Stability of a Time Delayed SIR Epidemic Model Along with Nonlinear Incidence Rate and Holling Type-II Treatment Rate

Abstract

In this paper, we present a mathematical study of a deterministic model for the transmission and control of epidemics. The incidence rate of susceptible being infected is very crucial in the spread of disease. The delay in the incidence rate is proved fatal. In the present study, we propose an SIR mathematical model with the delay in the infected population. We are taking nonlinear incidence rate for epidemics along with Holling type II treatment rate for understanding the dynamics of the epidemics. Model stability has been done by the basic reproduction number [Formula: see text]. The model is locally asymptotically stable for disease-free equilibrium [Formula: see text] when the basic reproduction number [Formula: see text] is less than one ([Formula: see text]). We investigated the stability of the model for disease-free equilibrium at [Formula: see text] equals to one using center manifold theory. We also investigated the stability for endemic equilibrium [Formula: see text] at [Formula: see text]. Further, numerical simulations are presented to exemplify the analytical studies.

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

“…Our analysis showed that when R 0 ≤ 1, the disease-free equilibrium is globally asymptotically stable, and when R 0 > 1, then there is a unique disease-endemic equilibrium, which is globally asymptotically stable. To put this result in context, we chose the treatment function T(I) = aI(t) 1+ξ I(t) (see [23]).…”

“…where a represents the maximal medical resources supplied per united time and is half-saturation constant, which measures the effect of being delayed for treatment. 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.…”

Mathematical analysis of a delayed epidemic model with nonlinear incidence and treatment rates

“…To control or remove a disease, complete understanding of the dynamics of its progression is required. Based on the observed characteristics of infectious diseases, epidemiologists [1][2][3][4][5][6][7][8][9][10][11][12][13] have attempted to construct mathematical models that make it possible to understand various aspects of many diseases and suggest methods for their control. A crucial issue in the study of the spread of an infectious disease is how it is transmitted.…”

“…There is therefore a need to modify the classical linear incidence rate in order to study the dynamics of infection among a large population. Many researchers [2][3][4][5]7,10,11,[15][16][17][18] have proposed transmission laws that include nonlinearity, such as the Holling type II functional, Crowley-Martin functional, Beddington-DeAngelis functional, etc., to study the dynamics of infectious diseases. The general incidence rate g(I )S = k I p S 1 + α I q , was suggested by Liu et al [19] and used by numerous authors in their models (see, for example, [9,15,[20][21][22]).…”

“…In classical epidemic models, the treatment rate of infected individuals is assumed to be either constant or proportional to the number of infected individuals. However, it is known that there are limited treatment resources available in the community [2]. In the absence of effective therapeutic treatments and vaccines, epidemic control strategies are based on taking appropriate preventive measures.…”