Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions
Carrying simplex in the Lotka-Volterra competition model with seasonal succession with applications
Modeling within-host viral dynamics: The role of CTL immune responses in the evolution of drug resistance
Qi Deng 1, , Zhipeng Qiu 1,, , Ting Guo 2, and Libin Rong 2,
Department of Mathematics, Nanjing University of Science and Technology, Nanjing 210094, China
Department of Mathematics, Nanjing University of Science and Technology, Nanjing 210094, China, Department of Mathematics, University of Florida, Gainesville, FL 32611, USA
* Corresponding author: Zhipeng Qiu
Received February 2020 Revised June 2020 Published August 2020
Fund Project: T. Guo was supported by the CSC (201806840119) and the NSFC (11971232). Z. Qiu was supported by the NSFC (11671206). L. Rong was supported by the NSF Grant DMS-1758290
To study the emergence and evolution of drug resistance during treatment of HIV infection, we study a mathematical model with two strains, one drug-sensitive and the other drug-resistant, by incorporating cytotoxic T lymphocyte (CTL) immune response. The reproductive numbers for each strain with and without the CTL immune response are obtained and shown to determine the stability of the steady states. By sensitivity analysis, we evaluate how the changes of parameters influence the reproductive numbers. The model shows that CTL immune response can suppress the development of drug resistance. There is a dynamic relationship between antiretroviral drug administration, the prevalence of drug resistance, the total level of viral production, and the strength of immune responses. We further investigate the scenario under which the drug-resistant strain can outcompete the wild-type strain. If drug efficacy is at an intermediate level, the drug-resistant virus is likely to arise. The slower the immune response wanes, the slower the drug-resistant strain grows. The results suggest that immunotherapy that aims to enhance immune responses, combined with antiretroviral drug treatment, may result in a functional control of HIV infection.
Keywords: Drug resistance, antiretroviral therapy, CTL immune response, sensitivity analysis, Stability.
Mathematics Subject Classification: 03C25, 34D20, 34D08, 37C75.
Citation: Qi Deng, Zhipeng Qiu, Ting Guo, Libin Rong. Modeling within-host viral dynamics: The role of CTL immune responses in the evolution of drug resistance. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020245
B. M. Adams, H. T. Banks, M. Davidian, H.-D. Kwon, H. T. Tran, S. N. Wynne and E. S. Rosenberg, HIV dynamics: Modeling, data analysis, and optimal treatment protocols, J. Comput. Appl. Math., 184 (2005), 10-49. doi: 10.1016/j.cam.2005.02.004. Google Scholar
K. Allali, J. Danane and Y. Kuang, Global analysis for an HIV infection model with CTL immune response and infected cells in eclipse phase, Appl. Sci., 7 (2017), 861. Google Scholar
R. A. Arnaout, N. Martin A and D. Wodarz, HIV-1 dynamics revisited: Biphasic decay by Cytotoxic T Lymphocyte killing?, Proc. R. Soc. Lond. B, 267 (2000), 1347-1354. doi: 10.1098/rspb.2000.1149. Google Scholar
N. P. Bhatia and G. P. Szegö, Stability Theory of Dynamical Systems, Springer Science & Business Media, 2002. Google Scholar
S. M. Blower, D. Hartel, H. Dowlatabadi, R. M. Anderson and R. M. May, Drugs, sex and HIV: A mathematical model for New York City, Proc. R. Soc. Lond. B, 331 (1991), 171-187. Google Scholar
