Modeling the effect of non-cytolytic immune response on viral infection dynamics in the presence of humoral immunity

  • Mausumi Dhar
  • Shilpa Samaddar
  • Paritosh BhattacharyaEmail author
Original paper


In this paper, a mathematical model describing the viral infection dynamics with non-cytolytic effect of humoral immune response is presented and analyzed. The effect of non-cytolytic immune response on the process of viral infectivity has been basically described by the non-cytolytic cure of infected cells and inhibition of viral replication, i.e., the non-lytic immune response. The sufficient criteria for the local and global stability of the equilibria, namely disease-free equilibrium, immune-free equilibrium and chronic equilibrium with humoral response, have been determined in terms of two threshold parameters, viz., the basic reproduction number, \(R_0\), and the humoral immune response reproduction number, \(R_1\). The condition governing the occurrence of Hopf bifurcation around the chronic equilibrium with humoral response has been obtained using the rate of infection as a bifurcation parameter. The obtained results indicate that the infection gets eradicated for \(R_0 \le 1\) and persists in the body for \(R_0 > 1\). Numerical simulations are presented to support our analytical findings. The comparison of various viral dynamic models suggests that the incorporation of the non-cytolytic immune response increases the concentration of uninfected cells, but causes a depletion of humoral immune response. Further, the effect of non-cytolytic immune response on the dynamical behavior of the system has been demonstrated.


