Optimal control for system of stochastic differential equations with Lévy jumps

We have addressed the challenge of designing robust vaccination and quarantine strategies in the presence of uncertainty and random perturbations, particularly in epidemic scenarios such as COVID-19. To realistically model the dynamics of disease spread, we develop a stochastic Susceptible–Infected–Recovered (SIR) model that incorporates both Brownian motion to capture continuous, small-scale fluctuations and Lévy jumps to represent rare events. This jump effectively captures key features of real-world epidemics, such as superspreading events, the sudden emergence of new variants, and mass gatherings, which are not captured by Poisson noise or Markov jumps. The model includes time-dependent vaccination and isolation control strategies under parameter uncertainty. We solve the optimal control problem using Pontryagin’s Maximum Principle and perform numerical simulations to assess the influence of different noise sources on infection dynamics and control performance. The results show that the incorporation of Lévy jumps significantly affects epidemic outcomes. In the case of negative Lévy jumps (representing sudden quarantine or lockdown), the maximum number of infected individuals is reduced by approximately 13.4%, and the total control cost is reduced by 31.9%. The positive jump significantly amplifies infection peaks and alters optimal control paths, underscoring its critical role in epidemic modeling. The findings highlight the need to incorporate jump-driven stochasticity when designing adaptive and resilient vaccination policies in the face of extreme and unpredictable epidemic events.