Exact Dynamics and Optimization for a Discrete-Time SIR Model Using Time-Dependent Parameters

This study extends the discrete version of the Susceptible–Infected–Recovered (SIR) model to the non-autonomous case in which the rates of infection and cure are time-dependent. This generalization addresses a critical open problem in the literature. We develop a non-standard finite difference approach that maintains essential biological properties, such as population conservation and non-negativity, even under parameter variation. The main novelty here is the derivation of an exact semi-analytical solution to the non-autonomous nonlinear system. We also leverage the exact solution to develop an optimal control problem, which illustrates the potential for rigorous optimization of treatment strategies to minimize infection costs. The numerical simulations verify the theoretical results, showing not only the stability of the model when parameters change seasonally, as well as the efficiency of the optimal controls, but also the predictive power of the non-autonomous approach compared to classical autonomous models.