We consider a delayed SIR epidemic model in which the susceptibles are assumed to satisfy the logistic equation and the incidence term is of saturated form with the susceptible. We investigate the qualitative behaviour of the model and find the conditions that guarantee the asymptotic stability of corresponding steady states. We present the conditions in the time lag

Epidemics have ever been a great concern of human kind, because the impact of infectious diseases on human and animal is enormous, both in terms of suffering and social and economic consequences. Mathematical modeling is an essential tool in studying a diverse range of such diseases to gain a better understanding of transmission mechanisms, and make predictions; determine and evaluate control strategies. Many authors have proposed various kinds of epidemic models to understand the mechanism of disease transmission (see [

Epidemiological models with latent or incubation period have been studied by many authors, because many diseases, such as influenza and tuberculosis, have a latent or incubation period, during which the individual is said to be infected but not infectious. Delay differential equations (DDEs) have been successfully used to model varying infectious period in a range of SIR, SIS, and SIRS epidemic models. Hethcote and van den Driessche [

The contact rate is often a function of population density, reflecting the fact that contacts take time and saturation occurs. In this paper, we consider a delayed SIR epidemic model with time-delay and incidence rate of saturated form with the susceptibles. Qualitative analysis of the model with constant infectious period is carried out. We present the conditions in the time lag

Let the SIR model be based on the following assumptions: (i) susceptible individuals are born at a rate

If a disease is not of short duration, then several changes need to be made to the SIR model. Saturating contact rate of individual contacts is very important in an epidemiology model. For more convenience and a practical point of view, model (

Assuming that the incubation period

For the model system (

The Jacobian matrix of the linearized system of model (

If

Here we investigate the linear stability of (

Assume that

the endemic equilibrium

if

If

However, if

A bifurcation analysis of a dynamical system is used to understand how the solutions and their stability change as the parameters in the system vary. In particular, it can be used for the stability, analysis, and continuation of equilibria (steady-state solutions), and periodic and quasi-periodic oscillations; see [

Suppose that (ii) of Theorem

We already showed in Theorem

System (

Figures

Solution of delayed SIR model (

Solution of delayed SIR model (

Solution of delayed SIR model (

This paper investigates the role of time delays in the stability of an SIR model with a nonlinear incidence function. The dynamical behavior of the model is studied and the basic reproductive number

The support of the United Arab Emirates University to execute this work through an NRF Grant (Project no. 20886) is highly acknowledged and appreciated. The authors would like to thank the referees and Professor Ephraim Agyingi for their constructive comments.

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