Dynamics analysis of an SVEIR epidemic model in a patchy environment

Maoxing Liu; Yuhang Li; Maoxing Liu; Yuhang Li

doi:10.3934/mbe.2023756

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 9: 16962-16977. doi: 10.3934/mbe.2023756

Previous Article Next Article

Research article

Dynamics analysis of an SVEIR epidemic model in a patchy environment

Maoxing Liu ^{1,2
,
,},
Yuhang Li ²

1.
College of Science, Beijing University of Civil Engineering and Architecture, Beijing 102616, China
2.
School of Mathematics, North University of China, Taiyuan 030051, China
Academic editor: Hao Wang

Received: 09 June 2023 Revised: 01 August 2023 Accepted: 14 August 2023 Published: 28 August 2023

In this paper, we propose a multi-patch SVEIR epidemic model that incorporates vaccination of both newborns and susceptible populations. We determine the basic reproduction number $ R_{0} $ and prove that the disease-free equilibrium $ P_{0} $ is locally and globally asymptotically stable if $ R_{0} < 1, $ and it is unstable if $ R_{0} > 1. $ Moreover, we show that the disease is uniformly persistent in the population when $ R_{0} > 1. $ Numerical simulations indicate that vaccination strategies can effectively control disease spread in all patches while population migration can either intensify or prevent disease transmission within a patch.
- patchy environment,
- vaccination,
- migration,
- asymptotically stable,
- basic reproduction number
Citation: Maoxing Liu, Yuhang Li. Dynamics analysis of an SVEIR epidemic model in a patchy environment[J]. Mathematical Biosciences and Engineering, 2023, 20(9): 16962-16977. doi: 10.3934/mbe.2023756

Related Papers:

Abstract

In this paper, we propose a multi-patch SVEIR epidemic model that incorporates vaccination of both newborns and susceptible populations. We determine the basic reproduction number $ R_{0} $ and prove that the disease-free equilibrium $ P_{0} $ is locally and globally asymptotically stable if $ R_{0} < 1, $ and it is unstable if $ R_{0} > 1. $ Moreover, we show that the disease is uniformly persistent in the population when $ R_{0} > 1. $ Numerical simulations indicate that vaccination strategies can effectively control disease spread in all patches while population migration can either intensify or prevent disease transmission within a patch.

