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

2023;
: pp. 101–118

Received: June 01, 2022
Revised: January 01, 2023
Accepted: January 02, 2023
1
Laboratory of Analysis, Modeling, and Simulation (LAMS), Department of Mathematics and Computer Science, Faculty of Sciences Ben M'Sick, Hassan II University of Casablanca, Morocco
2
Laboratory of Analysis, Modeling, and Simulation (LAMS), Department of Mathematics and Computer Science, Faculty of Sciences Ben M'Sick, Hassan II University of Casablanca, Morocco
3
Laboratory of Information Technology and Modeling, Department of Mathematics and Computer Science, Faculty of Sciences Ben M'Sick, Hassan II University of Casablanca, Morocco
4
Laboratory of Dynamical Systems, Mathematical Engineering Team (INMA), Department of Mathematics, Faculty of Sciences El Jadida, Chouaib Doukkali University, El Jadida, Morocco
5
Laboratory of Mathematics and Applications, ENS, Hassan II University of Casablanca, Morocco
6
Laboratory of Analysis, modeling and simulation, Department of mathematics and computer sciences, Faculty of sciences Ben M'Sik, University Hassan II of Casablanca

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.

Mathematical Modeling and Computing, Vol. 10, No. 1, pp. 101–118 (2023)