Epidemic Models On Dynamic Networks With Demographics

Infectious diseases not only threaten the physical and mental health of human beings,but also hinder the development of society seriously.It is vitally important and urgent to have a comprehensive understanding of the pathogenesis and transmission mechanism of infectious disease,and make effective prevention and control strategies.Traditional infectious disease models have mainly focused on an assumption that the population is well-mixed.However,this assumption is not realistic.The contact patterns in real populations tend to be heterogeneous.Considering individuals as nodes and the contacts among individuals as links between nodes,then the population can be regarded as a network.Therefore,it is more realistic to construct network-based models than using traditional well-mixed models when studying the dynamics of infectious disease.So far,great progress has been made in the study of network-based disease transmission models.However,most of them focus on static networks,ignoring the effect of demographics,which plays a critical role in the spread of infectious diseases.In fact,the demographics will change the network structure,which can in turn affect the disease propagation on networks.This leads to a natural question: how does the dynamics of networks interact with the disease dynamics on networks? In order to address this issue,we establish a series of dynamic network models of infectious diseases with demographics,investigate the effects of immigration and emigration of nodes,the degree distribution of newcomers,the mechanism that newcomers contact to existing nodes,and the distributions of infection time and recover time on the dynamics of disease propagation.The research will enrich the mathematical modeling methods of infectious diseases,and provide theoretical supports for the prevention and control of infectious diseases.The main works and innovations of this paper are summarized as follows:(1)For the problem that the dimensions of most epidemic models on networks are high,an SIR pairwise model with demographics is established.The dimension of this model is reduced by applying the generating function and two moment closure methods.Then we derive the basic reproduction number of the two low-dimensional models,and assess the rationality of two moment closure methods by comparing numerical results with stochastic simulations.Finally,we numerically analyze the effect of the immigration and emigration rates of nodes on the basic reproduction number and prevalence,and the effect of newcomer degree distribution on infectious disease spreading.(2)Considering that newcomers prefer to contact with large-degree nodes and avoid contacting with infectious nodes,we construct an SIS pairwise model with demographics and adaptive behaviors.The model is closed under two different assumptions for the network structure,respectively.Then,we reduce the dimensions of models by using the log-normal moment closure method,and derive the basic reproduction number of two low-dimensional models.The rationality of using log-normal moment closure method to reduce model and the accuracies of two models are evaluated by comparing the stochastic and numerical simulations.Finally,the effects of demographics and adaptive behaviors of newcomers on network structure and disease spreading are investigated through performing numerical simulations.(3)In order to study the effects of demographics,and different distributions of infection and recovery times on infectious disease propagation,we propose an SIR pairwise model with demographics and infection age.We obtain the basic reproduction number of this model,and prove the global stability of disease-free equilibrium and the existence and uniqueness of positive equilibrium.By performing numerical simulations,we analyze the effect of variance in infection time on infectious disease spreading when the infection time follows Gamma,Weibull or Uniform distributions,and the effect of variance in recovery time on infectious disease spreading when the recover time follows Gamma,Weibull or Uniform distributions.Furthermore,the effects of different recovery distributions with the same mean and variance on disease spreading are compared.Finally,the influence of demographics on disease spreading is investigated.(4)Most disease spreading models on networks with demographics assume that newcomers attach to existing nodes in networks randomly or according to degree preferential attachment,neglecting the local world and clustering properties of networks.For this problem,a new model with demographics is proposed.In the model,a new node is firstly linked to m existing nodes in the network according to local preferential attachment.Meanwhile,when a node in the network receives a link emanating form a newcomer,a randomly chosen neighbor of this node is also connected to the newcomer with probability p.We explore the impacts of different local world sizes and the ”triad formation” probability on the degree distribution and clustering of networks,and on the spread of infectious diseases by simulations.