# American Institute of Mathematical Sciences

2014, 19(10): 3379-3396. doi: 10.3934/dcdsb.2014.19.3379

## The effect of immune responses in viral infections: A mathematical model view

 1 School of Biomedical Engineering, Third Military Medical University, Chongqing, 400038, China 2 Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588, United States 3 Chongqing Academy of Science & Technology, Chongqing, 401123, China

Received  February 2013 Revised  July 2013 Published  October 2014

To study the effect of immune response in viral infections, a new mathematical model is proposed and analyzed. It describes the interactions between susceptible host cells, infected host cells, free virus, lytic and nonlytic immune response. Using the LaSalle's invariance principle, we establish conditions for the global stability of equilibria. Uniform persistence is obtained when there is a unique endemic equilibrium. Mathematical analysis and numerical simulations indicate that the basic reproduction number of the virus and immune response reproductive number are sharp threshold parameters to determine outcomes of infection. Lytic and nonlytic antiviral activities play a significant role in the amount of susceptible host cells and immune cells in the endemic steady state. We also present potential applications of the model in clinical practice by introducing antiviral effects of antiviral drugs.
Citation: Kaifa Wang, Yu Jin, Aijun Fan. The effect of immune responses in viral infections: A mathematical model view. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3379-3396. doi: 10.3934/dcdsb.2014.19.3379
##### References:
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Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6. [33] K. Wang and Y. Kuang, Fluctuation and extinction dynamics in host-microparasite systems,, Commun. Pur. Appl. Anal., 10 (2011), 1537. doi: 10.3934/cpaa.2011.10.1537. [34] K. Wang, W. Tan, Y. Tang and G. Deng, Numerical diagnoses of superinfection in chronic hepatitis B viral dynamics,, Intervirology, 54 (2011), 349. doi: 10.1159/000321454. [35] K. Wang, W. Wang and X. Liu, Global stability in a viral infection model with lytic and nonlytic immune responses,, Comput. Math. Appl., 51 (2006), 1593. doi: 10.1016/j.camwa.2005.07.020. [36] K. Wang, W. Wang, H. Pang and X. Liu, Complex dynamic behavior in a viral model with delayed immune response,, Physica D, 226 (2007), 197. doi: 10.1016/j.physd.2006.12.001. [37] D. Wodarz, J. P. Christensen and A. R. Thomsen, The importance of lytic and nonlytic immune responses in viral infections,, TRENDS Immunol., 23 (2002), 194. doi: 10.1016/S1471-4906(02)02189-0. [38] J. S. Yi, M. A. Cox and A. J. Zajac, T-cell exhaustion: Characteristics, causes and conversion,, Immunology, 129 (2010), 474. doi: 10.1111/j.1365-2567.2010.03255.x. [39] N. Yousfi, K. Hattaf and A. Tridane, Modeling the adaptive immune response in HBV infection,, J. Math. Biol., 63 (2011), 933. doi: 10.1007/s00285-010-0397-x.

