Optimal control and evolutionary behavior in a two-layer epidemic model with travel restriction and social distancing

Large-scale outbreaks of infectious diseases, often spread through person-to-person contact, have historically caused significant morbidity and mortality. In this study, we develop a two-layer SIR (Susceptible-Infected-Recovered) model that accounts for individual mobility within and between populations. We explore two complementary approaches to disease mitigation: (i) an optimal control framework and (ii) an evolutionary behavior model. The optimal control approach minimizes the disease burden by coordinating three controls: travel restrictions, social distancing, and antiviral treatment, guided by predefined cost functions. In contrast, the behavioral model captures adaptive individual responses based on infection prevalence, interpopulation infection disparities, and socioeconomic trade-offs, following evolutionary game theory. We find that for high-severity epidemics, combining travel restrictions with social distancing significantly reduces infection peaks and total cases, while these measures become less effective for lower-severity outbreaks. Across all scenarios, a combined control strategy is most effective. However, a key finding is that antiviral treatment alone can rival the effectiveness of combined travel and distancing measures, offering a streamlined alternative when the societal costs of nonpharmaceutical interventions are prohibitive. Furthermore, optimally coordinated policies consistently outperform adaptive behavioral responses, yielding a significant reduction in the population.