Global behavior of solutions to an SI epidemic model with nonlinear diffusion in heterogeneous environment

Shenghu Xu; Xiaojuan Li; Shenghu Xu; Xiaojuan Li

doi:10.3934/math.2022377

AIMS Mathematics

2022, Volume 7, Issue 4: 6779-6791. doi: 10.3934/math.2022377

Previous Article Next Article

Research article

Global behavior of solutions to an SI epidemic model with nonlinear diffusion in heterogeneous environment

Shenghu Xu ^{1,2
,
,},
Xiaojuan Li ^{1,2
,
,}

1.
School of Mathematics and Information Sciences, North Minzu University, Yinchuan, Ningxia 750021, China
2.
College of Mathematics and Information Science, Neijiang Normal University, Neijiang, Sichuan 641112, China

Received: 28 November 2021 Revised: 07 January 2022 Accepted: 18 January 2022 Published: 26 January 2022
MSC : 35J60, 35B32, 92D25

In this paper, a nonlinear diffusion SI epidemic model with a general incidence rate in heterogeneous environment is studied. Global behavior of classical solutions under certain restrictions on the coefficients is considered. We first establish the global existence of classical solutions of the system under heterogeneous environment by energy estimate and maximum principles. Based on such estimates, we then study the large-time behavior of the solution of system under homogeneous environment. The model and mathematical results in [M. Kirane, S. Kouachi, Global solutions to a system of strongly coupled reaction-diffusion equations, Nonlinear Anal., 26 (1996), 1387-1396.] are generalized.
- SI epidemic model,
- general incidence rate,
- cross-diffusion,
- global solution
Citation: Shenghu Xu, Xiaojuan Li. Global behavior of solutions to an SI epidemic model with nonlinear diffusion in heterogeneous environment[J]. AIMS Mathematics, 2022, 7(4): 6779-6791. doi: 10.3934/math.2022377

Related Papers:

Abstract

In this paper, a nonlinear diffusion SI epidemic model with a general incidence rate in heterogeneous environment is studied. Global behavior of classical solutions under certain restrictions on the coefficients is considered. We first establish the global existence of classical solutions of the system under heterogeneous environment by energy estimate and maximum principles. Based on such estimates, we then study the large-time behavior of the solution of system under homogeneous environment. The model and mathematical results in [M. Kirane, S. Kouachi, Global solutions to a system of strongly coupled reaction-diffusion equations, Nonlinear Anal., 26 (1996), 1387-1396.] are generalized.

