The Dynamical Analysis Of Two Viruses And Age-structured Epidemic Models On Complex Networks

Epidemiology mainly study the rules of occurrence,development,and the control measures to eliminate infectious diseases.The transmission of infectious diseases not only depends on the biological characteristics of diseases,but also on the contact be-tween populations,i.e.,the number of contact is heterogeneous for different people in unit time.However,the traditional models are based on uniformly mixing population-s,which are unable to describe the local contact of population.With the development of complex network theory,many real problems can be described as complex network models.Therefore,the combination of complex network theory and epidemiology has become the hot topic of epidemic research.In order to understand and control epidem-ic better,in this doctoral thesis,different epidemic models are established on complex networks based on existing researches,and profound researches on them are preformed.The main research results are as follows:Firstly,the classical epidemic models based on mean-field theory are only related to nodes,which cannot reflect the role of clustering coefficient in transmission dynamics.The necessary factor of disease spread is the contact between susceptible nodes and infected nodes.Therefore,the pair-wise approximation constructing models based on edges is more realistic.Meanwhile,a variety of strains with different infections,toxins and mobilities can exist together and compete for the same susceptible population.We propose a model where two strains compete with each other on heterogeneous networks by pair-wise approximation closed by the probability generating function(PGF),and the basic reproduction number is obtained via two methods.Then we discuss the conditions for the local stability of equilibria related to clustering coefficient.Secondly,infection age is an important factor in epidemic dynamics since the het-erogeneity of infection during the period of disease spread.In order to realistically ana-lyze the dynamical behavior of epidemic diseases,a generalized SIS transmission model with infection age and birth and death is discussed on a heterogeneous network.The model allows the infectious and the recovery rates to vary and depend on the age of in-fection,the time since individuals get infected.We address the uniform persistence and the sharp threshold property of the model,that is,for the basic reproduction number less than one,the disease-free equilibrium is globally asymptotically stable,while for the basic reproduction number is above one,a Lyapunov functional is used to show that the endemic equilibrium is globally stable.Finally,some numerical simulations are carried out to complement the main results.The disease dynamics rely not only on the network structure,but also on an age dependent factor.Thirdly,public health services are constantly searching for new ways to reduce the spread of diseases such as public vaccination of asymptomatic individuals,quarantine(isolation)and treatment of symptomatic individuals.However,some vaccines are not completely efficient,namely,vaccines rarely cover the entire population and only pro-vide finite-time immunity against infection.We constructed SIR model which includes imperfect vaccination and quarantine on scale-free networks to study these diseases with permanent natural immunity to infection.We show that the system exhibits a 'forward'bifurcation.Meanwhile,the disease-free and unique endemic equilibria are shown to be globally asymptotically stable by constructing suitable Lyapunov functions.Finally,we study the quarantine measures and show the target quarantine is better.Finally,we continue studying the effect of imperfect vaccination to the spread of these diseases without permanent natural immunity to infection(e.g.,encephalitis,flu,gonorrhea)on complex networks.The conditions ensuring the occurrence of multiple endemic equilibria are derived.Under certain conditions,this system cannot undergo a backward bifurcation.The global asymptotical stability of disease-free equilibrium,and the persistence of the disease are proved.The endemic equilibrium is globally attractive by using monotone iterative technique.The inefficiency of vaccination to susceptible people shows the linear positive relationship with basic reproductive number,while the inefficiency of vaccination to infected people reflects nonlinear positive relationship to basic reproductive number.