A new concept of optimal control for epidemic spreading by vaccination: Technique for assessing social optimum employing Pontryagin’s maximum principle

This research introduces a new approach utilizing optimal control theory to assess the Social Optimum (SO) of a vaccination game, navigating the intricate considerations of cost, availability, and distribution policies. By integrating an SIRS/V (Susceptible-Infected-Recovered-Susceptible/Vaccinated) epidemic model with a behavior model, the study analyzes individual vaccination strategies. A unique optimal control framework, centered on vaccination costs, is proposed, diverging significantly from previous methods. Our findings confirm the effectiveness and feasibility of this approach in managing vaccination strategies. Moreover, we examine the underlying social dilemma of the vaccination game, investigating key parameters. By calculating the Nash equilibrium (NE) through the behavior model and determining the SO using our approach, we measure the social efficiency deficit, quantifying the overall cost gap between the NE and SO. The results indicate that an increased waning immunity rate exacerbates the social dilemma, although higher vaccination costs partially mitigate it. This research provides valuable insights into optimizing vaccination strategies amid complex societal dynamics.