Stability Analysis of a Multigroup Epidemic Model with General Exposed Distribution and Nonlinear Incidence Rates

and Applied Analysis 3 implying that h j (u) = h(u) for j = 1, 2, . . . , m. We choose the gamma distribution: h (u) = h n,b (u) = u n−1 (n − 1)!b n e −u/b , (7) where b > 0 is a real number and n > 1 is an integer, which is widely used and can approximate several frequently used distributions. For example, when b → 0, h n,b (s) will approach the Dirac delta function, and when n = 1, h n,b (s) is an exponentially decaying function. The main object of this paper is to carry out the wellknown “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 R 0 and show that R 0 completely determines the global dynamics of system (6).More specifically, ifR 0 ≤ 1, the disease-free equilibrium is globally asymptotically stable and the disease dies out; if R 0 > 1, a unique endemic equilibrium exists and is globally asymptotically stable, and the disease persists at the endemic equilibrium. The global stability of P ∗ 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 [15], 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 R 0 . The global stability of the corresponding equilibria for R 0 ≤ 1 and R 0 > 1 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. 2. Preliminaries We make the following basic assumptions for the intrinsic growth rate of susceptible individuals in the ith group φ i (S i ) and the transmission functions f ij (S i , I j ). (A 1 ) φ i are C non-increasing functions on [0,∞) with φ i (0) > 0, and there is a unique positive solution ξ = S 0 i for the equation φ i (ξ) = 0. φ i (S) > 0 for 0 ≤ S < S 0 i , and φ i (S) < 0 for S > S i ; that is [φ i (S i ) − φ i (S 0 i )] (S i − S 0 i ) < 0, for S i ̸ = S 0 i , i = 1, 2, . . . , m. (8) (A 2 ) f ij (S i , I j ) ≤ c ij (S i )I j for all I j > 0. (A 3 ) c ij (S i ) ≤ c ij (S 0 i ), 0 < S i < S 0 i , i, j = 1, . . . , m. Following the technique and method in [14], define ?̂? ≡ b 1 + δb , (9) which can absorb the exponential term e into the delay kernel. The second equation in (6) can be rewritten as


Introduction
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 [1].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. [2] 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][3][4][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 ( ≥ 1).For 1 ≤  ≤ , 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 [6] 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 [6], 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 1 ≤ ,  ≤ ,   ≥ 0 denotes the coefficient of transmission between compartments   and   .It is assumed that -square matrix (  ) 1≤,≤ is irreducible [7].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: continuous with possibly finitely many jumps and satisfies   (0 + ) = 1, and lim  → ∞   () = 0 with ∫ ∞ 0   () 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    ( − ) ≤ 0 due to the aformentioned property) from the exposed class; hence Let ℎ  () = −   () 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   (),  = 1, . . ., , 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  = 1, () =   , 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 [10].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  = 1, 2, . . ., .Further, we assume that the functions ℎ  () are disease specific only, implying that ℎ  () = ℎ() for  = 1, 2, . . ., .We choose the gamma distribution: where  > 0 is a real number and  > 1 is an integer, which is widely used and can approximate several frequently used distributions.For example, when  → 0 + , ℎ , () will approach the Dirac delta function, and when  = 1, ℎ , () is an exponentially decaying function.
The main object of this paper is to carry out the wellknown "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  0 and show that  0 completely determines the global dynamics of system (6).More specifically, if  0 ≤ 1, the disease-free equilibrium is globally asymptotically stable and the disease dies out; if  0 > 1, 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 [15], 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  0 .The global stability of the corresponding equilibria for  0 ≤ 1 and  0 > 1 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.

Preliminaries
We make the following basic assumptions for the intrinsic growth rate of susceptible individuals in the th group   (  ) and the transmission functions   (  ,   ).
( 1 )   are  1 non-increasing functions on [0, ∞) with   (0) > 0, and there is a unique positive solution  =  0  for the equation   () = 0.   () > 0 for 0 ≤  <  0  , and   () < 0 for  >  0  ; that is Following the technique and method in [14], define which can absorb the exponential term  − into the delay kernel.The second equation in ( 6) can be rewritten as For  = 1, . . ., , let Thus, for  ∈ {2, . . ., }, we obtain For  = 1, 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 (  ,  ,1 ,  ,2 , . . .,  , ,   ) of system (15) follow from the standard theory of Volterra integro-differential equation [16].It can also be verified that every solution of (15) with nonnegative initial condition remains nonnegative.It follows from ( 1 ) and the first equation of ( 15) that lim sup  → ∞   () ≤  0  for all  = 1, 2, . . ., .Let    be the maximum of the function   on R + and let  be a positive real number such that  > b   .Denote by Υ  the th tube for system (15); that is, It follows from a similar argument to that in [14] that we can show that the set   defined by is a forward invariant compact absorbing set for system (15) for  > 0 and that the set Γ 0 (i.e., when  = 0) is a forward invariant compact set.
Under the assumption ( 1 ), we know that system (15) always has the disease-free equilibrium  0 = ( 0 1 , 0, . . ., 0,  0 1 ,  0 2 , 0, . . ., 0,  0 2 , . . .,  0  , 0, . . ., 0,  0 An equilibrium  * of ( * in the interior of Γ 0 is called an endemic equilibrium, and it satisfies the following equilibrium equations: The basic reproduction number  0 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  0 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 ( 1 )-( 5 ) of Theorem 2 of [17], we have the following proposition.
Lemma 1.For system (15), the disease-free equilibrium  0 is locally asymptotically stable if  0 < 1, while it is unstable if Following the method of [2], one defines a matrix whose spectral radius has a similar threshold property to that of  0 , since both of the nonnegative matrices FV −1 and  0 are irreducible, and hence from the Perron-Frobenius theorem [7] that their spectral radii are given by each of their simple eigenvalues.Thus, we obtain  0 = (FV −1 ) = (V −1 F) = ( 0 ).Then the following lemma immediately follows.

Main Results
The following main theorems are summarized in terms of system (15).Theorem 3. Assume that the functions   and   satisfy assumptions ( 1 )-( 3 ), and the matrix  = (  ) × is irreducible and  0 is defined by (24).
If  0 = ( 0 ) > 1, then and then, by continuity, we can obtain in a neighborhood of  0 in Γ 0 ; then  0 is unstable.Assume  0 = ( 0 ) > 1.By the uniform persistence result from [19] and a similar argument as in the proof of [2], the instability of  0 implies the uniform persistence of (15).This together with the dissipativity of ( 15) resulted from the forward invariant and compact property of Γ 0 stated previously, implies which that ( 15) has an equilibrium in Γ 0 , denoted by  * (see, e.g., Theorem D.3 in [20]).
In what follows we prove that the endemic equilibrium  * of system ( 15) is globally asymptotically stable when  0 > 1.
Throughout the paper, we denote Then () ≥ 0 for  > 0 and has global minimum at  = 1.
For convenience of notations, set   =     ( *  ,  *  ), 1 ≤ ,  ≤ , 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   > 0,  = 1, 2, . . ., , is the cofactor of the th diagonal entry of .To get the global stability of  * , the following assumptions in [15] are proposed: A difficult mathematical question for system (15) is that of whether the endemic equilibrium  * is unique when  0 > 1 and whether  * is globally asymptotically stable when it is unique.Our main global stability result is given.
then there is a unique endemic equilibrium  * for system (15), and  * is globally asymptotically stable in Γ 0 .