The Global Behavior of a Periodic Epidemic Model with Travel between Patches

and Applied Analysis 3 1st patch, respectively. B t ,C t , d t , β t , γ t , and ρij t are continuous, positiveω-periodic functions of t. We have the following system: dS11 dt ( B t N p 1 C t ) N p 1 ρ12 t S12 − σ1 t d t S11 − β t S11 I11 I21 γ t I11, dS12 dt σ1 t S11 − ( ρ12 t d t ) S12 − β t S12 I12 I22 γ t I12, dS21 dt σ2 t S22 − ( ρ21 t d t ) S21 − β t S21 I11 I21 γ t I21, dS22 dt ( B t N p 2 C t ) N p 2 ρ21 t S21 − σ2 t d t S22 − β t S22 I12 I22 γ t I22, dI11 dt ρ12 t I12 β t S11 I11 I21 − ( σ1 t d t γ t ) I11, dI12 dt σ1 t I11 β t S12 I12 I22 − ( ρ12 t d t γ t ) I12. dI21 dt σ2 t I22 β t S21 I11 I21 − ( ρ21 t d t γ t ) I21, dI22 dt ρ21 t I21 β t S22 I12 I22 − ( σ2 t d t γ t ) I22. 1.2 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 R, R be the standard ordered n-dimensional Euclidian space with a norm ‖ · ‖. For u, v ∈ R, we write u ≥ v if u − v ∈ R , u > v, if u − v ∈ R \ {0}, and u v if u − v ∈ Int R . Let A t be a continuous, cooperative, irreducible, and ω-periodic n × nmatrix function, and ΦA t is the fundamental solution matrix of the linear ordinary differential system


Introduction
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 9 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 10 presented a periodic SIS epidemic model with individuals immigration among n 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 11 , 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 12 and Bai and Zhou 13 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 n patches and the transmission rate is a periodic function. Our SIS epidemic model with mobility and seasonality is as follows: where S ij t and I ij t are the number of susceptible and infected individuals whose current location is the jth patch and home location is the ith patch at time t, respectively. Denote N ij S ij t I ij t . N p i n j 1 N ji , N p i , is the number of individuals who are physically present in the ith patch at time t. N n i,j 1 S ij I ij . B t, N p i is the birth rate of the population in the ith patch. d ij t is the death rate of the individuals whose current location is the jth patch and home location is the ith patch at time t. Individuals are assumed to leave a patch i at a certain constant rate, σ i t . The probability that a person travels from patch i to any other patch j is given by V ij t . So σ i t V ij t is the travel rate of individuals from the ith patch to the jth patch at time t. A person from patch i who travels to patch j returns home at a rate ρ ij t . β ikj t is the disease transmission coefficient in patch k that a susceptible individual from patch i contacts with an infectious individual from patch j. The recovery rate of infectious individuals from ith patch who are present in region j is γ ij t . In 14 , the birth rate B i t, N p i satisfies the following basic assumptions for N p i ∈ 0, ∞ : 1st patch, respectively. B t , C t , d t , β t , γ t , and ρ ij t are continuous, positive ω-periodic functions of t. We have the following system: 1.2 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.

The Basic Reproduction Ratio
Let R n , R n be the standard ordered n-dimensional Euclidian space with a norm · . For Let A t be a continuous, cooperative, irreducible, and ω-periodic n × n matrix function, and Φ A t is the fundamental solution matrix of the linear ordinary differential system and r Φ A ω be the spectral radius of Φ A ω . By the Perron-Frobenius theorem, r Φ A ω is the principal eigenvalue of Φ A ω in the sense that it is simple and admits an eigenvector v * 0. 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 Γ 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

2.8
Define function matrix

2.9
Then 2.8 can be rewritten as where I is a 2 × 2 identity matrix. Let C ω be the ordered Banach space of all ω-periodic function R → R 4 , which is equipped with norm · ∞ and the positive cone C ω {φ ∈ C ω : φ t ≥ 0, ∀t ∈ R}.
Consider the following operator L : We can define the basic reproduction ratio R 0 r L , the spectral of radius of L.

The Threshold Dynamics
In this section, we show R 0 as a threshold parameter between the extinction and the uniform persistence of the disease.