S. Bonhoeffer, R. M. May, G. M. Shaw and M. A. Nowak, Virus dynamics and drug therapy, P. Natl. A. Sci., 94 (1997), 6971-6976. doi: 10.1073/pnas.94.13.6971. Google Scholar
J. Cao, J. McNevin, S. Holte, L. Fink, L. Corey and M. J. McElrath, Comprehensive analysis of human immunodeficiency virus type 1 (HIV-1)-specific gamma interferon-secreting CD8+ T cells in primary HIV-1 infection, J. Virol., 77 (2003), 6867-6878. doi: 10.1128/JVI.77.12.6867-6878.2003. Google Scholar
H. Y. Chen, M. Di Mascio, A. S. Perelson, D. D. Ho and L. Zhang, Determination of virus burst size in vivo using a single-cycle SIV in rhesus macaques, P. Natl. A. Sci., 104 (2007), 19079-19084. doi: 10.1073/pnas.0707449104. Google Scholar
M. Ciupe, B. Bivort, D. Bortz and P. Nelson, Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models, Math. Biosci., 200 (2006), 1-27. doi: 10.1016/j.mbs.2005.12.006. Google Scholar
F. Clavel and A. J. Hance, HIV drug resistance, New. Engl. J. Med., 350 (2004), 1023-1035. doi: 10.1056/NEJMra025195. Google Scholar
R. V. Culshaw, S. Ruan and R. J. Spiteri, Optimal HIV treatment by maximising immune response, J. Math. Biol., 48 (2004), 545-562. doi: 10.1007/s00285-003-0245-3. Google Scholar
M. P. Davenport, R. M. Ribeiro and A. S. Perelson, Kinetics of virus-specific CD8+ T cells and the control of human immunodeficiency virus infection, J. Virol., 78 (2004), 10096-10103. doi: 10.1128/JVI.78.18.10096-10103.2004. Google Scholar
M. P. Davenport, R. M. Ribeiro, L. Zhang, D. P. Wilson and A. S. Perelson, Understanding the mechanisms and limitations of immune control of HIV, Immunlo. Rev., 216 (2007), 164-175. doi: 10.1017/CBO9780511818097. Google Scholar
M. P. Davenport, High-potency human immunodeficiency virus vaccination leads to delayed and reduced CD8+ T-cell expansion but improved virus control, J. Virol., 79 (2005), 10059-10062. doi: 10.1128/JVI.79.15.10059-10062.2005. Google Scholar
S. G. Deeks, M. Smith, M. Holodniy and J. O. Kahn, HIV-1 protease inhibitors: A review for clinicians, Jama, 277 (1997), 145-153. doi: 10.1001/jama.1997.03540260059037. Google Scholar
P. Dubey, U. S. Dubey and B. Dubey, Modeling the role of acquired immune response and antiretroviral therapy in the dynamics of HIV infection, Math. Comput. Simulat., 144 (2018), 120-137. doi: 10.1016/j.matcom.2017.07.006. Google Scholar
M. A. Gilchrist, D. Coombs and A. S. Perelson, Optimizing within-host viral fitness: Infected cell lifespan and virion production rate, J. Theor. Biol., 229 (2004), 281-288. doi: 10.1016/j.jtbi.2004.04.015. Google Scholar
T. Guo and Z. Qiu, The effects of CTL immune response on HIV infection model with potent therapy, latently infected cells and cell-to-cell viral transmission, doi: 10.3934/mbe.2019341. Google Scholar
T. Guo, Z. Qiu and L. Rong, Analysis of an hiv model with immune responses and cell-to-cell transmission, Bull. Malays. Math. Sci. So., 43 (2020), 581-607. doi: 10.1007/s40840-018-0699-5. Google Scholar
S. A. Kalams, P. J. Goulder, A. K. Shea, N. G. Jones, A. K. Trocha, G. S. Ogg and B. D. Walker, Levels of human immunodeficiency virus type 1-specific cytotoxic T-lymphocyte effector and memory responses decline after suppression of viremia with highly active antiretroviral therapy, J. Virol., 73 (1999), 6721-6728. doi: 10.1128/JVI.73.8.6721-6728.1999. Google Scholar
D. E. Kirschner and G. Webb, Understanding drug resistance for monotherapy treatment of HIV infection, Bull. Math. Biol., 59 (1997), 763-785. doi: 10.1007/BF02458429. Google Scholar
