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

2023;
: pp. 101–118
https://doi.org/10.23939/mmc2023.01.101
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.

  1. El Youssoufi L., Khajji B., Balatif O., Rachik M.  A discrete mathematical modeling for drinking alcohol model resulting in road accidents and violence: an optimal control approach.  Communications in Mathematical Biology and Neuroscience.  2021, 88 (2021).
  2. Kouidere A., Labzai A., Khajji B., Ferjouchia H., Balatif O., Boutayeb A., Rachik M.  Optimal control strategy with multi-delay in state and control variables of a discrete mathematical modeling for the dynamics of diabetic population.  Communications in Mathematical Biology and Neuroscience.  2020, 14 (2020).
  3. Kouidere A., Balatif O., Ferjouchia H., Boutayeb A., Rachik M.  Optimal control strategy for a discrete time to the dynamics of a population of diabetics with highlighting the impact of living environment.  Discrete Dynamics in Nature and Society.  2019, 6342169 (2019).
  4. Birkhoff G., Rota G. C.  Ordinary differential equations.  New York, John Wiley & Sons (1989).
  5. Kouidere A., Balatif O., Rachik M.  Analysis and optimal control of a mathematical modeling of the spread of African swine fever virus with a case study of South Korea and cost-effectiveness.  Chaos, Solitons & Fractals.  146, 110867 (2021).
  6. Zhang S., Xu X. Dynamic analysis and optimal control for a model of hepatitis C with treatment.  Communications in Nonlinear Science and Numerical Simulation.  46, 14–25 (2016).
  7. Sadki M., Harroudi S., Allali K.  Dynamical analysis of an HCV model with cell-to-cell transmission and cure rate in the presence of adaptive immunity.  Mathematical Modeling and Computing.  9 (3), 579–593 (2022).
  8. Khajji B., Boujallal L., Elhia M., Balatif O., Rachik M.  A fractional-order model for drinking alcohol behaviour leading to road accidents and violence.  Mathematical Modeling and Computing.  9 (3), 501–518 (2022).
  9. Diekmann O., Heesterbeek J. A. P., Metz J. A.  On the definition and the computation of the basic reproduction ratio $R_{0}$ in models for infectious diseases in heterogeneous populations.  Journal of Mathematical Biology.  28 (4), 365–382 (1990).
  10. La Salle J. P.  The stability of dynamical systems.  CBMS-NSF Regional Conference Series in Applied Mathematics.  CB25, SIAM, Philadelphia, PA, USA (1976).
  11. Gumel A. B., Shivakumar P. N., Sahai B. M.  A mathematical model for the dynamics of HIV-1 during the typical course of infection.  Nonlinear Analysis: Theory, Methods & Applications.  47 (3), 1773–1783 (2001).
  12. Kouidere A., Youssoufi L. E., Ferjouchia H., Balatif O., Rachik M.  Optimal Control of Mathematical modeling of the spread of the COVID-19 pandemic with highlighting the negative impact of quarantine on diabetics people with Cost-effectiveness.  Chaos, Solitons & Fractals.  145, 110777 (2021).
  13. Lhous M., Rachik M., Laarabi H., Abdelhak A.  Discrete mathematical modeling and optimal control of the marital status: the monogamous marriage case.  Advances in Difference Equations.  2017, 339 (2017).
  14. World Health Organization, Global Health Sector Strategy on Viral Hepatitis 2016–2021.  https://apps.who.int/iris/bitstream/handle/10665/246177/WHO-HIV-2016.06-eng.pdf (2016).
  15. World Health Organization.  https://www.who.int/news-room/fact-sheets/detail/hepatitis-c (2021).
  16. El Youssoufi L., Moutamanni H., Labzai A., Balatif O., Rachik M.  Optimal control for a discrete model of hepatitis C with latent, acute and chronic stages in the presence of treatment.  Communications in Mathematical Biology and Neuroscience.  2020, 82 (2020).
  17. Communiqué de presse, Association de lutte contre le sida (ALCS), membre de Coalition PLUS.  30 mai 2018 \'a Casablanca, Marocco.
  18. Bani-Yaghoub M., Gautam R., Shuai Z., van den Driessche P., Ivanek R.  Reproduction numbers for infections with free-living pathogens growing in the environment.  Journal of Biological Dynamics.  6 (2), 923–940 (2012).
  19. Van den Driessche P., Watmough J.  Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission.  Mathematical Biosciences.  180 (1–2), 29–48 (2002).
  20. Fleming W. H., Rishel R. W.  Deterministic and stochastic optimal control.  Springer, New York, NY, USA (1975).
  21. Boyce W. E., DiPrima R. C., Meade D. B.  Elementary Differential Equations and Boundary Value Problems.  John Wiley & Sons (2021).
  22. Pontryagin L. S., Boltyanskii V. G., Gamkrelidze R. V., Mishchenko E. F.  The Mathematical Theory of Optimal processes.  Wiley, New York, NY, USA (1962).
  23. Zakary O., Rachik M, Elmouki I.  On the analysis of a multi-regions discrete SIR epidemic model: An optimal control approach.  International Journal of Dynamics and Control.  5, 917–930 (2016).
Mathematical Modeling and Computing, Vol. 10, No. 1, pp. 101–118 (2023)