3.1
Abstract and Applied Analysis 7 By the aforementioned conclusion, the above system has a unique positive fixed point S * t which is globally attractive in R 4 \ {0}. It then follows that for any ε 1 > 0, there exists T 1 > 1 such that

3.3
Denote By Theorem 2.3, we have r Φ F−V ω < 1. We restrict ε 1 > 0 such that r Φ F−V εM 1 ω < 1. 3.5 Applying Lemma 2.1 and the standard comparison principle, there exists a positive ω-positive function V 1 t such that I t ≤ V 1 t e p 1 t , where p 1 ln r Φ F−V εM 1 ω /ω < 0. Hence, we have that lim t → ∞ I ij t 0, i, j 1, 2 . Consequently, we obtain that where N t N 11 t , N 12 t , N 21 t , N 22 t .

Abstract and Applied Analysis
Hence, the disease free periodic solution S * , 0, 0, 0 is globally attractive and the proof is complete.
The following result shows that R 0 is the threshold parameter for the extinction and the uniform persistence of the disease.
We define Let P : R 8 → R 8 be the Poincare map associated with 1.2 , P x 0 μ ω, x 0 , ∀x 0 ∈ R 8 , where μ t, x 0 is the solution of 1.2 with μ 0, x 0 x 0 . It is obvious that both X and X 0 are positively invariant and ∂X 0 is relatively closed in X. Set We now show that Obviously

3.11
Then I 11 t > 0 for any t > 0. I 12 t > 0 can be proven similarly. So that I 11 t > 0, I 12 t > 0, So I 21 t > 0 for some small t. From 1.2 , using the method of variation of constant, it is clear that I 11 t > 0, I 12 t > 0, I 21 t > 0, I 22 t > 0. It follows that S t , I t / ∈ ∂X 0 , for 0 < t 1. Thus, the positive invariance of X 0 implies 3.9 . It is clear that there are two fixed points of P in M ∂ , which are M 0 0, 0 and M 1 S * 0 , 0 . Now we see R 0 as a threshold parameter between the extinction and the uniform persistence of the disease. Proof. First we prove that P is uniformly persistent with respect to X 0 , ∂X . By Theorem 2.3, we have that R 0 > 1 if and only if r Φ F−V ω > 1. Then we choose η > 0 small enough such that r Φ F−V −ηM 1 ω > 1. Note that the perturbed system of 2.4 ,
For any t ≥ 0, let t mω t , where t ∈ 0, ω and m t/m is the greatest integer less than or equal to t/m. Then we get μ t, S 0 , I 0 − μ t, M i μ t , P m S 0 , I 0 − μ t , M i < δ, ∀t ≥ 0.
We have

3.16
Since the fixed point S * 0, δ of the Poincare map associated with 3.13 is globally attractive and S * t, δ > S * t − η, there is T > 0, such that S t > S * t − η for t > T, there holds 3.17 Since r Φ F−V −ηM 1 ω > 1, by Lemma 2.1, it is obvious that lim t → ∞ I ij t ∞, ∀i, j 1, 2. This leads to a contradiction. Then 3.14 holds. Note that S * 0 is globally attractive in R 4 \ {0}. By the aforementioned claim, it follows that M 0 and M 1 are isolated invariance sets in X, W s M 0 ∩ X 0 ∅, and W s M 1 ∩ X 0 ∅. Clearly, every orbit in M ∂ converges to either M 0 or M 1 , M 0 and M 1 are acyclic in M ∂ . By 15, Theorem 1.3.1 , P is uniformly persistent with respect to X 0 , ∂X . This implies the uniform persistence of the solutions of system 1.2 with respect to X 0 , ∂X . By 6, Theorem 1.3.6 , P has a fixed point P S 0 , I 0 ∈ X 0 . Then, S 0 ∈ R 4 , I 0 ∈ Int R 4 . We further claim that S 0 ∈ R 4 \{0}, suppose that S 0 0, by 2.8 , we can obtain −4 d t γ t I 11 0 I 12 0 I 21 0 I 22 0 0. And hence I ij 0 0, i, j 1, 2, a contradiction. Thus, S 0 ≥ 0. Then S 0 , I 0 is a positive ω-periodic solution of 1.2 . The proof is complete.

Numerical Simulations
In this section, we give the numerical solutions 1.2 to clarify the correctness of our theoretical results. We set B t  Figure 1 shows the numerical solutions of 1.2 when β 0.4. Because basic reproduction ratio R 0 > 1, a positive periodic solution exists, and the disease is uniform persistence. In Figure 2, β 0.25, the disease dies out because R 0 < 1.