February  2020, 13(2): 105-117. doi: 10.3934/dcdss.2020006

Stabilization in a chemotaxis model for virus infection

1. 

Politecnico of Torino, Corso Duca degli Abruzzi 24, Torino 10129, Italy, Collegio Carlo Alberto, Torino, Italy

2. 

Department of Applied Mathematics, Dong Hua University, Shanghai 200051, China

#Corresponding author: Youshan Tao

Received  March 2017 Revised  October 2017 Published  January 2019

Fund Project: Youshan Tao acknowledges the support by National Natural Science Foundation of China, No. 11571070.

This paper presents a qualitative analysis of a model describing the time and space dynamics of a virus which migrates driven by chemotaxis. The initial-boundary value problem related to applications of the model to a real biological dynamics is studied in detail. The main result consists in the proof of global existence and asymptotic stability.

Citation: Nicola Bellomo, Youshan Tao. Stabilization in a chemotaxis model for virus infection. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 105-117. doi: 10.3934/dcdss.2020006
References:
[1]

R. M. AndersonR. M. May and S. Gupta, Non-linear phenomena in host-parasite interactions, Parasitology, 99 (1989), 59-79.  doi: 10.1017/S0031182000083426.  Google Scholar

[2]

J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol., 44 (1975), 331-340.  doi: 10.2307/3866.  Google Scholar

[3]

N. BellomoA. Bellouquid and N. Chouhad, From a multiscale derivation of nonlinear cross-diffusion models to Keller-Segel models in a Navier-Stokes fluid, Math. Models Methods Appl. Sci., 26 (2016), 2041-2069.  doi: 10.1142/S0218202516400078.  Google Scholar

[4]

N. Bellomo and M. Winkler, A degenerate chemotaxis system with flux limitation: Maximally extended solutions and absence of gradient blow-up, Commun. Part. Diff. Eq., 42 (2017), 436-473.  doi: 10.1080/03605302.2016.1277237.  Google Scholar

[5]

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

[6]

J. CalvoJ. CamposV. CasellesO. Sanchez and J. Soler, Flux-saturated porous media equations and applications, Surv. Math. Sciences, 2 (2015), 131-218.  doi: 10.4171/EMSS/11.  Google Scholar

[7]

D. CamposV. Méndez and S. Fedotov, The effects of distributed life cycles on the dynamics of viral infections, J. Theor. Biol., 254 (2008), 430-438.  doi: 10.1016/j.jtbi.2008.05.035.  Google Scholar

[8]

D. L. DeAngelisR. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892.   Google Scholar

[9]

O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases, John Wiley & Sons, Ltd., Chichester, 2000.  Google Scholar

[10]

V. DoceulM. HollinsheadL. van der Linden and G. L. Smith, Repulsion of superinfecting virions: A mechanism for rapid virus spread, Science, 327 (2010), 873-876.  doi: 10.1126/science.1183173.  Google Scholar

[11]

L. GibelliA. ElaiwM.-A. Alghamdi and A. Althiabi, Heterogeneous population dynamics of active particles: Progression, mutations and selection dynamics, Math. Models Methods App. Sci., 27 (2017), 617-640.  doi: 10.1142/S0218202517500117.  Google Scholar

[12]

A. T. Haase, Targeting early infection to prevent HIV-1 mucosal transmission, Nature, 464 (2010), 217-223.  doi: 10.1038/nature08757.  Google Scholar

[13]

A. T. HaaseK. HenryM. ZupancicG. SedgewickR. A. FaustH. MelroeW. CavertK. GebhardK. StaskusZ. Q. ZhangP. J. DaileyH. H. BalfourA. Erice and A. S. Perelson, Quantitative image analysis of HIV-1 infection in lymphoid tissue, Science, 274 (1996), 985-989.  doi: 10.1126/science.274.5289.985.  Google Scholar

[14]

