Analysis and Hopf bifurcation of an SIHR epidemic model with multiple delays and optimal control

This paper proposes a SIHR (Susceptible-Infected-Hospitalized-Recovered) mathematical model that incorporates incidence rates of Holling type II and multiple time delays to analyze disease dynamics. We establish that the unknown parameters of the model are both bounded and non-negative. At critical delay thresholds, Hopf bifurcation occurs, as revealed by stability analysis using time delay as a bifurcation parameter. We analyze the stability and direction of Hopf bifurcation, primarily influenced by delays, through normal form theory and the centre manifold theorem. In all delay scenarios, we examine the local stability of disease-free equilibrium (DFE) and endemic equilibrium points. We also conduct a sensitivity analysis of the basic reproduction number (BRN). The model incorporates control measures such as educational campaigns, hospital bed availability, and immunity enhancements. To formulate an optimal control problem aimed at identifying the most effective strategy for minimizing costs, we apply Pontryagin’s maximum principle. Finally, we propose the optimization problem and discuss the control strategy designed to minimize disease outbreaks and control costs. Numerical simulations validate and support the established theoretical results.