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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.

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 [

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 [

In this paper, we formulate a differential infectivity epidemic model with stage structure. The proof of global stability of the endemic equilibrium utilizes a graph-theoretical approach [

The organization of this paper is as follows. In Section

We assume the following.

(A1)

From our assumptions, it is clear that system (

In the section, we will study the global asymptotical stability of equilibria of system (

Assume that (A1) holds and

If

If

Let

If

By Theorem

For convenience of notations, set

We further make the following assumption.

(A2)

Assume that (A1) and (A2) hold,

Consider a Lyapunov functional

From (

In the section, numerical simulations are presented to support and complement the theoretical findings. We consider the following model:

If

Dynamical behavior of system (

If

Dynamical behavior of system (

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.

The author declares that there are no competing interests.

This work was supported by Key Laboratory of Statistical Information Technology and Data Mining, State Statistics Bureau (SDL201601).