Spatiotemporal dynamics for a diffusive HIV-1 infection model with distributed delays and CTL immune response

Zhijun Liu; Lianwen Wang; Ronghua Tan

doi:10.3934/dcdsb.2021159

Article Contents

2022, Volume 27, Issue 5: 2767-2790. Doi: 10.3934/dcdsb.2021159

This issue Previous Article Corrigendum: On the Abel differential equations of third kind Next Article The qualitative behavior of a plankton-fish interaction model with food limited growth rate and non-constant fish harvesting

Spatiotemporal dynamics for a diffusive HIV-1 infection model with distributed delays and CTL immune response

School of Mathematics and Statistics, Hubei Minzu University, Enshi, 445000, China

^* Corresponding author: Zhijun Liu
^* Corresponding author: Zhijun Liu

Received: July 31, 2020

Revised: March 31, 2021

Early access: June 2021

Published: May 2022

The work is supported by the National Natural Science Foundation of China (No.11871201), and Natural Science Foundation of Hubei Province, China (No.2019CFB241)

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Abstract

In this study, we develop a diffusive HIV-1 infection model with intracellular invasion, production and latent infection distributed delays, nonlinear incidence rate and nonlinear CTL immune response. The well-posedness, local and global stability for the model proposed are carefully investigated in spite of its strong nonlinearity and high dimension. It is revealed that its threshold dynamics are fully determined by the viral infection reproduction number $ \mathfrak{R}_0 $ and the reproduction number of CTL immune response $ \mathfrak{R}_1 $. We also observe that the viral load at steady state (SS) fails to decrease even if $ \mathfrak{R}_1 $ increases through unit to lead to a stability switch from immune-inactivated infected SS to immune-activated infected SS. Finally, some simulations are performed to verify the analytical conclusions and we explore the significant impact of delays and CTL immune response on the spatiotemporal dynamics of HIV-1 infection.

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Mathematics Subject Classification: Primary: 92D30; Secondary: 34D23, 34K20.

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Figure 1. (1)-(10): Spatial distribution of each solution variable of model (28) in two time points $ t = 5, 12 $ in the case that $ \mathfrak{R}_0 = 0.386<1 $ and the parameter values are from Data1 in Table 2; (a)-(e): The time evolutions of the corresponding solution variables in three different positions $ (x, y) = (1, 1.5) $, $ (2.5, 2.5) $ and $ (1.5, 2) $ illustrates that $ \mathcal{E}^0 = (T^0, 0, 0, 0, 0) = (80, 0, 0, 0, 0) $ is GAS

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Figure 2. (1)-(15): Spatial distribution of each solution variable of model (28) in three time points $ t = 14, 20, 25 $ when $ \mathfrak{R}_0 = 3.858>1 $, $ \mathfrak{R}_1 = 0.831<1 $ and the parameter values are from Data2 in Table 2

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Figure 3. In the case of $ \mathfrak{R}_0 = 3.858>1 $ and $ \mathfrak{R}_1 = 0.831<1 $, the graph trajectory of the solution to model (28) versus $ t $ is illustrated by (a)-(c) with three different positions $ (x, y) = (1, 1.5) $, $ (2.5, 2.5) $ and $ (1.5, 2) $ in $ \Omega $, respectively, such that $ \mathcal{E}^* = (T^*, E^*, I^*, V^*, 0) = (26.122, 0.004, 0.523, 51.972, 0) $ is GAS, where the parameter values are from Data2 in Table 2

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Figure 4. (1)-(15): Spatial distribution of each solution variable of model (28) in three time points $ t = 14, 20, 25 $ when $ \mathfrak{R}_0 = 3.858>1 $, $ \mathfrak{R}_1 = 1.843>1 $ and the parameter values are from Data3 in Table 2

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Figure 5. In the case of $ \mathfrak{R}_0 = 3.858>1 $ and $ \mathfrak{R}_1 = 1.843>1 $, the graph trajectory of the solution to model (28) versus $ t $ is illustrated by (a)-(c) with three different positions $ (x, y) = (1, 1.5) $, $ (2.5, 2.5) $ and $ (1.5, 2) $ in $ \Omega $, respectively, such that $ \mathcal{E}^† = (T^†, E^†, I^†, V^†, Z^†) = (41.888, 0.003, 0.201, 20.019, 3.485) $ is GAS, where the parameter values are from Data3 in Table 2

