Abstract

In this paper, we discuss the global dynamics of a general susceptible-infected-recovered-susceptible (SIRS) epidemic model. By using LaSalle’s invariance principle and Lyapunov direct method, the global stability of equilibria is completely established. If there is no input of infectious individuals, the dynamical behaviors completely depend on the basic reproduction number. If there exists input of infectious individuals, the unique equilibrium of model is endemic equilibrium and is globally asymptotically stable. Once one place has imported a disease case, then it may become outbreak after that. Numerical simulations are presented to expound and complement our theoretical conclusions.

1. Introduction

In 1927, Kermack and McKendrick [1] proposed the compartment dynamic model, which uses differential equations research the dissemination rule of the infectious diseases. Since then, infectious disease dynamics has been developing vigorously and produced some important theoretical and practical valuable results. In classical Kermack-Mckendrick model, the whole population is divided into three subclasses: susceptible, infected, and recovered, which is also called SIR-type model. This type of model always assumed that the recovered individuals have permanent immunity. In fact, a plenty of diseases just have only a short period of acquired immunity, such as flu, tuberculosis, Hepatitis B, schistosomiasis, and the common cold. When acquired immunity disappears after some time, the recovered individuals become susceptible again [2]. To incorporate this phenomenon, SIRS epidemic models have emerged.

According to the total population, the SIRS epidemic model can be divided into two kinds: one kind is with constant total population, and the other kind is with varying total population. Theoretically, the analysis on basic properties and global asymptotic stability of SIRS model with varying total population size is more difficult. A large portion of works focus on some special cases of this type of SIRS epidemic model (see [3–5] and the references therein). Mena-Lorca and Hethcote [2] discussed an SIRS epidemic model with simple mass action incidence and constant immigration. They proposed the following open question: when the basic reproduction number is greater than one, the unique endemic equilibrium is globally asymptotically stable. Ma et al. also proposed the same open question in [6].

Nowadays, these open questions were gradually being solved. Li et al. [7] tried to study the global dynamical behaviors of SIRS model with incidence rate and constant immigration. They did obtain the global asymptotic stability of endemic equilibrium; however, this conclusion is valid only in the cases satisfying certain conditions but is not generally true. It is worth celebrating that the global asymptotic stabilities for classical SIRS model with general incidence rate are completely derived in [3, 8]. The method they used is LaSalle’s invariance principle and Lyapunov direct method. This is the most common method to study the local and global asymptotic stability of epidemic models. By using this method, Ma et al. [9] studied the local and global asymptotic stability of a SIQR model. Cui et al. [10] investigated the transmission dynamics of an epidemic model with vaccination, treatment, and isolation. Therefore, we wonder whether the global stability of SIRS model with general incidence rate and constant immigration can be obtained through this method or not.

Motivated by these studies, in this paper, we considered an SIRS epidemic model with constant immigration, especially with immigration of infected individuals. Notice that transmission of infection occurs after adequate contacts between the susceptible and the infectious. To better simulate this phenomenon, we considered a general incidence rate . Like in [8], function is assumed as a real locally Lipschitz function on and satisfied:(i) and for (ii), and is continuous and monotonously nonincreasing for

The same hypotheses on function can be found in [11]. This type of incidence rate was first proposed by Capasso and Serio [12] to model the cholera epidemic spread in Bari in 1973. Since then, many authors studied the different special case of ; for more details, one can see literature [11, 13–15].

Due to the fact that the infectious individuals or the recovery individuals may move from one place to another, in this paper, we assume that the rate constant of input to the total population is , of which fractions , and attribute to susceptible, infected, and recovered compartments. That is, the infectious individuals immigrate to infectious class at time at rate , and the recovery individuals immigrate to recovery class at time at rate . The rate constant for nature death is ; thus, is the average lifetime. The disease-related death rate for infected individuals is . After a period, the infected individuals might be declared cured of the disease and enter to the recovered class at rate ; thus, is the average recovery time. Due to the fact that the recovered individuals can only have temporary immunity, thus, the recovered individuals might lose their immunity and move to the infected class at rate . Therefore, the objective model in this paper can be given as

In this paper, we mainly use LaSalle’s invariance principle and Lyapunov direct method to investigate the global dynamics of model (1). The whole paper is organized as follows. In section 2, the existence and local stability of each equilibrium will be stated. The globally asymptotically stable state of equilibria will be given in the next section. Numerical simulations and conclusion will be listed in the last two sections.

2. Basic Properties

The total population number and satisfieswhich implies that . Therefore, the following domainis the positively invariant set with respect to model (1).

