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The basic reproduction number of discrete SIR and SEIS models with periodic parameters

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  • Seasonal fluctuations have been observed in many infectious diseases. Discrete epidemic models with periodic epidemiological parameters are formulated and studied to take into account seasonal variations of infectious diseases. The definition and the formula of the basic reproduction number $R_0$ are given by following the framework in [1,2,3,4,5]. Threshold results for a general model are obtained which show that the magnitude of $R_0$ determines whether the disease will go extinct (when $R_0<1$) or not (when $R_0>1$) in the population. Applications of these general results to discrete periodic SIR and SEIS models are demonstrated. The disease persistence and the existence of the positive periodic solution are established. Numerical explorations of the model properties are also presented via several examples including the calculations of the basic reproduction number, conditions for the disease extinction or persistence, and the existence of periodic solutions as well as its stability.
    Mathematics Subject Classification: Primary: 39A10; Secondary: 92D30.


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