Global dynamic of spatio-temporal fractional order SEIR model

The global analysis of a spatio-temporal fractional order SEIR infection epidemic model is studied and analyzed in this paper.  The dynamics of the infection is described by four partial differential equations with a fractional derivative order and with diffusion.  The equations of our model describe the evolution of the susceptible, the exposed, the infected and the recovered  individuals with taking into account the spatial diffusion for each compartment.  At first, we will prove the existence and uniqueness of the solution using the results of the fixed point theorem, and the equilibrium points are established and presented according to ${\cal R}_{0}$.  Next, the bornitude and the positivity of the solutions of the proposed model are established.  Using the Lyapunov direct method it has been proved that the global stability of the each equilibrium depends mainly on the basic reproduction number ${\cal R}_{0}$.  Finally, numerical simulations are performed to validate the theoretical results.

  1. Mendel J. B., Lee J. T., Rosman D.  Current concepts imaging in COVID-19 and the challenges for low and middle income countries.  Journal of Global Radiology.  6 (1), 3 (2020).
  2. Fu H., Gray K. A.  The key to maximizing the benefits of antimicrobial and self-cleaning coatings is to fully determine their risks.  Current Opinion in Chemical Engineering.  34, 100761 (2021).
  3. https://www.who.int/news/item/01-01-1996-infectious-diseases-kill-over-17-million-people-a-year-who-warns-of-global-crisis.
  4. Bernoulli D.  Essai d'une nouvelle analyse de la mortalité causée par la petite vérole et des avantages de l'inoculation pour la prévenir.  Mémoires de mathématique et de physique, presentès à l'Académie royale des sciences et lus dans ses assemblées. Paris (1766).
  5. Kermack W. O., McKendrick A. G.  A contribution to the mathematical theory of epidemics.  Proceedings of the Royal Society of London. Series A.  115 (772), 700–721 (1927).
  6. Li M. Y., Muldowney J. S.  Global stability for the SEIR model in epidemiology.  Mathematical Biosciences.  125 (2), 155–164 (1995).
  7. Danane J., Allali K., Tine L. M., Volpert V.  Nonlinear Spatiotemporal Viral Infection Model with CTL Immunity: Mathematical Analysis.  Mathematics.  8 (1), 52 (2020).
  8. Fadugba S. E., Ali F., Abubakar A. B.  Caputo fractional reduced differential transform method for SEIR epidemic model with fractional order.  Mathematical Modeling and Computing. 8 (3), 537–548 (2021).
  9. Khaloufi I., Lafif M., Benfatah Y., Laarabi H., Bouyaghroumni J., Rachik M.  A continuous SIR mathematical model of the spread of infectious illnesses that takes human immunity into account.  Mathematical Modeling and Computing.  10 (1), 53–65 (2023).
  10. 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).
  11. Ilnytskyi J. M.  Modeling of the COVID-19 pandemic in the limit of no acquired immunity.  Mathematical Modeling and Computing.  8 (2), 282–303 (2021).
  12. El Youssoufi L., Kouidere A., Kada D., Balatif O., Daouia A., Rachik M.  On stability analysis study and strategies for optimal control of a mathematical model of hepatitis HCV with the latent state.  Mathematical Modeling and Computing.  10 (1), 101–118 (2023).
  13. Hattaf K., Yousfi N.  Global stability for reaction-diffusion equations in biology.  Computers & Mathematics with Applications.  66 (8), 1488–1497 (2013).
  14. Chang L., Gao S., Wang Z.  Optimal control of pattern formations for an SIR reaction–diffusion epidemic model.  Journal of Theoretical Biology.  536, 111003  (2022).
  15. Chinviriyasit S., Chinviriyasit W.  Numerical modelling of an SIR epidemic model with diffusion.  Applied Mathematics and Computation.  216 (2), 395–409 (2010).
  16. Deng K.  Asymptotic behavior of an SIR reaction-diffusion model with a linear source.  Discrete & Continuous Dynamical Systems – B.  24 (11), 5945–5957 (2019).
  17. Debnath L.  Recent applications of fractional calculus to science and engineering.  International Journal of Mathematics and Mathematical Sciences.  2003, 753601 (2003).
  18. Ding Y., Ye H.  A fractional-order differential equation model of HIV infection of CD4$^+$ T-cells.  Mathematical and Computer Modeling.  50 (3–4), 386–392 (2009).
  19. Danane J., Hammouch Z., Allali K., Rashid S., Singh J.  A fractional-order model of coronavirus disease 2019 (COVID-19) with governmental action and individual reaction.  Mathematical Methods in the Applied Sciences. 1–14 (2021).
  20. Wang X., Wang Z., Huang X., Li Y.  Dynamic analysis of a delayed fractional-order SIR model with saturated incidence and treatment functions.  International Journal of Bifurcation and Chaos.  28 (14), 1850180 (2018).
  21. Naik P. A.  Global dynamics of a fractional-order SIR epidemic model with memory.  International Journal of Biomathematics.  13 (08), 2050071 (2020).
  22. Sidi Ammi M. R., Tahiri M., Tilioua M., Zeb A., Khan I., Andualem M.  Global analysis of a time fractional order spatio-temporal SIR model.  Scientific Reports.  12 (1), 5751 (2022).
  23. Duduchava R.  The Green formula and layer potentials.  Integral Equations and Operator Theory.  41 (2), 127–178 (2001).
  24. Dubois F., Galucio A. C., Point N.  Introduction à la dérivation fractionnaire – Théorie et applications. AF510 v1 (2010).
  25. Kilbas A. A., Srivastava H. M., Trujillo J. J.  Theory and applications of fractional differential equations.  Elsevier Science (2006).
  26. Bebernes J. W.  The Stability of Dynamical Systems (J. P. Lasalle).  SIAM Review.  21 (3), 418–420 (1979).
  27. LaSalle J.  Some extensions of Liapunov's second method.  IRE Transactions on Circuit Theory.  7 (4), 520–527 (1960).
  28. El-Borai M. M.  Some probability densities and fundamental solutions of fractional evolutions equations.  Chaos, Solitons & Fractals.  14 (3), 433–440 (2002).
  29. 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).
  30. Vargas-De-León C.  Volterra–type Lyapunov functions for fractional-order epidemic systems.  Communications in Nonlinear Science and Numerical Simulation.  24 (1–3), 75–85 (2015).
  31. Li C., Zeng F.  Numerical Methods for Fractional Calculus.  Chapman & Hall/CRC (2015).
  32. Qureshi S., Jan R.  Modeling of measles epidemic with optimized fractional order under Caputo differential operator.  Chaos, Solitons & Fractals.  145, 110766 (2021).