Optimal Feedback Control of the Fokker-Planck Equation for an SIRS Epidemic Model

Abstract

We develop a Fokker--Planck feedback control framework for a stochastic susceptible--infected--recovered--susceptible (SIRS) epidemic model with three intervention mechanisms targeting disease transmission, loss of immunity, and recovery. The interventions are formulated as space- and time-dependent feedback controls that influence both the drift and diffusion coefficients of the underlying stochastic differential equations, allowing control of the probability distribution of epidemic states. We investigate the existence of optimal controls and derive first-order necessary optimality conditions involving a forward Fokker--Planck equation, a backward adjoint equation, and a Hamiltonian minimization condition. For numerical approximation, we combine a Chang--Cooper discretization of the diagonal drift--diffusion terms with a finite-difference treatment of cross-diffusion and a sequential quadratic Hamiltonian (SQH) algorithm for computing the optimal feedback policies. We also examine sufficient-decrease and convergence properties of the SQH iteration. Numerical experiments consider four epidemiological objectives involving cumulative infection burden, infection-threshold exceedance, terminal susceptible population, and a combination of running and terminal costs. The resulting feedback policies exhibit distinct spatial and temporal structures depending on the epidemiological objective. Monte Carlo simulations of the underlying stochastic system illustrate that the optimized interventions can substantially reduce infection growth and undesirable epidemic outcomes relative to uncontrolled dynamics. The proposed framework provides a distribution-based approach to stochastic epidemic control that integrates mathematical analysis, numerical optimization, and state-dependent intervention design.