Optimal Control of a Waterborne Disease Model With Dual Environmental Pathways and Hospitalization

Waterborne diseases remain a persistent public health crisis in low-resource regions, where inadequate sanitation infrastructure, limited access to clean water, and weak healthcare systems create conditions for recurrent outbreaks. While mathematical models have advanced our understanding of disease transmission through water and direct contact, two critical factors have received insufficient attention: the role of solid waste dumpsites as independent pathogen reservoirs that contaminate water sources, and the mitigating effect of hospitalization as a transmission-interrupting mechanism. This study addresses these gaps by developing a novel compartmental model that explicitly couples human infection dynamics with pathogen concentrations in both aquatic systems and solid waste dumpsites, connected through a leaching parameter that quantifies waste-to-water contamination. Through deterministic sensitivity analysis and uncertainty quantification, we identify clean water coverage and waterborne transmission rate as the dominant parameters governing outbreak potential. An optimal control framework evaluating five time-dependent interventions demonstrates that environmental interventions targeting water purification and sanitation outperform clinical measures by a substantial margin, with the triple combination of awareness campaigns, water purification, and sanitation achieving the most fundamental outbreak control. Critically, we identify an intervention threshold where achieving sufficient clean water coverage yields a high probability of disease elimination, providing an actionable target for public health policy. These findings establish that integrated environmental management, rather than clinical treatment alone, offers the most effective pathway for controlling waterborne outbreaks in resource-limited settings.