Modelling Cholera Transmission Dynamics with Logistic Growth Through Optimal Vaccination, Treatment and Sanitation

Abstract

A novel mathematical model for the time-evolution of cholera incorporating pre-exposure vaccination of healthy individuals and logistic bacteria growth is presented and analysed. The nonlinear model is shown to be well-posed in the framework of positivity and boundedness of solutions. With next generation matrix approach, the effective reproduction number, \(\mathcal{R}_e,\) measuring the extent of cholera spread in an environment comprising susceptible and vaccinated individuals is obtained. The model has a unique cholera-present steady state, indicating possibility of eliminating cholera disease in the environment. Key sensitive parameters fueling transmission dynamics of cholera are identified through sensitivity analysis. Furthermore, three control intervention strategies representing pre-exposure vaccination, \(\nu(t)\), treatment effort, \(u_1(t),\) and environmental sanitation effort, \(u_2(t),\) are considered in mitigating the transmission of cholera as a consequence of the insights gained from the sensitivity analysis. The existence of the optimal control triple is explicitly proved and characterized through Pontryagin's maximum principle. Consequently, the three control measures are further classified into four different intervention scenarios, namely Scenario \(Z_1\) (implementation of vaccination strategy); Scenario \(Z_2\) (implementation of treatment effort); Scenario \(Z_3\) (implementation of environmental sanitation effort); and Scenario \(Z_4\) (combination of all the three controls). In particular, in-depth efficiency and economic assessments show that Scenario \(Z_4\) is the most efficient and most cost-effective intervention scenario. Therefore, the intervention scenario that significantly averts the cases of cholera disease is strongly recommended.