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  3. Volume 11, Number 2, 2024
  4. Spatiotemporal dynamics of RNA viruses in the presence of immunity and treatment: case of SARS-CoV-2

Spatiotemporal dynamics of RNA viruses in the presence of immunity and treatment: case of SARS-CoV-2

MMC.
2024;
Volume 11, Number 2
: pp. 518–527
https://doi.org/10.23939/mmc2024.02.518
Received: December 18, 2023
Revised: May 21, 2024
Accepted: May 28, 2024

El Karimi M. I., Hattaf K., Yousfi N. Spatiotemporal dynamics of RNA viruses in the presence of immunity and treatment: case of SARS-CoV-2.  Mathematical Modeling and Computing. Vol. 11, No. 2, pp. 518–527 (2024)

Authors:
  1. M. I. El Karimi
  2. K. Hattaf
  3. N. Yousfi
1
Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M'Sick, Hassan II University of Casablanca; Centre Régional des Métiers de l'Education et de la Formation (CRMEF)
2
Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M'Sick, Hassan II University of Casablanca; Equipe de Recherche en Modélisation et Enseignement des Mathématiques (ERMEM), Centre Régional des Métiers de l'Education et de la Formation (CRMEF)
3
Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M'Sick, Hassan II University of Casablanca

In this paper, we develop a mathematical model using partial differential equations to investigate the behavior of RNA viruses in the presence of antiviral treatment.  The developed model includes both cell-to-cell and virus-to-cell modes of transmission.  Initially, we establish the well-posedness of the model by demonstrating the existence and uniqueness of solutions, as well as their positivity and boundedness.  Additionally, we identify and analyze the stable equilibrium states, their global stability depending on specific threshold parameters, using Lyapunov functions.  To corroborate our theoretical findings, we provide illustrations through numerical simulations.

reaction–diffusion
RNA viruses
modes of transmission
mathematical modeling
stability analysis
  1. WHO. Hepatitis C. https://www.who.int/news-room/fact-sheets/detail/hepatitis-c.
  2. WHO. HIV and AIDS.  https://www.who.int/news-room/fact-sheets/detail/hiv-aids.
  3. Rezaei N.  Coronavirus disease-COVID-19.  Springer (2021).
  4. WHO. EG.5 Initial Risk Evaluation.  https://www.who.int/docs/default-source/coronaviruse/09082023eg.5_ire_final.pdf?sfvrsn=2aa2daee_3.
  5. WHO. Coronavirus (COVID-19). https://covid19.who.int/.
  6. Perelson A. S., Kirschner D. E., Boer R. D.  Dynamics of HIV infection of CD4$^+$ T cells.  Mathematical Biosciences.  114 (1), 81–125 (1993).
  7. Nowak M., May R. M.  AIDS pathogenesis: mathematical models of HIV and SIV infections.  AIDS.  7, S3–S18 (1993).
  8. Hattaf K., Yousfi N.  Dynamics of SARS-CoV-2 infection model with two modes of transmission and immune response.  Mathematical Biosciences and Engineering .  17 (5), 5326–5340 (2020).
  9. Hattaf K., Karimi M. I. E., Mohsen A. A., Hajhouji Z., Younoussi M. E., Yousfi N.  Mathematical modeling and analysis of the dynamics of RNA viruses in presence of immunity and treatment: A case study of SARS-CoV-2.  Vaccines.  11 (2),  201 (2023).
  10. Hattaf K.  Spatiotemporal dynamics of a generalized viral infection model with distributed delays and CTL immune response.  Computation.  7 (2), 21 (2019).
  11. Pazy A.  Semigroups of Linear Operators and Applications to Partial Differential Equations.  Vol. 44 of Applied Mathematical Sciences, Springer, New York, USA (1983).
  12. Henry D.  Geometric Theory of Semilinear Parabolic Equations.  Springer-Verlag, Berlin, New York (1993).
  13. Hattaf K., Yousfi N.  Global stability for reaction-diffusion equations in biology.  Computers & Mathematics with Applications.  66 (8), 1488–1497 (2013).
  14. Hale J. K., Verduyn Lunel S. M.  Introduction to Functional Differential Equations.  Springer-Verlag (1993).