Research Under Varying Environmental Epidemic Models

Epidemiological dynamics is a major component of mathematical biology, which is now widely receivingattention from many domestic and overseas scholars. By proposing mathematical models，we study the dy-namic behavior of epidemic models. In this paper, based on previous researches, we make a deeper discussionon epidemiological dynamical model and the spread of infectious diseases. Combining the method of numericalsimulation, the main conclusions are verified and discussed.The main contents can be summarized as follows:In Section1, the background of epidemiology and research status of the epidemic models are introduced inthis paper. Finally, the main contents of this paper are given.In Section2, we study an impulsive epidemic model with time-varying contact rate and vaccinate rate. Byanalysing, we compute the reproductive number R0. The sufficient conditions for global asymptotical stabilitythe disease-free periodic solution and permanence of the system are obtained.In Section3, a nonautonomous SEIR epidemic model with impulse is analysed. By constructing appropriateLiapunov functions and using the comparison theorem of impulsive differential equations, we obtain that theinfection-free periodic solution of system is globally asymptotically stable and the disease is persistent. In addi-tion, the theoretical results are confirmed by numerical simulations and biological significance of the theoreticalresults are explained.In last section, we study the schistosomiasis model with diffusion effect. From the spectral radius theory,we obtain the threshold value R0. The global asymptotical stability of infection-free solution of system andthe local asymptotical stability of the endemic equilibrium are discussed. At last, in order to verify our mainconclutions, we give some numerical simulations and biological significance of the theoretical results.