Mathematical model for the dynamics of visceral leishmaniasis–malaria co-infection

A mathematical model to understand the dynamics of malaria–visceral leishmaniasis co-infection is proposed and analyzed. Results show that both diseases can be eliminated if R₀, the basic reproduction number of the co-infection, is less than unity, and the system undergoes a backward bifurcation where an endemic equilibrium co-exists with the disease-free equilibrium when one of R_m or R_l, the basic reproduction numbers of malaria-only and visceral leishmaniasis-only, is precisely less than unity. Results also show that in the case of maximum protection against visceral leishmaniasis (VL), the disease-free equilibrium is globally asymptotically stable if malaria patients are protected from VL infection; similarly, in the case of maximum protection against malaria, the disease-free equilibrium is globally asymptotically stable if VL and post-kala-azar dermal leishmaniasis patients and the recovered humans after VL are protected from malaria infection. Numerical results show that if R_m and R_l are greater than unity, then we have co-existence of both disease at an endemic equilibrium, and malaria incidence is higher than visceral leishmaniasis incidence at steady state.