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Table 1. List of parameters

Parameter	Description
$ \lambda $	Generation rate of uninfected target cells
$ \mu $	Death rate of uninfected target cells
$ \beta $	Infection rate of cells by free virus
$ \theta $	Activation rate of uninfected target cells
$ a $	The sum of activation and death rates of latently infected cells
$ b $	Death rate of productively infected cells
$ c $	Clearance rate of free virus
$ d $	Death rate of effector cell of CTLs
$ p $	CTL effectiveness
$ q_3 $	Released rate for free viral particles
$ \gamma $	Proliferation rate of CTLs by productively infected cells
$ q_1, q_2 $	Fraction of infection leading to latency and production, respectively
$ D_1, D_2, D_3 $	The diffusion coefficients
$ \alpha, \sigma $	Inhibitory rate from target cells and effector cell of CTLs, respectively

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Table 2. Parameters and their values

Para.	Units	Data1	Data2	Data3	Range	Source
$ \lambda $	cells$ \cdot $ml$ ^{-1} $day$ ^{-1} $	0.8	0.8	0.8	$ [0, 10] $	[24]
$ \mu $	day$ ^{-1} $	0.01	0.01	0.01	$ [10^{-4}, 0.2] $	[16]
$ \beta $	ml$ \cdot $virion$ ^{-1} $day$ ^{-1} $	$ 5\times10^{-5} $	$ 5\times10^{-4} $	$ 5\times10^{-4} $	$ [4.6\times10^{-8}, 0.5] $	[26]
$ \theta $	day$ ^{-1} $	0.01	0.01	0.01	$ [0.01, 0.3] $	[31,43,32]
$ a $	day$ ^{-1} $	0.014	0.014	0.014	$ [0.001, 0.2] $	[31,43,30]
$ b $	day$ ^{-1} $	1	1	1	$ [1.9\times10^{-4}, 1.4] $	[31,16]
$ c $	day$ ^{-1} $	2	2	2	$ [0.081, 36] $	[4]
$ d $	day$ ^{-1} $	0.2	0.2	0.2	$ [0.004, 8.087] $	[4]
$ p $	ml$ \cdot $cell$ ^{-1} $day$ ^{-1} $	0.24	0.24	0.24	$ [10^{-4}, 4.048] $	[16,26]
$ q_3 $	virion$ \cdot $cell$ ^{-1} $day$ ^{-1} $	200	200	200	$ [0.38, 2800] $	[16]
$ \gamma $	ml$ \cdot $cell$ ^{-1} $day$ ^{-1} $	1	0.3	1	$ [0.0051, 3.912] $	[16]
$ q_1 $	–	$ 10^{-4} $	$ 10^{-4} $	$ 10^{-4} $	$ [0, 1] $	[43,32]
$ q_2 $	–	$ 1-10^{-4} $	$ 1-10^{-4} $	$ 1-10^{-4} $	$ [0, 1] $	[43,32]
$ D_1 $	mm$ ^{2} $day$ ^{-1} $	$ 10^{-4} $	$ 10^{-4} $	$ 10^{-4} $	$ [0, 1] $	[29]
$ D_2 $	mm$ ^{2} $day$ ^{-1} $	$ 5\times10^{-4} $	$ 5\times10^{-4} $	$ 5\times10^{-4} $	$ [0, 1] $	[29]
$ D_3 $	mm$ ^{2} $day$ ^{-1} $	$ 2\times10^{-4} $	$ 2\times10^{-4} $	$ 2\times10^{-4} $	$ [0, 1] $	[29]
$ \alpha $	–	0.005	0.005	0.005	$ [0, 1] $	[37,36]
$ \sigma $	–	0.002	0.002	0.002	$ [0, 1] $	[46]
$ m_1 $	day$ ^{-1} $	0.05	0.05	0.05	$ [0, 1] $	[43]
$ \tau_1 $	day	0.3	0.3	0.3	$ [0, 0.5] $	[43]
$ m_2 $	day$ ^{-1} $	0.05	0.05	0.05	$ [0, 1] $	[43]
$ \tau_2 $	day	0.6	0.6	0.6	$ [0.5, 1] $	[43]
$ m_3 $	day$ ^{-1} $	0.01	0.01	0.01	$ [0, 1] $	[54]
$ \tau_3 $	day	0.6	0.6	0.6	$ [0, 10] $	[49]
$ \mathfrak{R}_0 $		0.386	3.858	3.858
$ \mathfrak{R}_1 $		0.322	0.831	1.843