R. Koup, J. T. Safrit, Y. Cao, C. A. Andrews, G. McLeod, W. Borkowsky, C. Farthing and D. D. Ho, Temporal association of cellular immune responses with the initial control of viremia in primary human immunodeficiency virus type 1 syndrome, J. Virol., 68 (1994), 4650-4655. doi: 10.1128/JVI.68.7.4650-4655.1994. Google Scholar
M. Louie, Determining the relative efficacy of highly active antiretroviral therapy, J.Infect. Dis., 187 (2003), 896-900. doi: 10.1086/368164. Google Scholar
L. M. Mansky and H. M. Temin, Lower in vivo mutation rate of human immunodeficiency virus type 1 than that predicted from the fidelity of purified reverse transcriptase, J. Virol., 69 (1995), 5087-5094. doi: 10.1128/JVI.69.8.5087-5094.1995. Google Scholar
S. Marino, I. B. Hogue, C. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, J. Theor. Biol., 254 (2009), 178-196. doi: 10.1016/j.jtbi.2008.04.011. Google Scholar
R. D. Mason, M. I. Bowmer, C. M. Howley, M. Gallant, J. C. Myers and M. D. Grant, Antiretroviral drug resistance mutations sustain or enhance CTL recognition of common HIV-1 pol epitopes, J. Immunol., 172 (2004), 7212-7219. doi: 10.4049/jimmunol.172.11.7212. Google Scholar
A. R. McLean and M. A. Nowak, Competition between zidovudine-sensitive and zidovudine-resistant strains of HIV, Aids, 6 (1992), 71-79. doi: 10.1097/00002030-199201000-00009. Google Scholar
S. H. Michaels, R. Clark and P. Kissinger, Declining morbidity and mortality among patients with advanced human immunodeficiency virus infection, New. Engl. J. Med., 339 (1998), 405-406. doi: 10.1056/NEJM199808063390612. Google Scholar
P. Ngina, R. W. Mbogo and L. S. Luboobi, HIV drug resistance: Insights from mathematical modelling, Appl. Math. Model., 75 (2019), 141-161. doi: 10.1016/j.apm.2019.04.040. Google Scholar
[30] M. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology: Mathematical Principles of Immunology and Virology, Oxford University Press, UK, 2000. Google Scholar
M. A. Nowak and C. R. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. doi: 10.1126/science.272.5258.74. Google Scholar
M. A. Nowak and R. M. May, Mathematical biology of HIV infections: Antigenic variation and diversity threshold, Math. Biosci., 106 (1991), 1-21. doi: 10.1016/0025-5564(91)90037-J. Google Scholar
M. A. Nowak and A. J. McMichael, How HIV defeats the immune system, Sci. AM., 273 (1995), 58-65. doi: 10.1038/scientificamerican0895-58. Google Scholar
A. S. Perelson, D. E. Kirschner and R. De Boer, Dynamics of HIV infection of CD4+ T cells, Math. Biosci., 114 (1993), 81-125. doi: 10.1016/0025-5564(93)90043-A. Google Scholar
A. S. Perelson, A. U. Neumann, M. Markowitz, J. M. Leonard and D. 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. Google Scholar
A. S. Perelson and R. M. Ribeiro, Modeling the within-host dynamics of HIV infection, BMC Biol., 11 (2013), 96. doi: 10.1186/1741-7007-11-96. Google Scholar
Z. Qiu and Z. Feng, The dynamics of an epidemic model with targeted antiviral prophylaxis, J. Biol. Dyn., 4 (2010), 506-526. doi: 10.1080/17513758.2010.498925. Google Scholar
S. M. Raimundo, H. M. Yang, E. Venturino and E. Massad, Modeling the emergence of HIV-1 drug resistance resulting from antiretroviral therapy: Insights from theoretical and numerical studies, BioSystems, 108 (2012), 1-13. doi: 10.1016/j.biosystems.2011.11.009. Google Scholar
B. Ramratnam and et al., Rapid production and clearance of HIV-1 and hepatitis C virus assessed by large volume plasma apheresis, The Lancet, 354 (1999), 1782-1785. doi: 10.1016/S0140-6736(99)02035-8. Google Scholar
