Abstract
Discrete time SI and SIS epidemic models with vertical transmission are presented in this paper. With regard to the SI model with constant or variable population size, we introduce an epidemic threshold parameter, the basic reproductive number R0, for predicting disease dynamics. R0 > 1 implies that the disease tends to an endemic equilibrium, while R0 < 1 implies disease extinction. On the other hand, for the SIS epidemic model with another form force of infection, the basic reproduction number R0 determines the persistence or extinction of the disease. In the same time, we also explore the relationship between the demographic equation and the epidemic process. In particular, we show that the epidemic model can exhibit bistability (alternative stable equilibria) over a wide range of parameter values.
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