Competitive Coexistence in a Two-Strain Epidemic Model with a Periodic Infection Rate

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


Introduction
Since the first work [1] about the mathematical epidemic model published, there were plenty of results of infectious diseases in modeling and dynamics. ese 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][3][4][5][6]. In the analysis of the epidemic model, it is found that infectious diseases often fluctuate over time and exhibit periodic behavior [7][8][9]. is fluctuation is also observed in the real data [10,11]. e 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. e 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. e 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. e 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 multiplestrain 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 rate where S is the number of susceptible individuals and I i (i � 1, 2) is the number of individuals that are infected by the disease strain i. 0 < β i (t) < M(i � 1, 2) 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 the infected individuals of strain i, and α i > 0 is the disease introduced death rate.
One of the most important concepts about the dynamics of epidemic models is the basic reproductive number R 0 , 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 R 0 < 1, and instead it will occur if R 0 > 1. In this manuscript, we introduce two basic reproductive numbers R 1 0 and R 2 0 for our two-strain model and use these two values as the thresholds to analyze the dynamic behaviors of model (1). is 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. e 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.
Next, we prove the positivity of S(t) with positive initial values. Assuming the contrary and letting t 0 > 0 be the first time such that S(t 0 ) � 0, by the first equation of model (1), we have dS(t) which implies S(t) < 0 for t ∈ (t 0 − ε, t 0 ) and sufficient small ε > 0. is contradicts that t 0 is the first time such that S(t 0 ) � 0. It follows that S(t) > 0 for t > 0.
Next, we prove the boundedness. Let N(t) � S(t)+ I 1 (t) + I 2 (t) and α, we have If the initial condition 0 ≤ N(0) ≤ (λ/d) holds, then by the comparison theorem is inequality shows that the solution of model (1) is bounded.
is completes the proof.
□ Remark 1. In the rest of this paper, we assume that (S(0), . en, we have the following theorem.
Proof. e solution of model (1) satisfies For any given t ≥ 0, there exists an integer number n and a real number s such that t � nT + s, where 0 ≤ s < T and Discrete Dynamics in Nature and Society which implies that lim t⟶∞ I i (t) � 0; that is, for any ε > 0, there exists a T i > 0, such that e comparison theorem implies that Since e three characteristic multipliers are , then we claim that the disease-free equilibrium is globally asymptotically stable.
We give the competitive exclusion conditions of model (1).
Proof. To prove statement (i), we let m � (k 1 /k 2 ) and consider the function We have which follows Let en D 0 � (S, I 1 , I 2 ) | I 1 � 0 is an invariant set of model (1). By Lasalle's principle, we know that all solutions of model (1) in D satisfy lim t⟶∞ I 1 (t) � 0.
Using similar argument, we can obtain statement (ii). ese two conditions are usually called the principle of competitive exclusion [24,25]. Proof. As S E � λ/d is a global attractor for (dS/dt) � λ − dS, we can choose small enough δ > 0 that for the system, If 0 < η < δ, then ∀ε > 0. ere exists a time T that when t > T, |S(t) − S E | < ε. Without loss of generality, from the conditions T 0 (β 2 (t)S E /1 + S E ) > 1, we can choose δ * > 0 and small ε that, for any t > T, We prove by contradiction. Assume both of these two strains I 1 and I 2 satisfy lim sup t⟶∞ I 1 (t) < δ * and Discrete Dynamics in Nature and Society lim sup t⟶∞ I 2 (t) < δ * . Without loss of generality, we assume I 1 (t) < δ * and I 2 (t) < δ * .
As a consequence, for all t ≥ T, there hold and it is easy to see that lim t⟶∞ I 1 (t) � +∞ and lim t⟶∞ I 2 (t) � +∞, which lead to a contradiction. Hence, the conclusion holds. Moreover, if (β 2 (t)/β 1 (t)) � (k 2 /k 1 ) � m hold for all t ≥ 0, then it is easy to get that dI 2 dI 1 � m erefore, where Q is a constant which is determined by the initial values I 1 (0), I 2 (0). en, the dynamical behavior of model (1) is equivalent to that of the following model: We can get the following theorem about the above model.  (21) with positive initial conditions will satisfy lim inf t⟶∞ I 1 (t) ≥ δ, and model (21) will admit at least one positive periodic solution.
Proof. Since R 1 0 � μ 1 (S E )/k 1 > 1, there exists a small enough η > 0 such that μ 1 (S E − η) > k 1 . Let us consider the following equation: and the solution of this equation (22) is where C 0 is a constant. As 0 < β 1 (t) < M, the following inequalities hold: us, we can fix a small enough number δ � δ * > 0 such that lim t⟶∞ x(t) > S E − η.

Uniqueness of the Solution on the Surface
In this section, we mainly talk about the uniqueness of the solution of model (21), and the solution is on the surface where β(t) � b 0 + ε cos(t) and b 0 � 8, λ � 1, c 1 � c 2 � 1, d � 1, k � 2, Q � 2, and m � 2.

Numerical Simulation
Numerical simulations are performed to illustrate the dynamic behaviors of model (1) and (21). Figures 1(a) and 1(b) show the solution of (21) under the conditions R 1 0 � 1.5 and R 1 0 � 2. From these two figures, we can see that there exists one stable periodic solution of model (21) under different values of R 0 , and when the value of R 0 becomes larger, the mean value of I(t) becomes larger.

Discussion
In this paper, we study the dynamical behaviors of a twostrain SIS model with a periodic infection rate and get its 6 Discrete Dynamics in Nature and Society  persistence, competitive exclusion, and coexistence conditions. We gain the global stability conditions of the diseasefree equilibrium E 0 and establish the competitive exclusion condition of the two strains. e coexistence and uniqueness are also discussed and proved. Our results lead to a new insight into the mechanism of two strains interaction and provide a new approach to investigate the inference of the periodic infection rate on the coexistence of two strains. It is  (1) under the conditions (a) β 1 (t)/k 1 � 3 + sin(t) and β 2 (t)/k 2 � 2.85 + 0.95 sin(t) and (b) β 1 (t)/k 1 � 2.72 + 0.91 sin(t) and β 2 (t)/k 2 � 3 + sin(t).
8 Discrete Dynamics in Nature and Society worth to mention that the analysis in this article can be applied to the n-strain epidemic model with a periodic infection rate and obtain similar results.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest. Discrete Dynamics in Nature and Society 9