doi: 10.3934/dcdsb.2021159
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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

Received  August 2020 Revised  April 2021 Early access June 2021

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

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.

Citation: Zhijun Liu, Lianwen Wang, Ronghua Tan. Spatiotemporal dynamics for a diffusive HIV-1 infection model with distributed delays and CTL immune response. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021159
References:
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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.  Google Scholar

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K. A. PawelekS. LiuF. 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.  Google Scholar

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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.  Google Scholar

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show all references

References:
[1]

S. BonhoefferR. M. MayG. 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.  Google Scholar

[2]

N. Chomont, 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.  Google Scholar

[3]

T.-W. ChunL. StuyverS. B. MizellL. A. EhlerJ. A. M. MicanM. BaselerA. L. LloydM. 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.  Google Scholar

[4]

M. C. CiupeB. L. BivortD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

G. A. FunkV. A. A. JansenS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

D. Henry, Geometric Theory of Semilinear Parabolic Equations Lecture Notes in Mathematics, Springer, Berlin, 840 1981.  Google Scholar

[11]

G. HuangY. 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.  Google Scholar

[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.  Google Scholar

[13]

C. JiangK. 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.  Google Scholar

[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.  Google Scholar

[15]

J. P. LaSalle, The stability of dynamical systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, (1976).  Google Scholar

[16]

B. LiY. ChenX. 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.  Google Scholar

[17]

X. LuL. HuiS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

H. MiaoZ. TengX. 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.  Google Scholar

[22]

J. E. MittlerB. SulzerA. 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.  Google Scholar

[23]

B. A. Mock, Longitudinal patterns of trypanosome infections in red-spotted newts, J. Parasitol., 73 (1987), 730-737.  doi: 10.2307/3282402.  Google Scholar

[24]

P. W. NelsonJ. 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.  Google Scholar

[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.  Google Scholar

[26]

K. A. PawelekS. LiuF. 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.  Google Scholar

[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.  Google Scholar

[28]

A. K. PerelsonD. 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

[29]

X. RenY. TianL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

S. G. Ruan and J. H. Wu, Reaction-diffusion equations with infinite delay, Can. Appl. Math. Q., 2 (1994), 485-550.   Google Scholar

[34]

H. ShuL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

S. TangZ. 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.  Google Scholar

[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.  Google Scholar

[39]

H. WangR. XuZ. 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.  Google Scholar

[40]

J. WangM. GuoX. 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.  Google Scholar

[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.  Google Scholar

[42]

S. WangJ. ZhangF. 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.  Google Scholar

[43]

X. WangS. TangX. 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.  Google Scholar

[44]

X. WangX. TangZ. 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.  Google Scholar

[45]

D. WodarzM. 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.  Google Scholar

[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.  Google Scholar

[47]

World Health Organization, HIV/AIDS, 2018-19-7, https://www.who.int/news-room/fact-sheets/detail/hiv-aids. Google Scholar

[48]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

[49]

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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">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
Table 2">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
Table 2">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
Table 2">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
Table 2">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
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
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
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
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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