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Article Contents

# Global dynamics of a delay virus model with recruitment and saturation effects of immune responses

• * Corresponding authorr: Kaifa Wang
• In this paper, we formulate a virus dynamics model with the recruitment of immune responses, saturation effects and an intracellular time delay. With the help of uniform persistence theory and Lyapunov method, we show that the global stability of the model is totally determined by the basic reproductive number $R_0$. Furthermore, we analyze the effects of the recruitment of immune responses on virus infection by numerical simulation. The results show ignoring the recruitment of immune responses will result in overestimation of the basic reproductive number and the severity of viral infection.

Mathematics Subject Classification: Primary: 92D30; Secondary: 34K20, 34K25.

 Citation:

• Figure 1.  Illustration of the proportion of infected cells ($I_1$) and virus load ($V_1$) at the endemic equilibrium $E_{1}$. Here parameters are $\lambda=50, \beta=5\times 10^{-7}, d_1=0.008, d_2=0.8,$$d_3=3, d_4=0.05, d_5=0.1,$$ p=0.05, r=2500, q=0.2, k_1=0.12, h_1=1200, k_2=1.5, s=0.001, \tau=1.5$

Table 1.  Parameter definitions and values used in numerical simulations

 Par. Value Description Ref. $\lambda$ 0-50 cells ml-day$^{-1}$ Recruitment rate of healthy cells [33,38] $d_1$ $0.007-0.1$day $^{-1}$ Death rate of healthy cells [38] $\beta$ $5\times10^{-7}-0.5$ ml virion-day$^{-1}$ Infection rate of target cells by virus [33,38] $d_2$ $0.2-0.8$ day$^{-1}$ Death rate of infected cells [41,46] $r$ $10-2500$ virions/cell Burst size of virus [38] $d_3$ $2.4-3$ day$^{-1}$ Clearance rate of free virus [38] $p$ $0.05-1$ day$^{-1}$ Killing rate of CTL cells [41,40] $q$ $0.1-1$ day$^{-1}$ Neutralizing rate of antibody [41] $k_1$ $0.1-0.12$ day$^{-1}$ Proliferation rate of CTL response [2,41] $k_2$ $1.5$ day$^{-1}$ Production rate of antibody response [41] $d_4$ $0.05-2$ day$^{-1}$ Mortality rate of CTL response [2,40] $d_5$ $0.1$ day$^{-1}$ Clearance rate of antibody [41] $s$ $0.001-1.4$ 1/s is the average time [32,47] $\tau$ $0-2$ days Virus replication time [38] $h_1$ 1200 Saturation constant Assumed $h_2$ 1500 Saturation constant Assumed $\lambda_1$ Varied Rate of CTL export from thymus [9] $\lambda_2$ Varied Recruitment rate of antibody [9]
•  [1] I. Al-Darabsah and Y. Yuan, A time-delayed epidemic model for Ebola disease transmission, Appl. Math. Comput., 290 (2016), 307-325.  doi: 10.1016/j.amc.2016.05.043. [2] G. Bocharov, B. Ludewig, A. Bertoletti, P. Klenerman, T. Junt, P. Krebs, T. Luzyanina, C. Fraser and R. Anderson, Underwhelming the immune response: Effect of slow virus growth on CD8+T lymphocytes responses, J. Virol., 78 (2004), 2247-2254.  doi: 10.1128/JVI.78.5.2247-2254.2004. [3] S. Chen, C. Cheng and Y. Takeuchi, Stability analysis in delayed within-host viral dynamics with both viral and cellular infections, J. Math. Anal. Appl., 442 (2016), 642-672.  doi: 10.1016/j.jmaa.2016.05.003. [4] R. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4+T cells, Math. Biosci., 165 (2000), 27-39.  doi: 10.1016/S0025-5564(00)00006-7. [5] R. De Boer, Which of our modeling predictions are robust? PLoS Comput. Biol., 8 (2012), e10002593, 5pp. doi: 10.1371/journal.pcbi.1002593. [6] O. Diekmann, J. Heesterbeek and J. Metz, On the definition and the computation of the basic reproduction ratio $R_{0}$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324. [7] T. Gao, W. Wang and X. Liu, Mathematical analysis of an HIV model with impulsive antiretroviral drug doses, Math. Comput. Simulation, 82 (2011), 653-665.  