Epidemic Models On Random Switching Networks

The study of infectious diseases on complex networks is a direction of infectious disease transmission dynamics and a new field of mathematics in infectious disease transmission in recent 20 years.In human behavior,if individuals are regarded as nodes of the network,and the contact between individuals is regarded as the edge between nodes of the network,then human contact behavior can be abstracted into a complex static network.In real life,due to the random movement of human individuals(shopping,traveling,work,etc.),the social contact mode of individuals changes with time,that is to say,the topological structure of human social network is time-varying.Generally speaking,time-varying network can be understood as thus: the number of individuals in the network or the connecting edges between individuals increases or decreases with time,or it can be understood as network switching between a series of static networks.Compared with static networks,it is more realistic and reasonable to study the spread of infectious diseases on time-varying switching networks.This paper aims to study the mechanism of epidemic spreading in switching networks;establish the epidemic models of switching network;employ mathematical theories such as stochastic process and stochastic differential equations to study the representation of epidemiological parameters of infectious diseases and network topology parameters such as degree and its distribution in the model;make analysis of the dynamic behavior of the network epidemic models under the stochastic switching mechanism:computer the threshold of random extinction and persistence of the epidemic,investigate the ergodicity of the epidemic spreading process as well as the asymptotic behavior of the model solution,etc.;and compare the differences between epidemic spreading in the random switching network and the epidemic spreading in the static network.The results of this paper can expand and enrich the research methods of epidemic dynamics,and provide scientific theoretical basis for the prevention and control of epidemics.The innovations of this paper are as follows:(1)According to the degree-based static network epidemic modelling and using Markov switching mechanism,SIS network infectious disease model with Markov switching is established,and the threshold conditions of epidemic transmission are studied.The results show that the threshold conditions are not only related to the epidemic model parameters,but also correlated with the steady-state distribution of Markov chain.What is interesting is that under the mechanism of network stochastic switching,if the epidemic is stochastically persistent in some networks but extinct in others,it may be stochastically persistent or extinct in the whole networks,and the final result depends on the steady-state distribution of Markov chain.This illustrates that Markov chain plays an essential role in the evolution of epidemic,which differs from the spread of epidemic in the static networks.In addition,based on the Lyapunov function method,the positive recurrence and ergodicity of random spreading process are discussed.(2)To be more general,it is assumed that the waiting time between adjacent switching networks follows an arbitrary probability distribution,that is,the dependence between successive switching networks is considered to follow a semi-Markov process of continuous time.The degreebased network epidemic SIS model and semi-Markov mechanism in static networks are employed in this paper,a network epidemic model with stochastic switching is established,and a feasible method for analyzing the basic reproduction number R0 and threshold dynamics is proposed.It is theoretically proved that R0 is the weighted average of the basic reproduction numbers of subnetworks.If R0> 1,then the epidemic becomes stochastically persistent in the sense of time mean;otherwise,the epidemic becomes stochastically extinct.The numerical simulation also verifies the theoretical results.In addition,it is worth noting that although we only study semi-Markov processes under two states,the results of this paper can be easily extended to any finite state.This part is the extension of the epidemic model under the Markov switching mechanism.(3)By virtue of individual-based the network epidemic model,a Markov switching network epidemic model is established.Using Lyapunov function,we prove the existence,uniqueness,boundedness and long-term behavior of the positive solution of the model.In addition,we also discuss the sufficient conditions for the random extinction and persistence of the epidemic,which are not only related to the epidemic model parameters and network topology,but also correlated with the steady-state distribution of Markov chain.It is interesting that when the epidemic is extinct randomly in some subnetworks but persistent randomly in others,it may be extinct or persistent randomly in the whole networks.The final result depends on the probability of the subnetworks' dwelling in each state.That is to say,the greater the probability of subnetworks' dwelling in a certain state,the greater that state's influence on the subnetworks.(4)In order to study the influence of white noise on the spreading of epidemic in the switching networks,SIS epidemic model with Markov switching and white noise is established.The threshold conditions of infectious disease transmission is analyzed.The results show that compared with SIS network epidemic model which only considers colored noise(or switching),white noise plays a critically important role in the process of epidemic spreading: white noise can inhibit the spread of epidemic and large white noise can even make the persistent infectious diseases in the network tend to die out.In addition,asymptotic behavior of model solutions with small white noise is also studied.