Abstract
Starting from an age structured partial differential model, constructed taking into account the mosquito life cycle and the main features of theWolbachia-infection, we derived a delay differential model using the method of characteristics, to study the colonization and persistence of theWolbachia-transinfectedAedes aegyptimosquito in an environment where the uninfected wild mosquito population is already established. Under some conditions, the model can be reduced to a Nicholson-type delay differential system; here, the delay represents the duration of mosquito immature phase that comprises egg, larva and pupa. In addition to mortality and oviposition rates characteristic of the life cycle of the mosquito, other biological features such as cytoplasmic incompatibility, bacterial inheritance, and deviation on sex ratio are considered in the model. The model presents three equilibriums: the extinction of both populations, the extinction ofWolbachia-infected population and persistence of uninfected one, and the coexistence. The conditions of existence for each equilibrium are obtained analytically and have been interpreted biologically. It is shown that the increase of the delay can promote, through Hopf bifurcation, stability switch towards instability for the nonzero equilibriums. Overall, when the delay increases and crosses predetermined thresholds, the populations go to extinction.
Highlights
Aedes aegypti is a widespread human blood-feeding mosquito responsible for the transmission of several arboviruses including Dengue, Yellow fever, Zika, Murray Valley, La Crosse, Chikungunya and Rift Valley fever
The traditional approach to diminishing the mosquito population includes the reduction of breeding sites and the use of larvicides and pesticides for adults
Mechanical control and the application of larvicides are carried out before the period favorable to the proliferation of mosquitoes, while pesticides for adults are applied during epidemics when the number of infected humans is high [41]
Summary
Aedes aegypti is a widespread human blood-feeding mosquito responsible for the transmission of several arboviruses including Dengue, Yellow fever, Zika, Murray Valley, La Crosse, Chikungunya and Rift Valley fever. In [22], a DDE phasestructured model (larva and adult populations) evaluated the suppression of the wild population of Aedes mosquitoes by releasing a continuous constant number of Wolbachia-infected male This is modeled by changing the growing rate of the population. Starting from an age structured PDE model that considers the mosquito entomological parameters and biological features associated to Wolbachia infection, a new two-population DDE model is obtained and carefully assessed. Analytical results such as positiveness, boundedness, and uniqueness of solutions are provided. The model appears for the first time in [15] where numerical results concerning population dynamics were obtained
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