Research On Modeling Analysis And Numerical Simulation For Several Classes Of Infectious Disease Models

In this paper,we investigate several classes of infectious disease models by using the methods of nonlinear functional analysis combined with the Lyapunov functionals and differential equation qualitative and stability theories.For the complexity of systems studied,on the one hand,we verify the theoretical results obtained by numerical simulations;on the other hand,some new dynamical behaviors are revealed by some numerical simulations.This paper include seven chapters.The main contents and results are as follows:In chapter 1,the background,research status and development of delay differential equation,delay reaction-diffusion equation and the infectious disease dynamics and the main research works are briefly introduced.In chapter 2,we study the dynamical behaviors of a type of multi-delay diffusive Nicholson's blowflies equation with Neumann boundary conditions and initial conditions.The discussions of the long-time behaviors of the solutions of partial differential functional equations with multiple delays are much more difficult compared with those equations with a single delay.By the method of lower-upper solution pair,we obtain the global attractivity of constant equilibriums of the model proposed in this chapter.We discover the system has periodic solution(hopf bifurcation)by numerical simulation when the parameter values are changed appropriately.According to some simulation results,some open problems are put forward.In chapter 3,we propose a population and infectious diseases dynamics model which is covered with patients,anopheles mosquitoes and GM mosquitoes based on the current international research results.The sufficient condition of locally asymptotic stability of equilibriums of the model is obtained by the differential equation qualitative and stability theory.Through the studies of dynamical behaviors of the model,we reveal the influence of changes of the values of various parameters in the model on malaria control.The actual control effect of the number of patients by releasing GM mosquitoes are relatively clearly presented.The effects of two methods of releasing GM mosquitoes on controlling the amount of malaria patients are compared.These works provide a theoretical basis for studying the effects of releasing GM mosquitoes to control malaria transmission.In chapter 4,we consider the pesticides which work on the older mosquitoes.These pesticides function based on the physiological changes and weakness resulted due to aging.Unlike previous only considering the mosquito population changes,we propose a malaria transmission model which involve the general adult mosquitoes,drug-resistant mosquitoes and persons by dividing mosquitoes into larvae phase(which includes all phases of egg,larva and pupae),adult phase and the phase of old age.At the same time,the sufficient conditions to ensure that three non-trivial equilibrium points are locally asymptotically stable are obtained.The numerical simulations in this chapter on the one hand verify the main results,on the other hand illustrate the killing-effect of pesticides on mosquitoes in different phases for malaria control.The results can provide the corresponding theoretical basis for the effective control of malaria.In chapter 5,we study the spread of disease due to the migration of rural migrant workers.Based on the SIR model proposed by Wang and Wang [121] which described the effect of disease transmission where rural migrant workers lived by the mobility of migrant workers,we investigate the global exponential stability of epidemic equilibrium and the corresponding sufficient condition.The global exponential stability of epidemic equilibrium implies that the disease can be controlled more quickly and the result provides the theoretical basis for more effective control of disease transmission.In chapter 6,a delay model with periodic coefficients or almost periodic coefficients for the transmission of infectious diseases in migrants' home residence is considered.Firstly,the existence of positive almost periodic solution of the system is verified by constructing a contraction mapping and using a new analytical technique.Secondly,when the coefficients are periodic,we verify the existence of periodic solution by the coincidence degree theory.Finally,almost periodic solution and periodic solution are shown to be globally asymptotically stable by Lyapunov functional method.A conjecture is proposed as the end of this chapter.In chapter 7,we briefly summarize the main research contents and innovations of this paper,point out some still unsolved problems and prospect the next research works.