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

Abstract

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 𝑅₀ and two theorems on the global behavior: if 𝑅₀ < 1 the disease-free periodic solution is globally asymptotically stable and if 𝑅₀ > 1, then it is unstable; if 𝑅₀ > 1, the disease is uniform persistence. Finally, two numerical examples are given to clarify the theoretical results.

1. 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: 𝑑𝑆_𝑖𝑖𝑑𝑡=𝐵_𝑖𝑡,𝑁^𝑝_𝑖𝑁^𝑝_𝑖+^𝑛_{𝑘=1,𝑘≠𝑖}𝜌_𝑖𝑘(𝑡)𝑆_𝑖𝑘−𝜎_𝑖(𝑡)+𝑑_𝑖𝑖(𝑡)𝑆_𝑖𝑖−^𝑛_𝑘=1𝛽_𝑖𝑖𝑘(𝑡)𝑆_𝑖𝑖𝐼_𝑘𝑖+𝛾_𝑖𝑖(𝑡)𝐼_𝑖𝑖,𝑑𝐼_𝑖𝑖=𝑑𝑡^𝑛_{𝑘=1,𝑘≠𝑖}𝜌_𝑖𝑘(𝑡)𝐼_𝑖𝑘+^𝑛_𝑘=1𝛽_𝑖𝑖𝑘(𝑡)𝑆_𝑖𝑖𝐼_𝑘𝑖−𝜎_𝑖(𝑡)+𝑑_𝑖𝑖(𝑡)+𝛾_𝑖𝑖(𝑡)𝐼_𝑖𝑖,𝑑𝑆_𝑖𝑗𝑑𝑡=𝜎_𝑖(𝑡)𝑉_𝑖𝑗(𝑡)𝑆_𝑖𝑖−𝜌_𝑖𝑗(𝑡)+𝑑_𝑖𝑗(𝑡)𝑆_𝑖𝑗−^𝑛_𝑘=1𝛽_𝑖𝑗𝑘(𝑡)𝑆_𝑖𝑗𝐼_𝑘𝑖+𝛾_𝑖𝑗(𝑡)𝐼_𝑖𝑗,𝑖≠𝑗𝑑𝐼_𝑖𝑗𝑑𝑡=𝜎_𝑖(𝑡)𝑉_𝑖𝑗(𝑡)𝐼_𝑖𝑖+^𝑛_𝑘=1𝛽_𝑖𝑗𝑘(𝑡)𝑆_𝑖𝑗𝐼_𝑘𝑖−𝜌_𝑖𝑗(𝑡)+𝑑_𝑖𝑗(𝑡)+𝛾_𝑖𝑗(𝑡)𝐼_𝑖𝑗,𝑖≠𝑗, (1.1) 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=1𝑁_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,∞):(A1)𝐵_𝑖(𝑡,𝑁^𝑝_𝑖)>0,  𝑖=1,2,…,𝑛;(A2)𝐵_𝑖(𝑡,𝑁^𝑝_𝑖) is continuously differentiable with 𝑑𝐵_𝑖(𝑡,𝑁^𝑝_𝑖)/𝑑𝑁^𝑝_𝑖<0,  𝑖=1,2,…,𝑛;(A3) 𝐵_𝑖(𝑡,∞)<𝑑_𝑖𝑖(𝑡),  𝑖=1,2,…,𝑛;and the birth function 𝐵_𝑖(𝑡,𝑁^𝑝_𝑖)=𝐵(𝑡)/𝑁^𝑝_𝑖+𝐶(𝑡) can be found in the biological literature.

