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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.
MSC:
34C60Qualitative investigation and simulation of models (ODE)
34C25Periodic solutions of ODE
92D30Epidemiology
47N20Applications of operator theory to differential and integral equations
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Full Text: DOI
References:
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