Dynamical Models On Wolbachia Infection Frequency In Mosquito Populations

A biological and effective method to block dengue epidemics is to release Wolbachia-infected mosquitoes.Wolbachia,widespread in arthropods,have the mechanisms of maternal transmission,cytoplasmic incompatibility(CI for short)and pathogen blocking of dengue viruses in mosquitoes.Once the dengue vectors Aedes albopictus and Aedes aegypti are infected with Wolbachia,they greatly reduce their ability to transmit dengue viruses.Recently,Wolbachia spread dynamics in mosquito populations have attracted much interest and become a hot research topic.The earliest study on Wolbachia spread dynamics in mosquito populations can be traced back to 1959 when Caspari and Watson established the first discrete model.The model used the infection frequency to characterize the infection degree of Wolbachia in mosquito populations.Considering the maternal transmission is imperfect,in 1978,Fine further improved Caspari and Watson’s work.All these models generate an infection frequency threshold(an unstable positive equilibrium),above which the stable establishment of Wolbachia in mosquito populations is guaranteed,and below which Wolbachia are eventually wiped out.In this thesis,a series of works on the dynamics of Wolbachia infection frequency in mosquito populations are carried out.These models cover all relevant existing models proposed and investigated by Caspari,Fine and others.By using basic theories and methods of difference equations,we first develop a discrete model to characterize the Wolbachia infection enhancing(or decaying)domains in which the infection frequency in the next generation is always bigger(or smaller)than that in the present generation.For the decaying domain where the infection frequency tends to zero,supplemental releases are needed to improve the possibility of a stable establishment of Wolbachia in mosquito populations.When the release amounts are equal,we compare the effects among three release strategies,including releasing both Wolbachia-infected females and males,only Wolbachia-infected females and only Wolbachia-infected males.This provides a theoretical guidance for designing an optimal release strategy.Finally,assuming that Wolbachia-infected mosquitoes are released impulsively and periodically,we construct a discrete switching model on Wolbachia infection frequency,and then analyze the existence and the stability of each periodic solution.In Chapter 1,we mainly introduce the biological background and some existing mathematical models.We first give a brief summary of the current situation of dengue,and summarize traditional dengue control methods as well as their weakness.We then introduce a Wolbachia-driven dengue control method,and recount its development history.We review the existing discrete models,which characterize the dynamics of Wolbachia infection frequency in caged mosquito populations.In Chapter 2,we formulate a discrete model for caged mosquito populations,and propose the concept of Wolbachia infection enhancing(or decaying)domain for the first time.We give a complete description on Wolbachia infection enhancing and decaying domains in terms of the maternal transmission leakage rate,CI intensity and the fitness cost.In the Wolbachia infection enhancing domains or the Wolbachia infection decaying domains where infection frequencies don’t tend to zero,Wolbachia are successfully established in mosquito populations,and it is not necessary to consider supplemental releases of Wolbachia-infected mosquitoes.However,in the decaying domain where the infection frequency goes to zero,Wolbachia are eventually wiped out.To change the fate of Wolbachia,supplemental releases of Wolbachia-infected mosquitoes are required so that the infection frequency can exceed the infection frequency threshold after several generations,and then enters the Wolbachia infection enhancing domain.In Chapter 3,we investigate how supplementary releases affect the stable establishment of Wolbachia in mosquito populations based on the decaying domains where the infection frequencies go to zero.With the notion of supplementary releases,there are three release strategies,namely releasing both Wolbachia-infected females and males,only Wolbachia-infected females and only Wolbachia-infected males.We compare the effectiveness among these three release strategies on the premises of the equal release amount and the stable establishment of Wolbachia in mosquito populations is guaranteed.Theoretical results show that the first two release strategies are more effective than the last release strategy.For the first two release strategies,the former release strategy is either less effective than the latter release strategy,or more effective than the latter one in previous generations,and then becomes less effective in late generations.In Chapter 4,assuming that Wolbachia-infected mosquitoes are released periodically and impulsively,we assume that Wolbachia-infected males are supplementarily released during the first generations,and the releases are terminated from the( + 1)-st generation to the -th generation.With imperfect maternal transmission and incomplete CI,we develop a discrete switching model on Wolbachia infection frequency in mosquito populations,aiming to investigate how supplementary releases of Wolbachia-infected males affect the stable establishment of Wolbachia in mosquito populations.For the case with = 1 and = 2,we obtain sufficient conditions for the existence of a unique and exact two 2-periodic solutions,and analyze the stability of each of the 2-periodic solutions,respectively.