From the locally sign-preserving property of limit, we know that implies in the small domain of zero. Then, the assumption is continuous and monotonously nonincreasing for implies that for , that is, for . As the assumption is continuous and monotonously nonincreasing and also implies that , a direct calculation derived that . Then, it follows from and that . Therefore, from the assumptions (a) and (b) on , we have

To obtain the expressions of equilibria, we shall consider model (1) in two cases: one is and the other is , which, respectively, imply model (1) without input of infectious individuals and with constant input of infectious individuals. Case 1..  Any equilibrium of model (1) must satisfy  One can easily derive that model (1) always has a disease-free equilibrium . From van den Driessche and Watmough [16], we can derive that the basic reproduction number is Biologically, the basic reproduction number presents the average number of secondary infectious produced by single infective individuals, which are introduced into an entirely susceptible population.  Except disease-free equilibrium (i.e., the solution of (5) with ), equations (5) may have one positive solution, which can be denoted by . Then, a simple calculation derives the following:  And is the nonzero solution of the function   It follows from assumption (ii) that  Moreover,  Therefore, has a unique positive root in if .  In sum, we have the following result.

Theorem 1. When , we have the following:(i)Model (1) always has a disease-free equilibrium (ii)If , besides , model (1) also has a unique endemic equilibrium , where are given in (7) and the expression of is the unique positive root of given in (8)By analyzing the Jacobian matrix, the local stability of equilibria can be given as follows.

Theorem 2. For model (1) with , the disease-free equilibrium is locally asymptotically stable when , and it is unstable when , while the unique endemic equilibrium is locally asymptotically stable when .

Proof. Based on and , the Jacobian matrix at equilibrium of model (1) can be given asOne can easily obtain that the eigenvalues of the matrix are all less than zero when . This implies that is locally asymptotically stable. If , the matrix has a positive eigenvalue, which implies that is unstable.
For the endemic equilibrium , the Jacobian matrix is given bywhere . The characteristic equation of iswhereIt follows from the expression of and the conditions (4) of that if . Hence, andBy Routh–Hurwitz criterion, we obtain that the endemic equilibrium is locally asymptotically stable if . Case 2..  In this case, the existence and local stability of equilibrium for model (1) are stated as follows.

Theorem 3. When , model (1) has only one equilibrium, which is the endemic equilibrium , wherein the domain .

Proof. Due to , any equilibrium of model (1) can be given asOne can easily verify that all the solutions of equation (17) can not contain , which means that model (1) does not have disease-free equilibrium. Denote one solution of (17) by . Then, one can derive thatSubstituting (18) into the first equation of (17), we have that is the solution ofNext, we prove that equation (19) has only one solution in domain . DenoteNotice that is continuous and monotone nonincreasing for , that is, . It follows from the expression of in (18) thatTherefore, , which means that is a nondecreasing function of in . Furthermore, it follows from the expression of and thatMeanwhile, one can verify that in , , and . Therefore, there exists only one solution in domain for equation (19). This completes the proof.
Similar to the proof of Theorem 2, the eigenvalues of Jacobian matrix are all negative, which implies the following theorem.

Theorem 4. When , the unique endemic equilibrium of model (1) is locally asymptotically stable.

3. Global Dynamics of Model (1)

For the global dynamical behaviors of model (1), we study them in two cases: and , as discussed in section 2. To obtain the globally asymptotic stability of equilibria, we can rewrite model (1) as the following equivalent model:The dynamical behaviors of model (1) and model (23) are consistent.

We first discuss the global dynamics of model (1) in the case of . Case 1..

 In this case, by considering Lyapunov function and using LaSalle’s invariance principle [17], the global stability of the disease-free equilibrium can be derived. We can conclude this with following theorem.

Theorem 5. When and , then the disease-free equilibrium of model (1) is globally asymptotic stability in .
For the global stability of endemic equilibrium , we have the following theorem.

Theorem 6. When and , the endemic equilibrium for model (1) is globally asymptotically stable in .

Proof. Notice that is the unique solution of (17) when . When , it follows from the first equation of (17) that satisfiesThen, we can rewrite in (23) asHence, the following modelis the equivalent model of (23).
Define a Lyapunov functionOne can testify that except at . The derivative of along solutions of (26) isThe assumption that is a continuous and monotonously nonincreasing function for , which means thatTherefore, is a negative definite in . Using the Lyapunov stability theorem in [18], we know that of (23) is globally asymptotically stable. This completes the proof.

Remark 1. If , then , and model (1) becomesFrom Theorems 5 and 6, one can directly obtain that disease-free equilibrium and endemic equilibrium for this model are both globally asymptotically stable. In fact, model (30) is also a special case of literature [8].  Case 2.   For , except assumptions (i) and (ii), we add one assumption (iii). That is, for this case, function satisfies.(i) and for (ii), where is continuous and monotonously nonincreasing for (iii) as a nondecreasing function of There are many incidence rates that satisfy assumptions (i)-(iii), such as bilinear incidence stated in [1], saturated incidence rate discussed in [12].
Similar to the proof of the case , the following theorem holds.

Theorem 7. When , the endemic equilibrium of model (1) is globally asymptotically stable in .

Proof. Similarly, we can rewrite model (23) as the following model:Define the Lyapunov function bywhich is the same as given in the case of .
Taking the derivative along (31) derives thatThe assumption as a nondecreasing function of implies that . Therefore, is a negative definite. By the Lyapunov stability theorem [18], the globally asymptotic stability of endemic equilibrium is proved.