| Summary: | In this paper, we analyse a dynamical system taking into account the asymptomatic infection and we consider optimal control strategies based on a regular network. We obtain basic mathematical results for the model without control. We compute the basic reproduction number ($ \mathcal{R} $) by using the method of the next generation matrix then we analyse the local stability and global stability of the equilibria (disease-free equilibrium (DFE) and endemic equilibrium (EE)). We prove that DFE is LAS (locally asymptotically stable) when $ \mathcal{R} < 1 $ and it is unstable when $ \mathcal{R} > 1 $. Further, the existence, the uniqueness and the stability of EE is carried out. We deduce that when $ \mathcal{R} > 1 $, EE exists and is unique and it is LAS. By using generalized Bendixson-Dulac theorem, we prove that DFE is GAS (globally asymptotically stable) if $ \mathcal{R} < 1 $ and that the unique endemic equilibrium is globally asymptotically stable when $ \mathcal{R} > 1 $. Later, by using Pontryagin's maximum principle, we propose several reasonable optimal control strategies to the control and the prevention of the disease. We mathematically formulate these strategies. The unique optimal solution was expressed using adjoint variables. A particular numerical scheme was applied to solve the control problem. Finally, several numerical simulations that validate the obtained results were presented.
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