A SIR epidemic model with incubation period

A simple SIR-type model with vaccination and a discounting factor is formulated in order to investigate the effect of the time lag on the disease reproduction threshold. It is observed that the discounting factor may be misleading in terms of public health control measures since long time lags tend to reduce the vaccine-induced reproduction number. This may lead to multiple transition cycles within the population if eventually the value of the reproduction number is reduced below unity, thereby biologically speaking, this means that such diseases have a robust structure which makes them more difficult to eradicate and their control complicated.