Advanced SIWR Model: Optimal Control of Multiple Transmission Pathways in Water-borne Diseases

In this paper, a controlled SIWR model was considered which was an extension of the simple SIR model by adjoining a compartment (W) that tracks the pathogen concentration in the water. New infections arise both through exposure to contaminated water, as well as by the classical SIR person-person transmission pathway. The objective of optimal control is to effectively manage and minimize the transmission of waterborne diseases through strategic interventions and management strategies. The two controls were employed which represent immune boosting and pathogen suppressing drugs. The objective function is based on a combination of minimizing the number of infected individuals and the cost of the drug dose. Pontryagin's maximum principle has been used to characterize the optimal levels of the two controls. The optimal control is obtained by solving the optimality system which was composed of four nonlinear ODEs with initial conditions and four nonlinear adjoint ODEs with transversality conditions. The results show that the optimal combination of immune-boosting and pathogen-suppressing drugs required to achieve the set objective will depend on the relative cost of each of the control measures. The results were analysed and interpreted numerically using MATLAB.