R. M. Ribeiro and S. Bonhoeffer, Production of resistant HIV mutants during antiretroviral therapy, P. Natl. A. Sci., 97 (2000), 7681-7686. doi: 10.1073/pnas.97.14.7681. Google Scholar
R. M. Ribeiro, S. Bonhoeffer and M. A. Nowak, The frequency of resistant mutant virus before antiviral therapy, Aids, 12 (1998), 461-465. doi: 10.1097/00002030-199805000-00006. Google Scholar
L. Rong, Z. Feng and A. S. Perelson, Emergence of HIV-1 drug resistance during antiretroviral treatment, Bull. Math. Biol., 69 (2007), 2027-2060. doi: 10.1007/s11538-007-9203-3. Google Scholar
L. Rong, Z. Feng and A. S. Perelson, Mathematical modeling of HIV-1 infection and drug therapy, Math. Model. Bios., 87-131. doi: 10.1007/978-3-540-76784-8_3. Google Scholar
L. Rong, M. A. Gilchrist, Z. Feng and A. S. Perelson, Modeling within-host HIV-1 dynamics and the evolution of drug resistance: Trade-offs between viral enzyme function and drug susceptibility, J. Theor. Biol., 247 (2007), 804-818. doi: 10.1016/j.jtbi.2007.04.014. Google Scholar
L. Rong and A. S. Perelson, Asymmetric division of activated latently infected cells may explain the decay kinetics of the HIV-1 latent reservoir and intermittent viral blips, Math. Biosci., 217 (2009), 77-87. doi: 10.1016/j.mbs.2008.10.006. Google Scholar
B. Sebastian and A. N. Martin, Pre-existence and emergence of drug resistance in HIV-1 infection, Proc. R. Soc. Lond. B, 264 (1997), 631-637. doi: 10.1098/rspb.1997.0089. Google Scholar
A. K. Sewell, D. A. Price, A. Oxenius, A. D. Kelleher and R. E. Phillips, Cytotoxic T Lymphocyte responses to human immunodeficiency virus: Control and escape, Stem Cells, 18 (2000), 230-244. doi: 10.1634/stemcells.18-4-230. Google Scholar
T. Shiri, W. Garira and S. D. Musekwa, A two-strain hiv-1 mathematical model to assess the effects of chemotherapy on disease parameters, Math. Biosci. Eng., 2 (2005), 811. doi: 10.3934/mbe.2005.2.811. Google Scholar
M. O. Souza and J. P. Zubelli, Global stability for a class of virus models with Cytotoxic T Lymphocyte immune response and antigenic variation, Bull. Math. Biol., 73 (2011), 609-625. doi: 10.1007/s11538-010-9543-2. Google Scholar
N. Tarfulea and P. Read, A mathematical model for the emergence of HIV drug resistance during periodic bang-bang type antiretroviral treatment, Involve, J. Math., 8 (2015), 401-420. doi: 10.2140/involve.2015.8.401. Google Scholar
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. Google Scholar
L. M. Wahl and M. A. Nowak, Adherence and drug resistance: Predictions for therapy outcome, Proc. Biol. Sci., 267 (2000), 835-843. doi: 10.1098/rspb.2000.1079. Google Scholar
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. Google Scholar
K. Wang, W. Wang, H. Pang and X. Liu, Complex dynamic behavior in a viral model with delayed immune response, Physica D: Nonlinear Phenomena, 226 (2007), 197-208. doi: 10.1016/j.physd.2006.12.001. Google Scholar
X. Wang, A. Elaiw and X. Song, Global properties of a delayed HIV infection model with CTL immune response, Appl. Math. Comput., 218 (2012), 9405-9414. doi: 10.1016/j.amc.2012.03.024. Google Scholar
X. Wang, Y. Tao and X. Song, Global stability of a virus dynamics model with beddington-deangelis incidence rate and CTL immune response, Nonlinear Dyn., 66 (2011), 825-830. doi: 10.1007/s11071-011-9954-0. Google Scholar
Y. Wang, F. Brauer, J. Wu and J. M. Heffernan, A delay-dependent model with HIV drug resistance during therapy, J. Math. Anal. Appl., 414 (2014), 514-531. doi: 10.1016/j.jmaa.2013.12.064. Google Scholar
