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Stability Analysis of a Multigroup SEIR Epidemic Model with General Latency Distributions
The global stability of a multigroup SEIR epidemic model with general latency distribution and general incidence rate is investigated. Under the given assumptions, the basic reproduction number is defined and proved as the role of a threshold; that is, the disease-free equilibrium is globally asymptotically stable if , while an endemic equilibrium exists uniquely and is globally asymptotically stable if . For the proofs, we apply the classical method of Lyapunov functionals and a recently developed graph-theoretic approach.
Mathematical models have become important tools in analyzing the spread and control of infectious diseases. The SIR model is one of the most popular ones in this field, for which the total population is subdivided into three compartments: susceptible, infectious, and removed. For some diseases, it is reasonable to include a latent (or exposed) class for those susceptible individuals who are infected with the disease but are not yet infectious, which leads to SEIR model [1–6]. Let , , , and be the numbers of individuals in the susceptible, exposed, infectious, and removed compartments, respectively, with the total population . Suppose that represents the constant recruitment rate and the natural mortality rate. Assuming mass action for the disease transmission and letting denote the effective contact rate, the rate of change of is Taking into consideration a general exposed distribution, van den Driessche et al.  formulated and studied the following model: where is the rate at which infective individuals recover. is constant total populations. It is assumed in  that individuals rarely die of the disease and the disease-induced death is negligible, which ensures a constant population; that is, denotes the probability (without taking death into account) that an exposed individual still remains in the exposed class time units after entering the exposed class and it satisfies the following.
() is nonincreasing, piecewise continuous with possibly finitely many jumps and satisfies , with being positive and finite.
In fact, the integral term in model (2) is in the sense of Riemann-Stieltjes integrals; the second equation of (2) takes the following form: where . It follows from total population size which is constant that the rate of change of is governed by Thus, model (2) can be written as the system Recently, a model of this type including the possibility of disease relapse has been proposed in [5, 6] to study the transmission and spread of some infectious diseases such as herpes, and its global dynamics have been completely investigated in [5, 7].
Heterogeneity in the host population can result from different contact modes such as those among children and adults for childhood diseases (e.g., measles and mumps) or different behaviors such as the numbers of sexual partners for some sexually transmitted infections (e.g., herpes and condyloma acuminatum). Taking into consideration different contact patterns, distinct number of sexual partners, or different geography and so forth, it is more proper to divide individual hosts into groups. Therefore, lots of multigroup models have been proposed in the literature to describe the transmission of infectious disease in heterogeneity environment (see [8–17] and references cited therein).
In multigroup epidemic models, a heterogeneous host population is divided into several homogeneous groups according to modes of transmission, contact patterns, or geographic distributions, so that within-group and intergroup interactions can be modeled separately. In this paper, we formulate a multigroup SEIR epidemic model with general exposed distribution and general incidence rates. The population is divided into distinct groups (). For , the th group is further partitioned into four compartments: susceptible, exposed, infectious, and recovered, whose numbers of individuals at time are denoted by , , , and , respectively. Within the th group, represents the growth rate of , which includes both the production and the natural death of susceptible individuals.
In , Zhang et al. studied a multigroup SEIR epidemic model with general exposed distribution and general incidence rates. By using the well-known “linear chain trick,” the authors reformulate the model into an equivalent ordinary differential equations system. The global stability results of equilibria are obtained by constructing suitable Lyapunov functionals for general incidence rate function . In , Hattaf et al. introduced a general incidence rate in a delayed SIR epidemic model.
Motivated by these facts, in this paper, we incorporate the general incidence rate presented in  to the following system of differential and integral equations: where , the nonlinear term represents the cross-infection from group to group , denotes the natural death rates of exposed and infectious classes in the th group, and denotes the production of the recovered individuals from infectious ones in the th group. All constants , , , are assumed to be positive.
The organization of this paper is as follows; in the next section, we give some preliminaries of our main model. In Section 3, we prove the global asymptotic stability of the disease-free equilibrium for using the classical method of Lyapunov. The existence of endemic equilibrium is also proved. In Section 4, we prove global asymptotic stability of an endemic equilibrium for using the graph-theoretic approach.
Since the variables and do not appear in the first and third equations of (6), we can only consider the reduced system as follows:
The incidence function in (7) is assumed to be continuously differentiable in the interior of and to satisfy the following hypotheses:(S1), for all ;(S2), for all and ;(S3), for all and ; assume that the functions satisfy the following conditions:(S4) are local Lipschitz on with , and there is a unique positive solution for the equation ; for , and for . Typical examples of satisfying include common incidence functions such as The class of that satisfies () includes both and , which have been widely used in the literature of population dynamics [1, 8].
