Optimal Control of a Cholera Model With Sensitivity Analysis: Balancing Sanitation and Treatment Under Resource Constraints

This paper develops a mathematical framework for optimal resource allocation during cholera outbreaks under limited public health budgets, addressing the critical gap between theoretical control strategies and practical implementation constraints. We construct an extended SEIR-B compartmental model that uniquely integrates three biologically realistic features: (i) waning immunity dynamics, (ii) environmental transmission with saturation incidence, and (iii) dual time-dependent control measures representing sanitation and treatment interventions. Using the next generation matrix method, we derive the basic reproduction number R0 and decompose it into human-to-human and environment-to-human transmission components. Sensitivity analysis reveals that human transmission rate βh, environmental transmission rate βe, and bacterial shedding rate ξ are the most influential parameters, providing clear targets for intervention prioritization. Applying Pontryagin's maximum principle, we characterize optimal control strategies and demonstrate through numerical simulation that a two-phase intervention approach-intensive early sanitation followed by sustained treatment-achieves superior outbreak suppression. The optimized strategy reduces cumulative infections by 77.8% and shortens outbreak duration from 92.5 to 35.2 days compared with no intervention, while reducing resource expenditure by 35% relative to constant high-effort deployment. Our framework provides evidence-based guidance for public health decision-makers by quantifying the trade-offs between intervention timing, intensity, and cost-effectiveness in resource-limited settings.