Y. Wang, Y. Zhou, F. Brauer and J. M. Heffernan, Viral dynamics model with CTL immune response incorporating antiretroviral therapy, J. Math. Biol., 67 (2013), 901-934. doi: 10.1007/s00285-012-0580-3. Google Scholar
R. A. Weiss, How does HIV cause AIDS?, Science, 260 (1993), 1273-1279. doi: 10.1126/science.8493571. Google Scholar
WHO, HIV/AIDS: Key facts, http://www.who.int/news-room/fact-sheets/detail/hiv-aids, 2018. Google Scholar
D. Wodarz and A. L. Lloyd, Immune responses and the emergence of drug-resistant virus strains in vivo, Proc. R. Soc. Lond. B, 271 (2004), 1101-1109. Google Scholar
D. Wodarz and M. A. Nowak, Specific therapy regimes could lead to long-term immunological control of HIV, P. Natl. A. Sci., 96 (1999), 14464-14469. doi: 10.1073/pnas.96.25.14464. Google Scholar
D. Wodarz and M. A. Nowak, Immune responses and viral phenotype: Do replication rate and cytopathogenicity influence virus load?, Comput. Math. Method. M., 2 (2000), 113-127. doi: 10.1080/10273660008833041. Google Scholar
D. Wodarz and M. A. Nowak, Mathematical models of HIV pathogenesis and treatment, BioEssays, 24 (2002), 1178-1187. doi: 10.1002/bies.10196. Google Scholar
J. Wu, P. Yan and C. Archibald, Modelling the evolution of drug resistance in the presence of antiviral drugs, BMC Public Health, 7 (2007), 300. doi: 10.1186/1471-2458-7-300. Google Scholar
J. Wu, R. Dhingra, M. Gambhir and J. V. Remais, Sensitivity analysis of infectious disease models: Methods, advances and their application, J. R. Soc. Interface, 10 (2013), 20121018. doi: 10.1098/rsif.2012.1018. Google Scholar
Y. Xiao, S. Tang, Y. Zhou, R. J. Smith, J. Wu and N. Wang, Predicting the HIV/AIDS epidemic and measuring the effect of mobility in mainland China, J. Theor. Biol., 317 (2013), 271-285. doi: 10.1016/j.jtbi.2012.09.037. Google Scholar
H. Zhu and X. Zou, Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 511-524. doi: 10.3934/dcdsb.2009.12.511. Google Scholar
Figure 1. The schematic diagram of HIV transmission with wild-type and resistant virus
Figure 2. The dynamics of infection before converging to steady states. $ (a)-(d) $ represent uninfected cells, wild-type virus, resistant virus, and CTL cells, respectively. The blue solid line is the case in which CTL cells are present. The parameters are from Table 1 and $ b = 0.1 $. We have $ R_{s} = 3.13>R_{r} = 1.74>1 $, $ R_{cs} = 68.1>1 $. Solution trajectories of the system converge to the positive equilibrium $ E^{\star} = (415257, 14994.3, 586870, 1.01219, 26.4110, 44.9845) $. The red dotted line refers to the case in which CTL cells are absent. The parameter $ c $ is 0. This yields $ R_{s} = 3.13>R_{r} = 1.74>1 $, $ R_{cs} = 0<1 $. Solution trajectories of the system converge to the CTL immune-free equilibrium $ E^{f} = (318355, 23056.2, 902393, 1.55640, 40.6104, 0) $
Figure 3. Sensitivity analysis for four reproduction numbers. (a) The PRCC of $ R_{s} $ for seven parameters, (b) the PRCC of $ R_{r} $ for six parameters, (c) the PRCC of $ R_{cs} $ for nine parameters, and (d) the PRCC of $ R_{cr} $ for eight parameters