doi: 10.1016/j.matcom.2011.10.007. [8] J. Hale and S. Verduyn Lunel, Introduction to Functional-Differential Equations, Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [9] M. Hellerstein, M. Hanley, D. Cesar, S. Siler, C. Parageorgopolous, E. Wieder, D. Schmidt, R. Hoh, R. Neese, D. Macallan, S. Deeks and J. M. McCune, Directly measured kinetics of circulating T lymphocytes in normal and HIV-1-infected humans, Nat. Med., 5 (1999), 83-89.  doi: 10.1038/4772. [10] M. Hirsh, H. Hanisch and P. Gabriel, Differential equation models of some parasitic infections: Methods for the study of asymptotic behavior, Commun. Pur. Appl. Math., 38 (1985), 733-753.  doi: 10.1002/cpa.3160380607. [11] Y. Ji and L. Liu, Global stability of a delayed viral infection model with nonlinear immune response and general incidence rate, Discrete Contin. Dyn. Syst. Ser. B., 21 (2016), 133-149.  doi: 10.3934/dcdsb.2016.21.133. [12] C. Jiang and W. Wang, Complete classification of global dynamics of a virus model with immune responses, Discrete Contin. Dyn. Syst. Ser. B., 19 (2014), 1087-1103.  doi: 10.3934/dcdsb.2014.19.1087. [13] T. Kepler and A. Perelson, Drug concentration heterogeneity facilitates the evolution of drug resistance, Proc. Natl. Acad. Sci., USA, 95 (1998), 11514–11519. doi: 10.1073/pnas.95.20.11514. [14] A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883.  doi: 10.1016/j.bulm.2004.02.001. [15] J. Li, Y. Yang and Y. Zhou, Global stability of an epidemic model with latent stage and vaccination, Nonlinear Anal. Real World Appl., 12 (2011), 2163-2173.  doi: 10.1016/j.nonrwa.2010.12.030. [16] B. Li, Y. Chen, X. Lu and S. Liu, A delayed HIV-1 model with virus waning term, Math. Biosci. Eng., 13 (2016), 135-157.  doi: 10.3934/mbe.2016.13.135. [17] J. Luo, W. Wang, H. Chen and R. Fu, Bifurcations of a mathematical model for HIV dynamics, J. Math. Anal. Appl., 434 (2016), 837-857.  doi: 10.1016/j.jmaa.2015.09.048. [18] Y. Nakata, Global dynamics of a cell mediated immunity in viral infection models with distributed delays, J. Math. Anal. Appl., 375 (2011), 14-27.  doi: 10.1016/j.jmaa.2010.08.025. [19] P. Nelson, J. Mittler and A. Perelson, Effect of drug efficacy and the eclipse phase of the viral life cycle on the estimates of HIV viral dynamic parameters, J. AIDS, 26 (2001), 405-412.  doi: 10.1097/00126334-200104150-00002. [20] M. Nowak and R. May, Virus Dynamics: Mathematical Principles of Immunology Virology, Oxford University Press, Oxford, 2000. [21] M. Nowak, S. Bonhoeffer, A. Hill, R. Boehme, H. Thomas and H. Mcdade, Viral dynamics in hepatitis B virus infection, Proc. Natl. Acad. Sci., USA, 93 (1996), 4398–4402. doi: 10.1073/pnas.93.9.4398. [22] K. Pawelek, S. Liu, F. Pahlevani and L. Rong, A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data, Math. Biosci., 235 (2012), 98-109.  doi: 10.1016/j.mbs.2011.11.002. [23] J. Pang, J. Cui and J. Hui, The importance of immune rsponses in a model of hepatitis B virus, Nonlinear Dyn., 67 (2012), 727-734.  doi: 10.1007/s11071-011-0022-6. [24] H. Pang, W. Wang and K. Wang, Global properties of virus dynamics model with immune response, J. Southeast Univ. Nat. Sci., 30 (2005), 796-799. [25] A. Perelson and P. Nelson, Mathematical models of HIV dynamics in vivo, SIAM Rev., 41 (1999), 3-44.  doi: 10.1137/S0036144598335107. [26] A. Perelson, A. Neumann, M. Markowitz, J. Leonard and D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time, Science, 271 (1996), 1582-1586.  doi: 10.1126/science.271.5255.1582. [27] L. Rong and A. Perelson, Modeling HIV persistence, the latent reservoir, and viral blips, J. Theoret. Biol., 260 (2009), 308-331.  