T. H. HarrisE. J. BaniganD. A. ChristianC. KonradtE. D. Tait WojnoK. NoroseE. H. WilsonB. JohnW. WeningerA. D. LusterA. J. Liu and C. A. Hunter, Generalized Levy walks and the role of chemokines in migration of effector CD8 + T cells, Nature, 486 (2012), 545-548.  doi: 10.1038/nature11098.  Google Scholar

[15]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[16]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Diff. Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[17]

E. Jones and P. Roemer, Analysis and simulation of the three-component model of HIV dynamics, SIAM Undergraduate Research Online, 7 (2014), 89-105.  doi: 10.1137/13S012698.  Google Scholar

[18]

E. F. Keller and L. A. Segel, Initiation of slide mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.   Google Scholar

[19]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[20]

N. L. Komarova, Viral reproductive strategies: How can lytic viruses be evolutionarily competitive?, J. Theor. Biol., 249 (2007), 766-784. doi: 10.1016/j.jtbi.2007.09.013.  Google Scholar

[21]

A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883.  doi: 10.1016/j.bulm.2004.02.001.  Google Scholar

[22]

F. Lin and E. C. Butcher, T cell chemotaxis in a simple microfluidic device, Lab. Chip., 11 (2006), 1462-1469.  doi: 10.1039/B607071J.  Google Scholar

[23] M. A. Nowak, Evolutionary Dynamics. Exploring the Equations of Life, Harvard University Press, Cambridge (MA), 2006.   Google Scholar
[24]

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

[25]

N. A. Nowak and R. May, Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, 2000.  Google Scholar

[26]

N. OutadaN. VaucheletT. Akrid and M. Khaladi, From kinetic theory of multicellular systems to hyperbolic tissue equations: Asymptotic limits and computing, Math. Models Methods Appl. Sci., 26 (2016), 2709-2734.  doi: 10.1142/S0218202516500640.  Google Scholar

[27]

A. S. PerelsonA. U. NeumannM. MarkowitzJ. 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

[28]

B. Perthame, Transport Equations in Biology, Birkäuser, Basel, 2007.  Google Scholar

[29]

M. M. Porzio and V. Vespri, Holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Diff. Equations, 103 (1993), 146-178.  doi: 10.1006/jdeq.1993.1045.  Google Scholar

[30]

O. StancevicC. N. AngstmannJ. M. Murray and B. I. Henry, Turing patterns from dynamics of early HIV infection, Bull. Math. Biol., 75 (2013), 774-795.  doi: 10.1007/s11538-013-9834-5.  Google Scholar

[31]

Y. Tao and M. Winkler, A chemotaxis-haptotaxis model: The roles of porous medium diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704.  doi: 10.1137/100802943.  Google Scholar

[32]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Diff. Equations, 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019.  Google Scholar

[33]

M. J. TindallP. K. MainiS. L. Porter and J. P. Armitage, Overview of mathematical approaches to model bacterial chemotaxis Ⅰ: The single cell, Bull. Math. Biol., 70 (2008), 1525-1569.  doi: 10.1007/s11538-008-9321-6.  Google Scholar

[34]

M. J. TindallP. K. MainiS. L. Porter and J. P. Armitage, Overview of mathematical approaches to model bacterial chemotaxis Ⅱ: Bacterial populations, Bull. Math. Biol., 70 (2008), 1570-1607.  doi: 10.1007/s11538-008-9322-5.  Google Scholar

[35]

M. VerbeniO. SánchezE. MollicaI. Siegli-CachedenierA. CarletonI. GuerreroA. Ruiz i Altaba and J. Soler, Morphogenetic action through flux-limited spreading, Phys. Life Rev., 10 (2013), 457-475.  doi: 10.1016/j.plrev.2013.06.004.  Google Scholar

[36]

W. WangW. Ma and X. Lai, Repulsion effect on superinfecting virions by infected cells for virus infection dynamic model with absorption effect and chemotaxis, Nonlinear Anal. RWA, 33 (2017), 253-283.  doi: 10.1016/j.nonrwa.2016.04.013.  Google Scholar

