Stability Analysis Of Several Epidemic Models With Incubation Period

In this paper,the stabilities of three types of epidemic models with incubation period were analyzed.Got the basic regeneration number R0 and the existence of equilibrium point was verified.Sufficient conditions for local asymptotic stability and global asymptotic stability of disease-free equilibrium and endemic equilibrium were obtained.Finally,the conclusion was verified by numerical simulationThe first chapter mainly introduced the research background、current situation of epidemic models with incubation period,and listed the necessary preparatory knowledge.In chapter two,an epidemic model of SEIVR with continuous vaccination and in-cubation period was studied.By calculating the next generation matrix,the basic regen-eration number R0 used to determine whether a disease will spread was obtained.Using Routh-Hurwitz criterion、Lyapunov function and LaSalle invariant set principle,it was proved that when R0<1,the model has a unique disease-free equilibrium point PO,and it is globally asymptotically stable.When R0>1,the model has two equilibrium points,the disease-free equilibrium point P0 is unstable,and the local endemic equilibrium point P*is globally asymptotically stable.Further analysis,the spread of disease can be pre-vented by increasing the ratio θ of vaccination to reduce the basic reproductive number in disease control and prevention.In chapter three,an epidemic model of SEIQR with incubation period and infection period was studied,both the incubation period and the infection period are infectious.The basic regeneration number R0 was defined.And the Routh-Hurwitz criterion、Lyapunov function、LaSalle invariant set principle and second additive complex matrix were used to prove that when R0<1,the model has a unique disease-free equilibrium point p0,and p0 is globally asymptotically stable;When R0>1,the model has two equilibrium points,the disease-free equilibrium point PO is unstable,and the local endemic equilibrium point P*is globally asymptotically stable.In chapter four,a class of epidemic model with latent period and treatment control was studied.The threshold to determine whether the disease is prevalent or not was given.Routh-Hurwitz criterion、Lyapunov function and LaSalle invariant set theory were used to prove that when Ro<1,the model has a unique disease-free equilibrium p0,and p0 is global asymptotic stability;When R0>1,the model has two equilibrium points,they are P0 and P*,disease-free equilibrium PO is not stable,endemic equilibrium P*is global asymptotic stability.Further,increasing the proportion p of patients receiving treatment in disease treatment control can reduce the number of basic regeneration R0.This in turn makes R0<1 to prevent the spread of the disease.