References

[1]	A. L. Lloyd, R. M. May, Spatial heterogeneity in epidemic models, J. Theor. Biol., 179 (1996), 1–11. https://doi.org/10.1006/jtbi.1996.0042 doi: 10.1006/jtbi.1996.0042
[2]	D. J. Rodríguez, L. Torres-Sorando, Models of infectious diseases in spatially heterogeneous environments, Bull. Math. Biol., 63 (2001), 547–571. https://doi.org/10.1006/bulm.2001.0231 doi: 10.1006/bulm.2001.0231
[3]	J. Arino, J. R. Davis, D. Hartley, R. Jordan, J. M. Miller, P. Van Den Driessche, A multi-species epidemic model with spatial dynamics, Math. Med. Biol., 22 (2005), 129–142. https://doi.org/10.1093/imammb/dqi003 doi: 10.1093/imammb/dqi003
[4]	W. Wang, G. Mulone, Threshold of disease transmission in a patch environment, J. Math. Anal. Appl., 285 (2003), 321–335. https://doi.org/10.1016/S0022-247X(03)00428-1 doi: 10.1016/S0022-247X(03)00428-1
[5]	C. Wolf, M. Langlais, F. Sauvage, D. Pontier, A multi-patch epidemic model with periodic demography, direct and indirect transmission and variable maturation rate, Math. Popul. Stud., 13 (2006), 153–177. https://doi.org/10.1080/08898480600788584 doi: 10.1080/08898480600788584
[6]	D. Gao, S. Ruan, A multipatch malaria model with logistic growth populations, SIAM J. Appl. Math., 72 (2012), 819–841. https://doi.org/10.1137/110850761 doi: 10.1137/110850761
[7]	Q. Liu, D. Jiang, Global dynamical behavior of a multigroup SVIR epidemic model with Markovian switching, Int. J. Biomath., 15 (2022), 2150080. https://doi.org/10.1142/S1793524521500807 doi: 10.1142/S1793524521500807
[8]	Z. Qiu, Q. Kong, X. Li, M. Martcheva, The vector–host epidemic model with multiple strains in a patchy environment, J. Math. Anal. Appl., 405 (2013), 12–36. https://doi.org/10.1016/j.jmaa.2013.03.042 doi: 10.1016/j.jmaa.2013.03.042
[9]	Y. Chen, M. Yan, Z. Xiang, Transmission dynamics of a two-city SIR epidemic model with transport-related infections, J. Appl. Math., 2014 (2014). https://doi.org/10.1155/2014/764278
[10]	H. L. Li, L. Zhang, Z. Teng, Y. L. Jiang, A. Muhammadhaji, Global stability of an SI epidemic model with feedback controls in a patchy environment, Appl. Math. Comput., 321 (2018), 372–384. https://doi.org/10.1016/j.amc.2017.10.057 doi: 10.1016/j.amc.2017.10.057
[11]	V. P. Bajiya, J. P. Tripathi, V. Kakkar, Y. Kang, Modeling the impacts of awareness and limited medical resources on the epidemic size of a multi-group SIR epidemic model, Int. J. Biomath., 15 (2022), 2250045. https://doi.org/10.1142/S1793524522500450 doi: 10.1142/S1793524522500450
[12]	H. W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci., 28 (1976), 335–356. https://doi.org/10.1016/0025-5564(76)90132-2 doi: 10.1016/0025-5564(76)90132-2
[13]	S. Ruan, W. Wang, S. A. Levin, The effect of global travel on the spread of SARS, Math. Biosci. Eng., 3 (2006), 205. https://doi.org/10.3934/mbe.2006.3.205 doi: 10.3934/mbe.2006.3.205
[14]	W. Wang, X. Q. Zhao, An epidemic model in a patchy environment, Math. Biosci., 190 (2004), 97–112. https://doi.org/10.1016/j.mbs.2002.11.001 doi: 10.1016/j.mbs.2002.11.001
[15]	M. Salmani, P. van den Driessche, A model for disease transmission in a patchy environment, Discrete Contin. Dyn. Syst. Ser. B., 6 (2006), 185–202. https://doi.org/10.3934/dcdsb.2006.6.185 doi: 10.3934/dcdsb.2006.6.185
[16]	M. Y. Li, Z. Shuai, Global stability of an epidemic model in a patchy environment, Canad. Appl. Math. Q., 17 (2009), 175–187. https://doi.org/10.1016/j.amc.2017.10.057 doi: 10.1016/j.amc.2017.10.057
[17]	D. Gao, Travel frequency and infectious diseases, SIAM J. Appl. Math., 79 (2019), 1581–1606. https://doi.org/10.1137/18M1211957 doi: 10.1137/18M1211957
