Abstract:
This study examines a susceptible-infected-temporary-permanent-recovered (SITHR) epidemic model incorporating the Holling type II incidence rate to prevent and control the disease with optimal use of hospital beds. Initially, the well-posedness and feasibility of the model are analyzed, and then valid biological equilibrium points are calculated. Subsequently, the stability of these equilibrium points is assessed and the basic reproduction number is calculated as a threshold value that controls the dynamics of the disease. The proposed model undergoes several bifurcations, including transcritical (backward and forward), saddle-node, Hopf, and Bogdanov–Takens bifurcations. The normal form is derived to demonstrate the presence of a Bogdanov–Takens bifurcation. Furthermore, parameter estimation is conducted using COVID-19 data from Italy to refine the model’s accuracy and boost the reliability of the study’s predictions. Using the normalized forward sensitivity index (NFSI), a sensitivity analysis of parameters associated with the basic reproduction number is performed, and the partial rank correlation coefficient (PRCC) is calculated to locate the key parameters affecting disease transmission dynamics. Moreover, the system is expanded to incorporate time-dependent control variables to reduce the infected population and the cost associated with implementing these controls. The developed optimal control system is employed to build the Hamiltonian function, which is solved using Pontryagin’s maximum principle. Also, a cost-effectiveness analysis is performed to evaluate the economic efficiency of various intervention strategies. Beyond the deterministic framework, the study includes formulations for continuous-time Markov chains and stochastic differential equations to assess the impact of environmental noise on the system. Moreover, the Galton–Watson branching process determines the extinction threshold for the stochastic model and sets the parameters that govern disease extinction or persistence. Finally, numerical simulations are demonstrated to illustrate the impact of changes in system parameters on the dynamic behavior of the model. These findings will enhance preparedness and enable more efficient responses to health emergencies, leading to better patient care and less pressure on healthcare systems.