Figure 4. Predicted dynamics of system (4) under treatment. $ (a)-(d) $ represent uninfected cells, wild-type virus, resistant virus, and CTL cells, respectively. The blue solid line represents the case in which CTL cells are present. The parameters given in Table 1 are used and $ b=0.1 $. This yields $ R_{s}^{\prime}=1.57>R_{r}^{\prime}=1.54>1 $ and $ R_{cs}^{\prime}=36.30>1 $. Thus, solutions trajectories of the system converge to the interior equilibrium $ E^{\star\prime}=(733912, 7673.78, 243257, 13.8008, 346.384, 23.0628) $. The red dotted line refers to the case in which CTL cells are absent. Parameters from Table 1 are used and $ b=0.1, c=0 $. In this case, $ R_{s}^{\prime}=1.57>R_{r}^{\prime}=1.54>1 $ and $ R_{cs}^{\prime}=0<1 $. Solutions trajectories of the system converge to the CTL immune-free equilibrium $ E^{f\prime}=(635966, 12174.6, 476875, 21.8671, 571.018, 0) $
Figure 5. Invasion of drug-resistant virus. (a) Effects of the infection rate $ \beta_{s} $ and overall drug efficacy $ \eta_{s} $ on $ R_{s}^{\prime} $ and $ R_{r}^{\prime} $. (b) The projection of (a) on the $ \beta_{s}-\eta_{s} $ plane. (c) Part of (b) when $ \eta_{s} $ is from 0.4 to 0.6. The vertical line in magenta represents $ \eta_{s} = 0.51 $. In the simulation, we assume that $ \eta_{r} = 0.23\eta_{s}, \beta_{r} = 0.83\beta_{s} $. The other parameters are given in Table 1
Figure 6. Predicted dynamics of system (4) when drug resistance invades during treatment. Figure $ (a)-(b) $ shows the wild-type virus and the resistant virus, respectively. $ \eta_{s} = 0.6876, \eta_{r} = 0.1635 $, $ R_{s}^{\prime} = 0.9778<1<R_{r}^{\prime} = 1.455, R_{cr} = 312.7>1 $. The solutions of the system converge to the infected equilibrium with drug-resistant strain and CTL immune response $ E^{rc\prime} = (928780, 0, 0, 1756.85, 45817.3, 52.7080) $. Figure $ (c)-(d) $ is similar to $ (a)-(b) $ except $ \eta_{s} = 0.4330, \eta_{r} = 0.0957 $. In this case, $ R_{r}^{\prime} = 1.573<R_{s}^{\prime} = 1.775, R_{cs}^{\prime} = 436.6>1 $. Both strains of virus persist and the solutions of the system converge to the interior equilibrium $ E^{\star\prime} = (872574, 2744.46, 86963.0, 0.722593, 18.1288, 82.2710) $. The parameters are from Table 1
Figure 7. Dynamics of system (4) with different values of $ c $ (the proliferation rate of CTLs). Green dashed-dot line is for $ c = 0 $, black solid line is for $ c = 0.00003 $, blue dashed line is for $ c = 0.0002 $, and red dot line is for $ c = 0.001 $. We chose $ \eta_{s} = 0.5328, \eta_{r} = 0.1221 $ and the other parameters can be found in Table 1
Figure 8. Drug resistance dynamics under different drug efficacies. (a) The immune clearance rate is $ b = 0.1 $ and (b) $ b = 0.01 $. Green curve is for $ \eta_{s} = 0.52, \eta_{r} = 0.12 $, red curve is for $ \eta_{s} = 0.69, \eta_{r} = 0.16 $, and blue curve is for $ \eta_{s} = 0.96, \eta_{r} = 0.29 $. $ c $ is fixed at $ 0.001 $ and other parameters are given in Table 1
Figure 9. (a) The difference between non-infectious and infectious wild-type virus concentrations. (b) The difference between non-infectious and infectious drug-resistant virus concentrations. Shortly after initiation of potent antiretroviral therapy, the difference between non-infectious and infectious viral levels (logarithm with base 10) approaches a constant. The parameters can be found in Table 1
Table 1. Parameter descriptions and sources for their values
Parameter Value Ranges Description Reference
$ \lambda $ $ 10^{4}ml^{-1}day^{-1} $ $ 100\sim200000 $ Production rate of uninfected cells [42]
$ d $ $ 0.01day^{-1} $ $ 0.0001\sim 0.8 $ Death rate of uninfected cells [34]
$ \beta_{s} $ $ 2.4\times10^{-8}ml\cdot day^{-1} $ $ 10^{-6}\sim2.4\times10^{-6} $ Infection rate of uninfected cells by wild-type virus [34]
$ \beta_{r} $ $ 2.0\times10^{-8}ml\cdot day^{-1} $ $ 10^{-6}\sim2.0\times10^{-6} $ Infection rate of uninfected cells by drug-resistant virus [42]