[18]	X. Wang, S. Liu, L. Wang, W. Zhang, An epidemic patchy model with entry–exit screening, Bull. Math. Biol., 77 (2015), 1237–1255. https://doi.org/10.1007/s11538-015-0084-6 doi: 10.1007/s11538-015-0084-6
[19]	M. El Hajji, A. H. Albargi, A mathematical investigation of an "SVEIR" epidemic model for the measles transmission, Math. Biosci. Eng., 19 (2022), 2853–2875. https://doi.org/10.3934/mbe.2022131 doi: 10.3934/mbe.2022131
[20]	Z. Wang, Q. Zhang, Optimal vaccination strategy for a mean-field stochastic susceptible-infected-vaccinated system, Int. J. Biomath., 16 (2023), 2250061. https://doi.org/10.1142/S1793524522500619 doi: 10.1142/S1793524522500619
[21]	X. Liu, Y. Takeuchi, S. Iwami, SVIR epidemic models with vaccination strategies, J. Theor. Biol., 253 (2008), 1–11. https://doi.org/10.1016/j.jtbi.2007.10.014 doi: 10.1016/j.jtbi.2007.10.014
[22]	J. Li, Y. Yang, SIR-SVS epidemic models with continuous and impulsive vaccination strategies, J. Theor. Biol., 280 (2011), 108–116. https://doi.org/10.1016/j.jtbi.2011.03.013 doi: 10.1016/j.jtbi.2011.03.013
[23]	X. Duan, S. Yuan, Z. Qiu, J. Ma, Global stability of an SVEIR epidemic model with ages of vaccination and latency, Comput. Math. Appl., 68 (2014), 288–308. https://doi.org/10.1016/j.camwa.2014.06.002 doi: 10.1016/j.camwa.2014.06.002
[24]	L. M. Cai, Z. Li, X. Song, Global analysis of an epidemic model with vaccination, J. Appl. Math. Comput., 57 (2018), 605–628. https://doi.org/10.1007/s12190-017-1124-1 doi: 10.1007/s12190-017-1124-1
[25]	Q. Cui, Z. Qiu, L. Ding, An SIR epidemic model with vaccination in a patchy environment, Math. Biosci. Eng., 14 (2017), 1141–1157. https://doi.org/10.3934/mbe.2017059 doi: 10.3934/mbe.2017059
[26]	M. De la Sen, A. Ibeas, S. Alonso-Quesada, R. Nistal, On a SIR model in a patchy environment under constant and feedback decentralized controls with asymmetric parameterizations, Symmetry, 11 (2019), 430. https://doi.org/10.3390/sym11030430 doi: 10.3390/sym11030430
[27]	M. De la Sen, A. Ibeas, S. Alonso-Quesada, R. Nistal, Infectious diseases of humans: dynamics and control, Oxford University Press, 1991. https://doi.org/10.1016/0169-5347(91)90048-3
[28]	L. Zhang, Z. C. Wang, X. Q. Zhao, Threshold dynamics of a time periodic reaction–diffusion epidemic model with latent period, J. Differ. Equ., 258 (2015), 3011–3036. https://doi.org/10.1016/j.jde.2014.12.032 doi: 10.1016/j.jde.2014.12.032
[29]	Y. Lou, X. Q. Zhao, A reaction–diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543–568. https://doi.org/10.1007/s00285-010-0346-8 doi: 10.1007/s00285-010-0346-8
[30]	X. F. San, Z. C. Wang, Traveling waves for a two-group epidemic model with latent period in a patchy environment, J. Math. Anal. Appl., 475 (2019), 1502–1531. https://doi.org/10.1016/j.jmaa.2019.03.029 doi: 10.1016/j.jmaa.2019.03.029
[31]	J. Li, X. Zou, Dynamics of an epidemic model with non-local infections for diseases with latency over a patchy environment, J. Math. Biol., 60 (2010), 645–686. https://doi.org/10.1007/s00285-009-0280-9 doi: 10.1007/s00285-009-0280-9
[32]	A. Berman, R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979. https://doi.org/10.1016/C2013-0-10361-3
[33]	P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48. https://doi.org/10.1016/S0025-5564(02)00108-6 doi: 10.1016/S0025-5564(02)00108-6
[34]	J. P. LaSalle, The Stability of Dynamical Systems, SIAM, 1976. https://doi.org/10.1137/1.9781611970432
[35]	X. Q. Zhao, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflows with applications, Canad. Appl. Math. Q., 3 (1995), 473–495.
[36]	H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407–435. https://doi.org/10.1137/0524026 doi: 10.1137/0524026

Reader Comments

Your name:*

Email:*
© 2023 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)