We assume that these coefficients are functions being continuous, positive 𝜔-periodic in 𝑡 and we can obtain a periodic SIS epidemic model, in which individuals can travel among 𝑛 patches. For simplicity, we consider an SIS model with travel among two patches, that is, 𝑛=2. In this paper, we assume that 𝐵_𝑖(𝑡,𝑁^𝑝_𝑖)=𝐵(𝑡)/(𝑁^𝑝_𝑖)+𝐶(𝑡),𝑑_𝑖𝑗(𝑡)=𝑑(𝑡),𝐶(𝑡)<𝑑(𝑡).𝛽_𝑖𝑘𝑗(𝑡)=𝛽(𝑡),𝛾_𝑖𝑗(𝑡)=𝛾(𝑡),𝑖,𝑗=1,2. Hence ∑²_{𝑗=1,𝑗≠𝑖}𝑉_𝑖𝑗(𝑡)=1, we have 𝑉_𝑖𝑗(𝑡)=1. 𝜎₁(𝑡) and 𝜎₂(𝑡) are the travel rate from the 1st patch to the 2nd patch and from the 2nd patch to the 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: 𝑑𝑆₁₁=𝐵(𝑡)𝑑𝑡𝑁^𝑝₁+𝐶(𝑡)𝑁^𝑝₁+𝜌₁₂(𝑡)𝑆₁₂−𝜎₁(𝑡)+𝑑(𝑡)𝑆₁₁−𝛽(𝑡)𝑆₁₁𝐼₁₁+𝐼₂₁+𝛾(𝑡)𝐼₁₁,𝑑𝑆₁₂𝑑𝑡=𝜎₁(𝑡)𝑆₁₁−𝜌₁₂(𝑡)+𝑑(𝑡)𝑆₁₂−𝛽(𝑡)𝑆₁₂𝐼₁₂+𝐼₂₂+𝛾(𝑡)𝐼₁₂,𝑑𝑆₂₁𝑑𝑡=𝜎₂(𝑡)𝑆₂₂−𝜌₂₁(𝑡)+𝑑(𝑡)𝑆₂₁−𝛽(𝑡)𝑆₂₁𝐼₁₁+𝐼₂₁+𝛾(𝑡)𝐼₂₁,𝑑𝑆₂₂=𝐵(𝑡)𝑑𝑡𝑁^𝑝₂+𝐶(𝑡)𝑁^𝑝₂+𝜌₂₁(𝑡)𝑆₂₁−𝜎₂(𝑡)+𝑑(𝑡)𝑆₂₂−𝛽(𝑡)𝑆₂₂𝐼₁₂+𝐼₂₂+𝛾(𝑡)𝐼₂₂,𝑑𝐼₁₁𝑑𝑡=𝜌₁₂(𝑡)𝐼₁₂+𝛽(𝑡)𝑆₁₁𝐼₁₁+𝐼₂₁−𝜎₁(𝑡)+𝑑(𝑡)+𝛾(𝑡)𝐼₁₁,𝑑𝐼₁₂𝑑𝑡=𝜎₁(𝑡)𝐼₁₁+𝛽(𝑡)𝑆₁₂𝐼₁₂+𝐼₂₂−𝜌₁₂(𝑡)+𝑑(𝑡)+𝛾(𝑡)𝐼₁₂.𝑑𝐼₂₁𝑑𝑡=𝜎₂(𝑡)𝐼₂₂+𝛽(𝑡)𝑆₂₁𝐼₁₁+𝐼₂₁−𝜌₂₁(𝑡)+𝑑(𝑡)+𝛾(𝑡)𝐼₂₁,𝑑𝐼₂₂𝑑𝑡=𝜌₂₁(𝑡)𝐼₂₁+𝛽(𝑡)𝑆₂₂𝐼₁₂+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^𝑛,𝑅^𝑛₊) be the standard ordered 𝑛-dimensional Euclidian space with a norm ‖⋅‖. For 𝑢,𝑣∈𝑅^𝑛, we write 𝑢≥𝑣 if 𝑢−𝑣∈𝑅^𝑛₊,𝑢>𝑣, if 𝑢−𝑣∈𝑅^𝑛₊⧵{0}, and 𝑢≫𝑣 if 𝑢−𝑣∈Int(𝑅^𝑛₊). 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 𝑑𝑋(𝑡)𝑑𝑡=𝐴𝑋, (2.1) 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.

Lemma 2.1 (see [10]). Let 𝑃=(1/𝜔)ln𝑟(Φ_𝐴(𝜔)). Then there exists a positive, ω-periodic function 𝑉(𝑡) such that 𝑒^𝑃𝑡𝑉(𝑡) is a solution of (2.1).

Lemma 2.2. Every forward solution of (1.2) eventually into Γ=(𝑆(𝑡),𝐼(𝑡))∈𝑅⁸₊∣0≤²_𝑖,𝑗=1𝑆_𝑖𝑗+𝐼_𝑖𝑗≤2𝑏𝑐, (2.2) where 𝑏=max_{𝑡∈[0,𝜔]}(𝐵(𝑡)),𝑐=min_{𝑡∈[0,𝜔]}(𝑑(𝑡)−𝐶(𝑡)), and for each 𝑁(𝑡)⩾2𝑏/𝑐, Γ is a positively invariant set for (1.2).

