Dynamics Of A Mathematical Model For The Propagation Of Wolbachia Bacteria In Mosquito Populations

In the world,dengue fever is one of the most serious mosquito-borne diseases.At present,all kinds of mosquito-borne diseases often break out,and the prevention and control of mosquito-borne diseases has become an issue of concern which has been paid more and more attention.A new way to control mosquito-borne diseases is to use specific endosymbiotic bacterium Wolbachia to block the transmission of dengue fever over the years.Wolbachia can induce cytoplasmic incompatibility(CI),that is,the fertilized eggs produced by the mating of Wolbachia-infected males and uninfected females can not hatch normally,which makes Wolbachia-infected females have reproductive advantages.In this paper,the infection dynamics of Wolbachia is analyzed by using the idea of mathematical modeling.In the second chapter,we establish a new ordinary differential equation model to study the transmission of Wolbachia in mosquito population.The birth and survival functions are Ricker functions,while the death functions are linear functions.By using the qualitative theory of ordinary differential equations,the nonnegative and boundedness of solutions of the model are proved,and the existence conditions of the equilibrium points are given.The globally asymptotic stability of the equilibrium is obtained in different situations.The conditions of Wolbachia successfully invading wild mosquito population were discussed.In special case for parameters,the threshold for initial values of Wolbachia invasion was given,which provides a new strategy for releasing mosquitoes carrying Wolbachia.Finally,numerical simulation is used to explain the conclusion.In the third chapter,considering that mosquitoes need some time from mating to producing offspring,we establish a new delay differential equation model on the basis of ordinary differential equation model with delay.The asymptotic stability of the equilibrium is obtained in different situations.Sufficient conditions are analyzed which ensures that Hopf bifurcation occurs.The computational formulas for direction and stability of Hopf bifurcation are given by applying the center manifold theorem and norm form theory.Finally,we use numerical simulation to verify the phenomenon of periodic oscillation of mosquito population in the process of releasing mosquitoes infected with Wolbachia.