Cost-effectiveness and optimal control analysis of an $ SIR-P $ model for rodent-transmitted infection in agricultural environments

Rodent populations in agricultural environments serve as critical reservoirs for zoonotic diseases, posing significant risks to food security and public health. Transmission in these settings is complex, thereby occurring through both direct rodent-to-rodent contacts and indirect exposures to environmental contamination. In this paper, we formulate and analyze a deterministic model that integrates rodent population dynamics with an environmental pathogen compartment. To identify resource-efficient mitigation policies, we develop an optimal control framework that incorporates three time-dependent interventions: contact prevention, environmental sanitation, and treatment. By applying Pontryagin's Maximum Principle, we derive the Hamiltonian, the adjoint system, and the characterization of the optimal controls, and subsequently solve the optimality system using a forward-backward sweep algorithm. We evaluate the epidemiological impact and economic viability of seven distinct intervention strategies using the Average Cost-Effectiveness Ratio (ACER) and the Incremental Cost-Effectiveness Ratio (ICER). Numerical simulations demonstrate that while the full combination of controls yields the maximum reduction in infection, the prevention-only strategy emerges as the most economically attractive option under restricted budgets. These findings suggest that prioritizing contact-reduction measures provides the most cost-effective basis for disease management, while integrated sanitation and treatments should be scaled based on resource availability. This study provides a quantitative framework that may assist agricultural stakeholders in developing resource-efficient control policies.