Hostname: page-component-669899f699-g7b4s Total loading time: 0 Render date: 2025-04-25T20:13:28.751Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic and transient dynamics of SEIR epidemic models on weighted networks

Part of: Parabolic equations and systems Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  26 April 2022

CANRONG TIAN ,
ZUHAN LIU  and
SHIGUI RUAN Open the ORCID record for SHIGUI RUAN [Opens in a new window]
CANRONG TIAN
Affiliation:
School of Mathematics and Physics, Yancheng Institute of Technology, Yancheng, Jiangsu 224003, China emails: tiancanrong@163.com; zhliu@yzu.edu.cn
ZUHAN LIU
Affiliation:
School of Mathematics and Physics, Yancheng Institute of Technology, Yancheng, Jiangsu 224003, China emails: tiancanrong@163.com; zhliu@yzu.edu.cn
SHIGUI RUAN
Affiliation:
Department of Mathematics, University of Miami, Coral Gables, FL 33146, USA email: ruan@math.miami.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the effect of population mobility on the transmission dynamics of infectious diseases by considering a susceptible-exposed-infectious-recovered (SEIR) epidemic model with graph Laplacian diffusion, that is, on a weighted network. First, we establish the existence and uniqueness of solutions to the SEIR model defined on a weighed graph. Then by constructing Liapunov functions, we show that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than unity and the endemic equilibrium is globally asymptotically stable if the basic reproduction number is greater than unity. Finally, we apply our generalized weighed graph to Watts–Strogatz network and carry out numerical simulations, which demonstrate that degrees of nodes determine peak numbers of the infectious population as well as the time to reach these peaks. It also indicates that the network has an impact on the transient dynamical behaviour of the epidemic transmission.

Keywords

Asymptotic and transient dynamicsgraph Laplacian operatorLiapunov functionnetworkSEIR model

MSC classification

Primary: 35R02: Partial differential equations on graphs and networks (ramified or polygonal spaces) 35K57: Reaction-diffusion equations
Secondary: 92D30: Epidemiology
Type
Papers
Information
European Journal of Applied Mathematics , Volume 34 , Issue 2 , April 2023 , pp. 238 - 261
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

†

This work was partially supported by NSFC grants (61877052 and 11871065), Jiangsu Province 333 Talent Project, Jiangsu Province Qinglan Project, and NSF grant (DMS-1853622).

