Dynamics Of SIS Epidemic Models Based On Adaptive Networks

Nowadays,human society is still facing a huge threat of infectious diseases.With the development and progress of science and technology,people will take different measures to curb the disease epidemic.In order to study the impact of these measures on the spread of the disease,a series of epidemic models based on the adaptive network are established in this thesis.In the first part of this thesis,considering that people will be disconnected from the infected person during the epidemic,the SIS epidemic model based on the adaptive network with non-linear rewiring rate is analyzed.In the second part,considering the means,such as wearing masks and frequent hands washing,we establish and analyze an epidemic model based on the adaptive network with variable contact rates.In the third part of this thesis,considering the spontaneous characteristics of the disease,the epidemic model based on the adaptive network with the spontaneous infection items is analyzed.The main research contents of this thesis are as follows.Firstly,an SIS epidemic model based on an adaptive network with a nonlinear rewiring rate is established.In this model,the classical constant rewiring rate is changed into a nonlinear rewiring rate which depends on the disease scale to reflect the feature that people's self-protection consciousness will increase with the increase of the disease scale.Through dynamic analysis,the discriminant conditions of the stability of endemic equilibrium point and the conditions of saddle node bifurcation and Hopf bifurcation are given.The type of BogdanovTakens bifurcation is discussed by using the bifurcation theory.It was found by numerical simulation that the system has complex dynamic behaviors,such as double limit cycles and homoclinic bifurcation.The rewiring behavior can effectively suppress the trend of epidemic disease.After comparing the random simulation results with the numerical results of the approximation model,it is found that the nonlinear rewiring rate has a significant impact on disease spread and network structure.Secondly,an epidemic model with a variable contact rate based on the adaptive network is established and studied.During an infectious disease epidemic,individuals will spontaneously reduce their contact with infected persons for the purpose of self-protection(or take some protective measures),thus the contact rate is reduced.For this reason,it is more reasonable to take the contact rate as a function dependent on the scale of infection and it becomes smaller as the scale of infection increases.The theoretical analysis results of the model show the system has complex bifurcation phenomena,and the numerical simulations also show its rich dynamic behaviors,such as bistable phenomena,double limit cycles,etc.By numerical simulations,we show that this contact rate has a significant impact on the final scale of epidemic disease.Finally,an epidemic model with spontaneous infection based on the adaptive network is established and studied.In real life,certain infectious disease cases are caused by factors except for the social network or spontaneous infection.For this reason,spontaneous infection is introduced into the classical epidemic model based on the adaptive network to observe its impact on the disease process.Through model analysis,it is found that the system does not have a disease-free equilibrium and there is always an endemic equilibrium,there exist bistable behavior,periodic solutions and coexistence endemic disease equilibria.Complex dynamic behaviors such as Hopf bifurcation and Bogdanov-Takens bifurcation are observed in this model.Through numerical simulation,it is found that network topology and self-infected items have an important influence on the spread of diseases.