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References

[1]	S. Bonhoeffer, R. M. May, G. M. Shaw and M. A. Nowak, Virus dynamics and drug therapy, Proc. Natl. Acad. Sci. USA, 94 (1997), 6971-6976. doi: 10.1073/pnas.94.13.6971.
[2]	N. Chomont, et al., HIV reservoir size and persistence are driven by T cell survival and homeostatic proliferation, Nat. Med., 15 (2009), 893-900. doi: 10.1038/nm.1972.
[3]	T.-W. Chun, L. Stuyver, S. B. Mizell, L. A. Ehler, J. A. M. Mican, M. Baseler, A. L. Lloyd, M. A. Nowak and A. S. Fauci, Presence of an inducible HIV-1 latent reservoir during highly active antiretroviral therapy, Proc. Natl. Acad. Sci., 94 (1997), 13193-13197. doi: 10.1073/pnas.94.24.13193.
[4]	M. C. Ciupe, B. L. Bivort, D. M. Bortz and P. W. 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.
[5]	A. M. Elaiw and A. D. Al Agha, A reaction-diffusion model for oncolytic M1 virotherapy with distributed delays, Eur. Phys. J. Plus, 135 (2020), 117. doi: 10.1140/epjp/s13360-020-00188-z. doi: 10.1140/epjp/s13360-020-00188-z.
[6]	W. E. Fitzgibbon, Semilinear functional differential equations in Banach space, J. Differ. Equ., 29 (1978), 1-14. doi: 10.1016/0022-0396(78)90037-2.
[7]	G. A. Funk, V. A. A. Jansen, S. Bonhoeffer and T. Killingback, Spatial models of virus-immune dynamics, J. Theor. Biol., 233 (2005), 221-236. doi: 10.1016/j.jtbi.2004.10.004.
[8]	K. Hattaf, Spatiotemporal dynamics of a generalized viral infection model with distributed delays and CTL immune response, Computation, 7 (2019), 1-16. doi: 10.3390/computation7020021.
[9]	K. Hattaf and N. Yousfi, Global stability for reaction-diffusion equations in biology, Comput. Math. Appl., 66 (2013), 1488-1497. doi: 10.1016/j.camwa.2013.08.023.
[10]	D. Henry, Geometric Theory of Semilinear Parabolic Equations Lecture Notes in Mathematics, Springer, Berlin, 840 1981.
[11]	G. Huang, Y. Takeuchi and A. Korobeinikov, HIV evolution and progression of the infection to AIDS, J. Theor. Biol., 307 (2012), 149-159. doi: 10.1016/j.jtbi.2012.05.013.
[12]	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.
[13]	C. Jiang, K. Wang and L. Song, Global dynamics of a delay virus model with recruitment and saturation effects of immune responses, Math. Biosci. Eng., 14 (2017), 1233-1246. doi: 10.3934/mbe.2017063.
[14]	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.
[15]	J. P. LaSalle, The stability of dynamical systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, (1976).
[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]	X. Lu, L. Hui, S. Liu and J. Li, A mathematical model of HTLV-I infection with two time delays, Math. Biosci. Eng., 12 (2015), 431-449. doi: 10.3934/mbe.2015.12.431.
[18]	R. H. Jr. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Am. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590.
[19]	R. H. Jr. Martin and H. L. Smith, Reaction-diffusion systems with time delays: monotonicity, invariance, comparison and convergence, J. Reine Angenw. Math., 413 (1991), 1-35. doi: 10.1515/crll.1991.413.1.
[20]	C. C. McCluskey and Y. Yang, Global stability of a diffusive virus dynamics model with general incidence function and time delay, Nonlinear Anal. RWA, 25 (2015), 64-78. doi: 10.1016/j.nonrwa.2015.03.002.
[21]	H. Miao, Z. Teng, X. Abdurahman and Z. Li, Global stability of a diffusive and delayed virus infection model with general incidence function and adaptive immune response, Comput. Appl. Math., 37 (2018), 3780-3805. doi: 10.1007/s40314-017-0543-9.
[22]	J. E. Mittler, B. Sulzer, A. U. Neumann and A. S. Perelson, Influence of delayed viral production on viral dynamics in HIV-1 infected patients, Math. Biosci., 152 (1998), 143-163. doi: 10.1016/S0025-5564(98)10027-5.
[23]	B. A. Mock, Longitudinal patterns of trypanosome infections in red-spotted newts, J. Parasitol., 73 (1987), 730-737. doi: 10.2307/3282402.
[24]	P. W. Nelson, J. D. Murray and A. S. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay, Math. Biosci., 163 (2000), 201-215. doi: 10.1016/S0025-5564(99)00055-3.
[25]	M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. doi: 10.1126/science.272.5258.74.
[26]	K. A. 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.
[27]	A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107.
[28]	A. K. 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.
[29]	X. Ren, Y. Tian, L. Liu and X. Liu, A reaction-diffusion within-host HIV model with cell-to-cell transmission, J. Math. Biol., 76 (2018), 1831-1872. doi: 10.1007/s00285-017-1202-x.
