Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays

Abstract

In this paper, we establish the global asymptotic stability of equilibria for an SIR model of infectious diseases with distributed time delays governed by a wide class of nonlinear incidence rates. We obtain the global properties of the model by proving the permanence and constructing a suitable Lyapunov functional. Under some suitable assumptions on the nonlinear term in the incidence rate, the global dynamics of the model is completely determined by the basic reproduction number R 0 and the distributed delays do not influence the global dynamics of the model.

On the global stability of a delayed epidemic model with transport-related infection

“…Remark 2.5. The property of (2.7) in Lemma 2.4 is also found in [18,19] and it is used to analyze global dynamics of epidemic models having a nonlinear incidence rate.…”

Global stability analysis for a generalized delayed SIR model with vaccination and treatment

“…We suppose, by contradiction, that there exists λ ∈ C such that (λ) ≥ 0, and it satisfies equality (10). Then…”

“…For simplicity, we assume that the recovered class stands also for the vaccinated state. Hence, susceptible individuals get vaccinated with rate d. The nonlinear incidence rate and distributed delay are considered to represent wide class epidemic model similarly as in [10,11]. More precisely, by taking β the disease transmission coefficient, individuals leave the susceptible class at a rate…”

“…It is known that the function forms of the incidence rate of the infection have a crucial role in the modeling of the infection dynamics, many forms of incidence function have been considered by the researchers in mathematical epidemiology, for example, the bilinear incidence rate βSI, where β is the transmission rate of infection, the saturated incidence rate βSI 1+αI , with α defined as the inhibitory coefficient, and many other forms (see [1][2][3][4][5][6][7]). To make a model more realistic, the introduction of the time delay is more interesting, and considerable attention has been paid by several authors to studying the dynamics of epidemic models with discrete or distributed time delay (see [3,4,[8][9][10][11]). Vaccination and treatment are the two main public health control strategies that help to minimize the burden of an infectious disease spread and to delay a possible outbreak.…”

“…Other works have investigated the effects of the treatment on an epidemic (see [19][20][21][22][23][24][25]) and also its optimal control (see [26]). The motivation of this work comes from [10,11], where the authors studied an SIR epidemic model with nonlinear incidence function, and from [19][20][21], where the authors considered a special type of treatment function. The present work would be a continuation and generalization of the above cited works.…”