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, Rexact,0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{R}_{\mathrm{{exact}},0}$$\end{document}, which is seen to be the stochastic version of the well-known basic reproduction number. We characterize the probability distribution function of Rexact,0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{R}_{\mathrm{{exact}},0}$$\end{document} 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.