In this study, we extend the classical SIRS (Susceptible-Infectious-Recovered-Susceptible) framework by including a vaccinated compartment (V) to capture waning vaccine efficacy and the effects of periodic re-vaccination. The resulting SIRSV model couples population dynamics with time-dependent vaccination efficacy, formulated as a PDE for continuous vaccination time and as an ODE system under discrete re-vaccination intervals. We analyze the equilibria and the stability of the disease-free state and develop an optimal control framework balancing vaccination rate, re-vaccination timing, and non-pharmaceutical interventions. Numerical continuation and bifurcation analyses reveal rich dynamics, including bistability and multiple bifurcation scenarios, underlining the importance of coordinated vaccination and contact control strategies for effective epidemic management.

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