  Aedes aegypti is the main vector of multiple diseases, such as Dengue, Zika,
and Chikungunya. Due to modifications in weather patterns, its geographical
range is continuously evolving. Temperature is a key factor for its expansion
into regions with cool winters, but rainfall can also have a strong impact on
the colonization of these regions, since larvae emerging after a rainfall are
likely to die at temperatures below $10^{\circ}$C. As climate change is
expected to affect rainfall regimes, with a higher frequency of heavy storms
and an increase in drought-affected areas, it is important to understand how
different rainfall scenarios may shape Ae. aegypti's range. We develop a model
for the population dynamics of Ae. aegypti, coupled with a rainfall model to
study the effect of the temporal distribution of rainfall on mosquito
abundance. Using a fracturing process, we then investigate the effect of a
higher variability in the daily rainfall. As an example, we show that rainfall
distribution is necessary to explain the geographic range of Ae. aegypti in
Taiwan, an island characterized by rainy winters in the north and dry winters
in the south. We also predict that a higher variability in the rainfall time
distribution will decrease the maximum abundance of Ae. aegypti during the
summer. An increase in daily rainfall variability will likewise enhance its
extinction probability. Finally, we obtain a nonlinear relationship between dry
season duration and extinction probability. These findings can have a
significant impact on our ability to predict disease outbreaks.
