Spatiotemporal Dynamics of a Class of Models Describing Infectious Diseases

  • Khalid HattafEmail author
  • Noura Yousfi
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 200)


In this chapter, we propose and analyze a class of three spatiotemporal models describing infectious diseases caused by viruses such as the human immunodeficiency virus (HIV) and the hepatitis B virus (HBV). The first model with cellular immunity, the second with humoral immunity and the third with cellular and humoral immune responses. In the three proposed models, the disease transmission process is modeled by a general incidence function which includes several forms existing in the literature. In addition, the global analysis of the proposed models is rigorously investigated. Furthermore, biological findings of our analytical results are presented. Moreover, mathematical virus models and results presented in many previous studies are extended and generalized.


Virus dynamics Immunity Diffusion Lyapunov functional Global stability 


  1. 1.
    Wang, K., Wang, W.: Propagation of HBV with spatial dependence. Math. Biosci. 210, 78–95 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Wang, K., Wang, W., Song, S.: Dynamics of an HBV model with diffusion and delay. J. Theor. Biol. 253, 36–44 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brauner, C.-M., Jolly, D., Lorenzi, L., Thiebaut, R.: Heterogeneous viral environment in a HIV spatial model. Discrete Continuous Dyn. Syst. Ser. S 15, 545–572 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Xu, R., Ma, Z.E.: An HBV model with diffusion and time delay. J. Theor. Biol. 257, 499–509 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chan Chí, N., Ávila Vales, E., García Almeida, G.: Analysis of a HBV model with diffusion and time delay. J. Appl. Math. 2012, 1–25 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Zhang, Y., Xu, Z.: Dynamics of a diffusive HBV model with delayed Beddington-DeAngelis response. Nonlinear Anal.: Real World Appl. 15, 118–139 (2014)Google Scholar
  7. 7.
    Hattaf, K., Yousfi, N.: Global dynamics of a delay reaction-diffusion model for viral infection with specific functional response. J. Comput. Appl. Math. 34(3), 807–818 (2015)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Hattaf, K., Yousfi, N.: A generalized HBV model with diffusion and two delays. Comput. Math. Appl. 69(1), 31–40 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Crowley, P.H., Martin, E.K.: Functional responses and interference within and between year classes of a dragonfly population. J. North Am. Benth. Soc. 8, 211–221 (1989)CrossRefGoogle Scholar
  10. 10.
    X. Zhuo, Analysis of a HBV infection model with noncytolytic cure process. In: IEEE 6th International Conference on Systems Biology (ISB), pp. 148–151 (2012)Google Scholar
  11. 11.
    Hattaf, K., Yousfi, N.: A class of delayed viral infection models with general incidence rate and adaptive immune response. Int. J. Dyn. Control 4(3), 254–265 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Riad, D., Hattaf, K., Yousfi, N.: Dynamics of capital-labour model with Hattaf-Yousfi functional response. Br. J. Math. Comput. Sci. 18(5), 1–7 (2016)CrossRefGoogle Scholar
  13. 13.
    Mahrouf, M., Hattaf, K., Yousfi, N.: Dynamics of a stochastic viral infection model with immune response. Math. Model. Nat. Phenom. 12(5), 15–32 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hattaf, K., Yousfi, N.: A numerical method for a delayed viral infection model with general incidence rate. J. King Saud Univ. Sci. 28(4), 368–374 (2016)CrossRefGoogle Scholar
  15. 15.
    Wang, X.-Y., Hattaf, K., Huo, H.-F., Xiang, H.: Stability analysis of a delayed social epidemics model with general contact rate and its optimal control. J. Ind. Manag. Optim. 12(4), 1267–1285 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Wang, S., Feng, X., He, Y.: Global asymptotical properties for a diffused HBV infection model with CTL immune response and nonlinear incidence. Acta Math. Sci. 31, 1959–1967 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Yang, Y., Xu, Y.: Global stability of a diffusive and delayed virus dynamics model with Beddington-DeAngelis incidence function and CTL immune response. Comput. Math. Appl. 71, 922–930 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kang, C., Miao, H., Chen, X., Xu, J., Huang, D.: Global stability of a diffusive and delayed virus dynamics model with Crowley-Martin incidence function and CTL immune response. Adv. Differ. Equ. 2017(1), 324 (2017).
  19. 19.
    Hattaf, K., Yousfi, N., Tridane, A.: Global stability analysis of a generalized virus dynamics model with the immune response. Can. Appl. Math. Q. 20(4), 499–518 (2012)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1993)CrossRefGoogle Scholar
  21. 21.
    Deans, J.A., Cohen, S.: Immunology of malaria. Annu. Rev. Microbiol. 37, 25–50 (1983)CrossRefGoogle Scholar
  22. 22.
    Murase, A., Sasaki, T., Kajiwara, T.: Stability analysis of pathogen-immune interaction dynamics. J. Math. Biol. 51, 247–267 (2005)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Wang, X., Liu, S.: A class of delayed viral models with saturation infection rate and immune response. Math. Methods Appl. Sci. 36(2), 125–142 (2013)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Wang, T., Hu, Z., Liao, F., Ma, W.: Global stability analysis for delayed virus infection model with general incidence rate and humoral immunity. Math. Comput. Simul. 89, 13–22 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Wang, S., Zou, D.: Global stability of in-host viral models with humoral immunity and intracellular delays. Appl. Math. Model. 36, 1313–1322 (2012)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Elaiw, A.M., AlShamrani, N.H.: Global properties of nonlinear humoral immunity viral infection models. Int. J. Biomath. 8(5), 1550058 (2015)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Elaiw, A.M., AlShamrani, N.H., Hattaf, K.: Dynamical behaviors of a general humoral immunity viral infection model with distributed invasion and production. Int. J. Biomath. 10(3), 1750035 (2017)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Hattaf, K., Yousfi, N.: Global stability for reaction-diffusion equations in biology. Comput. Math. Appl. 66, 1488–1497 (2013)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Boni, C., Fisicaro, P., Valdatta, C., Amadei, B., Di Vincenzo, P., Giuberti, T., Laccabue, D., Zerbini, A., Cavalli, A., Missale, G., Bertolli, A., Ferrari, C.: Characterization of hepatitis B virus (HBV)-specific T-cell dysfunction in chronic HBV infection. J. Virol. 81(8), 4215–4225 (2007)CrossRefGoogle Scholar
  30. 30.
    Hattaf, K., Khabouze, M., Yousfi, N.: Dynamics of a generalized viral infection model with adaptive immune response. Int. J. Dyn. Control 3(3), 253–261 (2015)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Wodarz, D.: Hepatitis C virus dynamics and pathology: the role of CTL and antibody responses. J. Gen. Virol. 84, 1743–1750 (2003)CrossRefGoogle Scholar
  32. 32.
    Yan, Y., Wang, W.: Global stability of a five-dimensionalmodel with immune responses and delay. Discret. Continuous Dyn. Syst. Ser. B 17(1), 401–416 (2012)CrossRefGoogle Scholar
  33. 33.
    Zhao, Y., Xu, Z.: Global dynamics for a delayed hepatitis C virus infection model. Electron. J. Differ. Equ. 2014(132), 1–18 (2014)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Su, Y., Sun, D., Zhao, L.: Global analysis of a humoral and cellular immunity virus dynamics model with the Beddington-DeAngelis incidence rate. Math. Methods Appl. Sci. 38(14), 2984–2993 (2015)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Centre Régional des Métiers de l’Education et de la Formation (CRMEF)CasablancaMorocco
  2. 2.Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M’sikHassan II UniversitySidi Othman, CasablancaMorocco

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