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 𝑑𝑁(𝑡)−(𝑑(𝑡)(𝑡(𝑡)≥𝑑𝑡=2𝐵−𝐶))𝑁≤2𝑏−𝑐𝑁≤0if𝑁2𝑏𝑐. (2.3) 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 𝑑𝑆₁₁(𝑡)(𝑡)𝑆𝑑𝑡=𝐵+𝐶₁₁+𝑆₂₁+𝜌₁₂(𝑡)𝑆₁₂−𝜎₁(𝑡)+𝑑(𝑡)𝑆₁₁,𝑑𝑆₁₂𝑑𝑡=𝜎₁(𝑡)𝑆₁₁−𝜌₁₂(𝑡)+𝑑(𝑡)𝑆₁₂,𝑑𝑆₂₁𝑑𝑡=𝜎₂(𝑡)𝑆₂₂−𝜌₂₁(𝑡)+𝑑(𝑡)𝑆₂₁,𝑑𝑆₂₂(𝑡)(𝑡)𝑆𝑑𝑡=𝐵+𝐶₁₂+𝑆₂₂+𝜌₂₁(𝑡)𝑆₂₁−𝜎₂(𝑡)+𝑑(𝑡)𝑆₂₂. (2.4) Denote 𝑀(𝑡)=𝐶(𝑡)−𝜎₁(𝑡)+𝑑(𝑡)𝜌₁₂(𝑡)𝐶(𝑡)0𝜎₁(𝑡)−𝜌₁₂(𝑡)(𝑡)𝜌+𝑑0000−₂₁(𝑡)+𝑑(𝑡)𝜎₂(𝑡)0𝐶(𝑡)𝜌₂₁(𝑡)𝐶(𝑡)−𝜎₂(𝑡)(𝑡.+𝑑) (2.5) Let Ψ∶𝑅¹₊×𝑅⁴₊→𝑅⁴ be defined by the right-hand side of (2.4). Ψ_𝑖(𝑡,𝑆)≥0 for every 𝑆≥0 with 𝑆_𝑖=0,𝑡∈𝑅¹₊,1≤𝑖≤4.Ψ(𝑡,𝑆) is strongly subhomogeneous for 𝑆∈𝑅⁴₊ in the sense that Ψ(𝑡,𝛼𝑆)≫𝛼Ψ(𝑡,𝑆) for any 𝑡≥0,𝑆∈𝑅⁴₊ and 𝛼∈(0,1).  𝑀(𝑡) is a continuous, cooperative, irreducible, and 𝜔-periodic 4 × 4 matrix function. By Lemma 2.2, the solution of (2.4) is ultimately bounded in 𝑅⁴₊. By Theorem 2.3.2 of [15], applying the Poincare map associated with (2.4), it follows that system (2.4) has a unique positive periodic solution 𝑆^∗(𝑡)=𝑆^∗₁₁(𝑡),𝑆^∗₁₂(𝑡),𝑆^∗₂₁(𝑡),𝑆^∗₂₂(𝑡). (2.6)

We need to assume that 𝑟(𝜙_𝑀(𝜔))>1,𝑟(Φ_𝑀(𝜔)) be the spectral radius of 𝑀(𝜔). By Theorem 2.1.2 of [15], it then follows that the unique positive periodic solution 𝑆^∗(𝑡) of (2.4) is globally attractive for 𝑆⁰∈𝑅⁴₊⧵{0}. Hence, (1.2) has a unique disease-free periodic state (𝑆^∗,0,0,0).

For convenience, we denote 𝑆(𝑡)=𝑆₁₁(𝑡),𝑆₁₂(𝑡),𝑆₂₁(𝑡),𝑆₂₂(𝑡),𝐼(𝑡)=𝐼₁₁(𝑡),𝐼₁₂(𝑡),𝐼₂₁(𝑡),𝐼₂₂(𝑡). (2.7)

Consider the following system: 𝑑𝐼₁₁𝑑𝑡=𝜌₁₂(𝑡)𝐼₁₂+𝛽(𝑡)𝑆₁₁𝐼₁₁+𝐼₂₁−𝜎₁(𝑡)+𝑑(𝑡)+𝛾(𝑡)𝐼₁₁,𝑑𝐼₁₂𝑑𝑡=𝜎₁(𝑡)𝐼₁₁+𝛽(𝑡)𝑆₁₂𝐼₁₂+𝐼₂₂−𝜌₁₂(𝑡)+𝑑(𝑡)+𝛾(𝑡)𝐼₁₂,𝑑𝐼₂₁𝑑𝑡=𝜎₂(𝑡)𝐼₂₂+𝛽(𝑡)𝑆₂₁𝐼₁₁+𝐼₂₁−𝜌₂₁(𝑡)+𝑑(𝑡)+𝛾(𝑡)𝐼₂₁,𝑑𝐼₂₂𝑑𝑡=𝜌₂₁(𝑡)𝐼₂₁+𝛽(𝑡)𝑆₂₂𝐼₁₂+𝐼₂₂−𝜎₂(𝑡)+𝑑(𝑡)+𝛾(𝑡)𝐼₂₂. (2.8) Define function matrix 𝐹(𝑡)=𝛽(𝑡)𝑆^∗₁₁(𝑡)0𝛽(𝑡)𝑆^∗₁₁(𝑡)00𝛽(𝑡)𝑆^∗₁₂(𝑡)0𝛽(𝑡)𝑆^∗₁₂(𝑡)𝛽(𝑡)𝑆^∗₂₁(𝑡)0𝛽(𝑡)𝑆^∗₂₁(𝑡)00𝛽(𝑡)𝑆^∗₂₂(𝑡)0𝛽(𝑡)𝑆^∗₂₂(𝑡),𝑉(𝑡)=𝜎₁(𝑡)+𝑑(𝑡)+𝛾(𝑡)−𝜌₁₂(𝑡)00−𝜎₁(𝑡)𝜌₁₂(𝑡)+𝑑(𝑡)+𝛾(𝑡)0000𝜌₂₁(𝑡)+𝑑(𝑡)+𝛾(𝑡)−𝜎₂(𝑡)00−𝜌₂₁(𝑡)𝜎₂(𝑡)(𝑡)(𝑡).+𝑑+𝛾 (2.9) Then (2.8) can be rewritten as 𝑑𝑍=(𝐹(𝑡)(𝑡𝑑𝑡−𝑉))𝑍, (2.10) where 𝑍=𝐼(𝑡)^𝑇.

