Optimal control and economic evaluation of diphtheria disease model with booster immunization and hospitalization

In this paper, we develop a deterministic mathematical model to study the transmission dynamics of diphtheria, incorporating optimal control strategies and cost-effectiveness analysis. The model consists of six compartments: susceptible, exposed, infectious, asymptomatic, hospitalized, and recovered individuals. We analyze the model's qualitative behavior, including the existence of an invariant region, the positivity of solutions, and the identification and stability (both local and global) of two equilibrium points: the disease-free equilibrium and the endemic equilibrium. The effective reproduction number is derived to assess the potential spread of the disease. To determine optimal control strategies, we apply Pontryagin's Maximum Principle to obtain the Hamiltonian, adjoint equations, control characterizations, and the resulting optimality system. Various combinations of control strategies are evaluated to assess their impact on diphtheria transmission. We use the incremental cost-effectiveness ratio (ICER) to identify the most effective and efficient intervention. Sensitivity analysis and numerical simulations are conducted to support the findings. The simulation results indicate that a combined strategy of prevention and vaccination is the most cost-effective approach. The implementation of control strategies is shown to play a critical role in reducing the burden of diphtheria in the community.