Global Existence of a Virus Infection Model with Saturated Chemotaxis

Ruijing Li

doi:doi:10.11648/j.ijssam.20210602.12

| Peer-Reviewed

Global Existence of a Virus Infection Model with Saturated Chemotaxis

Ruijing Li

Published in International Journal of Systems Science and Applied Mathematics (Volume 6, Issue 2)

Received: 26 April 2021 Accepted: 11 May 2021 Published: 27 May 2021

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

In this paper, a virus infection model with saturated chemotaxis is formulated and analyzed, where the chemotactic sensitivity for chemotactic movements of the cells is described. This model contains three state variables namely the population density of uninfected cells, the population density of infected cells and the concentration of virus particles, respectively. By virtue of regularized approximation technique and fixed point theorem, the local solvability of the regularized system corresponding to the original system is established. Then by extracting a suitable sequence along which the respective approximate solutions approach a limit in convenient topologies, with addition of Gagliardo-Nirenberg interpolation inequality as well as L^p-estimate techniques, we show that the original system describing the virus infection model exists at least one global weak solution. To illustrate the application of our theoretical results, an optimal control problem of the epidemic system is considered, where the admissible control domain is assumed to be a bounded closed convex subset. With the help of Aubin compactness theorem and lower semicontinuous of the cost functional, the existence of the optimal pair is proved. Our results generalize and improve partial previously known ones, and moreover, we first prove that the optimal control problem has at least one optimal pair.

Published in	International Journal of Systems Science and Applied Mathematics (Volume 6, Issue 2)
DOI	10.11648/j.ijssam.20210602.12
Page(s)	40-54
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Virus Infection, Global Existence, Chemotaxis, Optimal Control

References

[1]	X. Bai, M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016) 553-583.
[2]	V. Barbu, Nonlinear Differential Equations of Monotone Type in Banach Spaces, Springer-Verlag, Berlin, 2010.
[3]	N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015) 1663-1763.
[4]	H. Chang, A. Astolfi, Control of HIV infection dynamics, IEEE Control Syst., 28 (2008) 28-39.
[5]	M. Ghergu, V. Rˇ adulescu, Nonlinear PDEs: Mathematical Models in Biology, Chemistry and Population Genetics, Springer Monographs in Mathematics, Springer, Heidelberg, 2012.
[6]	M. Herrero, J. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997) 633-683.
[7]	H. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (4) (2000) 599-653.
[8]	T. Hillen, K. Painter, A user¡¯s guide to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009) 183-217.
[9]	D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences I, Jahresber. Deutsch. Math.-Verein., 105 (2003) 103-165.
[10]	D. Horstmann, M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005) 52-107.
[11]	B. Hu, J. Lankeit, Boundedness of solutions to a virus infection model with saturated chemotaxis, J. Math. Anal. Appl., 468 (2018) 344-358.
[12]	E. Keller, L. Segel, Initiation of slime mold aggregation viewed as an instability. J. Theoret. Biol., 26 (1970) 399- 415.
[13]	D. Kirschner, S. Lenhart, S. Serbin, Optimal control of the chemotherapy of HIV, J. Math. Biol., 35 (1997) 775- 792.
[14]	J. Lankeit, Long-term behaviour in a chemotaxis-fluid system with logistic source, Math. Models Methods Appl. Sci., 26 (11) (2016) 2071-2109.
[15]	B. Liu, G. Ren, Global existence and asymptotic behavior in a three-dimensional two-species chemotaxis-Stokes system with tensor-valued sensitivity, Journal of the Korean Mathematical Society, 57 (1) (2020) 215-247.
[16]	H. Liu, Optimal control problems with state constraint governed by Navier-Stokes equations, Nonlinear Anal., 73 (2010) 3924-3939.
[17]	H. Liu, Optimal control of fluid dynamic systems with state constraint of pointwise type, Nonlinear Anal., 93 (2013) 97-108.
[18]	M. Nowak, R. May, Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, UK, 2000.
[19]	G. Ren, B. Liu, Boundedness in a chemotaxis system under a critical parameter condition, Bull. Braz. Math. Soc., New Series, (2020) In press.
[20]	G. Ren, B. Liu, Boundedness of solutions for a quasilinear chemotaxis-haptotaxis model, Hakkaido Mathematical Journal, (2019) Accepted.
[21]	G. Ren, B. Liu, Global boundedness and asymptotic behavior in a two-species chemotaxis-competition system with two signals, Nonlinear Analysis: Real World Applications, 48 (2019) 288-325.
[22]	G. Ren, B. Liu, Global boundedness of solutions to a chemotaxis-fluid system with singular sensitivity and logistic source, Commun. Pure Appl. Anal., 19 (7) (2020) 3843-3883.
[23]	G. Ren, B. Liu, Global dynamics for an attraction- repulsion chemotaxis model with logistic source, J. Differential Equations, 268 (8) (2020) 4320-4373.
[24]	G. Ren, B. Liu, Global existence and asymptotic behavior in a two-species chemotaxis system with logistic source, J. Differential Equations, 269 (2) (2020) 1484-1520.
[25]	G. Ren, B. Liu, Global existence of bounded solutions for a quasilinear chemotaxis system with logistic source, Nonlinear Analysis: Real World Applications, 46 (2019) 545-582.
[26]	O. Stancevic, C. Angstmann, J. Murray, B.I. Henry, Turing patterns from dynamics of early HIV infection, Bull. Math. Biol., 75 (5) (2013) 774-795.
[27]	R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Stud. Math. Appl., North-Holland, Amsterdam, 1977.
[28]	W. Wang, W. Ma, X. Lai, A diffusive virus infection dynamic model with nonlinear functional response, absorption effect and chemotaxis, Commun. Nonlinear Sci. Numer. Simul., 42 (2017) 585-606.
[29]	W. Wang, W. Ma, X. Lai, Repulsion effect on superinfecting virions by infected cells for virus infection dynamic model with absorption effect and chemotaxis, Nonlinear Analysis: Real World Applications, 33 (2017) 253-283.
[30]	Y. Wang, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with subcritical sensitivity, Math. Models Methods Appl. Sci., 27 (2017) 2745-2780.
[31]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010) 2889-2905.
[32]	M. Winkler, Global large-data solutions in a chemotaxis- (Navier-)Stokes system modeling cellular swimming in fluid drops, Commun. Partial Differ. Equ., 37 (2012) 319-352.
[33]	M. Winkler, Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities, SIAM J. Math. Anal., 47 (4) (2015) 3092-3115.
[34]	H. Xiang, B. Liu, Solving the inverse problem of an SIS epidemic reaction-diffusion model by optimal control methods, Comput. Math. Appl., 70 (2015) 805-819.
[35]	H. Xiang, B. Liu, Z. Fang, Optimal control strategies for a new ecosystem governed by reaction-diffusion equations, J. Math. Anal. Appl., 467 (2018) 270-291.
[36]	M. Zhou, H. Xiang, Z. Li, Optimal control strategies for a reaction-diffusion epidemic system, Nonlinear Analysis: Real World Applications, 46 (2019) 446-464.

