Existence of positive periodic solutions of an SEIR model with periodic coefficients.
type: Item
instance of: scholarly article
title: Existence of positive periodic solutions of an SEIR model with periodic coefficients.
zbMATH Open document ID: 1274.34150
author: Tailei Zhang
author: Junli Liu
author: Zhidong Teng
published in: Applications of Mathematics
publication date: 2013-03-21T00:00:00Z
full work available at URL: http://hdl.handle.net/10338.dmlcz/143005
review text: The paper deals with the SEIR epidemic model \[ S'=\Lambda (t)-\beta (t)S\,I-\mu (t)S, \] \[ E'=\beta (t)S\,I-(\mu (t)+\varepsilon (t))\,E, \] \[ I'=\varepsilon (t)E-(\mu (t)+\alpha (t)+\gamma (t))\,I, \] \[ R'=\gamma (t)I-\mu (t)R, \] where \(\Lambda ,\alpha ,\beta ,\gamma ,\varepsilon ,\mu \) are positive \(T\)-periodic continuous functions. The model assumes that the population \(N\) is divided into 4 classes: \(S\) (susceptible), \(E\) (exposed), \(I\) (infective) and \(R\) (recovered). The main result provides conditions ensuring the existence of at least one positive \(T\)-periodic solution to the given problem. Main tools are the continuation theorem due to \textit{R. E. Gaines} and \textit{J. L. Mawhin} [Coincidence degree, and nonlinear differential equations. Berlin-Heidelberg-New York: Springer-Verlag (1977; Zbl 0339.47031)] and the degree theory. Furthermore, a sufficient condition for the global stability of this model is obtained and an example based on the transmission of respiratory syncytial virus (RSV) is included.
Mathematics Subject Classification ID: 34C60
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 92D30
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 6147977
zbMATH Keywords: epidemic model
zbMATH Keywords: coincidence degree
zbMATH Keywords: Fredholm mapping
reviewed by: Milan Tvrdý
MaRDI profile type: Publication
OpenAlex ID: W2008701670
cites work: Existence of periodic solutions in a model of respiratory syncytial virus RSV
cites work: On the definition and the computation of the basic reproduction ratio \(R_ 0\) in models for infectious diseases in heterogeneous populations
cites work: Q4255467
cites work: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission
cites work: Periodicity in a delayed ratio-dependent predator-prey system
cites work: Nonautonomous SEIRS and Thron models for epidemiology and cell biology
cites work: The mathematics of infectious diseases
cites work: Modeling the spread of seasonal epidemiological diseases: Theory and applications
cites work: Periodic solutions of periodic delay Lotka-Volterra equations and systems
cites work: Global stability for the SEIR model in epidemiology
cites work: Epidemic threshold conditions for seasonally forced SEIR models
cites work: Dynamics of Two-Strain Influenza with Isolation and Partial Cross-Immunity
cites work: On the periodic solutions of periodic multi-species competitive systems with delays.
cites work: Permanence and extinction of periodic predator--prey systems in a patchy environment with delay
cites work: Modeling epidemics caused by respiratory syncytial virus (RSV)
cites work: The periodic solution of a class of epidemic models
cites work: Stability of Hopf bifurcation of a delayed SIRS epidemic model with stage structure
cites work: Global dynamics of an SEIR epidemic model with saturating contact rate
cites work: On a nonautonomous SEIRS model in epidemiology
DOI: 10.1007/S10492-012-0036-5
description: scientific article; zbMATH DE number 6147977