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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 354287, 11 pages
Stability Analysis of a Multigroup Epidemic Model with General Exposed Distribution and Nonlinear Incidence Rates
1School of Science, Department of Fundamental Mathematics, Jiamusi University, Jiamusi 154007, China
2School of Mathematical Science, Heilongjiang University, Harbin 150080, China
Received 26 January 2013; Revised 16 June 2013; Accepted 22 July 2013
Academic Editor: Pagavathi Balasubramaniam
Copyright © 2013 Ling Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We investigate a class of multigroup epidemic models with general exposed distribution and nonlinear incidence rates. For a simpler case that assumes an identical natural death rate for all groups, and with a gamma distribution for exposed distribution is considered. Some sufficient conditions are obtained to ensure that the global dynamics are completely determined by the basic production number . The proofs of the main results exploit the method of constructing Lyapunov functionals and a graph-theoretical technique in estimating the derivatives of Lyapunov functionals.
Multigroup epidemic models have been used in the literature to describe the transmission dynamics of many different infectious diseases such as mumps, measles, gonorrhea, HIV/AIDS and vector borne diseases such as Malaria . In the models, heterogeneous host population can be 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. It is well known that global dynamics of multigroup models with higher dimensions, especially the global stability of the endemic equilibrium, are a very challenging problem. Guo et al.  proposed a graph-theoretic approach to the method of global Lyapunov functions and used it to resolve the open problem on the uniqueness and global stability of the endemic equilibrium of a multigroup SIR model with varying subpopulation sizes. Subsequently, a series of studies on the global stability of multigroup epidemic models were produced in the literature (see e.g., [2–5]).
In the present paper, a more general multigroup epidemic model is proposed and studied to describe the disease spread in a heterogeneous host population with general exposed distribution and nonlinear incidence rate. The host population is divided into distinct groups (). For , the th group is further partitioned into four disjoint classes: the susceptible individuals, exposed individuals, infectious individuals, and recovered individuals, whose numbers of individuals at time are denoted by , and , respectively. Susceptible individuals infected with the disease but not yet infective are in the exposed (latent) class.
It is pointed in  that a fixed latent period can be considered as an approximation of the mean latent period, and this would be appropriate for those diseases whose latent periods vary only relatively slightly. For example, poliomyelitis has a latent period of 1–3 days (comparing to its much longer infectious period of 14–20 days). However disease such as tuberculosis, including bovine tuberculosis (a disease spread from animal to animal mainly by direct contact), may take months to develop to the infectious stage and also can relapse. Since the time it takes from the moment of new infection to the moment of becoming infectious may differ from disease to disease, even for the same disease, it differs from individual to individual, and it is indeed a random variable. It is thus of interest from both mathematical and biological viewpoints to investigate whether sustained oscillations are the result of general exposed distribution.
Following the method of , we also assume that the disease does not cause deaths during the latent period, taking the natural death rate into consideration. Let denote the probability that an exposed individual remains in the time after entering the exposed class. For , denotes the coefficient of transmission between compartments and . It is assumed that -square matrix is irreducible . So the proportion of exposed individuals can be expressed by the integral where the sum takes into account cross-infections from all groups. Integrals in (1) are in the Riemann-Stieltjes sense. satisfies the following reasonable properties: is nonincreasing, piecewise continuous with possibly finitely many jumps and satisfies , and with is positive and finite.
Differentiating (1) gives The first term on the right hand side in (2) is the rate at which new infected individuals come into the exposed class, and the last term explains the natural deaths. The second term accounts for the rate at which the individuals move to the infectious class (noting that due to the aformentioned property) from the exposed class; hence Let be the probability density function for the time (a random variable) it takes for an infected individual in the th group to become infectious. Then (4) becomes
Within the th group, denotes the growth rate of , which includes both the production and the natural death of susceptible individuals. Therefore, under the assumptions, the model to be studied takes the following differential and integral equations form:
Since the variables and do not appear in the first and third equations of model (5), and , , can be decoupled from the and equations; we only need to consider the subsystem of (5) consisting of only the and equations: where denotes the natural death rates of compartments in the th group, is the death rate caused by disease in the th group, and is the rate of recovery of infectious individuals in the th group. In what follows we investigate the global stability of system (5).