[37]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985.  Google Scholar

[38]

X. WeiS. K. GhosnM. E. TaylorV. A. A. JohnsonE. A. EminiP. DeutschJ. D. LifsonS. BonhoefferM. A. NowakB. H. HahnM. S. Saag and G. M. Shaw, Viral dynamics in human immunodeficiency virus type 1 infection, Nature, 373 (1995), 117-122.  doi: 10.1038/373117a0.  Google Scholar

show all references

References:
[1]

R. M. AndersonR. M. May and S. Gupta, Non-linear phenomena in host-parasite interactions, Parasitology, 99 (1989), 59-79.  doi: 10.1017/S0031182000083426.  Google Scholar

[2]

J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol., 44 (1975), 331-340.  doi: 10.2307/3866.  Google Scholar

[3]

N. BellomoA. Bellouquid and N. Chouhad, From a multiscale derivation of nonlinear cross-diffusion models to Keller-Segel models in a Navier-Stokes fluid, Math. Models Methods Appl. Sci., 26 (2016), 2041-2069.  doi: 10.1142/S0218202516400078.  Google Scholar

[4]

N. Bellomo and M. Winkler, A degenerate chemotaxis system with flux limitation: Maximally extended solutions and absence of gradient blow-up, Commun. Part. Diff. Eq., 42 (2017), 436-473.  doi: 10.1080/03605302.2016.1277237.  Google Scholar

[5]

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

[6]

J. CalvoJ. CamposV. CasellesO. Sanchez and J. Soler, Flux-saturated porous media equations and applications, Surv. Math. Sciences, 2 (2015), 131-218.  doi: 10.4171/EMSS/11.  Google Scholar

[7]

D. CamposV. Méndez and S. Fedotov, The effects of distributed life cycles on the dynamics of viral infections, J. Theor. Biol., 254 (2008), 430-438.  doi: 10.1016/j.jtbi.2008.05.035.  Google Scholar

[8]

D. L. DeAngelisR. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892.   Google Scholar

[9]

O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases, John Wiley & Sons, Ltd., Chichester, 2000.  Google Scholar

[10]

V. DoceulM. HollinsheadL. van der Linden and G. L. Smith, Repulsion of superinfecting virions: A mechanism for rapid virus spread, Science, 327 (2010), 873-876.  doi: 10.1126/science.1183173.  Google Scholar

[11]

L. GibelliA. ElaiwM.-A. Alghamdi and A. Althiabi, Heterogeneous population dynamics of active particles: Progression, mutations and selection dynamics, Math. Models Methods App. Sci., 27 (2017), 617-640.  doi: 10.1142/S0218202517500117.  Google Scholar

[12]

A. T. Haase, Targeting early infection to prevent HIV-1 mucosal transmission, Nature, 464 (2010), 217-223.  doi: 10.1038/nature08757.  Google Scholar

[13]

A. T. HaaseK. HenryM. ZupancicG. SedgewickR. A. FaustH. MelroeW. CavertK. GebhardK. StaskusZ. Q. ZhangP. J. DaileyH. H. BalfourA. Erice and A. S. Perelson, Quantitative image analysis of HIV-1 infection in lymphoid tissue, Science, 274 (1996), 985-989.  doi: 10.1126/science.274.5289.985.  Google Scholar

[14]

T. H. HarrisE. J. BaniganD. A. ChristianC. KonradtE. D. Tait WojnoK. NoroseE. H. WilsonB. JohnW. WeningerA. D. LusterA. J. Liu and C. A. Hunter, Generalized Levy walks and the role of chemokines in migration of effector CD8 + T cells, Nature, 486 (2012), 545-548.  doi: 10.1038/nature11098.  Google Scholar

[15]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[16]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Diff. Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[17]

E. Jones and P. Roemer, Analysis and simulation of the three-component model of HIV dynamics, SIAM Undergraduate Research Online, 7 (2014), 89-105.  doi: 10.1137/13S012698.  Google Scholar

