SIS (susceptible-infected-susceptible) epidemic model, in which the travel between patches and the periodic transmission rate are considered. As an example, the global behavior of the model with two patches is investigated. We present the expression of basic reproduction ratio and two theorems on the global behavior: if < 1 the disease-free periodic solution is globally asymptotically stable and if > 1, then it is unstable; if > 1, the disease is uniform persistence. Finally, two numerical examples are given to clarify the theoretical results.
Epidemic models have been paid intensive attention for recent decades. In the models, population is divided into several compartments, for example, susceptible (S), infected (I), and recovery (R) by individual state. The classic epidemic models, including SIS model and SIR model, generally aim at the basic reproduction ratio (the epidemic threshold) and the global behavior [1–6].
With the development of transportation, the travel becomes more and more easy for people. It has been observed that the travel can affect the spread of infectious disease. In [7, 8], authors showed that international travel is one of the major factors associated with the global spread of infectious disease. Ruan et al. investigated the effect of global travel on the spread of SARS  and pointed out that the basic reproduction ratio is independent upon the travel but the travel increase the number of infected individuals.
On the other hand, many infectious diseases show seasonal behavior, such as measles, chickenpox, rubella, and influenza. Zhang and Zhao  presented a periodic SIS epidemic model with individuals immigration among patches. By employing the persistence theory, they gave the expression of the epidemic threshold and obtained the conditions under which the positive periodic solution is globally asymptotically stable. In , Wang and Zhao showed that the threshold parameter is the basic reproduction ratio for a wide class of compartmental epidemic model in periodic environments. Applying the method in [10, 11], Nakata and Kuniya  and Bai and Zhou  examined the threshold dynamics of a periodic SEIRS epidemic model.
Combining the mobility and seasonality, we consider an SIS epidemic model, in which people can travel among patches and the transmission rate is a periodic function. Our SIS epidemic model with mobility and seasonality is as follows: where and are the number of susceptible and infected individuals whose current location is the patch and home location is the patch at time , respectively. Denote . is the number of individuals who are physically present in the patch at time . . is the birth rate of the population in the patch. is the death rate of the individuals whose current location is the th patch and home location is the patch at time . Individuals are assumed to leave a patch at a certain constant rate, . The probability that a person travels from patch to any other patch is given by . So is the travel rate of individuals from the patch to the th patch at time . A person from patch who travels to patch returns home at a rate . is the disease transmission coefficient in patch that a susceptible individual from patch contacts with an infectious individual from patch . The recovery rate of infectious individuals from patch who are present in region is . In , the birth rate satisfies the following basic assumptions for :(A1), ;(A2) is continuously differentiable with , ;(A3) , ;and the birth function can be found in the biological literature.
We assume that these coefficients are functions being continuous, positive -periodic in and we can obtain a periodic SIS epidemic model, in which individuals can travel among patches. For simplicity, we consider an SIS model with travel among two patches, that is, . In this paper, we assume that . Hence , we have . and are the travel rate from the 1st patch to the 2nd patch and from the 2nd patch to the 1st patch, respectively. , , , , , and are continuous, positive -periodic functions of . We have the following system:
In this paper, we will study the basic reproduction ratio and global behavior of system (1.2). This paper is organized as follows. In Section 2, we show the existence of the disease-free periodic solution of (1.2) and define the basic reproduction ratio. In Section 3, we show the global asymptotical stability of the periodic disease-free solution and the uniform persistence of the disease. In Section 4, two numerical examples are given to clarify the theoretical results.
2. The Basic Reproduction Ratio
Let () be the standard ordered -dimensional Euclidian space with a norm . For , we write if , if , and if . Let be a continuous, cooperative, irreducible, and -periodic matrix function, and is the fundamental solution matrix of the linear ordinary differential system and be the spectral radius of . By the Perron-Frobenius theorem, is the principal eigenvalue of in the sense that it is simple and admits an eigenvector . The following result is useful for our subsequent comparison arguments.
Proof. By the method of variation of constant, it is obvious that any solution of (1.2) with nonnegative initial values is nonnegative. From (1.2), we have This implies that is a forward invariant compact absorbing set of (1.2). Hence, the proof is complete.
