Abstract

In this article, we study the global dynamical behavior of a two-strain SIS model with a periodic infection rate. The positivity and boundedness of solutions are established, and the competitive exclusion conditions are given for the model. The conditions for the global stability of the disease-free equilibrium and persistence of the model are obtained. The conditions of coexistence in this model are also found. Finally, the conditions of uniqueness of the solution are proved.

1. Introduction

Since the first work [1] about the mathematical epidemic model published, there were plenty of results of infectious diseases in modeling and dynamics. These epidemic models usually include two important parameters, the infection rate and recovery rate. Many of the epidemic models focus on the disease eradication and persistence. Among these models, some are applied successfully on infectious disease forecast and control [2–6]. In the analysis of the epidemic model, it is found that infectious diseases often fluctuate over time and exhibit periodic behavior [7–9]. This fluctuation is also observed in the real data [10, 11]. The influenza data from CDC [10] show that the number of influenza cases per week oscillates with a period between two peaks of one year and measles data [11] illustrate the period of two years. The periodic behavior of the incidence of many infectious diseases is caused by the influence of temperature and humidity on virus [12, 13]. In [14, 15], the authors introduced the epidemic models that had periodic coefficients to illustrate the periodical phenomenon. An epidemic SIS model with a periodic infection rate was first considered by Hethcote [15]. Furthermore, Dietz [14] considered SIR and SEIR models with a periodic infection rate.

The causative agents of many diseases are represented by multiple genetically distinct variants. Early autonomous multistrain models suggested that competitive exclusion is the only possible result of the competition of many strains [16]. However, these models disregard many mechanisms such as coinfection [17], mutation [18], cross immunity [19], and periodicity [20], which were proposed as possible mechanisms that could support diversity of causative agents. Castillo-Chavez et al. [21] established a sexually transmitted disease model with two competing strains and found the conditions of coexistence equilibrium and its global stability. The interesting work finished by Nuño et al. [19] considered a two-strain influenza model with cross immunity, and their simulation results showed that there might be up to four coexistence equilibria for their model. Another result about multiple-strain model was finished by Martcheva [20]. In this work, the author introduced a nonautonomous multistrain epidemic model without susceptible individuals and found the coexistence and persistence conditions of this model. All of these results also show that the analysis of multiple-strain model can extend our knowledge about the mechanism of the multiple-strain coexistence and competitive exclusion.

In this paper, we investigate a two-strain SIS model with a periodic infection ratewhere S is the number of susceptible individuals and is the number of individuals that are infected by the disease strain i. is the infection rate of strain i, and it is a periodic function with the period T. “M” here is the upper bound of . is the death rate of susceptible individuals. , where is the recovery rate of the infected individuals of strain i, . is the death rate of the infected individuals of strain i, and is the disease introduced death rate.

One of the most important concepts about the dynamics of epidemic models is the basic reproductive number , which represents the expected number of secondary cases produced by a typical infected individual in a fully susceptible population [22, 23]. According to the definition, it is easy to conclude that an epidemic will never occur when , and instead it will occur if . In this manuscript, we introduce two basic reproductive numbers and for our two-strain model and use these two values as the thresholds to analyze the dynamic behaviors of model (1).

This paper is organized as follows: In Section 2, we analyze the positivity and boundedness of model (1), as well as the global stability of the disease-free equilibrium. The competitive exclusion conditions are also gained in this section. In Section 3, we study the coexistence and stability of T-periodic positive solution of this model. In Section 4, the conditions of uniqueness of the solution are found. Finally, the numerical simulations are done to illustrate our results.

2. The Global Stability of the Disease-Free Equilibrium and the Competitive Exclusion

We first give the positivity and boundedness of model (1).

Theorem 1. Solutions of model (1) with positive initial conditions are positive and bounded.

Proof. Assume that the initial condition is with , then we haveThe solutions and are nonnegative, and they are positive when and .
Next, we prove the positivity of with positive initial values. Assuming the contrary and letting be the first time such that , by the first equation of model (1), we havewhich implies for and sufficient small . This contradicts that is the first time such that . It follows that for .
Next, we prove the boundedness. Let and α, we haveIf the initial condition holds, then by the comparison theoremThis inequality shows that the solution of model (1) is bounded.
This completes the proof.

