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

In the case of an outbreak of an epidemic, psychological or inhibitory effects and various limitations on treatment methods play a major role in controlling the impact of the epidemic on society. The Monod–Haldane functional-type incidence rate is taken to interpret the psychological or inhibitory effect on the population with time delay representing the incubation period of the disease. The Holling type III saturated treatment rate is considered to incorporate the limitation in treatment availability to infective individuals. This novel combination of the Monod–Haldane incidence rate and Holling type III treatment rate is applied herein to a time-delayed susceptible–infected–recovered epidemic model to incorporate these important aspects. The mathematical analysis shows that the model has two equilibrium points, namely disease-free and endemic. Detailed dynamical analysis of the model is performed using the basic reproduction number R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{0}$$\end{document}, center manifold theory, and Routh–Hurwitz criterion. The results show that that the disease can be eradicated when the basic reproduction number is less than unity, while the disease will persist when the basic reproduction number is greater than unity. The Hopf bifurcation at endemic equilibrium is addressed. Furthermore, the global stability behavior of the equilibria is discussed. Finally, numerical simulations are performed to support the analytical findings.