  Since the last century, deterministic compartmental models have emerged as
powerful tools to predict and control epidemic outbreaks, in many cases helping
to mitigate their impacts. A key quantity for these models is the so-called
Basic Reproduction Number, that measures the number of secondary infections
produced by an initial infected individual in a fully susceptible population.
Some methods have been developed to allow the direct computation of this
quantity provided that some conditions are fulfilled, such that the model has a
pre-pandemic disease-free equilibrium state. This condition is only fulfilled
when the populations are stationary. In the case of vector-borne diseases, this
implies that the vector birth and death rates need to be balanced, what is not
fulfilled in many realistic cases in which the vector population grow or
decrease. Here we develop a vector-borne epidemic model with growing and
decaying vector populations and study the conditions under which the standard
methods to compute $R_0$ work and discuss an alternative when they fail. We
also show that growing vector populations produce a delay in the epidemic
dynamics when compared to the case of the stationary vector population.
Finally, we discuss the conditions under which the model can be reduced to the
SIR model with fewer compartments and parameters, which helps in solving the
problem of parameter unidentifiability of many vector-borne epidemic models.
