Stability analysis of a reaction-diffusion HIV immune model with absorption effect. (English) Zbl 07920859
Summary: In this article, a reaction-diffusion model of HIV immunity with chemotaxis and absorption effect is constructed. The paper proves the existence and boundedness of the global classical solution of this mode when the chemotactic coefficient is kept in a suitable range. Five equilibrium points are established based on the different ranges of the basic regeneration number, two immune reproduction numbers and the immune competitive reproduction number. The global asymptotic stability of each equilibrium point is established by constructing an appropriate Lyapunov function when the chemotactic coefficient is kept in a small interval.
MSC:
| 35A09 | Classical solutions to PDEs |
| 35B35 | Stability in context of PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 35K57 | Reaction-diffusion equations |
| 92C17 | Cell movement (chemotaxis, etc.) |
Keywords:
immune response; Lyapunov function; reaction-diffusion equation; chemotaxis; global existenceReferences:
| [1] | Ahmed, N.; Rafiq, M.; Adel, W., Structure preserving numerical analysis of HIV and CD4+ T-cells reaction diffusion model in two space dimensions, Chaos Solitons Fractals, 139, 2020 · Zbl 1490.92002 · doi:10.1016/j.chaos.2020.110307 |
| [2] | Ahn, I.; Yoon, C., Global well-posedness and stability analysis of prey-predator model with indirect prey-taxis, J. Differ. Equ., 268, 4222-4255, 2020 · Zbl 1430.35120 · doi:10.1016/j.jde.2019.10.019 |
| [3] | Amann, H., Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differ. Integral. Equ., 3, 1, 13-75, 1990 · Zbl 0729.35062 |
| [4] | Anderson, RM; May, RM; Gupta, S., Non-linear phenomena in host-parasite interactions, Parasitology, 99, 59-79, 1989 · doi:10.1017/S0031182000083426 |
| [5] | Bai, X.; Winkler, M., Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 55, 553-583, 2016 · Zbl 1345.35117 |
| [6] | Bellomo, N.; Painter, KJ; Tao, Y.; Winkler, M., Occurrence versus absence of taxis-driven instabilities in a May-Nowak model for virus infection, SIAM J. Appl. Math., 79, 1990-2010, 2019 · Zbl 1428.35615 · doi:10.1137/19M1250261 |
| [7] | Bellomo, N.; Tao, Y., Stabilization in a chemotaxis model for virus infection, Discrete Contin. Dyn. Syst. Ser. S, 13, 105-117, 2020 · Zbl 1439.35052 |
| [8] | Cao, XR, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 2015 · Zbl 1515.35047 · doi:10.3934/dcds.2015.35.1891 |
| [9] | Driessche, PVD; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180, 29-48, 2002 · Zbl 1015.92036 · doi:10.1016/S0025-5564(02)00108-6 |
| [10] | Freitag, M., Global solutions to a higher-dimensional system related to crime modeling, Math. Methods Appl., 41, 6326-6335, 2018 · Zbl 1401.35203 · doi:10.1002/mma.5141 |
| [11] | Han, R.; Dai, B.; Chen, Y., Chemotaxis-driven stationary and oscillatory patterns in a diffusive HIV-1 model with CTL immune response and general sensitivity, Chaos, 33, 2023 · Zbl 1537.92122 · doi:10.1063/5.0150072 |
| [12] | Harris, TH; Banigan, EJ; Christian, DA, Generalized Levy walks and the role of chemokines in migration of effector CD8+ T cells, Nature, 486, 545-548, 2012 · doi:10.1038/nature11098 |
| [13] | Jung, MC; Pape, GR, Immunology of hepatitis B infection, Lancet Infect. Dis., 2, 43-50, 2002 · doi:10.1016/S1473-3099(01)00172-4 |
| [14] | Lai, X.; Zou, X., A reaction diffusion system modeling virus dynamics and CTLs response with chemotaxis, Disc. Cont. Dyn. Syst. B, 21, 8, 2567, 2016 · Zbl 1347.35131 · doi:10.3934/dcdsb.2016061 |
| [15] | Lieberman, G., Second order parabolic differential equations, 2003, River Edge, NJ: World Scientific, River Edge, NJ |
| [16] | Lin, F.; Butcher, EC, T cell chemotaxis in a simple microfluidic device, Lab. Chip., 11, 1462-1469, 2006 · doi:10.1039/B607071J |
| [17] | Nowak, MA; May, RM, Virus dynamics, mathematical principles of immunology and virology, 2000, Oxford: Oxford University Press, Oxford · Zbl 1101.92028 · doi:10.1093/oso/9780198504184.001.0001 |
| [18] | Pan, X.; Wang, L.; Hu, X., Boundedness and stabilization of solutions to a chemotaxis May-Nowak model, Z. Angew. Math. Phys., 72, 16, 2021 · Zbl 1466.35044 · doi:10.1007/s00033-021-01491-0 |
| [19] | Porzio, MM; Vespri, V., Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differ. Equ., 103, 146-178, 1993 · Zbl 0796.35089 · doi:10.1006/jdeq.1993.1045 |
| [20] | Rodríguez, N.; Winkler, M., On the global existence and qualitative behavior of one-dimensional solutions to a model for urban crime, Eur. J. Appl. Math., 33, 5, 1-41, 2019 |
| [21] | Short, MB; Bertozzi, AL; Brantingham, PJ, Nonlinear patterns in urban crime: hotspots, bifurcations, and suppression, SIAM J. Appl. Dyn. Syst., 9, 462-483, 2010 · Zbl 1282.91273 · doi:10.1137/090759069 |
| [22] | Stancevic, O.; Angstmann, CN; Murray, JM; Henry, BI, Turing patterns from dynamics of early HIV infection, Bull. Math. Biol., 75, 774-795, 2013 · Zbl 1273.92035 · doi:10.1007/s11538-013-9834-5 |
| [23] | Tao, Y.; Winkler, M., Global smooth solutions in a two-dimensional cross-diffusion system modeling propagation of urban crime, Commun. Math., 19, 829-849, 2021 · Zbl 1479.35120 · doi:10.4310/CMS.2021.v19.n3.a12 |
| [24] | Tao, Y.; Winkler, M., Taxis-driven formation of singular hotspots in a May-Nowak type model for virus infection, SIAM J. Math. Anal., 53, 1411-1433, 2021 · Zbl 1461.35201 · doi:10.1137/20M1362851 |
| [25] | Winkler, M., Aggregation versus global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equ., 248, 2889-2905, 2010 · Zbl 1190.92004 · doi:10.1016/j.jde.2010.02.008 |
| [26] | Winkler, M., Boundedness in a chemotaxis-May-Nowak model for virus dynamics with mildly saturated chemotactic sensitivity, Acta Appl. Math., 163, 1-17, 2019 · Zbl 1423.35172 · doi:10.1007/s10440-018-0211-0 |
| [27] | Wodarz, D., Killer cell dynamics, mathematical and computational approaches to immunology, 2007, New York: Springer, New York · Zbl 1125.92032 |
| [28] | Xu, R., Global dynamics of a delayed HIV-1 infection model with absorption and saturation infection, Int. J. Biomath., 5, 1260012, 2012 · Zbl 1280.92066 · doi:10.1142/S1793524512600121 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
