Optimal control of an SIR model with effect of vaccination and a generic nonlinear infection rate

This paper introduces a SIR model with a nonlinear incidence rate and incorporates a vaccination scenario, offering a more realistic framework for disease dynamics. Assuming a vaccination coverage of p\%, we analyze its influence on epidemic outcomes. The model features a disease-free equilibrium E_۰ from which we derive the basic reproduction number \mathcal{R}_۰, serving as the threshold for disease eradication. When \mathcal{R}_۰ > ۱, an endemic equilibrium E_۱ emerges; conversely, \mathcal{R}_۰< ۱ guarantees the global stability of E_۰, indicating disease elimination. A transcritical bifurcation at \mathcal{R}_۰=۱ captures the transition between disease extinction and persistence, with no evidence of Hopf bifurcations as shown by limit set analysis. Sensitivity analysis of \mathcal{R}_۰ highlights key parameters influencing transmission, informing intervention strategies. We also develop an optimal control framework to determine the most effective vaccination coverage, providing actionable insights for public health policies. Numerical simulations validate the theoretical results, illustrating how variations in p impact outbreak trajectories and underscoring the importance of sustained vaccination efforts. By integrating nonlinear transmission with vaccination dynamics, this study advances epidemic modeling and offers practical tools for disease management.

کلیدواژه ها:

SIR model ، Optimal control ، Vaccination