Kinetic Analysis Of A Class Of Infectious Disease Models

At present,COVID-19 continues to spread around the world,posing a serious threat to human health.Therefore,the study on the infection principle,transmission law and prevention and control strategy of epidemics has attracted the attention of all sectors of society.In this paper,the dynamics models of two kinds of SEIR epidemics were established considering the media reports and the epidemic in the incubation period.As is known to all,in the early stage of a disease outbreak,the effect of media reports could not be well displayed due to the small number of infected people and the small chance for susceptible people to contact the source of infection.However,as the number of infected people increases,the role of media reports will gradually become prominent.In the later stage of the epidemic,although there are a large number of infected people,the role of media reports will become saturated.That is,the value of media influence function will approach 1.Based on the above facts,we have for the first time used the form of x2/m+x2 to describe the influence of media coverage on the spread of disease,and established a class of SEIR epidemic model.Firstly,the existence of the equilibrium is verified.Secondly,by defining the basic regeneration number R0 and using Hurwitz criterion,Lasalle invariant principle and Bendixson criterion,it is found that when R0?1,the disease-free equilibrium P0 is globally asymptotically stable.When R0>1,even though P0 is unstable,its unique endemic equilibrium P*is globally asymptotically stable.Then we improved the above model,introduced the influence of environmental noise,and established a class of random SEIR epidemic model.Firstly,the existence and uniqueness of global positive solutions are verified,which makes our discussion have practical biological significance.Secondly,it is found that when the noise is with-in a certain range,although the random system does not have an equilibrium at this time,its solution will oscillate near the equilibrium of the previously determined sys-tem.Then,we found a threshold of Rs in the random model,which determines the outcome of disease transmission.At the same time,sufficient conditions for the ex-tinction and persistence in mean are obtained.We also found that the higher the noise,the shorter the average time for the disease to go extinct.It is found that media reports can effectively slow down the spread of the disease and reduce the number of infected people by observing the curve of the transfer probability density function of infected people.In addition,after observation for a certain time,the values of infected popu-lation presented a stable ergodic distribution.When the noise intensity increases,the peak value of the distribution becomes smaller and the prediction becomes worse.In the end,the theoretical results are verified by some numerical simulations.Finally,the numerical algorithm of stochastic dynamical systems is explored.With the help of Mathematica,the algorithm is implemented.Firstly,we study the transfer probability density function and use the numerical simulation method of exponential closure to verify the effectiveness of the method from two aspects:on the one hand,we compare the error of the numerical solution of FPK equation obtained by the method with that of the exact solution under the generalized stationary condition,and find that the error of the solution method is less than three parts per thousand.On the other hand,by comparing the steady-state regions determined by the transition probability density function corresponding to the numerical solution and the phase diagram of the system,it is found that the two regions are highly consistent.Therefore,this method is not only simple and accurate,but also an effective method to solve FPK equation.In addition,the numerical solution of the stochastic delay differential equation is programmed and applied to two kinds of common dynamical systems.