Remark 1. In the rest of this paper, we assume that , where .
Model (1) has a disease-free equilibrium . So, we define the basic reproductive number of strain i as , where . Then, we have the following theorem.

Theorem 2. For model (1), if and , then the disease-free equilibrium is globally asymptotically stable.

Proof. The solution of model (1) satisfiesFor any given , there exists an integer number n and a real number s such that , where and . The inequality implies thatwhich implies that ; that is, for any , there exists a , such that . Let and . If , the first equation of (1) satisfiesThe comparison theorem implies thatSince , we have .
The limits and show that is globally attractive.
The Jacobin matrix at the disease-free equilibrium isThe three characteristic multipliers areThe condition implies that if , then we claim that the disease-free equilibrium is globally asymptotically stable.

We give the competitive exclusion conditions of model (1).

Theorem 3. (i) If hold, then all solutions of model (1) with satisfies . (ii) If hold, then all solutions of model (1) with satisfies .

Proof. To prove statement (i), we let and consider the functionWe haveThe condition implies thatwhich followsLetThen is an invariant set of model (1). By Lasalle’s principle, we know that all solutions of model (1) in D satisfy .
Using similar argument, we can obtain statement (ii).

These two conditions are usually called the principle of competitive exclusion [24, 25].

3. Coexistence

This section discusses the persistence of model (1).

Theorem 4. For model (1), if and , then there exists a positive constant , such that for at least one strain satisfies .

Proof. As is a global attractor for , we can choose small enough that for the system,If , then . There exists a time T that when , . Without loss of generality, from the conditions and , we can choose and small that, for any , , , and .
We prove by contradiction. Assume both of these two strains and satisfy and . Without loss of generality, we assume and .
As a consequence, for all , there holdand it is easy to see that and , which lead to a contradiction. Hence, the conclusion holds.

Moreover, if hold for all , then it is easy to get that

Therefore,where Q is a constant which is determined by the initial values . Then, the dynamical behavior of model (1) is equivalent to that of the following model:

We can get the following theorem about the above model.

Theorem 5. If , then there exists a positive constant ; then (1) all solutions of model (21) with positive initial conditions will satisfy , and model (21) will admit at least one positive periodic solution.

Proof. Since , there exists a small enough such that . Let us consider the following equation:and the solution of this equation (22) iswhere is a constant. As , the following inequalities hold:Thus, we can fix a small enough number such that .
Suppose, by contradiction, that Without loss of generality, we can assume that , it follows . By the first equation of model (21), we have the following inequality:Thus, there exists a time ; for all , we have . As a consequence, for all , there holdsSince , it is easy to see that , which leads to a contradiction. Hence,
Definewhere and . Then, assume is the solution of model (21) and is the Poincare map.
is globally attractive in for p, and is isolated invariant sets in X with . Clearly, every orbit in converges to . By Theorem 1.3.1 in [26], for a stronger repelling property of , we conclude that p is uniformly persistent with respect to . Thus, Theorem 3.1.1 in [26] implies the uniform persistence of the solutions of model (21) with respect to . By Theorem 1.3.6 in [26], p has a fixed point . Then and . We further claim that . Suppose , then we have , which is a contradiction. So from model (21), we can get ,. Then is a component-wise positive fixed point of p. Thus, is a positive T-periodic solution of model (21).

Remark 2. It is easy to conclude that when the condition and hold, there exist at least one persistent solution with respect to and . Moreover, when , model (21) becomes the plane system that .
Then, we consider a more special case that the infection rate of model (21) satisfies , where is a small number. When and , model (21) becomes an autonomous modelThe positive equilibrium for model (28) satisfiesAs , and implies , so there exists unique positive equilibrium for model (28). It is easy to check the Jacobin matrix at equilibrium isand its two eigenvalues satisfywhen , which implies that the real parts of and are negative. So the unique equilibrium is asymptotically stable.
The right-hand side of model (21) can be illustrated asIt is easy to see is analytic with and , and when , model (21) becomes (28). Then by Theorem 1.1 in Chapter 14 in [27], there exists a T-periodic solution near equilibrium for small , and by Theorem 1.2 in Chapter 14 in [27] , this solution is asymptotically stable.
Thus, we have the following theorem.