Global behavior of a multi-group SEIR epidemic model with age structure and spatial diffusion

Pengyan Liu; Hong-Xu Li; Pengyan Liu; Hong-Xu Li

doi:10.3934/mbe.2020372

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 6: 7248-7273. doi: 10.3934/mbe.2020372

Previous Article Next Article

Research article

Global behavior of a multi-group SEIR epidemic model with age structure and spatial diffusion

Pengyan Liu ,
Hong-Xu Li ^,

College of Mathematics, Sichuan University, Chengdu, Sichuan 610065, China

Received: 15 August 2020 Accepted: 16 October 2020 Published: 23 October 2020

Different epidemic models with one or two characteristics of multi-group, age structure and spatial diffusion have been proposed, but few models take all three into consideration. In this paper, a novel multi-group SEIR epidemic model with both age structure and spatial diffusion is constructed for the first time ever to study the transmission dynamics of infectious diseases. We first analytically study the positivity, boundedness, existence and uniqueness of solution and the existence of compact global attractor of the associated solution semiflow. Based on some assumptions for parameters, we then show that the disease-free steady state is globally asymptotically stable by utilizing appropriate Lyapunov functionals and the LaSalle's invariance principle. By means of Perron-Frobenius theorem and graph-theoretical results, the existence and global stability of endemic steady state are ensured under appropriate conditions. Finally, feasibility of main theoretical results is showed with the aid of numerical examples for model with two groups which is important from the viewpoint of applications.
- global stability,
- multi-group epidemic model,
- age structure,
- spatial diffusion
Citation: Pengyan Liu, Hong-Xu Li. Global behavior of a multi-group SEIR epidemic model with age structure and spatial diffusion[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 7248-7273. doi: 10.3934/mbe.2020372

Related Papers:

Abstract

Different epidemic models with one or two characteristics of multi-group, age structure and spatial diffusion have been proposed, but few models take all three into consideration. In this paper, a novel multi-group SEIR epidemic model with both age structure and spatial diffusion is constructed for the first time ever to study the transmission dynamics of infectious diseases. We first analytically study the positivity, boundedness, existence and uniqueness of solution and the existence of compact global attractor of the associated solution semiflow. Based on some assumptions for parameters, we then show that the disease-free steady state is globally asymptotically stable by utilizing appropriate Lyapunov functionals and the LaSalle's invariance principle. By means of Perron-Frobenius theorem and graph-theoretical results, the existence and global stability of endemic steady state are ensured under appropriate conditions. Finally, feasibility of main theoretical results is showed with the aid of numerical examples for model with two groups which is important from the viewpoint of applications.