show all references

##### References:
 [1] B. S. Adiwijaya, T. L. Kieffer, J. Henshaw, K. Elsenhauer, H. Kimko, J. J. Alam, R. S. Kauffman and V. Garg, A viral dynamic model for treatment regimens with direct-acting antivirals for chronic hepatitis C infection,, PLoS Comput. Biol., 8 (2012). doi: 10.1371/journal.pcbi.1002339. [2] A. Bergthaler, L. Flatz, A. N. Hegazy, S. Johnson, E. Horvath, M. Löhning and D. D. Pinschewer, Viral replicative capacity is the primary determinant of lymphocytic choriomeningitis virus persistence and immunosuppression,, Proc. Natl. Acad. Sci. USA, 107 (2010), 21641. doi: 10.1073/pnas.1011998107. [3] S. M. Ciupe, R. M. Ribeiro, P. W. Nelson and A. S. Perelson, Modeling the mechanisms of acute hepatitis B virus infection,, J. Theor. Biol., 247 (2007), 23. doi: 10.1016/j.jtbi.2007.02.017. [4] H. Dahari, E. Shudo, R. M. Ribeiro and A. S. Perelson, Modeling complex decay profiles of hepatitis B virus during antiviral therapy,, Hepatology, 49 (2009), 32. doi: 10.1002/hep.22586. [5] R. J. De Boer and A. S. Perelson, Towards a general function describing T cell proliferation,, J. Theor. Biol., 175 (1995), 567. [6] R. J. De Boer and A. S. Perelson, Target cell limited and immune control models of HIV infection: A comparison,, J. Theor. Biol., 190 (1998), 201. [7] S. G. Deeks, Protease inhibitors as immunomodulatory drugs for HIV infection,, Clin. Pharmacol. Ther., 82 (2007), 248. doi: 10.1038/sj.clpt.6100205. [8] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations,, J. Math. Biol., 28 (1990), 365. doi: 10.1007/BF00178324. [9] C. Flexner, HIV drug development: The next 25 years,, Nat. Rev. Drug Discov., 6 (2007), 959. doi: 10.1038/nrd2336. [10] L. G. Guidotti, R. Rochford, J. Chung, M. Shapiro, R. Purcell and F. V. Chisari, Viral clearance without destruction of infected cells during acute HBV infection,, Science, 284 (1999), 825. doi: 10.1126/science.284.5415.825. [11] J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems,, SIAM J. Math. Anal., 20 (1989), 388. doi: 10.1137/0520025. [12] S. Hews, S. Eikenberry, J. D. Nagy and Y. Kuang, Rich dynamics of a hepatitis B viral infection model with logistic hepatocyte growth,, J. Math. Biol., 60 (2010), 573. doi: 10.1007/s00285-009-0278-3. [13] W. M. Hirsh, H. Hanisch and J. P. Gabriel, Differential equation models of some parasitic infections: Methods for the study of asymptotic behavior,, Commun. Pur. Appl. Math., 38 (1985), 733. doi: 10.1002/cpa.3160380607. [14] G. Huang, Y. Takeuchi and A. Korobeinikov, HIV evolution and progression of the infection to AIDS,, J. Theor. Biol., 307 (2012), 149. doi: 10.1016/j.jtbi.2012.05.013. [15] C. A. Janeway, J. P. Travers, M. Walport and M. J. Shlomchik, Immunobiology 5: The Immune System in Health and Disease,, Garland Publishing, (2001). [16] A. Korobeinikov, Global properties of basic virus dynamics models,, Bull. Math. Biol., 66 (2004), 879. doi: 10.1016/j.bulm.2004.02.001. [17] J. P. LaSalle, The Stability of Dynamical Systems,, in Regional Conference Series in Applied Mathematics, (1976). [18] Q. Li, F. Lu and K. Wang, Modeling of HIV-1 infection: Insights to the role of monocytes/macrophages, latently infected T4 cells, and HAART regimes,, PLoS ONE, 7 (2012). doi: 10.1371/journal.pone.0046026. [19] W. M. Liu, Nonlinear oscillation in models of immune responses to persistent viruses,, Theor. Popul. Biol., 52 (1997), 224. doi: 10.1006/tpbi.1997.1334. [20] H. Miao, X. Xia, A. S. Perelson and H. Wu, On identifiability of nonlinear ODE models and applications in viral dynamics,, SIAM Rev., 53 (2011), 3. doi: 10.1137/090757009. [21] S. N. Mueller and R. Ahmed, High antigen levels are the cause of T cell exhaustion during chronic viral infection,, Proc. Natl. Acad. Sci. USA, 106 (2009), 8623. doi: 10.1073/pnas.0809818106. [22] A. Murase, T. Sasaki and T. Kajiwara, Stability analysis of pathogen-immune interaction dynamics,, J. Math. Biol., 51 (2005), 247. doi: 10.1007/s00285-005-0321-y. [23] Y. Nakata, Global dynamics of a cell mediated immunity in viral infection models with distributed delays,, J. Math. Anal. Appl., 375 (2011), 14. doi: 10.1016/j.jmaa.2010.08.025. [24] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses,, Science, 272 (1996), 74. doi: 10.1126/science.272.5258.74. [25] M. A. Nowak and R. M. May, Virus Dynamics,, Oxford University Press, (2000). [26] J. Pang, J. A. Cui and J. Hui, The importance of immune responses in a model of hepatitis B virus,, Nonlinear Dynam., 67 (2012), 723. doi: 10.1007/s11071-011-0022-6. [27] A. S. Perelson, Modelling viral and immune system dynamics,, Nat. Rev. Immunol., 2 (2002), 28. doi: 10.1038/nri700. [28] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo,, SIAM Rev., 41 (1999), 3. doi: 10.1137/S0036144598335107. [29] L. Rong, Z. Feng and A. S. Perelson, Mathematical analysis of age-structured HIV-1 dynamics with combination antiretroviral therapy,, SIAM J. Appl. Math., 67 (2007), 731. doi: 10.1137/060663945. [30] L. Rong and A. S. Perelson, Modeling latently infected cell activation: Viral and latent reservoir persistence, and viral blips in HIV-infected patients on potent therapy,, PLoS Comput. Biol., 5 (2009). doi: 10.1371/journal.pcbi.1000533. [31] C. Vargas De León and A. Korobeinikov, Global stability of a population dynamics model with inhibition and negative feedback,, Math. Med. Biol., 30 (2013), 65. doi: 10.1093/imammb/dqr027. [32] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6. [33] K. Wang and Y. Kuang, Fluctuation and extinction dynamics in host-microparasite systems,, Commun. Pur. Appl. Anal., 10 (2011), 1537. doi: 10.3934/cpaa.2011.10.1537. [34] K. Wang, W. Tan, Y. Tang and G. Deng, Numerical diagnoses of superinfection in chronic hepatitis B viral dynamics,, Intervirology, 54 (2011), 349. doi: 10.1159/000321454. [35] K. Wang, W. Wang and X. Liu, Global stability in a viral infection model with lytic and nonlytic immune responses,, Comput. Math. Appl., 51 (2006), 1593. doi: 10.1016/j.camwa.2005.07.020. [36] K. Wang, W. Wang, H. Pang and X. Liu, Complex dynamic behavior in a viral model with delayed immune response,, Physica D, 226 (2007), 197. doi: 10.1016/j.physd.2006.12.001. [37] D. Wodarz, J. P. Christensen and A. R. Thomsen, The importance of lytic and nonlytic immune responses in viral infections,, TRENDS Immunol., 23 (2002), 194. doi: 10.1016/S1471-4906(02)02189-0. [38] J. S. Yi, M. A. Cox and A. J. Zajac, T-cell exhaustion: Characteristics, causes and conversion,, Immunology, 129 (2010), 474. doi: 10.1111/j.1365-2567.2010.03255.x. [39] N. Yousfi, K. Hattaf and A. Tridane, Modeling the adaptive immune response in HBV infection,, J. Math. Biol., 63 (2011), 933. doi: 10.1007/s00285-010-0397-x.
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