Vector-borne diseases with nonstationary vector populations: The case of growing and decaying populations

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, R_{0}, 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 prepandemic disease-free equilibrium state. This condition is fulfilled only 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. This is not fulfilled in many realistic cases in which the vector population grows or decreases. Here we develop a vector-borne epidemic model with growing and decaying vector populations that in the long term converge to an asymptotic stationary state, 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 Susceptible, Infectious, and/or Recovered (SIR) model with fewer compartments and parameters, which helps in solving the problem of parameter unidentifiability of many vector-borne epidemic models.