Next, we show the existence of the disease-free periodic solution of (1.2). To find the disease-free periodic solution of (1.2), we consider Denote Let be defined by the right-hand side of (2.4). for every with is strongly subhomogeneous for in the sense that for any and . is a continuous, cooperative, irreducible, and -periodic 4 × 4 matrix function. By Lemma 2.2, the solution of (2.4) is ultimately bounded in . By Theorem 2.3.2 of , applying the Poincare map associated with (2.4), it follows that system (2.4) has a unique positive periodic solution
We need to assume that be the spectral radius of . By Theorem 2.1.2 of , it then follows that the unique positive periodic solution of (2.4) is globally attractive for . Hence, (1.2) has a unique disease-free periodic state .
For convenience, we denote
Consider the following system: Define function matrix Then (2.8) can be rewritten as where .
Assume that is the evolution operator of the linear periodic system That is, for each , the matrix satisfies where is a identity matrix.
Let be the ordered Banach space of all -periodic function , which is equipped with norm and the positive cone .
Consider the following operator by
We can define the basic reproduction ratio the spectral of radius of .
Theorem 2.3 (see [11, Theorem 2.2]). The following statements are valid:(i) if and only if .(ii) if and only if .(iii) if and only if .Thus, of (1.2) is asymptotically stable if and it is unstable if .
3. The Threshold Dynamics
In this section, we show as a threshold parameter between the extinction and the uniform persistence of the disease.
Theorem 3.1. If , the disease-free periodic solution is globally asymptotically stable and if , it is unstable.
Proof. By Theorem 2.3, if , the disease-free periodic solution is unstable. If , the disease-free periodic solution is locally stable. Hence, it is sufficient to show the global attractivity of when .
By (1.2), we have
By the aforementioned conclusion, the above system has a unique positive fixed point which is globally attractive in . It then follows that for any , there exists such that Obviously, . Hence, we have Denote By Theorem 2.3, we have . We restrict such that .
Consider the system
Applying Lemma 2.1 and the standard comparison principle, there exists a positive -positive function such that , where . Hence, we have that . Consequently, we obtain that where .
Hence, the disease free periodic solution is globally attractive and the proof is complete.
The following result shows that is the threshold parameter for the extinction and the uniform persistence of the disease.
It is obvious that both and are positively invariant and is relatively closed in . Set We now show that Obviously, . To show that , we consider for any .
Firstly, if one element of is 0, say , that is, , then , for any . From (1.2) It is clear that , , , .
Secondly, if two elements of is 0, for example, . From (1.2), using the method of variation of constant, it is clear that , , , , for any , Then for any . can be proven similarly. So that , , , .
It follows that , for . Thus, the positive invariance of implies (3.9). It is clear that there are two fixed points of in , which are and .
Now we see as a threshold parameter between the extinction and the uniform persistence of the disease.
Proof. First we prove that is uniformly persistent with respect to . By Theorem 2.3, we have that if and only if . Then we choose small enough such that . Note that the perturbed system of (2.4),
As in our previous analysis of system (2.4), we can choose small enough such that the Poincare map associated with (3.13) admits a unique positive fixed point which is globally attractive in . By the implicit function theorem, it follows that is continuous in . Thus, we can fix a small number such that , where . By the continuity of solutions with respect to the initial values, there exists such that for all , with . We have . We now claim that Suppose, by contradiction, that , for some , and . Without loss of generality, we can assume that . Then, we have .
For any , let , where and is the greatest integer less than or equal to . Then we get Let . It then follows that .
We have Since the fixed point of the Poincare map associated with (3.13) is globally attractive and , there is , such that for , there holds Since , by Lemma 2.1, it is obvious that . This leads to a contradiction. Then (3.14) holds. Note that is globally attractive in . By the aforementioned claim, it follows that and are isolated invariance sets in , and . Clearly, every orbit in converges to either or , and are acyclic in . By [15, Theorem 1.3.1], is uniformly persistent with respect to . This implies the uniform persistence of the solutions of system (1.2) with respect to . By [6, Theorem 1.3.6], has a fixed point . Then, . We further claim that , suppose that , by (2.8), we can obtain . And hence , a contradiction. Thus, . Then is a positive -periodic solution of (1.2). The proof is complete.
4. Numerical Simulations
In this section, we give the numerical solutions (1.2) to clarify the correctness of our theoretical results. We set , , , , , , , , . The initial value of the model is , , , , , , , . Figure 1 shows the numerical solutions of (1.2) when . Because basic reproduction ratio , a positive periodic solution exists, and the disease is uniform persistence. In Figure 2, , the disease dies out because .
This work is supported by the Fundamental Research Funds for the Central Universities (CDJZR10100011) and the support of the Natural Science Foundation of Chongqing CSTC (2009BB2184).
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Copyright © 2012 Luosheng Wen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.