Research On Population And Infectious Disease Dynamics Models With State And Time-dependent Delays

Delay differential equation,as a description of development state,depends on both present and past differential equation,which can more accurately show the change law of objective things,so it is widely used in many fields.This thesis mainly studies the practical application of state and time-dependent delay differential equation in population and infectious disease modeling.It is worth noting that,the maturity time of individual is self-adjusting with the change of growth state,and its maturity delay reflects state dependence.Therefore,by studying the influence of state-dependent delay on single population and multi population model,we can better describe the growth process of species,and promote the adaptive development of species diversity based on this.Beyond that,the seasonal outbreak of infectious diseases is affected by the changes of temperature and rainfall in climate factors.Therefore,the study of the infectious diseases model with time-dependent delay is of great practical significance to predict the development trend of diseases and control the spread of viruses.This thesis is divided into six chapters.In Chapter 1,firstly,the basic case of delay differential equation is briefly introduced.Secondly,the research background and significance of state and time-dependent delay population and infectious disease model are introduced.Then,the preliminary knowledge used in this thesis is given.Finally,the main research content and innovation points of this thesis are given.In Chapter 2,a novel stage-structured single population model with state-dependent mature delay is formulated and analyzed.The delay is related to the number of total population and taken as a non-decreasing differential bounded function.The model is quite different from previous state-dependent delay models in the sense that a correction term,1-?^??z?t???z?t?,is included in the mature rate.Firstly,positivity and boundedness of solutions are proved without additional conditions.Secondly,existence of all equilibria and uniqueness of a positive equilibrium are discussed.Thirdly,local stabilities of the equilibria are obtained.Finally,permanence of the system is analyzed and explicit bounds for the eventual behaviors of the immature and mature populations are established.In Chapter 3,based on a predator-prey model with correction factors and state-dependent delay,a new predator-prey competition model with a state-dependent maturity delay is developed,which incorporates one prey species and two stage-structured predator species.The main innovation is that the model directly manifests the relationship between prey and maturity time of predators through a correction term,1-?^??x?t??x^??t?.Firstly,the well-posedness of the solution is studied.At the same time,the existence and uniqueness of all equi-libria are discussed.Then,the linearized stabilities of the equilibria are achieved.Finally,some sufficient conditions which ensure the global attractivity of the coexistence equilibrium are obtained.In Chapter 4,in order to study the effect of temperature on the mosquito population dynamics,we propose a Chikungunya transmission model with time-varying parameters and time-dependent extrinsic incubation period?EIP?.The model includes the larval and mature stages of mosquitoes.We deduce the vector reproduction number R_vand the basic reproduction number R₀,and then show that these two threshold values completely determine the global dynamics of the model system.More precisely,?i?if R_v<1,then mosquito population will become extinct ultimately;?ii?if R_v>1 and R₀<1,then Chikungunya disease will be eliminated;?iii?if R_v>1 and R₀>1,then the disease will persist and fluctuate periodically.Numerically,we explore the spread of Chikungunya disease in Delhi,India.The analytic results are in good consistence with our numerical simulations.Further,an interesting finding is that if the time-averaged EIP is used,then R₀may be underestimated,and the number of infectious humans and mosquitoes may be underestimated or overestimated.In Chapter 5,a periodic Chikungunya model with temperature and rain-fall effects is proposed and studied,which incorporates time-dependent extrinsic incubation period,time-dependent maturation delay,asymptomatic and symp-tomatic infectious humans.The two threshold parameters for the extinction and persistence of mosquitos and the virus are derived,respectively:the mosquito re-production number R_mand the basic reproduction number R₀.Then the results of theoretical analysis are verified by numerical simulation with the temperature and rainfall data of the state of Cear?a,where the largest outbreak in Brazil's his-tory occurred in 2017.And the effects of rainfall,seasonality,and asymptomatic infection in humans on mosquito population and the Chikungunya transmission are explored.The simulations show that if these factors are not taken into ac-count,the number of mosquito population and people infected may be overesti-mated.Finally,the relationships between R_mand R₀and some parameters are performed.In the last chapter,the research contents are briefly summarized and dis-cussed,and the future research direction is prospected.