Dynamic Analysis Of Several Classes Complex Network-based Epidemic Models

In this thesis,by employing the theory of complex network and differential equation,three classes epidemic models on complex networks are constructed and their dynamics are studied.The dissertation consists of four chapters.Chapter one,firstly,the research background of mathematical modeling and the dy-namics analysis of infectious diseases are described.Then,the research status of epidemic models considered in this thesis is summarized.Finally,the research contents of this thesis are addressed briefly.Chapter two,by incorporating demographics and transfer from infectious to suscep-tible individuals into network-based epidemic models,we construct an improved SIRS epidemic model on complex heterogeneous networks and study its dynamics.Using the next generation matrix method,the basic reproduction number R0 is obtained.By using the methods of Jacobian matrix and Lyapunov function,we investigate the stability of equilibria,i.e.,if R0<1,then the disease-free equilibrium is globally asymptotically sta-ble;if R0>1,then there exists a unique endemic equilibrium and the permanence of the disease is shown in detail.When R0>1 and ?>?,the global stability of the endemic equilibrium is proved as well.Moreover,the effects of three major immunization strate-gies are investigated.Finally,some numerical simulations are carried out to demonstrate the correctness and validness of the theoretical results.Chapter three,an improved SIRS epidemic model with feedback mechanism and vaccination on complex heterogeneous networks is established and investigated.By using the next generation matrix method,we obtain the basic reproduction number R0 which has no concern with the feedback mechanism.Meanwhile,we prove that if R0<1,then the disease-free equilibrium is globally asymptotically stable;if R0>1,then the disease-free equilibrium is unstable and there exists a unique endemic equilibrium.By adopting an appropriate Lyapunov function,the globally asymptotical stability of the endemic equilibrium is proved.Although the feedback mechanism can not change the basic reproduction number R0,it still plays a key role in lowering the level of endemic diseases.Finally,some numerical experiments are conducted to verify and extend the theoretical results.Chapter four,an improved SIRS epidemic model on weighted networks is investigated and we obtain that the basic reproduction number R0 determines whether a disease persists,i.e.,if R0<1,then the disease-free equilibrium E0 is globally asymptotically stable;if R0>1,there exists a unique endemic equilibrium E*.Furthermore,the model is quasi-persistent if R0>1.Finally,some simulations are presented to demonstrate the correctness of the theoretical results.