A co-infection model of malaria and cholera diseases with optimal control

•We formulate a mathematical model for malaria–cholera co-infection.•We calculate the basic reproduction number and investigate the existence and stability of equilibria.•The co-infection model is found to exhibit backward bifurcation.•Increased risk of cholera enhances malaria infection.•The combination of all control strategy produced the best strategy. In this paper we formulate a mathematical model for malaria–cholera co-infection in order to investigate their synergistic relationship in the presence of treatments. We first analyze the single infection steady states, calculate the basic reproduction number and then investigate the existence and stability of equilibria. We then analyze the co-infection model, which is found to exhibit backward bifurcation. The impact of malaria and its treatment on the dynamics of cholera is further investigated. Secondly, we incorporate time dependent controls, using Pontryagin’s Maximum Principle to derive necessary conditions for the optimal control of the disease. We found that malaria infection may be associated with an increased risk of cholera but however, cholera infection is not associated with an increased risk for malaria. Therefore, to effectively control malaria, the malaria intervention strategies by policy makers must at the same time also include cholera control.