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South Africa.


In this paper a mathematical model for trypanosomiasis-cryptosporidium co-infection dynamics is investigated to give a theoretical mathematical account of the impact of cryptosporidiosis on trypanosomiasis dynamics. The model steady states are analyzed. The disease-free equilibrium is shown to be locally asymptotically stable when the associated epidemic basic reproduction number for the model is less than unity. The trypanosomiasis only and the cryptosporidiosis only model are each found to exhibit transcritical and backward bifurcation phenomena respectively. While the co-infection model exhibits the possibility of multiple endemic equilibria. From the sensitivity analysis, the trypanosomiasis reproductive number Rlt 0 is more sensitive to d (death due to insecticides) and crypto parameters whenever Rcr 0 > 1 (crypto reproductive number). While the cryptosporidiosis reproductive number Rcr 0 is less sensitive to trypanosomiasis parameters whenever Rlt 0 > 1 (trypanosomiasis reproductive number). This is an indication that cryptosporidiosis infection may be associated with an increased risk of trypanosomiasis, while trypanosomiasis infection is not associated with an increased risk for cryptosporidiosis.We incorporate time dependent controls, using Pontryagin’sMaximum Principle to derive necessary conditions for the optimal control of the disease. Furthermore, the effect of the presence of each infection on the endemicity of the other is investigated and presented numerically.

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