Optimal control analysis of malaria and typhoid fever co-dynamics

Malaria is an infectious vector-borne disease spread by infected mosquitoes, while typhoid fever is contracted by either drinking water or eating food contaminated with the Salmonella typhoid bacteria. Both diseases affect millions of individuals every year, causing a great deal of morbidity and mortality. Previous mathematical models of the codynamics of these two diseases have not considered the interplay between symptomatic and asymptomatic individuals infected with typhoid. To fill this gap, we formulate a malaria and typhoid co-infection model explicitly including both of these classes and use standard theory of dynamical systems to analyze the model. The sub-models reproduction numbers are derived. Theoretical results show that the disease-free and endemic equilibria could co-exist (backward bifurcation) for both the typhoid only and malaria only sub-models when the respective reproduction number is less than unity. The potential impact of malaria on typhoid reveals that the increase in the number of cases due to malaria could lead to a decrease of the number of typhoid fever cases. To mitigate the spread of both malaria and typhoid fever, the model is extended to include three control measures: malaria prevention, typhoid vaccination and treatment. Numerical simulations are carried out and graphically depicted, and it is noted that reduction in the spread of typhoid greatly impacts the decrease in the number of malaria infectious individuals. Also, as expected, the most effective combination control strategy is the simultaneous implementation of malaria prevention, typhoid treatment and vaccination.