Modeling within-host viral dynamics: The role of CTL immune responses in the evolution of drug resistance

Qi Deng; Zhipeng Qiu; Ting Guo; Libin Rong

doi:10.3934/dcdsb.2020245

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July 2021, 26(7): 3543-3562. doi: 10.3934/dcdsb.2020245

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,

1.	Department of Mathematics, Nanjing University of Science and Technology, Nanjing 210094, China
2.	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, 2021, 26 (7) : 3543-3562. doi: 10.3934/dcdsb.2020245

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Figure 1. The schematic diagram of HIV transmission with wild-type and resistant virus

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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) $

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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

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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) $

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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

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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

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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

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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

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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

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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

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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_{s}<R_{r} $	$ R_{cr>1} $	$ E^{rc} $ $ is $ $ L.A.S $
	$ R_{s}>R_{r} $	$ R_{cs<1} $	$ E^{f} $ $ is $ $ L.A.S $
	$ R_{s}>R_{r} $	$ R_{cs>1} $	$ E^{\star} $ $ is $ $ L.A.S $ $ (\hbox{by simulation}) $

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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_{s}<R_{r} $	$ R_{cr>1} $	$ E^{rc} $ $ is $ $ L.A.S $
	$ R_{s}>R_{r} $	$ R_{cs<1} $	$ E^{f} $ $ is $ $ L.A.S $
	$ R_{s}>R_{r} $	$ R_{cs>1} $	$ E^{\star} $ $ is $ $ L.A.S $ $ (\hbox{by simulation}) $

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American Institute of Mathematical Sciences

Modeling within-host viral dynamics: The role of CTL immune responses in the evolution of drug resistance

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