In this paper we propose and analyze a pathogen dynamics model with antibody and Cytotoxic T Lymphocyte (CTL) immune responses. We incorporate latently infected cells and three distributed time delays into the model. We show that the solutions of the proposed model are nonnegative and ultimately bounded. We derive four threshold parameters which fully determine the existence and stability of the five steady states of the model. Using Lyapunov functionals, we established the global stability of the steady states of the model. The theoretical results are confirmed by numerical simulations.
AcevedoH.G. and LiM.Y., Backward bifurcation in a model for HTLV-I infection of CD4+ T cells, Bulletin of Mathematical Biology67(1) (2005), 101–114. doi:10.1016/j.bulm.2004.06.004.
2.
AliN., ZamanG. and AlgahtaniO., Stability analysis of HIV-1 model with multiple delays, Advances in Difference Equations2016 (2016), 88. doi:10.1186/s13662-016-0808-4.
3.
AlshormanA., WangX., MeyerJ. and RongL., Analysis of HIV models with two time delays, Journal of Biological Dynamics11(1s) (2017), 40–64. doi:10.1080/17513758.2016.1148202.
4.
CallawayD.S. and PerelsonA.S., HIV-1 infection and low steady state viral loads, Bulletin of Mathematical Biology64 (2002), 29–64. doi:10.1006/bulm.2001.0266.
5.
ElaiwA.M., Global properties of a class of HIV models, Nonlinear Analysis: Real World Applications11 (2010), 2253–2263. doi:10.1016/j.nonrwa.2009.07.001.
6.
ElaiwA.M. and AlmuallemN.A., Global dynamics of delay-distributed HIV infection models with differential drug efficacy in cocirculating target cells, Mathematical Methods in the Applied Sciences39 (2016), 4–31. doi:10.1002/mma.3453.
7.
ElaiwA.M. and AlShamraniN.H., Global stability of humoral immunity virus dynamics models with nonlinear infection rate and removal, Nonlinear Analysis: Real World Applications26 (2015), 161–190. doi:10.1016/j.nonrwa.2015.05.007.
8.
ElaiwA.M. and AlShamraniN.H., Stability of a general delay-distributed virus dynamics model with multi-staged infected progression and immune response, Mathematical Methods in the Applied Sciences40(3) (2017), 699–719. doi:10.1002/mma.4002.
9.
ElaiwA.M., AlShamraniN.H. and AlofiA.S., Stability of CTL immunity pathogen dynamics model with capsids and distributed delay, AIP Advances7(12) (2017), 125111.
10.
ElaiwA.M., AlShamraniN.H. and HattafK., Dynamical behaviors of a general humoral immunity viral infection model with distributed invasion and production, International Journal of Biomathematics10(3) (2017), Article ID 1750035.
11.
ElaiwA.M. and AzozS.A., Global properties of a class of HIV infection models with Beddington-DeAngelis functional response, Mathematical Methods in the Applied Sciences36 (2013), 383–394. doi:10.1002/mma.2596.
12.
ElaiwA.M., ElnaharyE.K. and RaezahA.A., Effect of cellular reservoirs and delays on the global dynamics of HIV, Advances in Difference Equations2018 (2018), 85. doi:10.1186/s13662-018-1523-0.
13.
ElaiwA.M., HassanienI.A. and AzozS.A., Global stability of HIV infection models with intracellular delays, Journal of the Korean Mathematical Society49(4) (2012), 779–794. doi:10.4134/JKMS.2012.49.4.779.
14.
ElaiwA.M. and RaezahA.A., Stability of general virus dynamics models with both cellular and viral infections and delays, Mathematical Methods in the Applied Sciences40(16) (2017), 5863–5880. doi:10.1002/mma.4436.
15.
ElaiwA.M., RaezahA.A. and AlofiA.S., Stability of a general delayed virus dynamics model with humoral immunity and cellular infection, AIP Advances7(6) (2017), Article ID 065210.
16.
ElaiwA.M., RaezahA.A. and AlofiB.S., Dynamics of delayed pathogen infection models with pathogenic and cellular infections and immune impairment, AIP Advances8(2) (2018), Article ID 025323. doi:10.1063/1.5023752.
17.
ElaiwA.M., RaezahA.A. and HattafK., Stability of HIV-1 infection with saturated virus-target and infected-target incidences and CTL immune response, International Journal of Biomathematics10(5) (2017), Article ID 1750070.
18.
GibelliL., ElaiwA., AlghamdiM.A. and AlthiabiA.M., Heterogeneous population dynamics of active particles: Progression, mutations, and selection dynamics, Mathematical Models and Methods in Applied Sciences27 (2017), 617–640. doi:10.1142/S0218202517500117.