Humoral immune response Non-cytolytic immune response Global stability Hopf bifurcation 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Srivastava, P., Chandra, P.: Hopf bifurcation and periodic solutions in a dynamical model for HIV and immune response. Differ. Equ. Dyn. Syst. 16(1–2), 77–100 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Yang, Y., Zou, L., Ruan, S.: Global dynamics of a delayed within-host viral infection model with both virus-to-cell and cell-to-cell transmissions. Math. Biosci. 270, 183–191 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Douam, F., Lavillette, D., Cosset, F.L.: The mechanism of HCV entry into host cells. In: Progress in Molecular Biology and Translational Science, vol. 129, pp. 63–107. Elsevier (2015)Google Scholar
  4. 4.
    Nowak, M.A., Bangham, C.R.: Population dynamics of immune responses to persistent viruses. Science 272(5258), 74–79 (1996)CrossRefGoogle Scholar
  5. 5.
    Bonhoeffer, S., May, R.M., Shaw, G.M., Nowak, M.A.: Virus dynamics and drug therapy. Proc. Natl. Acad. Sci. 94(13), 6971–6976 (1997)CrossRefGoogle Scholar
  6. 6.
    Murase, A., Sasaki, T., Kajiwara, T.: Stability analysis of pathogen–immune interaction dynamics. J. Math. Biol. 51(3), 247–267 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Pugliese, A., Gandolfi, A.: A simple model of pathogen-immune dynamics including specific and non-specific immunity. Math. Biosci. 214(1–2), 73–80 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Zhou, X., Shi, X., Zhang, Z., Song, X.: Dynamical behavior of a virus dynamics model with CTL immune response. Appl. Math. Comput. 213(2), 329–347 (2009)Google Scholar
  9. 9.
    Kajiwara, T., Sasaki, T.: Global stability of pathogen-immune dynamics with absorption. J. Biol. Dyn. 4(3), 258–269 (2010)Google Scholar
  10. 10.
    Wodarz, D., Christensen, J.P., Thomsen, A.R.: The importance of lytic and nonlytic immune responses in viral infections. Trends Immunol. 23(4), 194–200 (2002)CrossRefGoogle Scholar
  11. 11.
    Wang, K., Jin, Y., Fan, A.: The effect of immune responses in viral infections: a mathematical model view. Discrete Contin. Dyn. Syst. Ser. B 19, 3379–3396 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dhar, M., Samaddar, S., Bhattacharya, P., Upadhyay, R.K.: Viral dynamic model with cellular immune response: a case study of HIV-1 infected humanized mice. Physica A Stat. Mech. Appl. 524, 1–14 (2019)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Montefiori, D.C.: Role of complement and Fc receptors in the pathogenesis of HIV-1 infection. In: Fauci, A.S., Pantaleo, G. (eds.) Immunopathogenesis of HIV Infection, pp. 119–138. Springer-Verlag, Berlin, Heidelberg (1997) Google Scholar
  14. 14.
    Parren, P.W., Burton, D.R.: The Antiviral Activity of Antibodies In Vitro and In Vivo. Elsevier, Amsterdam (2001)CrossRefGoogle Scholar
  15. 15.
    Wodarz, D.: Mathematical models of immune effector responses to viral infections: virus control versus the development of pathology. J. Comput. Appl. Math. 184(1), 301–319 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Yousfi, N., Hattaf, K., Rachik, M.: Analysis of a HCV model with CTL and antibody responses. Appl. Math. Sci. 3(57), 2835–2845 (2009)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Obaid, M.A., Elaiw, A.: Stability of virus infection models with antibodies and chronically infected cells. In: Abstract and Applied Analysis, vol. 2014. Hindawi (2014)Google Scholar
  18. 18.
    Pan, S., Chakrabarty, S.P.: Threshold dynamics of HCV model with cell-to-cell transmission and a non-cytolytic cure in the presence of humoral immunity. Commun. Nonlinear Sci. Numer. Simul. 61, 180–197 (2018)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Ciupe, S.M., Ribeiro, R.M., Nelson, P.W., Perelson, A.S.: Modeling the mechanisms of acute hepatitis B virus infection. J. Theor. Biol. 247(1), 23–35 (2007)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Zhou, X., Song, X., Shi, X.: A differential equation model of HIV infection of CD4+ T-cells with cure rate. J. Math. Anal. Appl. 342(2), 1342–1355 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Liu, X., Wang, H., Hu, Z., Ma, W.: Global stability of an HIV pathogenesis model with cure rate. Nonlinear Anal. Real World Appl. 12(6), 2947–2961 (2011)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Hattaf, K., Yousfi, N.: A delay differential equation model of HIV with therapy and cure rate. Int. J. Nonlinear Sci. 12(4), 503–512 (2011)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Hattaf, K., Yousfi, N., Tridane, A.: Mathematical analysis of a virus dynamics model with general incidence rate and cure rate. Nonlinear Anal. Real World Appl. 13(4), 1866–1872 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Tian, Y., Liu, X.: Global dynamics of a virus dynamical model with general incidence rate and cure rate. Nonlinear Anal. Real World Appl. 16, 17–26 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Guidotti, L.G., Rochford, R., Chung, J., Shapiro, M., Purcell, R., Chisari, F.V.: Viral clearance without destruction of infected cells during acute HBV infection. Science 284(5415), 825–829 (1999)CrossRefGoogle Scholar
  26. 26.
    Hale, J.K., Lunel, S.V.: Introduction to functional differential equations. Appl. Math. Sci. 99, 191–238 (1993)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Banerjee, S., Keval, R., Gakkhar, S.: Modeling the dynamics of Hepatitis C virus with combined antiviral drug therapy: interferon and ribavirin. Math. Biosci. 245(2), 235–248 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180(1–2), 29–48 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Hethcote, H.W.: The mathematics of infectious diseases. SIAM Rev. 42(4), 599–653 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Dubey, B., Dubey, P., Dubey, U.S.: Modeling the intracellular pathogen-immune interaction with cure rate. Commun. Nonlinear Sci. Numer. Simul. 38, 72–90 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Dahari, H., Lo, A., Ribeiro, R.M., Perelson, A.S.: Modeling hepatitis C virus dynamics: liver regeneration and critical drug efficacy. J. Theor. Biol. 247(2), 371–381 (2007)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Reyes-Silveyra, J., Mikler, A.R.: Modeling immune response and its effect on infectious disease outbreak dynamics. Theor. Biol. Med. Model. 13(1), 10 (2016)CrossRefGoogle Scholar
  33. 33.
    Shi, X., Zhou, X., Song, X.: Dynamical behavior of a delay virus dynamics model with CTL immune response. Nonlinear Anal. Real World Appl. 11(3), 1795–1809 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Song, X., Wang, S., Dong, J.: Stability properties and Hopf bifurcation of a delayed viral infection model with lytic immune response. J. Math. Anal. Appl. 373(2), 345–355 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Wang, Z., Xu, R.: Stability and Hopf bifurcation in a viral infection model with nonlinear incidence rate and delayed immune response. Commun. Nonlinear Sci. Numer. Simul. 17(2), 964–978 (2012)Google Scholar
  36. 36.
    Miao, H., Teng, Z., Abdurahman, X.: Stability and Hopf bifurcation for a five-dimensional virus infection model with Beddington–DeAngelis incidence and three delays. J. Biol. Dyn. 12(1), 146–170 (2018)MathSciNetCrossRefGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsNational Institute of Technology AgartalaTripuraIndia

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