Assume that 𝑌(𝑡,𝑠),𝑡≥𝑠 is the evolution operator of the linear periodic system 𝑑𝑦(𝑡)𝑑𝑡=−𝑉𝑦. (2.11) That is, for each 𝑠∈𝑅, the 4×4 matrix 𝑌(𝑡,𝑠) satisfies 𝑑𝑌(𝑡,𝑠)(𝑡)𝑌()()𝑑𝑡=−𝑉𝑡,𝑠,∀𝑡≥𝑠,𝑌𝑠,𝑠=𝐼, (2.12) where 𝐼 is a 2×2 identity matrix.

Let 𝐶_𝜔 be the ordered Banach space of all 𝜔-periodic function 𝑅→𝑅⁴, which is equipped with norm ‖⋅‖_∞ and the positive cone 𝐶⁺_𝜔={𝜙∈𝐶_𝜔∶𝜙(𝑡)≥0,∀𝑡∈𝑅}.

Consider the following operator 𝐿∶𝐶_𝜔→𝐶_𝜔 by 𝐿𝜙(𝑡)=₀^+∞𝑌(𝑡,𝑡−𝑎)𝜙(𝑡−𝑎)𝑑𝑎,∀𝑡∈𝑅,𝜙∈𝐶_𝜔. (2.13)

We can define the basic reproduction ratio 𝑅₀=𝑟(𝐿), the spectral of radius of 𝐿.

Theorem 2.3 (see [11, Theorem 2.2]). The following statements are valid:(i)𝑅₀=1 if and only if 𝑟(Φ_𝐹−𝑉(𝜔))=1.(ii)𝑅₀>1 if and only if 𝑟(Φ_𝐹−𝑉(𝜔))>1.(iii)𝑅₀<1 if and only if 𝑟(Φ_𝐹−𝑉(𝜔))<1.Thus, (𝑆^∗,0,0,0) of (1.2) is asymptotically stable if 𝑅₀<1 and it is unstable if 𝑅₀>1.

3. The Threshold Dynamics

In this section, we show 𝑅₀ as a threshold parameter between the extinction and the uniform persistence of the disease.

Theorem 3.1. If 𝑅₀<1, the disease-free periodic solution (𝑆^∗,0,0,0) is globally asymptotically stable and if 𝑅₀>1, it is unstable.