19.
HaleJ.K. and Verduyn LunelS., Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.
20.
HattafK. and YousfiN., A class of delayed viral infection models with general incidence rate and adaptive immune response, International Journal of Dynamics and Control4(3) (2016), 254–265. doi:10.1007/s40435-015-0158-1.
21.
HuangD., ZhangX., GuoY. and WangH., Analysis of an HIV infection model with treatments and delayed immune response, Applied Mathematical Modelling40(4) (2016), 3081–3089. doi:10.1016/j.apm.2015.10.003.
22.
KajiwaraT. and SasakiT., A note on the stability analysis of pathogen-immune interaction dynamics, Discrete and Continuous Dynamical Systems-Series B4 (2004), 615–622. doi:10.3934/dcdsb.2004.4.615.
23.
LiB., ChenY., LuX. and LiuS., A delayed HIV-1 model with virus waning term, Mathematical Biosciences and Engineering13 (2016), 135–157. doi:10.3934/mbe.2016.13.135.
24.
LiL. and XuR., Global dynamics of an age-structured in-host viral infection model with humoral immunity, Advances in Difference Equations2016 (2016), 6. doi:10.1186/s13662-015-0733-y.
25.
LiM.Y. and ShuH., Global dynamics of a mathematical model for HTLV-I infection of CD4+ T cells with delayed CTL response, Nonlinear Analysis: Real World Applications13 (2012), 1080–1092. doi:10.1016/j.nonrwa.2011.02.026.
26.
LiM.Y. and WangL., Backward bifurcation in a mathematical model for HIV infection in vivo with anti-retroviral treatment, Nonlinear Analysis: Real World Applications17 (2014), 147–160. doi:10.1016/j.nonrwa.2013.11.002.
27.
LiX. and FuS., Global stability of a virus dynamics model with intracellular delay and CTL immune response, Mathematical Methods in the Applied Sciences38 (2015), 420–430. doi:10.1002/mma.3078.
28.
LvC., HuangL. and YuanZ., Global stability for an HIV-1 infection model with Beddington–DeAngelis incidence rate and CTL immune response, Communications in Nonlinear Science and Numerical Simulation19 (2014), 121–127. doi:10.1016/j.cnsns.2013.06.025.
29.
MiaoH., TengZ., KangC. and MuhammadhajiA., Stability analysis of a virus infection model with humoral immunity response and two time delays, Mathematical Methods in the Applied Sciences39(12) (2016), 3434–3449. doi:10.1002/mma.3790.
30.
MonicaC. and PitchaimaniM., Analysis of stability and Hopf bifurcation for HIV-1 dynamics with PI and three intracellular delays, Nonlinear Analysis: Real World Applications27 (2016), 55–69. doi:10.1016/j.nonrwa.2015.07.014.
31.
MuraseA., SasakiT. and KajiwaraT., Stability analysis of pathogen-immune interaction dynamics, J. Math. Biol.51 (2005), 247–267. doi:10.1007/s00285-005-0321-y.
32.
NeumannA.U., LamN.P., DahariH., GretchD.R., WileyT.E., LaydenT.J. and PerelsonA.S., Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon-alpha therapy, Science282 (1998), 103–107. doi:10.1126/science.282.5386.103.
33.
NowakM.A. and BanghamC.R.M., Population dynamics of immune responses to persistent viruses, Science272 (1996), 74–79. doi:10.1126/science.272.5258.74.
34.
NowakM.A. and MayR.M., Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford Uni., Oxford, 2000.
35.
PangJ. and CuiJ., Analysis of a hepatitis B viral infection model with immune response delay, International Journal of Biomathematics10(2) (2017), Article ID 1750020. doi:10.1142/S1793524517500206.
36.
PangJ., CuiJ.-A. and HuiJ., The importance of immune responses in a model of hepatitis B virus, Nonlinear Dynamics67(1) (2012), 723–734. doi:10.1007/s11071-011-0022-6.
37.
PerthameB., Transport Equations in Biology, Birkhauser, Basel, 2007.
38.
RezounenkoA., Continuous solutions to a viral infection model with general incidence rate, discrete state-dependent delay, CTL and antibody immune responses, Electronic Journal of Qualitative Theory of Differential Equations79 (2016), 1–15.
39.
RezounenkoA., Stability of a viral infection model with state-dependent delay, CTL and antibody immune responses, Discrete and Continuous Dynamical Systems-Series B22(4) (2016). doi:10.3934/dcdsb.2017074.
