Mathematical Modelling of Cholera Transmission Incorporating Open Defecation as an Environmental Driver: An Optimal Control Approach

Open defecation-referring to the public and unrestricted exposure of defecation practices remains a persistent driver of waterborne diseases like cholera in many low- and middle-income regions. Despite various interventions, cholera continues to re-emerge due to the sustained presence of vibrio cholerae in contaminated environments. In this study, we formulate and analyse a deterministic compartmental model that incorporates open defecation as a major environmental factor influencing the transmission of cholera. The model integrates exposed and infectious human compartments, a dynamic bacterial population in the environment, and a control effort that responds to infection levels. The numerical simulations show that optimal control measures, which aim to reduce environmental contamination through sanitation efforts and behavioural changes, result in a 60% reduction in peak infections and a 40% decrease in open defecation (D) compared to uncontrolled situations. The basic reproduction number ( ) falls from 2.3 (uncontrolled) to 1.1, indicating a critical threshold for disease eradication. Sensitivity analyses indicate that transmission rate ( ), defecation contribution ( ), and pathogen decay ( ) have an impact on epidemic severity. These findings advocate for policies integrating hygiene education, sanitation infrastructure, and dynamic resource allocation to sustainably mitigate disease burden in resource-constrained settings.