When , , and with bilinear incidence rate, system (5) will reduce to the standard SEIR ordinary differential equation (ODE) model studied in [8, 9], and if we further assume that is a step function, system (5) becomes the SEIR model with a discrete delay studied in . Recently, a model of this type, but including the possibility of disease relapse, has been proposed in [11, 12] to investigate the transmission of herpes, and its global dynamics have been completely investigated in [5, 13].
To express the main idea and the approaches more clearly, we consider a simpler case in which all groups share the same natural death rate: for . Further, we assume that the functions are disease specific only, implying that for . We choose the gamma distribution: where is a real number and is an integer, which is widely used and can approximate several frequently used distributions. For example, when , will approach the Dirac delta function, and when , is an exponentially decaying function.
The main object of this paper is to carry out the well-known “linear chain trick” to system (6), transfer system into an equivalent ordinary differential equations system, and establish its global dynamics. We derive the basic reproductive number and show that completely determines the global dynamics of system (6). More specifically, if , the disease-free equilibrium is globally asymptotically stable and the disease dies out; if , a unique endemic equilibrium exists and is globally asymptotically stable, and the disease persists at the endemic equilibrium. The global stability of rules out any possibility for Hopf bifurcations and the existence of sustained oscillations. We should point out here that this work is motivated by Yuan and Zou [11, 12, 14]. In the proof we demonstrate that the graph-theoretic approach developed in [2, 3] can be successfully applied to construct suitable Lyapunov functionals and thus prove the global stability of the endemic equilibrium for model (6) with general exposed distribution and nonlinear incidence rate. Our work is also based on a recent work by Sun and Shi , which resolved the dynamics of multigroup SEIR epidemic models with nonlinear incidence of infection and nonlinear removal functions between compartments.
In Section 2, we first give the model, preliminaries and the basic reproduction number . The global stability of the corresponding equilibria for and is shown, respectively, in Section 3—the key results of this paper. And in Section 4, some numerical simulations are shown to illustrate the effectiveness of the proposed result.
We make the following basic assumptions for the intrinsic growth rate of susceptible individuals in the th group and the transmission functions . are non-increasing functions on with , and there is a unique positive solution for the equation . for , and for ; that is for all . , , .
Following the technique and method in , define which can absorb the exponential term into the delay kernel. The second equation in (6) can be rewritten as For , let Thus, for , we obtain For , we have It follows that
Thus the integro-differential system (6) is equivalent to the ordinary differential equations
For initial condition the existence, uniqueness, and continuity of the solution () of system (15) follow from the standard theory of Volterra integro-differential equation . It can also be verified that every solution of (15) with nonnegative initial condition remains nonnegative.
It follows from () and the first equation of (15) that for all . Let be the maximum of the function on and let be a positive real number such that . Denote by the th tube for system (15); that is, It follows from a similar argument to that in  that we can show that the set defined by is a forward invariant compact absorbing set for system (15) for and that the set (i.e., when ) is a forward invariant compact set.
Under the assumption , we know that system (15) always has the disease-free equilibrium An equilibrium of (6) has the form with , , . Translating to the equivalent system (15), is corresponding to in the interior of is called an endemic equilibrium, and it satisfies the following equilibrium equations: The basic reproduction number is defined as the expected number of secondary cases produced by single infectious individual during its entire period of infectiousness in a completely susceptible population. For system (15), we can calculate it as the spectral radius of a matrix called the next generation matrix. Let Then the next generation matrix is and hence, the basic reproduction number is where and denote the spectral radius and the set of eigenvalues of a matrix, respectively. Since it can be verified that system (15) satisfies conditions of Theorem 2 of , we have the following proposition.
Lemma 1. For system (15), the disease-free equilibrium is locally asymptotically stable if , while it is unstable if .
Following the method of , one defines a matrix whose spectral radius has a similar threshold property to that of , since both of the nonnegative matrices and are irreducible, and hence from the Perron-Frobenius theorem  that their spectral radii are given by each of their simple eigenvalues. Thus, we obtain . Then the following lemma immediately follows.
Lemma 2. if and only if .
