SIR Dynamics In Random Networks With Communities

Control of epidemic spreading has always attracted interest in the biological mathe-matics field.Much research on infectious disease transmission in complex populations has focused on understanding the implications of network structure on the epidemic processes supported by the research.Recent works have shown that community structure is a univer-sal feature of contact networks.Although many previous works have studied many ways on community structure,there exists some problems whether it is calculation or results.For example,although the model of Miller assumes the same distribution of within and between-group partnerships in both populations,which implies the same community structure in the two populations,the total degree distribution in this process is also changed.Consequently,their work ignores the impact of community structure on the epidemic dynamics.In Koch,the random edge removal alters not only the community structure,but also the total degree distribution.Therefore,they cannot discern whether their dynamical behaviors are altered by changing the total degree distribution or by changing the community structure.For the reasons given above,the paper not only considers degree heterogeneous and arbitrary joint degree distribution,but also introduces the community structure Q into this paper.Further more,it studies the effects of the community structure of a network on the spread of an epidemic with the fixing quantities as well as the calculation is relatively simple.We first establish a SIR model in a two-community network with an arbitrary joint degree distribution.Due to heterogeneous and astringency,the network is formulated as a probability generating function with joint degree distribution.We also obtain the sufficient conditions for disease outbreak and extinction.This paper is divided into five chapters.In Chapter 1,we introduce some preliminaries,research background and main content.In Chapter 2,introduce PGF with community structure as well as generating of com-munity networks.In Chapter 3,the present paper extends the Volz network SIR,model to disease spread-ing in two-community complex networks with an arbitrary joint degree distribution,and established a 12 dimensional model.Our model is consistent with the model of Miller with n = 2.Although the model of Miller has fewer dimensions than our model,the basic reproduction number of their model requires finding the root of a quartic equation,which greatly complicates the disease threshold analysis.In Chapter 4,we obtain threshold for SIR model with community structure.Analysing PGF with Poisson joint degree distribution.confirming that the model is reliable and accurate,we obtain threshold condition theorem for epidemic spread in a complex network with a Poisson joint degree distribution.In Chapter 5,analyze effect of community structure on spread of disease.The conclusion is that,by strengthening the community structure in the simulation,i.e.fixing the total degree distribution and reducing the number ratio of the external edges,we can increase or decrease the final cumulative epidemic incidence depending on the transmissibility of the virus between humans and the community structure at that point and obtain a result that disease transmission is most obviously affected by the community structure when the human-to-human transmissibility of the virus is near the threshold.Why community structure can affect disease dynamics in a complicated way is also discussed.In any case,for large-scale epidemics,strengthening the community structure to reduce the size of disease is undoubtedly an effective way.