References

[1]	W. O. Kermack, A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. London, Ser. A, 115 (1927), 700-721. doi: 10.1098/rspa.1927.0118
[2]	P. A. Naik, Global dynamics of a fractional-order SIR epidemic model with memory, Int. J. Biomath., 13 (2020), 1-24.
[3]	P. A. Naik, J. Zu, M. Ghoreishi, Stability analysis and approximate solution of SIR epidemic model with Crowley-Martin type functional response and holling type-II treatment rate by using homotopy analysis method, J. Appl. Anal. Comput., 10 (2020), 1482-1515.
[4]	P. A. Naik, J. Zu, K. M. Owolabi, Global dynamics of a fractional order model for the transmission of HIV epidemic with optimal control, Chaos Solitons Fractals, 138 (2020), 109826.
[5]	P. A. Naik, J. Zu, K. M. Owolabi, Modeling the mechanics of viral kinetics under immune control during primary infection of HIV-1 with treatment in fractional order, Phys. A, 545 (2020), 123816.
[6]	J. Zu, M. Li, Y. Gu, S. Fu, Modelling the evolutionary dynamics of host resistance-related traits in a susceptible-infected community with density-dependent mortality, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 3049-3086.
[7]	J. Zu, G. Zhuang, P. Liang, F. Cui, F. Wang, H. Zheng, et al., Estimating age-related incidence of HBsAg seroclearance in chronic hepatitis B virus infections of China by using a dynamic compartmental model, Sci. Rep., 7 (2017), 2912.
[8]	J. Zu, M. Li, G. Zhuang, P. Liang, F. Cui, F. Wang, et al., Estimating the impact of test-and-treat strategies on hepatitis B virus infection in China by using an age-structured mathematical model, Medicine, 97 (2018), e0484.
[9]	A. Chekroun, M. N. Frioui, T. Kuniya, T. M. Touaoula, Global stability of an age-structured epidemic model with general Lyapunov functional, Math. Biosci. Eng., 16 (2019), 1525-1553. doi: 10.3934/mbe.2019073
[10]	Z. Feng, W. Huang, C. Castillo-Chavez, Global behavior of a multi-group SIS epidemic model with age structure, J. Differ. Equations, 218 (2005), 292-324. doi: 10.1016/j.jde.2004.10.009
[11]	T. Kuniya, Global stability analysis with a discretization approach for an age-structured multigroup SIR epidemic model, Nonlinear Anal. Real World Appl., 12 (2011), 2640-2655. doi: 10.1016/j.nonrwa.2011.03.011
[12]	J. Wang, H. Shu, Global analysis on a class of multi-group SEIR model with latency and relapse, Math. Biosci. Eng., 13 (2016), 209-225. doi: 10.3934/mbe.2016.13.209
[13]	J. Yang, R. Xu, X. Luo, Dynamical analysis of an age-structured multi-group SIVS epidemic model, Math. Biosci. Eng., 16 (2019), 636-666. doi: 10.3934/mbe.2019031
[14]	R. S. Cantrell, C. Cosner, Spatial ecology via reaction-diffusion equations, Wiley Series in Mathematical and Computational Biology, John Wiley and Sons Ltd, Chichester, 2004.
[15]	J. Wu, Theory and applications of partial functional differential equations, Springer-Verlag, New York, 1996.
[16]	A. Chekroun, T. Kuniya, An infection age-space structured SIR epidemic model with Neumann boundary condition, Appl. Anal., 1-14.
[17]	R. Cui, Y. Lou, A spatial SIS model in advective heterogeneous environments, J. Differ. Equations, 261 (2016), 3305-3343. doi: 10.1016/j.jde.2016.05.025
[18]	A. Ducrot, P. Magal, S. Ruan, Travelling wave solutions in multigroup age-structured epidemic models, Arch. Ration. Mech. Anal., 195 (2010), 311-331. doi: 10.1007/s00205-008-0203-8
[19]	W. E. Fitzgibbon, J. J. Morgan, G. F. Webb, Y. Wu, A vector-host epidemic model with spatial structure and age of infection, Nonlinear Anal. Real World Appl., 41 (2018), 692-705. doi: 10.1016/j.nonrwa.2017.11.005
[20]	T. Kuniya, R. Oizumi, Existence result for an age-structured SIS epidemic model with spatial diffusion, Nonlinear Anal. Real World Appl., 23 (2015), 196-208. doi: 10.1016/j.nonrwa.2014.10.006
[21]	Y. Luo, S. Tang, Z. Teng, Global dynamics in a reaction-diffusion multi-group SIR epidemic model with nonlinear incidence, Nonlinear Anal. Real World Appl., 50 (2019), 365-385. doi: 10.1016/j.nonrwa.2019.05.008
[22]	X. Wang, X. Q. Zhao, J. Wang, A cholera epidemic model in a spatiotemporally heterogeneous environment, J. Math. Anal. Appl., 468 (2018), 893-912. doi: 10.1016/j.jmaa.2018.08.039
[23]	J. Yang, Z. Jin, F. Xu, Threshold dynamics of an age-space structured SIR model on heterogeneous environment, Appl. Math. Lett., 96 (2019), 69-74. doi: 10.1016/j.aml.2019.03.009
[24]	J. Yang, R. Xu, J. Li, Threshold dynamics of an age-space structured brucellosis disease model with Neumann boundary condition, Nonlinear Anal. Real World Appl., 50 (2019), 192-217. doi: 10.1016/j.nonrwa.2019.04.013
[25]	L. Zhao, Z. C. Wang, S. Ruan, Traveling wave solutions in a two-group SIR epidemic model with constant recruitment, J. Math. Biol., 77 (2018), 1871-1915. doi: 10.1007/s00285-018-1227-9
[26]	L. Liu, J. Wang, X. Liu, Global stability of an SEIR epidemic model with age-dependent latency and relapse, Nonlinear Anal. Real World Appl., 24 (2015), 18-35. doi: 10.1016/j.nonrwa.2015.01.001
[27]	L. Liu, X. Feng, A multigroup SEIR epidemic model with age-dependent latency and relapse, Math. Methods Appl. Sci., 41 (2018), 6814-6833. doi: 10.1002/mma.5193
[28]	X. Ren, Y. Tian, L. Liu, X. Liu, A reaction-diffusion within-host HIV model with cell-to-cell transmission, J. Math. Biol., 76 (2018), 1831-1872. doi: 10.1007/s00285-017-1202-x
[29]	H. L. Smith, Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, vol. 41 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1995.
[30]	K. Yosida, Functional analysis, Springer, New York, 1978.
[31]	Y. Lou, X. Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568. doi: 10.1007/s00285-010-0346-8
[32]	J. K. Hale, Asymptotic behavior of dissipative systems, American Mathematical Society, Providence, RI, 1988.
[33]	A. Berman, R. J. Plemmons, Nonnegative matrices in mathematical sciences, Academic Press, New York, 1979.
[34]	M. A. Krasnosel'skiǐ, Positive solutions of operator equations, Noordhoff, Groningen, 1964.
[35]	M. Y. Li, Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differ. Equations, 248 (2010), 1-20. doi: 10.1016/j.jde.2009.09.003

Reader Comments

Your name:*

Email:*
© 2020 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)