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A theoretical study on schistosomiasis infection model: Application of biological optimal control
Mouhamadou Diaby
Keywords:Schistosomiasis Model, Stability Analysis, Basic Reproduction Number, Optimal control
      This paper is a continuation and an extension of the study undertaken in Diaby et~al~\cite{diaby2014global} concerning the evolution of a schistosomiasis infection. In the first part of our analysis we describe and propose a complete mathematical analysis of a new mathematical model for schistosomiasis infection with fixed control for both drug and biological treatment. This model is based on the same derivations as the model of Allen et al. \cite{allen2003modelling} but it considers the miracidia and cercariae dynamics. It also includes a net inflow of competitor snails into the aquatic region at the rate $u$ per unit of time as control term. Schistosomiasis is associated with water resource development such as dams and irrigation schemes, where the snail is the intermediate host of the parasite breeds. The snail intermediate host breeds in slow-flowing or stagnant water. We establish a deterministic model to explore the role of biological control strategy. We derive the basic reproduction number $\mathcal{R}_0$ and establish that the global dynamics are completely determined by the values of $\mathcal{R}_0$. It is shown that the disease can be eradicated when $\mathcal{R}_0 \leq 1$. In the case where $\mathcal{R}_0 >1$, we prove the existence, uniqueness and global asymptotic stability of an endemic equilibrium. We also show how the control $u$ can be chosen in order to eradicate the disease. In the second part, we take the controls as time dependent and obtain the optimal control strategy to minimize both infected humans and snails populations. All the analytical results are verified by simulations works. Some important conclusions are given at the end of the paper.