Optimal Vaccination Strategy for a Time-Fractional Reaction–Diffusion SIR Model with Generalized Incidence Rate in a Heterogeneous Environment

Abstract

In this paper, we propose a novel fractional-order SIR epidemic model with a generalized nonlinear incidence rate that incorporates various epidemic transmission patterns. The model accounts for spatial heterogeneity via diffusion coefficients that reflect regional differences in population mobility. We employ the Atangana–Baleanu fractional derivative to effectively capture memory effects and non-local interactions, essential features for modeling realistic epidemic dynamics. Within an optimal control framework, we formulate vaccination as a control strategy designed to minimize both disease spread and intervention costs. Theoretical analysis establishes the existence and uniqueness of the solution to the state system, along with a proof of optimal control existence and the derivation of corresponding optimality conditions. Numerical simulations demonstrate the model’s effectiveness in quantifying the influence of spatial heterogeneity on outbreak progression and in evaluating optimized vaccination strategies across different transmission scenarios. Special emphasis is placed on analyzing the interplay between fractional-order memory effects, saturated incidence rates, and spatially varying diffusion parameters in shaping epidemic dynamics.