Existence of positive periodic solutions of an SEIR model with periodic coefficients.
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.
scientific article; zbMATH DE number 6147977