  In this paper, we propose a simplified bidimensional Wolbachia infestation
model in a population of Aedes aegypti mosquitoes, preserving the main features
associated with the biology of this species that can be found in
higher-dimensional models. Namely, our model represents the maternal
transmission of the Wolbachia symbiont, expresses the reproductive phenotype of
cytoplasmic incompatibility, accounts for different fecundities and mortalities
of infected and wild insects, and exhibits the bistable nature leading to the
so-called principle of competitive exclusion. Since Wolbachia-based biocontrol
is now accepted as an ecologically friendly and potentially cost-effective
method for prevention and control of dengue and other arboviral infections, it
is essential to have reduced models with the main biological characteristics of
Aedes aegypti in the presence of Wolbachia-carriers because such models help to
simplify the mathematical analysis for determining appropriate biocontrol
strategies. Using tools borrowed from monotone dynamical system theory, in the
proposed model, we prove the existence of an invariant threshold manifold that
allows us to provide practical recommendations for performing single and
periodic releases of Wolbachia-carrying mosquitoes, seeking the eventual
elimination of wild insects that are capable of transmitting infections to
humans. We illustrate these findings with numerical simulations using parameter
values corresponding to the wMelPop strain of Wolbachia that is considered the
best virus blocker but induces fitness loss in its carriers.