$ \delta $ $ 0.3day^{-1} $ $ 0.1\sim0.9 $ Death rate of infected cells [42]
$ k $ $ 0.002mm^{3}day^{-1} $ - Clearance rate of infected cells by CTL killing [48, 1]
$ p_{s} $ $ 900day^{-1} $ $ 90\sim1000 $ Generation rate of wild-type virus [8, 17]
$ p_{r} $ $ 600day^{-1} $ $ 60\sim1000 $ Generation rate of drug-resistant virus [8, 17]
$ d_{1} $ $ 23day^{-1} $ $ 10\sim50 $ Clearance rate of wild-type and resistant virus [39]
$ c $ $ 0.0003ml^{-1}day^{-1} $ $ 0\sim0.001 $ generation rate of CTL [55, 61]
$ b $ $ 0.01day^{-1} $ $ 0.01\sim1 $ death rate of CTL [48, 61]
$ \mu $ $ 3\times10^{-5} $ $ 10^{-6}\sim10^{-4} $ Single mutation rate [24]
$ \varepsilon_{RTI} $ Varied $ 0\sim1 $ Efficacy of RTI see text
$ \varepsilon_{PI} $ Varied $ 0\sim1 $ Efficacy of PI see text
$ \sigma_{RTI} $ Varied $ 0\sim1 $ Resistance ratio of RTI see text
$ \sigma_{PI} $ Varied $ 0\sim1 $ Resistance ratio of PI see text
Table 2. Summary of the stability results of system (1)
Conditions System (1)
$ R_{0<1} $ $ E^{0} $ $ is $ $ L.A.S $
$ R_{0}>1 $ $ R_{s}<R_{r} $ $ R_{cr<1} $ $ E^{r} $ $ is $ $ L.A.S $
$ R_{cr>1} $ $ E^{rc} $ $ is $ $ L.A.S $
$ R_{s}>R_{r} $ $ R_{cs<1} $ $ E^{f} $ $ is $ $ L.A.S $
$ R_{cs>1} $ $ E^{\star} $ $ is $ $ L.A.S $ $ (\hbox{by simulation}) $
Xin Zhao, Tao Feng, Liang Wang, Zhipeng Qiu. Threshold dynamics and sensitivity analysis of a stochastic semi-Markov switched SIRS epidemic model with nonlinear incidence and vaccination. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021010
Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020341
Do Lan. Regularity and stability analysis for semilinear generalized Rayleigh-Stokes equations. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021002
Shujing Shi, Jicai Huang, Yang Kuang. Global dynamics in a tumor-immune model with an immune checkpoint inhibitor. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1149-1170. doi: 10.3934/dcdsb.2020157
Adam Glick, Antonio Mastroberardino. Combined therapy for treating solid tumors with chemotherapy and angiogenic inhibitors. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020343
Shigui Ruan. Nonlinear dynamics in tumor-immune system interaction models with delays. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 541-602. doi: 10.3934/dcdsb.2020282
Yi-Hsuan Lin, Gen Nakamura, Roland Potthast, Haibing Wang. Duality between range and no-response tests and its application for inverse problems. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020072
He Zhang, John Harlim, Xiantao Li. Estimating linear response statistics using orthogonal polynomials: An RKHS formulation. Foundations of Data Science, 2020, 2 (4) : 443-485. doi: 10.3934/fods.2020021
George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003
Yi-Ming Tai, Zhengyang Zhang. Relaxation oscillations in a spruce-budworm interaction model with Holling's type II functional response. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021027
Skyler Simmons. Stability of Broucke's isosceles orbit. Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021015
Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251
Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020 doi: 10.3934/jgm.2020032
Meihua Dong, Keonhee Lee, Carlos Morales. Gromov-Hausdorff stability for group actions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1347-1357. doi: 10.3934/dcds.2020320
Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020346
Qi Deng Zhipeng Qiu Ting Guo Libin Rong