Section: New Results

Modeling the use of Wolbachia for controlling the incidence of dengue

We continued research on modeling the introduction of Wolbachia in a population of Aedes Aegypti. This research is done in collaboration with FGV (Fundação Getulio Vargas), Fiocruz (Fondation Oswaldo Cruz) and UFF (Universidade Federal Fluminense) in Rio de Janeiro (Brazil) [16] .

Wolbachia is a bacteria which infects arthropod species, including a high proportion of insects (60% of species). Its interactions with its hosts are often complex, and in some cases it is considered as an endosymbiont. The unique biology of Wolbachia has attracted a growing number of researchers interested in questions ranging from the evolutionary implications of infection through to the use of this agent for pest and disease control: a public web site has been funded by the National Science Foundation of Australia, and a research in pubmeb (http://www.ncbi.nlm.nih.gov/pubmed ) typing wolbachia gives 1889 results.

While Wolbachia is commonly found in many mosquitoes it is absent from the species that are considered to be of major importance for the transmission of human pathogens. The successful introduction of a life-shortening strain of Wolbachia into the dengue vector Aedes aegypti that decreases adult mean life has recently been reported.

Moreover it is estimated that the population of mosquitoes harboring Wolbachia is less efficient to transmit dengue [18] , [21] , [22] , [25] . Then it is considered that using Wolbachia can be a viable option for controlling the incidence of the dengue.

We consider an alternative infection (by Wolbachia) model which exhibits monotonous properties. This model is designed to take into account both the biology of this infection and any available data. The objective is to use this model for predicting the sustainable introduction of this bacteria. We provide a complete mathematical analysis of the model proposed and give the basic reproduction ratio ${ℛ}_0$ for Wolbachia. We observe a bistability phenomenon. Two equilibria are asymptotically stable: an equilibrium where all the population is uninfected and an equilibrium where all the population is infected. A third unstable equilibrium exists. We provide a lower bound for the basin of attraction of the desired infected equilibrium. We are in a backward bifurcation situation. The bistable situation occurs with natural biological values for the parameters. Our model is an example of an epidemiological model with only vertical transmission.

This infection model is then connected with a classical dengue model. We prove that for the complete model the equilibrium with Wolbachia for the mosquitoes and without dengue for the human is asymptotically stable. We prove that, if a sufficiently large population of infected (by Wolbachia) mosquitoes is introduced, dengue will disappear.

These results have been obtained in collaboration with Pierre-Alexandre Bliman (FGV, Inria); Moacyr Silva (FGV), Claudia Codeço (Fiocruz), Max Souza (UFF) and Jair Koiller (FGV).