Dynamical Model of Epidemic Along with Time Delay; Holling Type II Incidence Rate and Monod–Haldane Type Treatment Rate

Abstract

The present study aims to control the infectious diseases and epidemics in the human population. Therefore, in the present article, we have proposed a delayed SIR epidemic model along with Holling type II incidence rate and treatment rate as Monod–Haldane type. Model stability has been established in the three regions of the basic reproduction number $$ {\text{R}}_{0} $$ i.e. $$ {\text{R}}_{0} $$ equals to one, greater than one and less than one. The model is locally asymptotically stable for disease-free equilibrium $$ {\text{Q}} $$ when the basic reproduction number $$ {\text{R}}_{0} $$ is less than one ( $$ {\text{R}}_{0} < 1) $$ and unstable when $$ {\text{R}}_{0} > 1 $$ for time lag $$ \tau \ge 0 $$ . We investigated the stability of the model for disease-free equilibrium at $$ {\text{R}}_{0} $$ equals to one using central manifold theory. Using center manifold theory, we proved that at $$ {\text{R}}_{0} = 1 $$ , disease-free equilibrium changes its stability from stable to unstable. We also investigated the stability for endemic equilibrium $$ {\text{Q}}^{ *} $$ for time lag $$ \tau \ge 0 $$ . Further, numerical simulations are presented to exemplify the analytical studies.