  A wide range of infectious diseases are both vertically and horizontally
transmitted. Such diseases are spatially transmitted via multiple species in
heterogeneous environments, typically described by complex meta-population
models. The reproduction number is a critical metric predicting whether the
disease can invade the meta-population system. This paper presents the
reproduction number for a generic disease vertically and horizontally
transmitted among multiple species in heterogeneous networks, where nodes are
locations, and links reflect outgoing or incoming movement flows. The
metapopulation model for vertically and horizontally transmitted diseases is
gradually formulated from two species, two-node network models. We derived an
explicit expression of the reproduction number, which is the spectral radius of
a matrix reduced in size with respect to the original next generation matrix.
The reproduction number is shown to be a function of vertical and horizontal
transmission parameters, and the lower bound is the reproduction number for
horizontal transmission. As an application, the reproduction number and its
bounds for the Rift Valley fever zoonosis, where livestock, mosquitoes, and
humans are the involved species are derived. By computing the reproduction
number for different scenarios through numerical simulations, we found the
reproduction number is affected by livestock movement rates only when
parameters are heterogeneous across nodes. To summarize, our study contributes
the reproduction number for vertically and horizontally transmitted diseases in
heterogeneous networks. This explicit expression is easily adaptable to
specific infectious diseases, affording insights into disease evolution.
