Research On Modeling And Global Dynamics Of Disease Transmission On Complex Heterogeneous Networks

Throughout the human history,infectious diseases have been a serious threat to the public health and social development.So it is crucial to study the dynamical mechanism of disease spread,and then develop control strategies toward disease outbreaks.In the study of epidemiol-ogy,mathematical models have become an important tool in studying epidemic dynamics.How-ever,traditional models are based on the homogeneous mixing assumption.In fact,individual behaviors and disease transmissions exhibit heterogeneity.Fortunately,the shortcoming can be greatly overcome by the theory of complex heterogeneous networks.And it also leads to a burst of research on network-based epidemic models.To better understand and control the spread of infectious diseases,in this thesis,based on the work of predecessors,we propose several epidemic models with different characteristics and study the global dynamics of these models on complex heterogeneous networks.The main findings are presented as follows:Firstly,we present an SEIRS epidemic model with nonlinear infectivity on complex het-erogeneous networks,and mainly focus on the global dynamics of this model.By using the next generation matrix method,the basic reproduction number is calculated.By employing Lya-punov functional approach and LaSalle invariance principle,the global asymptotical stability of the disease-free equilibrium is obtained.By utilizing the uniform persistence theory,the perma-nence of the disease is proved.Further,by applying a monotone iteration scheme and the com-parison principle,we find sufficient conditions under which the endemic equilibrium is globally attractive.In addition,vaccination is a very powerful controlling strategy to reduce the spread of diseases.So the effects of two major immunization schemes are studied and compared.Secondly,we propose an SIS epidemic model with a general infection rate on complex het-erogeneous networks,and analyze its global stability.We calculate the basic reproduction number,which is closely related to the heterogeneous infection rate.With the help of the Lyapunov's di-rect method,the global stability of the disease-free equilibrium is shown.By using the uniform persistence theory and the theory of cooperative system,the permanence of the disease and the global stability of the endemic equilibrium are proved,respectively.Based on the heterogene-ity of contact patterns,the effects of various immunization schemes are discussed and compared.Meanwhile,we explore the relation between the vaccination rate and the recovery rate.Thirdly,we study the global dynamics of an SIQRS epidemic model with vaccination on complex heterogeneous networks.We analytically derive the basic reproduction number,which determines not only the existence of endemic equilibrium but also the global dynamics of the mod-el.The permanence of the disease and the global asymptotical stability of disease-free equilibrium are rigorously proved.By constructing a series of iterative sequences,we show that the unique endemic equilibrium is globally attractive under certain conditions.Furthermore,we discuss the effectiveness of quarantine and vaccination against infectious diseases.Fourthly,we investigate the global dynamics of an SIRS epidemic model with a general feed-back mechanism on complex heterogeneous networks.In contrast to previous models,we further consider the different fear levels of individuals with different potential number of contacts when an epidemic prevails.The basic reproduction number is obtained by mathematical analysis,and the comparison principle can be used to show the global asymptotical stability of the disease-free equilibrium.Using the generalized Lajmanovich-Yorke theorem,the persistence of the model is proved.Furthermore,applying the monotone iterative technique,we study the global attractivity of the endemic equilibrium.Although the general feedback mechanism cannot change the ba-sic reproduction number,theoretical and numerical results indicate that it plays an active role in reducing the occurrence of disease.Fifthly,we develop an SIS epidemic model with a general nonlinear incidence rate on com-plex heterogeneous networks,and study its global stability.By constructing Lyapunov func-tion and using LaSalle invariance principle,the disease-free equilibrium is proved to be globally asymptotically stable.According to the idea of the proof of Lajmanovich-Yorke theorem,the per-sistence of the disease is obtained.Furthermore,by applying an iteration scheme and the theory of cooperative system,we obtain sufficient conditions under which the endemic equilibrium is globally asymptotically stable.Significantly,the basic reproduction number is independent of the specific form of the nonlinear incidence rate,but our simulations show that the nonlinear incidence rate does affect the dynamical behaviors of epidemic spreading.Finally,we intensively study the global stability of a malware propagation model with weakly-protected and strongly-protected susceptible nodes on heterogeneous networks.Based on this model,a control parameter that completely determines the global dynamics of mobile malware propagation is verified.By a simple comparison method,the global asymptotical stabil-ity of the malware-free equilibrium is proved.With the help of Lyapunov theorem and LaSalle invariance principle,we obtain the global asymptotical stability conditions for the malware equi-librium.Moreover,the permanence of mobile malware is proved by using the uniform persistence theory.Interestingly,increasing the recovery rate of infected nodes can result in the increase of strongly-protected susceptible nodes and the decrease of the basic reproduction number.