Global Dynamics for a Novel Differential Infectivity Epidemic Model with Stage Structure

A novel differential infectivity epidemic model with stage structure is formulated and studied. Under biological motivation, the stability of equilibria is investigated by the global Lyapunov functions. Some novel techniques are applied to the global dynamics analysis for the differential infectivity epidemic model. Uniform persistence and the sharp threshold dynamics are established; that is, the reproduction number determines the global dynamics of the system. Finally, numerical simulations are given to illustrate the main theoretical results.


Introduction
Mathematical model that reflects the characteristics of an epidemic to some extent can help us to understand better how the disease spreads in the community and can investigate how changes in the various assumptions and parameter values affect the course of epidemic.In [1], Hyman et al. proposed a differential infectivity model that accounted for differences in infectiousness between individuals during the chronic stages and the correlation between viral loads and rates of developing AIDS.They assumed that the susceptible population was homogeneous and neglected variations in susceptibility, risk behavior, and many other factors associated with the dynamics of HIV spread.Ma et al. [2] presented several differential infectivity epidemic models under different assumptions.
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 disease transmission in adult population such as typhus and diphtheria.In recent years, epidemic models with stage structure have been studied in many papers [3][4][5][6][7][8][9][10].
In this paper, we formulate a differential infectivity epidemic model with stage structure.The proof of global stability of the endemic equilibrium utilizes a graphtheoretical approach [11][12][13][14][15][16][17][18][19][20][21][22] to the method of Lyapunov functions.Let  1 and  2 denote the immature susceptible and mature susceptible populations, respectively.The infectious population  was subdivided into  subgroups  1 ,  2 , . . .,   . 1 and  2 denote the probabilities of an immature infectious individual and a mature infectious individual enter subgroup , respectively, where ∑  =1  1 = ∑  =1  2 = 1.The disease incidence in the th subgroup can be calculated as , where   is the transmission coefficient between compartments   and   .  (  ) includes some special incidence functions in the literature.For instance,   (  ) =   /(1 +     ) (saturation effect).Since we do not assume that recovered individuals return into the susceptible class, the recovered class does not need to be explicitly modeled.Then, we obtain the following model: where ( 1 ) =  −  1  1 with  being the recruitment constant and  1 being the natural death rate. is the conversion rate from an immature individual to a mature individual. 2 is the natural death rate of the mature susceptible class.  =    +   , where    is the death rate of  population in subgroup  and   is the recovery rate in the th subgroup.All parameter values are assumed to be nonnegative and , ,   ,  1 ,  2 > 0.
The organization of this paper is as follows.In Section 2, we prove some preliminary results for system (1).In Section 3, the main theorem of this paper is stated and proved.In Section 4, numerical simulations which support our theoretical analysis are given.

Preliminaries
We assume the following.

Main Results
In the section, we will study the global asymptotical stability of equilibria of system (1).
By Theorem 1, we have the idea that if B = [∑ 2 =1     ] is irreducible, (A1) holds and  0 > 1, and then system (1) exists in endemic equilibrium , and then the components of  * satisfy Since  is strictly decreasing on [0, +∞), we have For convenience of notations, set Then,  is also irreducible.It follows from Lemma 2.1 of [11] that the solution space of linear system, has dimension 1, with a basis where   denotes the cofactor of the th diagonal entry of .
Note that from (12) we have From ( 14), we have We further make the following assumption.(A2)   is strictly increasing on [0, +∞), and where   > 0 is chosen in an arbitrary way and equality holds if   =   .
Theorem 2. Assume that (A1) and (A2) hold, , then  * is globally asymptotically stable in Γ ∘ and thus is the unique endemic equilibrium.

Numerical Examples
In the section, numerical simulations are presented to support and complement the theoretical findings.We consider the following model: where   (  ) =   /(1 +     ).Clearly, (A1) and (A2) hold.We fix the parameters as follows:

Conclusions
A differential infectivity epidemic model with stage structure has been used to describe the spreading of such a disease.We have focused on the theoretical analysis of the equilibriums.Using a graph-theoretic approach to the method of Lyapunov functions, we have proved the global stability of the endemic equilibrium.We have established uniform persistence and the sharp threshold.The work has potential extensions and improvements, which remains to be discussed in the future.