Optimal Vaccination Strategies in an SIR Epidemic Model: A Mathematical Analysis

Vaccination is one of the most effective interventions for controlling the spread of infectious diseases; however, in practice, vaccination programs are often constrained by limited resources such as vaccine supply, logistics, and cost. Many existing epidemic models assume unlimited vaccination capacity, which reduces their applicability to realistic public health settings. In this work, an SIR (Susceptible–Infected–Recovered) epidemic model incorporating vaccination is studied under explicit resource limitations. Vaccination is modeled as a time-dependent control variable subject to both pointwise bounds and an integral constraint representing limited total vaccine availability. An optimal control problem is formulated with the objective of minimizing the number of infected individuals while accounting for the cost of vaccination over a finite time horizon. The analysis begins with establishing the positivity and boundedness of solutions of the controlled system. The existence of an optimal vaccination strategy is then proven using standard results from optimal control theory. Necessary conditions for optimality are derived using Pontryagin’s Maximum Principle, leading to an explicit characterization of the optimal control in terms of the state and adjoint variables, as well as a Lagrange multiplier associated with the resource constraint. The qualitative structure of the optimal vaccination strategy is discussed and interpreted from an epidemiological perspective. The results provide rigorous mathematical insight into how limited vaccination resources can be optimally allocated over time to reduce disease burden, and they offer a foundation for further extensions to more complex epidemic models