For model (7), the existence, uniqueness, and continuity of solutions follow from the theory for integrodifferential equations in . It can be easily verified that every solution of (7) with nonnegative initial conditions remains nonnegative. It follows from and the first equation in (7) that , and thus From the biological significance, we only need to consider (7) in the following region: Indeed, one can easily verify that the set is positively invariant with respect to (7).
It is clear that system (7) has a disease-free equilibrium in . Next, we will give some notations which will be useful for our main results.
Let It can be verified that .
Since the limiting system (12) contains an infinite delay, its associated initial condition needs to be restricted in an appropriate fading memory space. For any , define the following Banach space of fading memory type (see [23, 24] and references therein): with norm . Let and be such that , .
Let such that for all . We consider solutions of system (12), , with initial conditions The standard theory of functional differential equations  implies for all . We study system (12) in the following phase space: It can be verified that solutions of (12) in with initial conditions (14) remain nonnegative.
An equilibrium in the interior of is called an endemic equilibrium of system (12), where satisfy the equilibrium equations Set to denote the special radius of the matrix , where The parameter is defined as the basic reproduction number [25, 26]. Since it can be verified that system (7) satisfies conditions of Theorem 2 of , we have the following lemma.
Lemma 1. For system (7), the disease-free equilibrium is locally asymptotically stable if while it is unstable if .
3. Global Stability of the Disease-Free Equilibrium
Theorem 2. Assume that the functions and satisfy , and is irreducible. (i)If , then is the unique equilibrium of system (7), and is globally asymptotically stable in .(ii)If , then is unstable and system (7) is uniformly persistent.
Proof. It follows from the Perron-Frobenius theorem (see Theorem in ) that the nonnegative irreducible matrix has a positive eigenvector such that
Now, we construct a Lyapunov functional
Differentiating along the solution of system (7) and under () and (), we obtain
where . Suppose that . Then, if and only if . Suppose that . Then, it follows from (20) that implies
If , then
which implies that (21) has only the trivial solution . Therefore, if and only if or provided . It can be verified that the only compact invariant subset of the set where is the singleton . By LaSalle’s Invariance Principle, is globally asymptotically stable in if .
If and , it is easy to see that It follows from the continuity that holds in a small neighborhood of . This implies that is unstable. Using a uniform persistence result from  and similar arguments as in [4, 10, 13, 16, 17], we know that, if , the instability of implies the uniform persistence of (7) in ; that is, there exists a positive constant such that The uniform persistence of system (7) together with the uniform boundedness of solutions in , which follows from the positive invariance of , implies the existence of an endemic equilibrium in (see Theorem of  or Theorem D.3 of ). Summarizing the statements above, if , system (7) is uniformly persistent and there exists at least one endemic equilibrium in . This completes the proof.
4. Global Stability of an Endemic Equilibrium
Denote Obviously, attains its strict global minimum at and .
To get the global stability of , we make the following assumptions:(S5) for ;(S6) for ;(S7))) − /(, for .
Theorem 3. Assume that the functions and satisfy , and the matrix is irreducible. If , then there is a unique endemic equilibrium for system (12), and is globally asymptotically stable in the interior of .
Proof. Define a Lyapunov functional as where First, we calculate the derivative of ; then, we have Calculating the time derivative of along the solution of system (12), we have Using equilibrium equations (16), we have Using , we rewrite (30) as Therefore, Furthermore, under ()–(), we have Obviously, the equalities in (33) hold if and only if and , . Therefore, the functional as defined in Theorem 3.1 of  is a Lyapunov function for system (12). Using similar arguments as in [4, 8–13, 16, 17], one can show that the largest invariant subset where is the singleton . By LaSalle’s Invariance Principle, is globally asymptotically stable in the interior of . This completes the proof of Theorem 3.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the anonymous referees and the editor for very helpful suggestions and comments which led to improvements of our original paper. J. Wang is supported by National Natural Science Foundation of China (no. 11201128), the Science and Technology Research Project of the Department of Education of Heilongjiang Province (no. 12531495), the Natural Science Foundation of Heilongjiang Province (no. A201211), and the Science and Technology Innovation Team in Higher Education Institutions of Heilongjiang Province.
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