Bifurcation-aware optimal control of discrete SIRS epidemic models: suppressing period-doubling and multi-wave outbreaks

Abstract

Discrete-time epidemic models may display rich dynamics, including oscillatory outbreaks and chaotic behavior arising through period-doubling (Flip) bifurcations when key parameters, such as transmission or recruitment rates, vary. In most optimal control studies, the main objective is to reduce infection prevalence and intervention cost, while the possible impact of bifurcation-induced qualitative changes is often left aside. In this paper, we propose a bifurcation-aware optimal control framework for a discrete-time SIRS epidemic model that aims not only to reduce infection levels and control effort, but also to suppress multi-wave epidemic behavior. We first study the equilibria of the controlled system and derive the corresponding controlled basic reproduction number. We then analyze the local stability of the endemic equilibrium through the Jacobian spectrum and introduce a practical Flip-proximity indicator, based on the distance of eigenvalues to -1, in order to identify regimes that are close to period-doubling. Motivated by this analysis, we formulate a discrete-time optimal control problem with two intervention mechanisms: modulation of population recruitment and reduction of disease transmission. To discourage persistent oscillatory behavior in the infected population, we include an additional oscillation-penalty term in the objective functional, which serves as a control-oriented surrogate for limiting higher-period outbreak patterns. The resulting optimality system is derived by means of a discrete Pontryagin maximum principle and solved numerically using a projected forward-backward algorithm under bounded control constraints. Numerical results indicate that the proposed strategy leads to smoother infection profiles, lower peak prevalence, and a greater tendency to avoid flip-prone regimes than standard quadratic-cost control formulations.