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In this paper we propose the global dynamics of an SIRI epidemic model with latency and a general nonlinear incidence function. The model is based on the susceptible-infective-recovered (SIR) compartmental structure with relapse (SIRI). Sufficient conditions for the global stability of equilibria (the disease-free equilibrium and the endemic equilibrium) are obtained by means of Lyapunov-LaSalle theorem. Also some numerical simulations are given to illustrate this result.

Epidemic models have long been an important tool for understanding and controlling the spread of infectious diseases. Most of them are described by delayed differential equations. The introduction of time delay is often used to model the latent period, that is, the time from the acquisition of infection to the time when the host becomes infectious [

Recently, considerable attention has been paid to model the relapse phenomenon, that is, the return of signs and symptoms of a disease after a remission. Hence, the recovered individual can return to the infectious class (see [

For malaria, Bignami [

For tuberculosis, relapse can be caused by incomplete treatment or by latent infection, being observed that HIV-positive patients are significantly more likely to relapse than HIV-negative patients, although it is often difficult to differentiate relapse from reinfection (see [

In this paper, we propose the following epidemic model with time delay and relapse (delayed SIRI epidemic model) as follows (see [

Here

In model (

This incidence function includes different forms presented in literature (see, e.g., [

System (

In [

In [

In [

In [

In [

In [

In [

In [

In [

In [

In [

In [

In [

In this paper we extend the global stability results presented in [

In this section, we discuss the global stability of a disease-free equilibrium

If

Define a Lyapunov functional

Furthermore, It follows from the hypothesis

Therefore,

If

To prove global stability of the endemic equilibrium, we define a Lyapunov functional

In this section, we give a numerical simulation supporting the theoretical analysis given in Section

Solutions (

In this paper, we presented a mathematical analysis and numerical simulations for an SIRI epidemiological model applied to the evolution of the spread of disease with relapse in a given population. We denote

In the following, we present the method of Lyapunov functionals in the context of a delay differential equations:

We say

The following result is the Lyapunov-LaSalle type theorem for (

If

The authors declare that there is no conflict of interests regarding the publication of this paper.