Global dynamics of an seir epidemic model with vertical transmission

We study a population model for an infectious disease that spreads in the host population through both horizontal and vertical transmission. The total host population is assumed to have constant density and the incidence term is of the bilinear mass-action form. We prove that the global dynamics are completely determined by the basic reproduction number R₀(p, q), where p and q are fractions of infected newborns from the exposed and infectious classes, respectively. If R₀(p, q) ≤ 1, the disease-free equilibrium is globally stable and the disease always dies out. If R₀(p, q) > 1, a unique endemic equilibrium exists and is globally stable in the interior of the feasible region, and the disease persists at an endemic equilibrium state if it initially exists. The contribution of the vertical transmission to the basic reproduction number is also analyzed.