Proof. By Theorem 2.3, if 𝑅₀>1, the disease-free periodic solution (𝑆^∗,0,0,0) is unstable. If 𝑅₀<1, the disease-free periodic solution (𝑆^∗,0,0,0) is locally stable. Hence, it is sufficient to show the global attractivity of (𝑆^∗,0,0,0) when 𝑅₀<1.
By (1.2), we have 𝑑𝑁₁₁(𝑡)(𝑡)𝑁𝑑𝑡=𝐵+𝐶₁₁+𝑁₂₁+𝜌₁₂(𝑡)𝑁₁₂−𝜎₁(𝑡)+𝑑(𝑡)𝑁₁₁,𝑑𝑁₁₂𝑑𝑡=𝜎₁(𝑡)𝑁₁₁−𝜌₁₂(𝑡)+𝑑(𝑡)𝑁₁₂,𝑑𝑁₂₁𝑑𝑡=𝜎₂(𝑡)𝑁₂₂−𝜌₂₁(𝑡)+𝑑(𝑡)𝑁₂₁,𝑑𝑁₂₂(𝑡)(𝑡)𝑁𝑑𝑡=𝐵+𝐶₁₂+𝑁₂₂+𝜌₂₁(𝑡)𝑁₂₁−𝜎₂(𝑡)+𝑑(𝑡)𝑁₂₂. (3.1)
By the aforementioned conclusion, the above system has a unique positive fixed point 𝑆^∗(𝑡) which is globally attractive in 𝑅⁴₊⧵{0}. It then follows that for any 𝜀₁>0, there exists 𝑇₁>1 such that 𝑁_𝑖𝑗(𝑡)=𝑆_𝑖𝑗(𝑡)+𝐼_𝑖𝑗(𝑡)≤𝑆^∗_𝑖𝑗(𝑡)+𝜀₁,∀𝑡>𝑇₁. (3.2) Obviously, 𝑆_𝑖𝑗(𝑡)≤𝑆^∗_𝑖𝑗(𝑡)+𝜀₁,(𝑖,𝑗=1,2). Hence, we have 𝑑𝐼₁₁𝑑𝑡≤𝜌₁₂(𝑡)𝐼₁₂+𝛽(𝑡)𝑆^∗₁₁+𝜀₁𝐼₁₁+𝐼₂₁−𝜎₁(𝑡)+𝑑(𝑡)+𝛾(𝑡)𝐼₁₁,𝑑𝐼₁₂𝑑𝑡≤𝜎₁(𝑡)𝐼₁₁+𝛽(𝑡)𝑆^∗₁₂+𝜀₁𝐼₁₂+𝐼₂₂−𝜌₁₂(𝑡)+𝑑(𝑡)+𝛾(𝑡)𝐼₁₂,𝑑𝐼₂₁𝑑𝑡≤𝜎₂(𝑡)𝐼₂₂+𝛽(𝑡)𝑆^∗₂₁+𝜀₁𝐼₁₁+𝐼₂₁−𝜌₂₁(𝑡)+𝑑(𝑡)+𝛾(𝑡)𝐼₂₁,𝑑𝐼₂₂𝑑𝑡≤𝜌₂₁(𝑡)𝐼₂₁+𝛽(𝑡)𝑆^∗₂₂+𝜀₁𝐼₁₂+𝐼₂₂−𝜎₂(𝑡)+𝑑(𝑡)+𝛾(𝑡)𝐼₂₂. (3.3) Denote 𝑀₁(𝑡)=𝛽(𝑡)(𝑡)0(𝑡)(𝑡)𝛽(𝑡)(𝑡)0(𝑡)(𝑡)0𝛽0𝛽0𝛽0𝛽0𝛽0𝛽. (3.4) By Theorem 2.3, we have 𝑟(Φ_𝐹−𝑉(𝜔))<1. We restrict 𝜀₁>0 such that 𝑟(Φ_{𝐹−𝑉+𝜀𝑀1}(𝜔))<1.
Consider the system 𝑑𝐼̃₁₁𝑑𝑡=𝜌₁₂(𝑡)𝐼₁₂+𝛽(𝑡)𝑆^∗₁₁+𝜀₁𝐼₁₁+𝐼₂₁−𝜎₁(𝑡)+𝑑(𝑡)+𝛾(𝑡)𝐼₁₁,𝑑𝐼̃₁₂𝑑𝑡=𝜎₁(𝑡)𝐼₁₁+𝛽(𝑡)𝑆^∗₁₂+𝜀₁𝐼₁₂+𝐼₂₂−𝜌₁₂(𝑡)+𝑑(𝑡)+𝛾(𝑡)𝐼₁₂,𝑑𝐼̃₂₁𝑑𝑡=𝜎₂(𝑡)𝐼₂₂+𝛽(𝑡)𝑆^∗₂₁+𝜀₁𝐼₁₁+𝐼₂₁−𝜌₂₁(𝑡)+𝑑(𝑡)+𝛾(𝑡)𝐼₂₁,𝑑𝐼̃₂₂𝑑𝑡=𝜌₂₁(𝑡)𝐼₂₁+𝛽(𝑡)𝑆^∗₂₂+𝜀₁𝐼₁₂+𝐼₂₂−𝜎₂(𝑡)+𝑑(𝑡)+𝛾(𝑡)𝐼₂₂. (3.5)
Applying Lemma 2.1 and the standard comparison principle, there exists a positive 𝜔-positive function 𝑉₁(𝑡) such that 𝐼(𝑡)≤𝑉₁(𝑡)𝑒^𝑝1𝑡, where 𝑝₁=ln𝑟(Φ_{𝐹−𝑉+𝜀𝑀1}(𝜔))/𝜔<0. Hence, we have that lim_𝑡→∞𝐼_𝑖𝑗(𝑡)=0,(𝑖,𝑗=1,2). Consequently, we obtain that lim_𝑡→∞𝑆(𝑡)−𝑆^∗(𝑡)=lim_𝑡→∞𝑁(𝑡)−𝐼(𝑡)−𝑆^∗(𝑡)=0, (3.6) where 𝑁(𝑡)=(𝑁₁₁(𝑡),𝑁₁₂(𝑡),𝑁₂₁(𝑡),𝑁₂₂(𝑡)).
Hence, the disease free periodic solution (𝑆^∗,0,0,0) is globally attractive and the proof is complete.

The following result shows that 𝑅₀ is the threshold parameter for the extinction and the uniform persistence of the disease.

We define 𝑋=𝑅⁸₊,𝑋₀=𝑅⁴₊×𝐼𝑛𝑡𝑅⁴₊,𝜕𝑋₀=𝑋⧵𝑋₀. (3.7) Let 𝑃∶𝑅⁸₊→𝑅⁸₊ be the Poincare map associated with (1.2), 𝑃(𝑥⁰)=𝜇(𝜔,𝑥⁰),∀𝑥⁰∈𝑅⁸₊, where 𝜇(𝑡,𝑥⁰) is the solution of (1.2) with 𝜇(0,𝑥⁰)=𝑥⁰.