Cite This Article

Plain Text BibTeX RIS

APA Style

Ruijing Li. (2021). Global Existence of a Virus Infection Model with Saturated Chemotaxis. International Journal of Systems Science and Applied Mathematics, 6(2), 40-54. https://doi.org/10.11648/j.ijssam.20210602.12

Copy | Download

ACS Style

Ruijing Li. Global Existence of a Virus Infection Model with Saturated Chemotaxis. Int. J. Syst. Sci. Appl. Math. 2021, 6(2), 40-54. doi: 10.11648/j.ijssam.20210602.12

Copy | Download

AMA Style

Ruijing Li. Global Existence of a Virus Infection Model with Saturated Chemotaxis. Int J Syst Sci Appl Math. 2021;6(2):40-54. doi: 10.11648/j.ijssam.20210602.12

Copy | Download

@article{10.11648/j.ijssam.20210602.12,
  author = {Ruijing Li},
  title = {Global Existence of a Virus Infection Model with Saturated Chemotaxis},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {6},
  number = {2},
  pages = {40-54},
  doi = {10.11648/j.ijssam.20210602.12},
  url = {https://doi.org/10.11648/j.ijssam.20210602.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20210602.12},
  abstract = {In this paper, a virus infection model with saturated chemotaxis is formulated and analyzed, where the chemotactic sensitivity for chemotactic movements of the cells is described. This model contains three state variables namely the population density of uninfected cells, the population density of infected cells and the concentration of virus particles, respectively. By virtue of regularized approximation technique and fixed point theorem, the local solvability of the regularized system corresponding to the original system is established. Then by extracting a suitable sequence along which the respective approximate solutions approach a limit in convenient topologies, with addition of Gagliardo-Nirenberg interpolation inequality as well as Lp-estimate techniques, we show that the original system describing the virus infection model exists at least one global weak solution. To illustrate the application of our theoretical results, an optimal control problem of the epidemic system is considered, where the admissible control domain is assumed to be a bounded closed convex subset. With the help of Aubin compactness theorem and lower semicontinuous of the cost functional, the existence of the optimal pair is proved. Our results generalize and improve partial previously known ones, and moreover, we first prove that the optimal control problem has at least one optimal pair.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Global Existence of a Virus Infection Model with Saturated Chemotaxis
AU  - Ruijing Li
Y1  - 2021/05/27
PY  - 2021
N1  - https://doi.org/10.11648/j.ijssam.20210602.12
DO  - 10.11648/j.ijssam.20210602.12
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 40
EP  - 54
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20210602.12
AB  - In this paper, a virus infection model with saturated chemotaxis is formulated and analyzed, where the chemotactic sensitivity for chemotactic movements of the cells is described. This model contains three state variables namely the population density of uninfected cells, the population density of infected cells and the concentration of virus particles, respectively. By virtue of regularized approximation technique and fixed point theorem, the local solvability of the regularized system corresponding to the original system is established. Then by extracting a suitable sequence along which the respective approximate solutions approach a limit in convenient topologies, with addition of Gagliardo-Nirenberg interpolation inequality as well as Lp-estimate techniques, we show that the original system describing the virus infection model exists at least one global weak solution. To illustrate the application of our theoretical results, an optimal control problem of the epidemic system is considered, where the admissible control domain is assumed to be a bounded closed convex subset. With the help of Aubin compactness theorem and lower semicontinuous of the cost functional, the existence of the optimal pair is proved. Our results generalize and improve partial previously known ones, and moreover, we first prove that the optimal control problem has at least one optimal pair.
VL  - 6
IS  - 2
ER  -