References

Allen, L. J. S., Bolker, B. M., Lou, Y. & Nevai, A. L. (2008) Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete Contin. Dyn. Syst. 21, 1–20.CrossRefGoogle Scholar
Bauer, F., Horn, P., Lin, Y., Lippner, G., Mangoubi, D. & Yau, S. T. (2015) Li-Yau inequality on graphs. J. Differ. Geom. 99, 359–405.CrossRefGoogle Scholar
Chung, S.-Y. & Choi, M.-J. (2017) A new condition for blow-up solutions to discrete semilinear heat equations on networks. Comput. Math. Appl. 74, 2929–2939.CrossRefGoogle Scholar
Chung, Y., Lee, Y. & Chung, S. (2011) Extinction and positivity of the solutions of the heat equations with absorption on networks. J. Math. Anal. Appl. 380, 642–652.CrossRefGoogle Scholar
Du, Y., Lou, B., Peng, R. & Zhou, M. (2020) The Fisher-KPP equation over simple graphs: varied persistence states in river networks. J. Math. Biol. 80, 1559–1616.CrossRefGoogle ScholarPubMed
Fitzgibbon, W. E. & Langlais, M. (2008) Simple models for the transmission of microparasites between host populations living on non coincident spatial domain. In: P. Magal and S. Ruan (editors), Structured Population Models in Biology and Epidemiology, Lecture Notes in Mathematics, Vol. 1936, Springer-Verlag, Berlin, pp. 115–164.Google Scholar
Grigoryan, A., Lin, Y. & Yang, Y. (2016) Yamabe type equations on graphs. J. Differ. Equations 261, 4924–4943.CrossRefGoogle Scholar
Grigoryan, A., Lin, Y. & Yang, Y. (2016) Kazdan-Warner equation on graph. Calc. Var. Partial Differ. Equations 55, 92.CrossRefGoogle Scholar
Hale, J. K. (1980) Ordinary Differential Equations, Krieger, Malabar, FL.Google Scholar
Lei, C., Li, F. & Liu, J. (2018) Theoretical analysis on a diffusive SIR epidemic model with nonlinear incidence in a heterogeneous environment. Discrete Contin. Dyn. Syst. Ser. B 23, 4499–4517.Google Scholar
Li, H., Peng, R. & Wang, Z.-A. (2018) On a diffusive susceptible-infected-susceptible epidemic model with mass action mechanism and birth-death effect: analysis. simulations, and comparison with other mechanisms. SIAM J. Appl. Math. 78, 2129–2153.CrossRefGoogle Scholar
Li, M. & Shuai, Z. (2010) Global-stability problem for coupled systems of differential equations on networks. J. Differ. Equations 248, 1–20.CrossRefGoogle Scholar
Lin, Z. & Zhu, H. (2017) Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary. J. Math. Biol. 75, 1381–1409.CrossRefGoogle ScholarPubMed
Magal, P., Webb, G. F. & Wu, Y. (2019) On the basic reproduction number of reaction-diffusion epidemic models. SIAM J. Appl. Math. 79, 284–304.CrossRefGoogle Scholar
Martcheva, M. (2015) An Introduction to Mathematical Epidemiology, Springer, New York.CrossRefGoogle Scholar
Murray, J. D. (2003) Mathematical Biology II: Spatial Models and Biomedical Applications, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Pao, C. V. (1992) Nonlinear Parabolic and Elliptic Equations, Plenum, New York.Google Scholar
Pastor-Satorras, R., Castellano, C., Van Mieghem, P. & Vespignani, A. (2015) Epidemic processes in complex networks. Rev. Mod. Phys. 87, 925–986.CrossRefGoogle Scholar
Ruan, S. (2017) Spatiotemporal epidemic models for rabies among animals. Infect. Dis. Model. 2, 277–287.Google ScholarPubMed
Ruan, S. & Wu, J. (2009) Modeling spatial spread of communicable diseases involving animal hosts. In: R. S. Cantrell, C. Cosner and S. Ruan (editors), Spatial Ecology, Chapman and Hall/CRC, Boca Raton, FL, pp. 293–316.Google Scholar
Smith, H. L. (2008) Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI.CrossRefGoogle Scholar
Song, P., Lou, Y. & Xiao, Y. (2019) A spatial SEIRS reaction-diffusion model in heterogeneous environment. J. Differ. Equations 267, 5084–5114.CrossRefGoogle Scholar
Tian, C. & Ruan, S. (2019) Pattern formation and synchronism in an allelopathic plankton model with delay in a network. SIAM J. Appl. Dyn. Syst. 18, 531–557.CrossRefGoogle Scholar
World Health Organization (WHO). Coronavirus disease (COVID-19) pandemic. https://www.who.int/emergencies/diseases/novel-coronavirus-2019.Google Scholar
WHO. Coronavirus disease (COVID-2019) situation reports. https://www.who.int/emergencies/diseases/novel-coronavirus-2019/situation-reports/.Google Scholar
Yang, F., Li, W. & Ruan, S. (2019) Dynamics of a nonlocal dispersal SIS epidemic model with Neumann boundary conditions. J. Differ. Equations 267, 2011–2051.CrossRefGoogle Scholar
Zhang, H., Small, M., Fu, X., Sun, G. & Wang, B. (2012) Modeling the influence of information on the coevolution of contact networks and the dynamics of infectious diseases. Physica D 241, 1512–1517.CrossRefGoogle Scholar