Optimal control analysis of cryptosporidiosis disease

In Sub-Saharan Africa, HIV/AIDS and chronic parasitic diseases (such as trypanosomiasis, schistosomiasis, malaria, etc) have profound effect on the immune system and alter the host’s immune response to infections, thereby paving way for opportunistic parasitic infections (such as cryptosporidiosis, isosporiasis, microsporidiosis, etc) to attack. Hence, mathematical models provide a quantitative and potentially valuable tool for this purpose. This paper therefore is to derive and analyze a deterministic model for the transmission of cryptosporidiosis disease. The basic reproduction number, and the existence and stability of equilibria were investigated. Firstly, we analyzed the model and investigated its stability and bifurcation behaviour. In our findings, model exhibited a backward bifurcation phenomenon. This has epidemiological implication, which means that for effective eradication and control of the disease, bringing R0 less than unity was no longer sufficient, but rather, that R0 < R0c. Furthermore, we modified the model to incorporate susceptible individuals with weakened immune system. We then analyzed the model and investigated its stability property, in order to determine the effect and impact of individuals with weakened immune systems in the disease transmission. From the sensitivity analysis, it was found that increase in the number of susceptible individuals with weak immune system increases the basic reproduction number.