Copy | Download

Author Information

Ruijing Li

School of Statistics and Mathematics, Guangdong University of Finance and Economics, Guangzhou, China

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Ruijing Li. (2021). Global Existence of a Virus Infection Model with Saturated Chemotaxis. International Journal of Systems Science and Applied Mathematics, 6(2), 40-54. https://doi.org/10.11648/j.ijssam.20210602.12

Copy | Download

ACS Style

Ruijing Li. Global Existence of a Virus Infection Model with Saturated Chemotaxis. Int. J. Syst. Sci. Appl. Math. 2021, 6(2), 40-54. doi: 10.11648/j.ijssam.20210602.12

Copy | Download

AMA Style

Ruijing Li. Global Existence of a Virus Infection Model with Saturated Chemotaxis. Int J Syst Sci Appl Math. 2021;6(2):40-54. doi: 10.11648/j.ijssam.20210602.12

Copy | Download

@article{10.11648/j.ijssam.20210602.12,
  author = {Ruijing Li},
  title = {Global Existence of a Virus Infection Model with Saturated Chemotaxis},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {6},
  number = {2},
  pages = {40-54},
  doi = {10.11648/j.ijssam.20210602.12},
  url = {https://doi.org/10.11648/j.ijssam.20210602.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20210602.12},
  abstract = {In this paper, a virus infection model with saturated chemotaxis is formulated and analyzed, where the chemotactic sensitivity for chemotactic movements of the cells is described. This model contains three state variables namely the population density of uninfected cells, the population density of infected cells and the concentration of virus particles, respectively. By virtue of regularized approximation technique and fixed point theorem, the local solvability of the regularized system corresponding to the original system is established. Then by extracting a suitable sequence along which the respective approximate solutions approach a limit in convenient topologies, with addition of Gagliardo-Nirenberg interpolation inequality as well as Lp-estimate techniques, we show that the original system describing the virus infection model exists at least one global weak solution. To illustrate the application of our theoretical results, an optimal control problem of the epidemic system is considered, where the admissible control domain is assumed to be a bounded closed convex subset. With the help of Aubin compactness theorem and lower semicontinuous of the cost functional, the existence of the optimal pair is proved. Our results generalize and improve partial previously known ones, and moreover, we first prove that the optimal control problem has at least one optimal pair.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Global Existence of a Virus Infection Model with Saturated Chemotaxis
AU  - Ruijing Li
Y1  - 2021/05/27
PY  - 2021
N1  - https://doi.org/10.11648/j.ijssam.20210602.12
DO  - 10.11648/j.ijssam.20210602.12
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 40
EP  - 54
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20210602.12
AB  - In this paper, a virus infection model with saturated chemotaxis is formulated and analyzed, where the chemotactic sensitivity for chemotactic movements of the cells is described. This model contains three state variables namely the population density of uninfected cells, the population density of infected cells and the concentration of virus particles, respectively. By virtue of regularized approximation technique and fixed point theorem, the local solvability of the regularized system corresponding to the original system is established. Then by extracting a suitable sequence along which the respective approximate solutions approach a limit in convenient topologies, with addition of Gagliardo-Nirenberg interpolation inequality as well as Lp-estimate techniques, we show that the original system describing the virus infection model exists at least one global weak solution. To illustrate the application of our theoretical results, an optimal control problem of the epidemic system is considered, where the admissible control domain is assumed to be a bounded closed convex subset. With the help of Aubin compactness theorem and lower semicontinuous of the cost functional, the existence of the optimal pair is proved. Our results generalize and improve partial previously known ones, and moreover, we first prove that the optimal control problem has at least one optimal pair.
VL  - 6
IS  - 2
ER  -

Copy | Download