40.
RoyP.K., ChatterjeeA.N., GreenhalghD. and KhanQ.J.A., Long term dynamics in a mathematical model of HIV-1 infection with delay in different variants of the basic drug therapy model, Nonlinear Analysis: Real World Applications14 (2013), 1621–1633. doi:10.1016/j.nonrwa.2012.10.021.
41.
ShiX., ZhouX. and SonX., Dynamical behavior of a delay virus dynamics model with CTL immune response, Nonlinear Analysis: Real World Applications11 (2010), 1795–1809. doi:10.1016/j.nonrwa.2009.04.005.
42.
ShuH., WangL. and WatmoughJ., Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL imune responses, SIAM Journal of Applied Mathematics73(3) (2013), 1280–1302. doi:10.1137/120896463.
43.
WangJ., PangJ., KuniyaT. and EnatsuY., Global threshold dynamics in a five-dimensional virus model with cell-mediated, humoral immune responses and distributed delays, Applied Mathematics and Computation241(15) (2014), 298–316. doi:10.1016/j.amc.2014.05.015.
44.
WangK., FanA. and TorresA., Global properties of an improved hepatitis B virus model, Nonlinear Analysis: Real World Applications11 (2010), 3131–3138. doi:10.1016/j.nonrwa.2009.11.008.
45.
WangL., LiM.Y. and KirschnerD., Mathematical analysis of the global dynamics of a model for HTLV-I infection and ATL progression, Mathematical Biosciences179 (2002), 207–217. doi:10.1016/S0025-5564(02)00103-7.
46.
WangS. and ZouD., Global stability of in host viral models with humoral immunity and intracellular delays, Applied Mathematical Modeling36 (2012), 1313–1322. doi:10.1016/j.apm.2011.07.086.
47.
WangT., HuZ. and LiaoF., Stability and Hopf bifurcation for a virus infection model with delayed humoral immunity response, Journal of Mathematical Analysis and Applications411 (2014), 63–74. doi:10.1016/j.jmaa.2013.09.035.
48.
WangT., HuZ., LiaoF. and MaW., Global stability analysis for delayed virus infection model with general incidence rate and humoral immunity, Mathematics and Computers in Simulation89 (2013), 13–22. doi:10.1016/j.matcom.2013.03.004.
49.
WangX., ElaiwA.M. and SongX., Global properties of a delayed HIV infection model with CTL immune response, Applied Mathematics and Computation218(18) (2012), 9405–9414. doi:10.1016/j.amc.2012.03.024.
50.
WangX. and LiuS., A class of delayed viral models with saturation infection rate and immune response, Mathematical Methods in the Applied Science36 (2013), 125–142. doi:10.1002/mma.2576.
51.
WodarzD., Hepatitis C virus dynamics and pathology: The role of CTL and antibody responses, Journal of General Virology84 (2003), 1743–1750. doi:10.1099/vir.0.19118-0.
52.
XuJ., ZhouY., LiY. and YangY., Global dynamics of a intracellular infection model with delays and humoral immunity, Mathematical Methods in the Applied Sciences39(18) (2016), 5427–5435. doi:10.1002/mma.3927.
53.
YanY. and WangW., Global stability of a five-dimensional model with immune responses and delay, Discrete and Continuous Dynamical Systems-Series B17 (2012), 401–416. doi:10.3934/dcdsb.2012.17.401.
54.
YousfiN., HattafK. and TridaneA., Modeling the adaptive immune response in HBV infection, Journal of Mathematical Biology63 (2011), 933–957. doi:10.1007/s00285-010-0397-x.
55.
ZhangF., LiJ., ZhengC. and WangL., Dynamics of an HBV/HCV infection model with intracellular delay and cell proliferation, Communications in Nonlinear Science and Numerical Simulation42 (2017), 464–476. doi:10.1016/j.cnsns.2016.06.009.
56.
ZhangS. and XuX., Dynamic analysis and optimal control for a model of hepatitis C with treatment, Communications in Nonlinear Science and Numerical Simulation46 (2017), 14–25. doi:10.1016/j.cnsns.2016.10.017.
57.
ZhaoY., DimitrovD.T., LiuH. and KuangY., Mathematical insights in evaluating state dependent effectiveness of HIV prevention interventions, Bulletin of Mathematical Biology75(4) (2013), 649–675. doi:10.1007/s11538-013-9824-7.
58.
ZhaoY. and XuZ., Global dynamics for a delyed hepatitis C virus, infection model, Electronic Journal of Differential Equations2014(132) (2014), 1–18.