References

[1]	N. T. J. Bailey, The mathematical theory of infectious diseases and its applications, J. R. Stat. Soc. C-Appl., 26 (1977), 85–87. https://doi.org/10.2307/2346882 doi: 10.2307/2346882
[2]	C. Holing, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Can., 45 (1965), 1–65. https://doi.org/10.4039/entm9745fv doi: 10.4039/entm9745fv
[3]	R. Arditi, L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio dependence, J. Theor. Biol., 139 (1989), 311–326. https://doi.org/10.1016/S0022-5193(89)80211-5 doi: 10.1016/S0022-5193(89)80211-5
[4]	J. R. Beddington, Mutual interference between parasites or predators and its effect on searhing efficiency, J. Anim. Ecodogy, 1975,331–340.
[5]	D. L. DeAngelis, R. A. Goldstein, R. V. A. O'Neill, A model for tropic interaction, Ecology, 56 (1975), 881–892. https://doi.org/10.2307/1936298 doi: 10.2307/1936298
[6]	P. Crowley, E. Martin, Functional responses and interference within and between year classes of a dragonfly population, J. North. Am. Benth. Soc., 8 (1989), 211–221. https://doi.org/10.2307/1467324 doi: 10.2307/1467324
[7]	M. Kirane, S. Kouachi, Global solutions to a system of strongly coupled reaction-diffusion equations, Nonlinear Anal., 26 (1996), 1387–1396. https://doi.org/10.1016/0362-546x(94)00337-h doi: 10.1016/0362-546x(94)00337-h
[8]	Y. Choi, R. Lui, Y. Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion, Discrete Con. Dyn., 10 (2004), 719–730. https://doi.org/10.3934/dcds.2004.10.719 doi: 10.3934/dcds.2004.10.719
[9]	P. Tuoc, On global existence of solutions to a cross-diffusion system, J. Math. Anal. Appl., 343 (2008), 826–834. https://doi.org/10.1016/j.jmaa.2008.01.089 doi: 10.1016/j.jmaa.2008.01.089
[10]	S. Fu, S. Cui, Global existence and stability of solution of a reaction-diffusion model for cancer invasion, Nonlinear Anal. RWA, 10 (2009), 1362–1369. https://doi.org/10.1016/j.nonrwa.2008.01.011 doi: 10.1016/j.nonrwa.2008.01.011
[11]	H. Xu, S. Xu, Existence and stability of global solutions for a cross-diffusion predator-prey model with sex-structure, Nonlinear Anal. RWA, 13 (2012) 999–1009. https://doi.org/10.1016/j.nonrwa.2010.08.029
[12]	S. Fu, L. Zhang, P. Hu, Global behavior of solutions in a Lotka-Volterra predator-prey model with prey-stage structure, Nonlinear Anal. RWA, 14 (2013) 2027–2045. https://doi.org/10.1016/j.nonrwa.2013.02.007
[13]	X. Li, S. Fu, Global stability of the virus dynamics model with intracellular delay and Crowley-Martin functional response, Math. Meth. Appl. Sci., 37 (2014), 1405–1411. https://doi.org/10.1002/mma.2895 doi: 10.1002/mma.2895
[14]	X. Li, S. Fu, Global stability of a virus dynamics model with intracellular delay and CTL immune response, Math. Meth. Appl. Sci., 38 (2015), 420–430. https://doi.org/10.1002/mma.3078 doi: 10.1002/mma.3078
[15]	H. Amann, Dynamic theory of quasilinear parabolic equations: Reaction-diffusion, Differ. Integral Equ., 3 (1990), 13–75. https://doi.org/10.1007/BF02571246
[16]	H. Amann, Dynamic theory of quasilinear parabolic equations: Global existence, Math. Z., 202 (1989), 219–250. https://doi.org/10.1007/BF01215256
[17]	O. A. Ladyzenskaja, V. A. Solonnikov, N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, AMS, 1968.
[18]	A. Haraux, M. Kirane, Estimations $C^{1}$ pour des problemes paraboliques semi-linéaires, Annales de la Faculte des Sciences de Toulouse, University Paul Sabatier, 1983. https://doi.org/10.5802/afst.598
[19]	M. Protter, H. Weinerger, Maximum principles in differential equations, (Section Editions), Springer-Verlag, New York, 1984. https://doi.org/10.1007/978-1-4612-5282-5
[20]	P. Pang, M. Wang, Strategy and stationary pattern in a three-species predator-prey model, J. Differ. Eq., 200 (2004), 245–273. https://doi.org/10.1016/j.jde.2004.01.004 doi: 10.1016/j.jde.2004.01.004
[21]	D. Jiang, Z. C. Wang, L. Zhang, A reaction-diffusion-advection SIS epidemic model in a spatially-temporally heterogeneous environment, Discrete Con. Dyn. Sys.-B, 23 (2018), 4557–4578. https://doi.org/10.3934/dcdsb.2018176 doi: 10.3934/dcdsb.2018176
[22]	H. Li, R. Peng, T. Xiang, Dynamics and asymptotic profiles of endemic equilibrium for two frequency-dependent SIS epidemic models with cross-diffusion, Eur. J. Appl. Math., 2017, 1–31. https://doi.org/10.1017/S0956792518000463
[23]	F. Y. Yang, W. T. Li, Dynamics of a nonlocal dispersal SIS epidemic model, Commun. Pure Appl. Anal., 16 (2017), 781–798. https://doi.org/10.3934/cpaa.2017037 doi: 10.3934/cpaa.2017037
