On stability analysis study and strategies for optimal control of a mathematical model of hepatitis HCV with the latent state

In this work, we analyze a viral hepatitis C model.  This epidemic remains a major problem for global public health, in all communities, despite the efforts made.  The model is analyzed using the stability theory of systems of nonlinear differential equations.  Based on the results of the analysis, the proposed model has two equilibrium points: a disease-free equilibrium point $E_0$ and an endemic equilibrium point $E^{*}$.  We investigate the existence of equilibrium point of the model.  Furthermore, based on the indirect Lyapunov method, we study the local stability of each equilibrium point of the model.  Moreover, by constructing the appropriate Lyapunov function and by using LaSalle invariance principle, we get some information on the global stability of equilibrium points under certain conditions.  The basic reproduction number $R_0$ is calculated using the Next Generation method.  The positivity of the solutions and their bornitude have been proven, the existence of the solutions has also been proven.  Optimal control of the system was studied by proposing three types of intervention: awareness program, early detection, isolation and treatment.  The maximum principle of Pontryagin was used to characterize the optimal controls found.  Numerical simulations were carried out with a finite numerical difference diagram and using MATLAB to confirm acquired results.