In this paper, we present an optimal control analysis of a susceptible-infected (SI) epidemic model governed by a single objective function. The main purpose of this study is to minimize the number of infected individuals while reducing the cost associated with the control strategy. A dynamic control variable \(u(t)\) is introduced to represent preventive measures, such as vaccination, awareness, or treatment. The model dynamics are formulated as a system of nonlinear differential equations with bilinear incidence rate. The existence of an optimal control is established using Pontryagin’s Maximum Principle, and the corresponding adjoint system and optimality conditions are derived explicitly. A numerical scheme based on the forward-backward sweep method is implemented to compute the optimal trajectories of the state and control variables. Simulation results demonstrate that the optimal control significantly reduces the infection level and maintains a high proportion of susceptible individuals over time. To demonstrate the practical applicability of the theoretical framework, we present a numerical application using measles-like epidemiological parameters, including an illustrative cost-effectiveness assessment and epidemiological interpretation. The study highlights the effectiveness of incorporating optimal control theory into epidemiological modeling for efficient disease management.