It is obvious that both 𝑋 and 𝑋₀ are positively invariant and 𝜕𝑋₀ is relatively closed in 𝑋. Set 𝑀_𝜕=𝑆⁰,𝐼⁰∈𝜕𝑋₀∶𝑃^𝑚𝑆⁰,𝐼⁰∈𝜕𝑋₀,∀𝑚≥0. (3.8) We now show that 𝑀_𝜕={(𝑆,0)∶𝑆≥0}. (3.9) Obviously, {(𝑆,0)∶𝑆≥0}⊆𝑀_𝜕. To show that 𝑀_𝜕⧵{(𝑆,0)∶𝑆≥0}=∅, we consider for any (𝑆⁰,𝐼⁰)∈𝜕𝑋₀⧵{(𝑆,0)∶𝑆≥0}.

Firstly, if one element of 𝐼⁰=(𝐼⁰₁₁,𝐼⁰₁₂,𝐼⁰₂₁,𝐼⁰₂₂) is 0, say 𝐼⁰₁₁, that is, 𝐼⁰₁₁=0,𝐼⁰₁₂>0,𝐼⁰₂₁>0,𝐼⁰₂₂>0, then 𝐼₂₁(𝑡)>0, 𝐼₂₂(𝑡)>0 for any 𝑡>0. From (1.2) 𝑑𝐼₁₁𝑑𝑡_𝑡=0=𝜌₁₂(0)𝐼₁₂(0)+𝛽(0)𝑆₁₁(0)𝐼₂₁(0)>0. (3.10) It is clear that 𝐼₁₁(𝑡)>0, 𝐼₁₂(𝑡)>0, 𝐼₂₁(𝑡)>0, 𝐼₂₂(𝑡)>0.

Secondly, if two elements of 𝐼⁰=(𝐼⁰₁₁,𝐼⁰₁₂,𝐼⁰₂₁,𝐼⁰₂₂) is 0, for example, 𝐼⁰₁₁=0,𝐼⁰₁₂=0,𝐼⁰₂₁>0,𝐼⁰₂₂>0. From (1.2), using the method of variation of constant, it is clear that 𝑆₁₁(𝑡)>0, 𝑆₁₂(𝑡)>0, 𝑆₂₁(𝑡)>0, 𝑆₂₂(𝑡)>0, for any 𝑡>0, 𝐼₁₁=𝐼⁰₁₁+^𝑡₀𝜌₁₂(𝑠)𝐼₁₂(𝑠)+𝛽(𝑠)𝑆₁₁(𝑠)𝐼₂₁(𝑠)𝑒^{∫𝑠0(𝜎1(𝑢)+𝑑(𝑢)+𝛾(𝑢)−𝛽(𝑠)𝑆11(𝑠))𝑑𝑢}𝑑𝑠×𝑒^{−∫𝑡0(𝜎1(𝑠)+𝑑(𝑠)+𝛾(𝑠)−𝛽(𝑠)𝑆11(𝑠))𝑑𝑠.} (3.11) Then 𝐼₁₁(𝑡)>0 for any 𝑡>0. 𝐼₁₂(𝑡)>0 can be proven similarly. So that 𝐼₁₁(𝑡)>0, 𝐼₁₂(𝑡)>0, 𝐼₂₁(𝑡)>0, 𝐼₂₂(𝑡)>0.

Thirdly, if three elements of 𝐼⁰=(𝐼⁰₁₁,𝐼⁰₁₂,𝐼⁰₂₁,𝐼⁰₂₂) are 0, for example, we chose 𝐼⁰₁₁=0,𝐼⁰₁₂=0,𝐼⁰₂₁=0,𝐼⁰₂₂>0. From (1.2), 𝑑𝐼₂₁𝑑𝑡_𝑡=0=𝜎₂(0)𝐼₂₂(0)>0. (3.12) So 𝐼₂₁(𝑡)>0 for some small 𝑡. From (1.2), using the method of variation of constant, it is clear that 𝐼₁₁(𝑡)>0,𝐼₁₂(𝑡)>0,𝐼₂₁(𝑡)>0,𝐼₂₂(𝑡)>0.

It follows that (𝑆(𝑡),𝐼(𝑡))∉𝜕𝑋₀, for 0<𝑡≪1. Thus, the positive invariance of 𝑋₀ implies (3.9). It is clear that there are two fixed points of 𝑃 in 𝑀_𝜕, which are 𝑀₀=(0,0) and 𝑀₁=(𝑆^∗(0),0).

Now we see 𝑅₀ as a threshold parameter between the extinction and the uniform persistence of the disease.

