Several Types Of Model Stability Analysis Of Infectious Diseases With A Time Lag

Infectious diseases has been threatening human health, and many scientists have studiedthe method to prevent infectious diseases spreading. And the diferential equation is usedto study the spread of infectious diseases,which has been developed rapidly. In this thesis,by analyzing the dynamic stability of diferential equation and the mechanism of the spreadof infectious diseases, People can prevent the spread of infectious diseases efectively.Firstly,we study a delayed SIR epidemic model and get the threshold value which de-termines the global dynamics and outcome of the disease in the system. For any τ, weshow the disease-free equilibrium is globally asymptotically stable when the basic reproduc-tion number R₀<1, the disease will die out. We prove the endemic equilibrium is locallyasymptotically stable for any τ=0when R₀>1, the disease will persist. For any τ=0,the existence conditions for Hopf bifurcations at the endemic equilibrium are obtained. Wecompare the delayed SIR epidemic model with nonlinear incidence rate with the one withbilinear incidence rate. At last, numerical simulations are performed to illustrate and verifythe conclusions.Secondly,a kind of a delayed SEIQR epidemic model with the latent and quarantineis studied. Using Hurwitz criterion, the local stability of the disease-free equilibrium andendemic equilibrium of system （3.2） is proved. For any time delay τ, we prove the disease-free equilibrium is globally asymptotically stable when the basic reproduction number R₀isless than unity and the endemic equilibrium is globally asymptotically stable when the basicreproduction number R₀is greater than unity by means of suitable Lyapunov functionsand LaSalle’s invariance principle. So the delay is harmless to system （3.2）. From thebiological point of view, the delay here has no infuence on the the transmission of diseases.Above all, we consider that E（t） is quarantined and can recovery in this model, which willefect changing trends of S, E, I, Q, R. Here, we take k3as an example to explain that.Meanwhile, the simulation image which R₀changes as τ can be obtained and we can fndout τ0which the basic reproduction number is unity. Those are useful for us to control epidemics. At last, the conclusions above are verifed by numerical simulations.Finally, we study a kind of delayed SIQR epidemic model with nonlinear incidencerate and the quarantine measure and get the threshold value which determines the globaldynamics and the outcome of the disease. For any τ, we show the disease-free equilibrium islocally and globally asymptotically stable when the R₀<1, the disease will die out. And theendemic equilibrium is proved to be locally and globally asymptotically stable when R₀>1,the disease will persist in the population. Numerical simulations are performed to illustrateand verify the conclusions. Meanwhile, we fnd out it is useful for controlling the spread ofthe infectious disease to quarantine infected individuals.