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Existence of positive periodic solutions of an SEIR model with periodic coefficients. (English) Zbl 1274.34150
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 {\it R. E. Gaines} and {\it 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.
34C60Qualitative investigation and simulation of models (ODE)
34C25Periodic solutions of ODE
47N20Applications of operator theory to differential and integral equations
Full Text: DOI
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