[30]	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.
[31]	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), e1000533, 18 pp. doi: 10.1371/journal.pcbi.1000533. doi: 10.1371/journal.pcbi.1000533.
[32]	L. Rong and A. S. 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.
[33]	S. G. Ruan and J. H. Wu, Reaction-diffusion equations with infinite delay, Can. Appl. Math. Q., 2 (1994), 485-550.
[34]	H. Shu, L. Wang and J. Watmough, Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL immune responses, SIAM J. Appl. Math., 73 (2013), 1280-1302. doi: 10.1137/120896463.
[35]	R. P. Sigdel and C. C. McCluskey, Global stability for an SEI model of infectious disease with immigration, Appl. Math. Comput., 243 (2014), 684-689. doi: 10.1016/j.amc.2014.06.020.
[36]	H. Sun and J. Wang, Dynamics of a diffusive virus model with general incidence function, cell-to-cell transmission and time delay, Comput. Math. Appl., 77 (2019), 284-301. doi: 10.1016/j.camwa.2018.09.032.
[37]	S. Tang, Z. Teng and H. Miao, Global dynamics of a reaction-diffusion virus infection model with humoral immunity and nonlinear incidence, Comput. Math. Appl., 78 (2019), 786-806. doi: 10.1016/j.camwa.2019.03.004.
[38]	C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Trans. Am. Math. Soc., 200 (1974), 395-418. doi: 10.1090/S0002-9947-1974-0382808-3.
[39]	H. Wang, R. Xu, Z. Wang and H. Chen, Global dynamics of a class of HIV-1 infection models with latently infected cells, Nonlinear Anal. Model. Control, 20 (2015), 21-37. doi: 10.15388/NA.2015.1.2.
[40]	J. Wang, M. Guo, X. Liu and Z. Zhao, Threshold dynamics of HIV-1 virus model with cell-to-cell transmission, cell-mediated immune responses and distributed delay, Appl. Math. Comput., 291 (2016), 149-161. doi: 10.1016/j.amc.2016.06.032.
[41]	K. Wang and W. Wang, Propagation of HBV with spatial dependence, Math. Biosci., 210 (2007), 78-95. doi: 10.1016/j.mbs.2007.05.004.
[42]	S. Wang, J. Zhang, F. Xu and X. Song, Dynamics of virus infection models with density-dependent diffusion, Comput. Math. Appl., 74 (2017), 2403-2422. doi: 10.1016/j.camwa.2017.07.019.
[43]	X. Wang, S. Tang, X. Song and L. Rong, Mathematical analysis of an HIV latent infection model including both virus-to-cell infection and cell-to-cell transmission, J. Biol. Dynam., 11 (2017), 1-29. doi: 10.1080/17513758.2016.1242784.
[44]	X. Wang, X. Tang, Z. Wang and X. Li, Global dynamics of a diffusive viral infection model with general incidence function and distributed delays, Ricerche Mat., 69 (2020), 683-702. doi: 10.1007/s11587-020-00481-0.
[45]	D. Wodarz, M. A. Nowak and C. R. M. Bangham, The dynamics of HTLV-I and the CTL response, Immunol. Today, 20 (1999), 220-227. doi: 10.1016/S0167-5699(99)01446-2.
[46]	D. Wodarz and M. A. Nowak, Immune responses and viral phenotype: do replication rate and cytopathogenicity influence virus load?, Comput. Math. Methods Med., 2 (2000), 113-127. doi: 10.1080/10273660008833041.
[47]	World Health Organization, HIV/AIDS, 2018-19-7, https://www.who.int/news-room/fact-sheets/detail/hiv-aids.
[48]	J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1. doi: 10.1007/978-1-4612-4050-1.
[49]	J. Xu, Y. Geng and J. Hou, Global dynamics of a diffusive and delayed viral infection model with cellular infection and nonlinear infection rate, Comput. Math. Appl., 73 (2017), 640-652. doi: 10.1016/j.camwa.2016.12.032.
[50]	J. Xu and Y. Zhou, Bifurcation analysis of HIV-1 infection model with cell-to-cell transmission and immune response delay, Math. Biosci. Eng., 13 (2016), 343-367. doi: 10.3934/mbe.2015006.
[51]	Y. Yang, Y. Dong and Y. Takeuchi, Global dynamics of a latent HIV infection model with general incidence function and multiple delays, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 783-800. doi: 10.3934/dcdsb.2018207.
[52]	Y. Yang, T. Zhang, Y. Xu and J. Zhou, A delayed virus infection model with cell-to-cell transmission and CTL immune response, Int. J. Bifurcat. Chaos, 27 (2017), 1750150. doi: 10.1142/S0218127417501504. doi: 10.1142/S0218127417501504.
[53]	Z. Yuan and X. Zou, Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays, Math. Biosci. Eng., 10 (2013), 483-498. doi: 10.3934/mbe.2013.10.483.
[54]	H. Zhu and X. Zou, Impact of delays in cell infection and virus production on HIV-1 dynamics, Math. Med. Biol., 25 (2008), 99-112. doi: 10.1093/imammb/dqm010.