[24]	D. Hnaien, K. Ferdaous, L. Rafika, Asymptotic behavior of global solutions of an anomalous diffusion system, J. Math. Anal. Appl., 421 (2015), 1519–1530. https://doi.org/10.1016/j.jmaa.2014.07.083 doi: 10.1016/j.jmaa.2014.07.083
[25]	Y. Wang, J. Wang, L. Zhang, Cross diffusion-induced pattern in an SI model, Appl. Math. Comput., 217 (2010), 1965–1970. https://doi.org/10.1016/j.amc.2010.06.052 doi: 10.1016/j.amc.2010.06.052
[26]	Y. Fan, Pattern formation of an epidemic model with cross diffusion, Appl. Math. Comput., 228 (2014), 311–319. https://doi.org/10.1016/j.amc.2013.11.090 doi: 10.1016/j.amc.2013.11.090
[27]	C. S. Ruoja, C. Surulescu, A. Zhigun, On a model for epidemic spread with interpopulation contact and repellent taxis, arXiv: 1902.02171, 2019.
[28]	D. Le, L. V. Nguyen, T. T. Nguyen, Shigesada-Kawasaki-Teramoto model on higher dimen-sional domains, Electron J. Differ. Eq., 72 (2003). https://doi.org/10.1023/A:1022197004856
[29]	L. T. Hoang, T. V. Nguyen, P. V. Tuoc, Gradient estimates and global existence of smooth solutions to a cross-diffusion system, SIAM J. Math. Anal., 47 (2015), 2122–2177. https://doi.org/10.1137/140981447 doi: 10.1137/140981447
[30]	P. V. Tuoc, Global existence of solutions to Shigesada-Kawasaki-Teramoto cross-diffusion systems on domains of arbitrary dimensions, Proc. Amer. Math. Soc., 135 (2007), 3933–3941. https://doi.org/10.1090/S0002-9939-07-08978-2 doi: 10.1090/S0002-9939-07-08978-2
[31]	D. Le, T. T. Nguyen, Global existence for a class of triangular parabolic systems on domains of arbitrary dimension, Proc. Amer. Math. Soc., 133 (2005), 1985–1992. https://doi.org/10.1090/S0002-9939-05-07867-6 doi: 10.1090/S0002-9939-05-07867-6
[32]	Y. Lou, W. M. Ni, Y. Wu, On the global existence of a cross-diffusion system, Discrete Cont. Dyn-A, 4 (1998), 193–203. https://doi.org/10.3934/dcds.1998.4.193 doi: 10.3934/dcds.1998.4.193
[33]	Y. Yamada, Global solutions for the Shigesada-Kawasaki-Teramoto model with cross- diffusion, Recent progress on reaction-diffusion systems and viscosity solutions, World Scientific River Edge, NJ, 2009,282–299. https://doi.org/10.1142/9789812834744_0013
[34]	Y. Tao, M. Winkler, Boundedness and stabilization in a population model with cross-diffusion for one species, Proc. London Math. Soc., 119 (2019), 1598–1632. https://doi.org/10.1112/plms.12276 doi: 10.1112/plms.12276
[35]	Y. Liu, Y. Tao, Dynamics in a parabolic elliptic two species population competition model with cross-diffusion for one species, J. Math. Anal. Appl., 456 (2017), 1–15. https://doi.org/10.1016/j.jmaa.2017.05.058 doi: 10.1016/j.jmaa.2017.05.058
[36]	A. A. Khan, R. Amin, S. Ullah, Numerical simulation of a Caputo fractional epidemic model for the novel coronavirus with the impact of environmental transmission, J. Alex. Eng., 61 (2022), 729–746. https://doi.org/10.1016/j.aej.2021.10.008 doi: 10.1016/j.aej.2021.10.008
[37]	A. Alshabanat, M. Jleli, S. Kumar, B. Samet, Generalization of Caputo-Fabrizio fractional derivative and applications to electrical circuits, Front. Phy., 8 (2020). https://doi.org/10.3389/fphy.2020.00064
[38]	P. Veeresha, D. G. Prakasha, S. Kumar, A fractional model for propagation of classical optical solitons by using nonsingular derivative, Math. Meth. Appl. Sci., 2020, 1–15. https://doi.org/10.1002/mma.6335
[39]	H. Mohammadi, S. Kuma, S. Rezapour, S. Etemad, A theoretical study of the Caputo-Fabrizio fractional modeling for hearing loss due to Mumps virus with optimal control, Chaos Soliton. Fract., 144 (2021), 110668. https://doi.org/10.1016/j.chaos.2021.110668 doi: 10.1016/j.chaos.2021.110668
[40]	E. F. D. Goufo, S. Kumar, S. B. Mugisha, Similarities in a fifth-order evolution equation with and with no singular kernel, Chaos Soliton. Fract., 130 (2020), 109467. https://doi.org/10.1016/j.chaos.2019.109467 doi: 10.1016/j.chaos.2019.109467
[41]	B. Ghanbari, S. Kumar, R. Kumar, A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative, Chaos Soliton. Fract., 133 (2020), 109619. https://doi.org/10.1016/j.chaos.2020.109619 doi: 10.1016/j.chaos.2020.109619
[42]	S. Kumar, A. Kumar, B. Samet, J. F. Gomez-Aguilar, M. S. Osman, A chaos study of tumor and effector cells in fractional tumor-immune model for cancer treatment, Chaos Soliton. Fract., 141 (2020), 110321. https://doi.org/10.1016/j.chaos.2020.110321 doi: 10.1016/j.chaos.2020.110321
[43]	S. Kumar, R. Kumar, R. P. Agarwal, B. Samet, A study of fractional Lotka-Volterra population model using Haar wavelet and Adam-Bashforth-Moulton methods, Math. Meth. Appl. Sci., 43 (2020), 5564–5578. https://doi.org/10.1002/mma.6297 doi: 10.1002/mma.6297

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)