Optimal Control of Endemic Epidemic Diseases With Behavioral Response

Behavioral factors play a crucial role in the emergence, spread, and containment of human diseases, significantly influencing the effectiveness of intervention measures. However, the integration of such factors into epidemic models is still limited, hindering the possibility of understanding how to optimally design interventions to mitigate epidemic outbreaks in real life. This paper aims to fill in this gap. In particular, we propose a parsimonious model that couples an epidemic compartmental model with a population game that captures the behavioral response, obtaining a nonlinear system of ordinary differential equations. Grounded on prevalence-elastic behavior—the empirically proven assumption that the disease prevalence affects the adherence to self-protective behavior—we consider a nontrivial negative feedback between contagions and adoption of self-protective behavior. We characterize the asymptotic behavior of the system, establishing conditions under which the disease is quickly eradicated or a global convergence to an endemic equilibrium is attained. In addition, we elucidate how the behavioral response affects the endemic equilibrium. Then, we formulate and solve an optimal control problem to plan cost-effective interventions for the model, accounting for their healthcare and social-economical implications. Numerical simulations on a case study calibrated on sexually transmitted diseases demonstrate and validate our findings.