Global stability of an epidemic model with stage structure and nonlinear incidence rates in a heterogeneous host population

In this paper, we study an epidemic model with stage structure and latency spreading in a heterogeneous host population. We show that if the disease-free equilibrium exists, then the global dynamics are determined by the basic reproduction number R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R_{0}$\end{document}. We prove that the disease-free equilibrium is globally asymptotically stable when R0≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R_{0}\leq1$\end{document}; and there exists a unique endemic equilibrium which is globally asymptotically stable when R0>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R_{0}>1$\end{document}. The global stability of the endemic equilibrium is also proved by using a graph-theoretic approach to the method of Lyapunov functions. Finally, numerical simulations are given to illustrate the main theoretical results.


Introduction
A heterogeneous host population can be divided into several homogeneous groups according to models of transmission, contact patterns, or geographic distributions. Multigroup epidemic models have been proposed in the literature of mathematical epidemiology to describe the transmission dynamics of infectious diseases in heterogeneous host populations, such as measles, mumps, gonorrhea, HIV/AIDS, West-Nile virus and vectorborne diseases such as malaria. Various forms of multi-group epidemic models have subsequently been studied to understand the mechanism of disease transmission. One of the most important subjects in this field is to obtain a threshold that determines the persistence or extinction of a disease. Guo et al. in [] developed a graph-theoretic approach to prove the global asymptotic stability of a unique endemic equilibrium of a multi-group epidemic model. By applying the idea, global stability of endemic equilibrium for several classes of multi-group epidemic models was investigated in [-].
In the real world, some epidemics, such as malaria, dengue, fever, gonorrhea and bacterial infections, may have a different ability to transmit the infections in different ages. For example, measles and varicella always occur in juveniles, while it is reasonable to consider the transmission of diseases such as typhus, diphtheria in adult population. In recent years, epidemic models with stage structure have been studied in many papers [-]. For some disease (for example, tuberculosis, influenza, measles), on adequate contact with an infective, a susceptible individual becomes exposed, that is, infected but not infective. This individual remains in the exposed class for a certain period before becoming infective (see, for example, [-]).
In this paper, we formulate an epidemic model with latency spreading in a heterogeneous host population. Let S () k , S () k , E k , I k and R k denote the immature susceptible, mature susceptible, infected but non-infectious, infectious and recovered population in the kth group, respectively. The disease incidence in the kth group can be calculated as where the sum takes into account cross-infections from all groups and β (i) kj is the transmission coefficient between compartments S (i) k and I j . G j (I j ) includes some special incidence functions in the literature. For instance, G j (I j ) = I j +α j I j (saturation effect). Let d () k and d () k represent death rates of S () k and S () k populations, respectively. Then we obtain the following model for a disease with latency: where ϕ k (S () k ) denotes the net growth rate of the immature susceptible class in the kth group (a typical form of k the natural death rate). a k is the conversion rate from an immature individual to a mature individual in group k. η k represents the rate of becoming infectious after a latent period in the kth group. d k , μ k and γ k are the natural death rate, the disease-related death rate and the recovery rate in the kth group, respectively. All parameter values are assumed to be nonnegative and a k , η k , d (i) k , d k > .

Remark
The model () can be regarded as an SVEIR model such that S () k is unvaccinated and S () k is vaccinated with vaccination rate a k . References studied on the SVEIR model can be seen in [, ] and so on.
Since the variable R k does not appear in the remaining four equations of (), if we denote m k := d k + μ k + γ k , then we can obtain the following reduced system: The initial conditions for system () are The organization of this paper is as follows. In the next section, we prove some preliminary results for system (). In Section , the main theorem of this paper is stated and proved. In the last section, a brief discussion and numerical simulations which support our theoretical analysis are given.

Preliminaries
We assume: (A) ϕ k and G k are Lipschitz on [, +∞); (A) ϕ k is strictly decreasing on [, +∞), and there exists S () k >  such that x is nonincreasing on (, +∞) and From our assumptions, it is clear that system () has a unique solution for any given initial conditions () and the solution remains nonnegative. If (A) holds, then we see that system () has a disease-free equilibrium For two nonnegative n-square matrices A = (a kj ) and B = (a kj ), we write A ≤ B if a kj ≤ b kj for all k and j, and A < B if A ≤ B and A = B. Following [], we set matrices The next generation matrix for system () is Thus, we obtain the basic reproduction number R  for system () as where ρ denotes the spectral radius.
We derive from () that the region is positively invariant with respect to (). Let • denote the interior of .

Main results
In the section, we study the global stability of equilibria of system ().
Notice that B is irreducible, then Q(S, I) and Q are irreducible. By (A), we have  ≤ Q(S, I) ≤ Q. Hence Q(S, I) + Q is also irreducible. That is,  ≤ Q(S, I) < Q and Q(S, I) + Q is irreducible provided that S = S  . Thus, by [], Corollary ., p., ρ(Q(S,

Consider a Lyapunov functional
Differentiating L along the solution of system (), we obtaiṅ Since ϕ k is strictly decreasing on [, +∞), we have where equality holds if and only if S () k = S () * k , k = , , . . . , n.
We further make the following assumption: (A) G k is strictly increasing on [, +∞), and where x k >  is chosen in an arbitrary way and equality holds if I k = x k .
Then B is also irreducible. It follows from Lemma . of [] that the solution space of the linear system Differentiating V along the solution of system (), we obtaiṅ

By () and (), we obtaiṅ
From (), we know that . By () and the arithmetic-geometric mean, we easily see that We can rewrite B  as .
By the arithmetic-geometric mean, we have that We can rewrite B  as .
Using the fact that x + ln x ≤ , where equality holds if and only if x = , we obtain .

(   )
In the following, we will show that We first give the proof of () for n = , which would give a reader the basic yet clear ideas without being hidden by the complexity of terms caused by larger values of n. When n = , we have .
Formula () gives v  = β  and v  = β  in this case. Expanding H  yields For more general n, by a similar argument as in the proof of n k,j= v k β kj ln , we obtain that If () holds, it follows from () that Then we obtain thaṫ This implies that By Theorem ., the disease dies out in both groups. Numerical simulation illustrates this fact (see Figure ).
By Theorem ., the disease persists in both groups. Numerical simulation illustrates this fact (see Figure ).