| Infectious diseases are caused by many pathogens,which can be transmitted between humans,animals or humans and animals by various means.Most pathogens are germs,bacteria,fungi and so on,and very few are parasites.Infectious diseases are epidemic and infectious,and may also be immune after infecting.In view of the development of Mathematical Modeling Research on infectious diseases,the modeling method of Kermack and Mekendrick is the beginning of the study of infectious disease dynamics,which opens the door to the important method of using mathematical model to study infectious diseases.Some diseases,such as,tuberculosis,AIDS,hepatitis virus,are often transmitted by means of vertical infection,that is vertical transmission.Therefore,when modeling and discussing infectious diseases,it is more practical meaning to consider the infectious mode of infectious diseases.In this paper,an infectious disease model with vertical infection is first studied.The existence of the disease free equilibrium point and the equilibrium point of endemic disease is analyzed.The local asymptotical stability of the disease free equilibrium point and the endemic equilibrium point is proved by the Hurwitz principle.The global stability of the disease free equilibrium point and the equilibrium point of endemic disease is proved by constructing the Lyapunov function and combining the theory of the LaSalle invariant set.Secondly,the model of the time delay infectious disease with vertical infection is studied,and the threshold for judging the epidemic or the extinction of the disease is determined.It is proved by the stability theory of the delay differential equation that when o<1,for any time delay,the disease free equilibrium point are locally asymptotically stable.And when R0>1 and 0<?<?0,the equilibrium point of endemic disease is locally asymptotically stable.When R0>1 and ? = ?0,the system experiences Hopf bifurcation.Finally,numerical simulations verify the conclusions. |
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