Theorem 3.2. If 𝑅₀>1, then there exists some 𝜀>0 such that any solution (𝑆(𝑡),𝐼(𝑡)) of (1.2) with initial value (𝑆(0),𝐼(0))=(𝑆⁰,𝐼⁰)∈𝑅⁴₊×Int(𝑅⁴₊), satisfies lim_𝑡→∞inf𝐼(𝑡)≥𝜀. Furthermore, (1.2) admits at least one positive periodic solution.

Proof. First we prove that 𝑃 is uniformly persistent with respect to (𝑋₀,𝜕𝑋). By Theorem 2.3, we have that 𝑅₀>1 if and only if 𝑟(Φ_𝐹−𝑉(𝜔))>1. Then we choose 𝜂>0 small enough such that 𝑟(Φ_{𝐹−𝑉−𝜂𝑀1}(𝜔))>1. Note that the perturbed system of (2.4), 𝑑𝑆₁₁(𝑡)(𝑡)𝑆𝑑𝑡=𝐵+𝐶₁₁+𝑆₂₁+𝜌₁₂(𝑡)𝑆₁₂−𝜎₁(𝑡)+𝑑(𝑡)+𝛽(𝑡)𝛿𝑆₁₁,𝑑𝑆₁₂𝑑𝑡=𝜎₁(𝑡)𝑆₁₁−𝜌₁₂(𝑡)+𝑑(𝑡)+𝛽(𝑡)𝛿𝑆₁₂,𝑑𝑆₂₁𝑑𝑡=𝜎₂(𝑡)𝑆₂₂−𝜌₂₁(𝑡)+𝑑(𝑡)+𝛽(𝑡)𝛿𝑆₂₁,𝑑𝑆₂₂(𝑡)(𝑡)𝑆𝑑𝑡=𝐵+𝐶₁₂+𝑆₂₂+𝜌₂₁(𝑡)𝑆₂₁−𝜎₂(𝑡)+𝑑(𝑡)+𝛽(𝑡)𝛿𝑆₂₂. (3.13)
As in our previous analysis of system (2.4), we can choose 𝛿>0 small enough such that the Poincare map associated with (3.13) admits a unique positive fixed point 𝑆^∗(0,𝛿) which is globally attractive in 𝑅⁴₊⧵{0}. By the implicit function theorem, it follows that 𝑆^∗(0,𝛿) is continuous in 𝛿. Thus, we can fix a small number 𝛿>0 such that 𝑆^∗(𝑡,𝛿)>𝑆^∗(𝑡)−𝜂, where 𝜂={𝜂,𝜂,𝜂,𝜂}. By the continuity of solutions with respect to the initial values, there exists 𝛿^∗₀>0 such that for all (𝑆⁰,𝐼⁰)∈𝑋₀, with ‖(𝑆⁰,𝐼⁰)−𝑀‖≤𝛿^∗₀. We have ‖𝜇(𝑡,(𝑆⁰,𝐼⁰))−𝜇(𝑡,𝑀_𝑖)‖<𝛿,∀𝑡∈[0,𝜔],𝑖=0,1. We now claim that lim_𝑚→∞sup𝑑𝑃^𝑚𝑆⁰,𝐼⁰,𝑀_𝑖≥𝛿^∗₀. (3.14) Suppose, by contradiction, that lim_𝑚→∞sup𝑑(𝑃^𝑚(𝑆⁰,𝐼⁰),𝑀_𝑖)<𝛿^∗₀, for some (𝑆⁰,𝐼⁰)∈𝑋₀, and 𝑖=0,1. Without loss of generality, we can assume that 𝑑(𝑃^𝑚(𝑆⁰,𝐼⁰),𝑀_𝑖)<𝛿^∗₀,∀𝑚≥0. Then, we have ‖𝜇(𝑡,𝑃^𝑚(𝑆⁰,𝐼⁰))−𝜇(𝑡,𝑀_𝑖)‖<𝛿,∀𝑚≥0,∀𝑡∈[0,𝜔].
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 𝜇𝑡,𝑆⁰,𝐼⁰−𝜇𝑡,𝑀_𝑖=𝜇𝑡^′,𝑃^𝑚𝑆⁰,𝐼⁰−𝜇𝑡^′,𝑀_𝑖<𝛿,∀𝑡≥0. (3.15) Let (𝑆(𝑡),𝐼(𝑡))=𝜇(𝑡,(𝑆⁰,𝐼⁰)). It then follows that 0≤𝐼_𝑖𝑗(𝑡)≤𝛿,∀𝑡≥0,∀𝑖,𝑗=1,2.
We have 𝑑𝑆₁₁(𝑡)(𝑡)𝑆𝑑𝑡≥𝐵+𝐶₁₁+𝑆₂₁+𝜌₁₂(𝑡)𝑆₁₂−𝜎₁(𝑡)+𝑑(𝑡)+𝛽(𝑡)𝛿𝑆₁₁,𝑑𝑆₁₂𝑑𝑡≥𝜎₁(𝑡)𝑆₁₁−𝜌₁₂(𝑡)+𝑑(𝑡)+𝛽(𝑡)𝛿𝑆₁₂,𝑑𝑆₂₁𝑑𝑡≥𝜎₂(𝑡)𝑆₂₂−𝜌₂₁(𝑡)+𝑑(𝑡)+𝛽(𝑡)𝛿𝑆₂₁,𝑑𝑆₂₂(𝑡)(𝑡)𝑆𝑑𝑡≥𝐵+𝐶₁₂+𝑆₂₂+𝜌₂₁(𝑡)𝑆₂₁−𝜎₂(𝑡)+𝑑(𝑡)+𝛽(𝑡)𝛿𝑆₂₂. (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 𝑑𝐼₁₁𝑑𝑡≥𝜌₁₂(𝑡)𝐼₁₂+𝛽(𝑡)𝑆^∗₁₁−𝜂𝐼₁₁+𝐼₂₁−𝜎₁(𝑡)+𝑑(𝑡)+𝛾(𝑡)𝐼₁₁,𝑑𝐼₁₂𝑑𝑡≥𝜎₁(𝑡)𝐼₁₁+𝛽(𝑡)𝑆^∗₁₂−𝜂𝐼₁₂+𝐼₂₂−𝜌₁₂(𝑡)+𝑑(𝑡)+𝛾(𝑡)𝐼₁₂,𝑑𝐼₂₁𝑑𝑡≥𝜎₂(𝑡)𝐼₂₂+𝛽(𝑡)𝑆^∗₂₁−𝜂𝐼₁₁+𝐼₂₁−𝜌₂₁(𝑡)+𝑑(𝑡)+𝛾(𝑡)𝐼₂₁,𝑑𝐼₂₂𝑑𝑡≥𝜌₂₁(𝑡)𝐼₂₁+𝛽(𝑡)𝑆^∗₂₂−𝜂𝐼₁₂+𝐼₂₂−𝜎₂(𝑡)+𝑑(𝑡)+𝛾(𝑡)𝐼₂₂. (3.17) Since 𝑟(Φ_{𝐹−𝑉−𝜂𝑀1}(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 𝑅⁴₊⧵{0}. By the aforementioned claim, it follows that 𝑀₀ and 𝑀₁ are isolated invariance sets in 𝑋,𝑊^𝑠(𝑀₀)∩𝑋₀=∅, and 𝑊^𝑠(𝑀₁)∩𝑋₀=∅. Clearly, every orbit in 𝑀_𝜕 converges to either 𝑀₀ or 𝑀₁, 𝑀₀ and 𝑀₁ are acyclic in 𝑀_𝜕. By [15, Theorem 1.3.1], 𝑃 is uniformly persistent with respect to (𝑋₀,𝜕𝑋). This implies the uniform persistence of the solutions of system (1.2) with respect to (𝑋₀,𝜕𝑋). By [6, Theorem 1.3.6], 𝑃 has a fixed point 𝑃(𝑆(0),𝐼(0))∈𝑋₀. Then, 𝑆(0)∈𝑅⁴₊,𝐼(0)∈Int(𝑅⁴₊). We further claim that 𝑆(0)∈𝑅⁴₊⧵{0}, suppose that 𝑆(0)=0, by (2.8), we can obtain −4(𝑑(𝑡)+𝛾(𝑡))(𝐼₁₁(0)+𝐼₁₂(0)+𝐼₂₁(0)+𝐼₂₂(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.