3. Main Results
The following main theorems are summarized in terms of system (15).
Theorem 3. Assume that the functions and satisfy assumptions , and the matrix is irreducible and is defined by (24). (i)If , then is the unique equilibrium of system (15), and is globally asymptotically stable in . (ii)If , then is unstable, and system (15) is uniformly persistent in .
Proof. Let us define , where . Note that . Since is irreducible, the matrix is also irreducible.
First we claim that there does not exist any endemic equilibrium in . Suppose that . Then we have . Since nonnegative matrix is irreducible, it follows from the Perron-Frobenius theorem (see Corollary 2.1.5 of ) that . This implies that has only the trivial solution , where . Hence the claim is true. Next we claim that the disease-free equilibrium is globally asymptotically stable in . From the Perron-Frobenius (see Theorem 2.1.4 of ), the nonnegative irreducible matrix has a strictly positive left eigenvector associated with the eigenvalue such that Using this positive eigenvector, we construct the following Lyapunov function: Computing the derivative of along the solutions of (15) in , we get Thus, under the assumption , , and if and only if and . Suppose that . Then it follows from the previous that implies Hence, if , then and thus is the only solution of (29). Summarizing the statements, we see that if and only if or , which implies that the compact invariant subset of the set where is only the singleton . Thus, by LaSalle’s invariance principle , it follows that the disease-free equilibrium is globally asymptotically stable in .
If , then and then, by continuity, we can obtain in a neighborhood of in ; then is unstable.
Assume . By the uniform persistence result from  and a similar argument as in the proof of , the instability of implies the uniform persistence of (15). This together with the dissipativity of (15) resulted from the forward invariant and compact property of stated previously, implies which that (15) has an equilibrium in , denoted by (see, e.g., Theorem D.3 in ).
In what follows we prove that the endemic equilibrium of system (15) is globally asymptotically stable when .
Throughout the paper, we denote Then for and has global minimum at .
For convenience of notations, set , and Then, is also irreducible. One knows that the solution space of the linear system has dimension 1 and gives a base of this space, where , is the cofactor of the th diagonal entry of . To get the global stability of , the following assumptions in  are proposed: : for , , : For , . : For , A difficult mathematical question for system (15) is that of whether the endemic equilibrium is unique when and whether is globally asymptotically stable when it is unique. Our main global stability result is given.
Proof. We show that is globally asymptotically stable in , which implies that there exists a unique endemic equilibrium.
Consider a Lyapunov function as
This function has a linear part expressed by First, calculating the derivatives of , we obtain
Calculating the time derivative of along the solutions of system (15) and using equilibrium equation (21), we have It follows from the assumptions - that can be estimated by From the assumption and (32), we know that where .
Obviously, the equalities in (41) and (42) hold if and only if That is, . We can show that and satisfy the assumptions of Theorem 3.1 and Corollary 3.3 in . Therefore, the function is a Lyapunov function for system (15); namely, for . One can only show that the largest invariant subset, where , is the singleton by the same argument as in [2–5, 13, 21]. By LaSalle’s invariance principle, is globally asymptotically stable in . This completes the proof of Theorem 4.
Remark 5. We show a complete proof for global asymptotic stability of unique endemic equilibrium of system (15). In the case of , system (15) will reduce to the system studied in [14, 22]. Here Theorem 4 extends related results in [14, 22] to a result to a more general case allowing a nonlinear incidence rate. Our result also cover the related results of single group model in  for the case of .
4. Numerical Example
Consider the system (15) when , , , and , . One then has a two-group model as follows: If we choose parameters as , , , and , we can compute , and hence is the unique equilibrium of system (45) and it is globally stable from Theorem 4 (see Figures 1 and 2).
On the other hand, if are chosen as , , , and , we can compute , and hence (0.347644, 0.0760948, 0.0760948, 4.51674, 0.330353, 0.0765909, 0.0765909, 4.4678) is the unique equilibrium of system (45) and it is globally stable from Theorem 4 (see Figures 3 and 4).
The authors wish to thank the reviewers for their valuable comments and suggestions that led to truly significant improvement of the paper. J. Wang is supported by the Science and Technology Research Project of the Department of Education of Heilongjiang Province (no. 12531495).
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