Periodic Dynamics Of A Mosquito Population Suppression Model Based On Incomplete Cytoplasmic Incompatibility

Mosquito-borne diseases are one of the serious challenges facing the World Health Organization in the prevention and control of infectious diseases.Dengue,as the most prevalent mosquito-borne disease,is of great concern to World Health Organization.The most direct and effective approach to prevent the transmission of dengue is to interrupt the transmission pathway,such as suppressing wild mosquitoes.To achieve the goal of a 25%reduction in the global burden of dengue by 2030,World Health Organization recommends the use of the Wolbachia-driven mosquito control technique to eliminate wild mosquitoes,which is environmentfriendly and longer sustainable compared to traditional mosquito control methods.Wolbachia is an intracellular bacterium that is widely present in arthropods.The bacterium has three important functions:inducing cytoplasmic incompatibility,maternal transmission and blocking pathogen replication in mosquitoes.By releasing Wolbachia-infected males into the wild,cytoplasmic incompatibility can be used to reduce the density of wild mosquitoes,which yields that mosquito-borne diseases will not be prevalent.Based on the above biological significance,we developed a mosquito population suppression mathematical model with incomplete cytoplasmic incompatibility in this paper.By introducing an important parameter,the sexually active lifespan (?) of Wolbachia-infected males,the waiting release period T between two consecutive releases of Wolbachia-infected males will result in three different release strategies with (?):T=(?),T>(?) and T<(?).Thus,based on the above three release strategies,this paper mainly discusses the dynamical behaviors of wild mosquitoes under different release strategies.In Chapter 1,we introduce the research background,research progresses,research contents and preliminary knowledge of this paper.First,we introduced the harm of mosquito-borne diseases to human society,especially dengue.Then,we introduced the Wolbachia-driven mosquito control technique,many mathematical models have been established and studied.Based on the above work,we established a non-autonomous mosquito population suppression model with incomplete cytoplasmic incompatibility.In Chapter 2,we consider the release strategy T=(?),in which case the model is a one-dimensional autonomous model.By analyzing the existence of the positive equilibria of the model,we get four parameter thresholds:cytoplasmic incompatibility intensity threshold sh*and sh**,and the release amount thresholds c0 and c1,which divide the parameter plane sh-c into three subregions.We prove the stability of the equilibria in these subregions and give numerical examples to verify the correctness of the main results.In Chapter 3,we consider the release strategy T>(?),where the model becomes a piecewise continuous equation with two sub-equations continuously switching over time.At this time,the periodic phenomenon may occur with high probability.By Poincare map,we transform the periodic solution problem of the model into the fixed point problem of Poincare map.On the basis of Chapter 2,we discuss the periodic dynamics in each parameter regions,and sufficient conditions for the existence and uniqueness,or the existence of exactly two periodic solutions are obtained,together with their stability analyses.Some numerical examples are given to verify the correctness of the theoretical results.In Chapter 4,we consider the release strategy T<(?).Based on Chapters 2 and 3,this chapter considers the influence of the wild environment on the death rate of wild mosquito population.Based on time switching,we establish a mosquito population suppression stochastic model by using stochastic differential equations,where stochastic noises are given by independent standard Brownian motions.First,the existence and uniqueness of global positive solutions and stochastically ultimate boundedness for the stochastic model are obtained.Next,some sufficient conditions for the extinction and the existence of periodic solutions are established.Furthermore,we assume that the release function is a general periodic function and obtain some stochastic dynamical behaviors.Finally,numerical examples are presented to illustrate the theoretical results.