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

Shakeri Barzoki, Mohamad Ali; Kazemi, Rasool; Asheghi, Rasoul

doi:10.22034/cmde.2025.65636.3030

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

Document Type : Research Paper

Authors

¹ Department of Mathematical Sciences, University of Kashan, Kashan, Iran.

² Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran.

10.22034/cmde.2025.65636.3030

Abstract

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 $E_0$ from which we derive the basic reproduction number $\mathcal{R}_0$, serving as the threshold for disease eradication. When $\mathcal{R}_0 > 1$, an endemic equilibrium $E_1$ emerges; conversely, $\mathcal{R}_0< 1$ guarantees the global stability of $E_0$, indicating disease elimination. A transcritical bifurcation at $\mathcal{R}_0=1$ captures the transition between disease extinction and persistence, with no evidence of Hopf bifurcations as shown by limit set analysis. Sensitivity analysis of $\mathcal{R}_0$ 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.

Keywords

Main Subjects

Dynamical systems and ergodic theory

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