[18]

E. F. Keller and L. A. Segel, Initiation of slide mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.   Google Scholar

[19]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[20]

N. L. Komarova, Viral reproductive strategies: How can lytic viruses be evolutionarily competitive?, J. Theor. Biol., 249 (2007), 766-784. doi: 10.1016/j.jtbi.2007.09.013.  Google Scholar

[21]

A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883.  doi: 10.1016/j.bulm.2004.02.001.  Google Scholar

[22]

F. Lin and E. C. Butcher, T cell chemotaxis in a simple microfluidic device, Lab. Chip., 11 (2006), 1462-1469.  doi: 10.1039/B607071J.  Google Scholar

[23] M. A. Nowak, Evolutionary Dynamics. Exploring the Equations of Life, Harvard University Press, Cambridge (MA), 2006.   Google Scholar
[24]

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

[25]

N. A. Nowak and R. May, Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, 2000.  Google Scholar

[26]

N. OutadaN. VaucheletT. Akrid and M. Khaladi, From kinetic theory of multicellular systems to hyperbolic tissue equations: Asymptotic limits and computing, Math. Models Methods Appl. Sci., 26 (2016), 2709-2734.  doi: 10.1142/S0218202516500640.  Google Scholar

[27]

A. S. PerelsonA. U. NeumannM. MarkowitzJ. 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

[28]

B. Perthame, Transport Equations in Biology, Birkäuser, Basel, 2007.  Google Scholar

[29]

M. M. Porzio and V. Vespri, Holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Diff. Equations, 103 (1993), 146-178.  doi: 10.1006/jdeq.1993.1045.  Google Scholar

[30]

O. StancevicC. N. AngstmannJ. M. Murray and B. I. Henry, Turing patterns from dynamics of early HIV infection, Bull. Math. Biol., 75 (2013), 774-795.  doi: 10.1007/s11538-013-9834-5.  Google Scholar

[31]

Y. Tao and M. Winkler, A chemotaxis-haptotaxis model: The roles of porous medium diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704.  doi: 10.1137/100802943.  Google Scholar

[32]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Diff. Equations, 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019.  Google Scholar

[33]

M. J. TindallP. K. MainiS. L. Porter and J. P. Armitage, Overview of mathematical approaches to model bacterial chemotaxis Ⅰ: The single cell, Bull. Math. Biol., 70 (2008), 1525-1569.  doi: 10.1007/s11538-008-9321-6.  Google Scholar

[34]

M. J. TindallP. K. MainiS. L. Porter and J. P. Armitage, Overview of mathematical approaches to model bacterial chemotaxis Ⅱ: Bacterial populations, Bull. Math. Biol., 70 (2008), 1570-1607.  doi: 10.1007/s11538-008-9322-5.  Google Scholar

[35]

M. VerbeniO. SánchezE. MollicaI. Siegli-CachedenierA. CarletonI. GuerreroA. Ruiz i Altaba and J. Soler, Morphogenetic action through flux-limited spreading, Phys. Life Rev., 10 (2013), 457-475.  doi: 10.1016/j.plrev.2013.06.004.  Google Scholar

[36]

W. WangW. Ma and X. Lai, Repulsion effect on superinfecting virions by infected cells for virus infection dynamic model with absorption effect and chemotaxis, Nonlinear Anal. RWA, 33 (2017), 253-283.  doi: 10.1016/j.nonrwa.2016.04.013.  Google Scholar

[37]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985.  Google Scholar

[38]

X. WeiS. K. GhosnM. E. TaylorV. A. A. JohnsonE. A. EminiP. DeutschJ. D. LifsonS. BonhoefferM. A. NowakB. H. HahnM. S. Saag and G. M. Shaw, Viral dynamics in human immunodeficiency virus type 1 infection, Nature, 373 (1995), 117-122.  doi: 10.1038/373117a0.  Google Scholar

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