doi: 10.1016/j.jtbi.2009.06.011. [28] L. Rong, M. Gilchristb, Z. Feng and A. Perelson, Modeling within-host HIV-1 dynamics and the evolution of drug resistance: Trade-offs between viral enzyme function and drug susceptibility, J. Theoret. Biol., 247 (2007), 804-818.  doi: 10.1016/j.jtbi.2007.04.014. [29] H. Shu and L. Wang, Role of CD4+T-cell proliferation in HIV infection under antiretroviral therapy, J. Math. Anal. Appl., 394 (2012), 529-544.  doi: 10.1016/j.jmaa.2012.05.027. [30] H. Smith, Monotone Dynamical System: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI, 1995. doi: 10.1090/surv/041. [31] H. Smith and X. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2. [32] X. Song, S. Wang and J. Dong, Stability properties and Hopf bifurcation of a delayed viral infection model with lytic immune response, J. Math. Anal. Appl., 373 (2011), 345-355.  doi: 10.1016/j.jmaa.2010.04.010. [33] M. Stafford, L. Corey, Y. Cao, E. Daar, D. Ho and A. Perelson, Modeling plasma virus concentration during primary HIV infection, J. Theoret. Biol., 203 (2000), 285-301.  doi: 10.1006/jtbi.2000.1076. [34] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6. [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), 1593-1610.  doi: 10.1016/j.camwa.2005.07.020. [36] K. Wang, W. Wang and X. Liu, Viral infection model with periodic lytic immune response, Chaos Solitons Fractals, 28 (2006), 90-99.  doi: 10.1016/j.chaos.2005.05.003. [37] X. Wang and W. Wang, An HIV infection model based on a vectored immunoprophylaxis experiment, J. Theoret. Biol., 313 (2012), 127-135.  doi: 10.1016/j.jtbi.2012.08.023. [38] Y. Wang, Y. Zhou, J. Wu and J. Heffernan, Oscillatory viral dynamics in a delayed HIV pathogenesis model, Math. Biosci., 219 (2009), 104-112.  doi: 10.1016/j.mbs.2009.03.003. [39] K. Wang, Y. Jin and A. Fan, The effect of immune responses in viral infections: A mathematical model view, Discrete Contin. Dyn. Syst. Ser. B., 19 (2014), 3379-3396.  doi: 10.3934/dcdsb.2014.19.3379. [40] Z. Wang and R. Xu, Stability and Hopf bifurcation in a viral infection modelwith nonlinear incidence rate and delayed immune response, Commun Nonlinear Sci Numer Simulat., 17 (2012), 964-978.  doi: 10.1016/j.cnsns.2011.06.024. [41] D. Wodarz, Hepatitis C virus dynamics and pathology: The role of CTL and antibody responses, J. Gen. Virol., 84 (2003), 1743-1750.  doi: 10.1099/vir.0.19118-0. [42] Y. Yan and W. Wang, Global stability of a five-dimensional model with immune response and delay, Discrete Contin. Dyn. Syst. Ser. B., 17 (2012), 401-416.  doi: 10.3934/dcdsb.2012.17.401. [43] Y. Yang and Y. Xiao, Threshold dynamics for an HIV model in periodic environments, J. Math. Anal. Appl., 361 (2010), 59-68.  doi: 10.1016/j.jmaa.2009.09.012. [44] Y. Yang, L. Zou and S. Ruan, Global dynamics of a delayed within-host viral infection model with both virus-to-cell and cell-to-cell transmissions, Math. Biosci., 270 (2015), 183-191.  doi: 10.1016/j.mbs.2015.05.001. [45] Y. Yang and Y. Xu, Global stability of a diffusive and delayed virus dynamics model with Beddington -DeAngelis incidence function and CTL immune response, Comput. Math. Appl., 71 (2016), 922-930.  doi: 10.1016/j.camwa.2016.01.009. [46] J. Zack, S. Arrigo, S. Weitsman, A. Go, A. Haislip and I. Chen, HIV-1 entry into quiescent primary lymphocytes: Molecular analysis reveals a labile latent viral structure, Cell, 61 (1990), 213-222.  doi: 10.1016/0092-8674(90)90802-L. [47] X. Zhou, X. Song and X. Shi, Analysis of stability and Hopf bifurcation for an HIV infection model with time delay, Appl. Math. Comput., 199 (2008), 23-38.  doi: 10.1016/j.amc.2007.09.030.

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