4. Numerical Simulations

In this section, we give the numerical solutions (1.2) to clarify the correctness of our theoretical results. We set 𝐵(𝑡)=0.4, 𝐶(𝑡)=0.12, 𝜌₂₁=0.16, 𝜌₁₂=0.051, 𝑑(𝑡)=0.3, 𝛾(𝑡)=0.365, 𝜎₁(𝑡)=0.65+0.04cos(𝜋𝑡/6), 𝜎₂(𝑡)=0.3+0.04cos(𝜋𝑡/6), 𝛽(𝑡)=𝛽+0.06588cos(𝜋𝑡/6). The initial value of the model is 𝑆₁₁(0)=0.8, 𝑆₁₂(0)=0.02, 𝑆₂₁(0)=0.03, 𝑆₂₂(0)=0.4, 𝐼₁₁(0)=0.1, 𝐼₁₂(0)=0.061, 𝐼₂₁(0)=0.03, 𝐼₂₂(0)=0.6. Figure 1 shows the numerical solutions of (1.2) when 𝛽=0.4. Because basic reproduction ratio 𝑅₀>1, a positive periodic solution exists, and the disease is uniform persistence. In Figure 2, 𝛽=0.25, the disease dies out because 𝑅₀<1.

Figure 1 

When 𝑅0>1, a positive periodic solution exists and the disease will be uniform persistence.

Figure 2 

When 𝑅0<1, a periodic disease-free solution exists and the disease dies out.

Acknowledgments

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).