Dynamic Study Of SIQS Epidemic Model With Random Factors And Media Coverage

The spread of infectious diseases hinders the development of society and economy.The control of infectious diseases is one of the eternal themes of human beings.Microscopically,the development of the drug to kill bacterium and viruses,as well as the vaccines to pre-vent the diseases is the fundamental means to control infectious diseases.In macroscopical sight,the formulation of efficient and comprehensive control measures is the primary strate-gic choice to prevent the spread of infectious diseases among the population.Mathematical modeling is a good method to verify the effects of these policies and measures on the control of infectious diseases.By appropriate mathematical models for infectious diseases,we can study the law of its propagation,analyse effect of various policies and measures,compare the differences between different control measures.Moreover,we can develop an optimal com-prehensive control strategy for infectious diseases.This paper mainly studies how different ways of isolation,infectious disease,health education and other external interventions,and different kinds of environmental random disturbances affect the spread of infectious diseases.Main tasks are as follows:1.Based on the clinical treatment standard of infectious diseases,we establish a time delay SIQS epidemic model with the same isolation time for the infected.Then,we analyze the existence and local stability of the equilibrium.We give the basic reproduction number.We prove the globally stability of the disease free equilibrium by the comparison principle and so on,and give the sufficient conditions of the extinction of the disease.We prove the globally stability of the endemic equilibrium by Lyapunov function and give the sufficient conditions of the persistence of the disease.In addition,we prove the existence of periodic solutions for systems by numerical simulation.Moreover,we consider the impact of random factors on infectious diseases,and establish a stochastic SIQS epidemic model with time delay.We prove the existence and uniqueness of the globally positive.We give the threshold conditions to determine the extinction or persistence of the disease.Our results indicate that random factors can restrain the spread of the disease.At last,we apply our results to a actual disease and carry out numerical simulation to support our results.2.We consider the impact of media coverage and health education on infectious dis-eases,and establish a SIQS epidemic model with media coverage and health education.We give the basic reproduction number.We prove the globally stability of the disease free e-quilibrium by the comparison principle.We prove the globally stability of the endemic equilibrium by Lyapunov function and give the sufficient conditions of the persistence of the disease.Then,we the random fluctuations in mortality and infection rate and establish a stochastic SIQS epidemic model with media coverage and health education.We prove the existence and uniqueness of the globally positive.We give the sufficient conditions to de-termine the extinction of the disease by It?o~′s formula.We prove the existence of stationary distribution by Lyapunov function and Has’Minskii theory.In addition,we apply our results to a actual disease and carry out numerical simulation to support our results.Our results indicate that random factors can restrain the spread of the disease,and media coverage and health education can’t eliminate infectious diseases but can reduce the number of infections.3.We consider the effect of continuous random factors on infectious diseases and es-tablish a Brown-driven SIQS epidemic model.We prove the existence and uniqueness of the globally positive.We give the threshold conditions to determine the extinction or persistence of the disease.Moreover,we consider the effect of discontinuous random factors on infec-tious diseases and establish a Lévy-driven SIQS epidemic model.We prove the existence and uniqueness of the globally positive.We give the threshold conditions to determine the extinction or persistence of the disease.By Lyapunov function,we analyze the asymptotic behavior of the stochastic system near the equilibrium of the corresponding deterministic system.Our results show that the continuous random factors or the discontinuous random factors,can all inhibit the spread of the diseases.