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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, 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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[9]

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

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

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

[25]

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

[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.

[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.

[28]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[37]

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

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[9]

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

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

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

[25]

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

[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.

[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.

[28]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[37]

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

[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.

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