Applied Mathematics & Information Sciences

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This paper presents a non-linear mathematical model of malaria by considering the human reservoir and larvivorous fishes. The different equilibria of the model are computed and stability of these equilibria is investigated in-detail. Also, the basic reproduction number R0 of the model is computed and we observe that the model exhibits backward bifurcation for some set of parameters implying the existence of multiple endemic equilibria for R0 < 1. This existence of multiple endemic equilibria emphasizes the fact that R0 < 1 is not sufficient to eradicate the disease from the population and the need is to lower R0 much below one to make the disease-free equilibrium to be globally stable. The numerical simulation is performed to support analytical findings and the presented results show meaningful agreement. Additionally, the model is extended to incorporate optimal control by introducing the ‘insecticide control’ to control the mosquito population and Pontryagin’s maximum principle[1] is used to analyze the optimal control model. Here too the numerical simulation is performed to demonstrate the effect of optimal control.