On the exact reproduction number in SIS epidemic models with vertical transmission

This paper proposes a bi-variate competition process to describe the spread of epidemics of SIS type through both horizontal and vertical transmission. The interest is in the exact reproduction number, ℛ_exact,0 , which is seen to be the stochastic version of the well-known basic reproduction number. We characterize the probability distribution function of ℛ_exact,0 by decomposing this number into two random contributions allowing us to distinguish between infectious person-to-person contacts and infections of newborns with infective parents. Numerical examples are presented to illustrate our analytical results.