The Comparison Of Several Different Pair-Approximation Methods For SIS Epidemic Model

Epidemics have been along with people's production and livelihood since ancient times.Some disrupt the regular life of the public and some have threat to human life.Therefore,it has been always focus to know the mechanism of the spread of infectious diseases and make corresponding prevention and controlling measures.As epidemics are not feasible by experiment in the crowd,it is a valid and effective method to research and forecast epidemics by dynamical models.Taking into account the heterogeneity of contacts between individuals,network-based dynamical epidemic models are models which are more in tune with the real life.Pair-approximation model is one of network-based epidemic models,which looks up nodes and edges of networks as variables to explore the laws of the spread of epidemics.The accuracy and rationality of this kind of models depend largely on the pair-approximation methods.In this paper,we compare the accuracy,advantages and disadvantages of different pair-approximation methods by SIS epidemic models on homogeneous,heterogeneous and clustering networks.Chapter 1,firstly,gives the research background ? significance of epidemic models on network.Then,it introduces several topological network parameters,four traditional network models and the classification of network-based epidemic models.Finally,the progress of pair-approximation models and the current research of approximating methods are depicted,further leading to the focus of our works.In Chapter 2,via SIS pair-approximation models(P-PW,B-PW and ER-PW),the accuracy of three approximation methods in the homogeneous networks is compared.The first method is based on the number of infected neighborhoods of individuals following Poisson distribution.The second is based on the number of infected neighborhoods of individuals following multinomial distribution.The last one is based on the mean-field theory.Then we find the error of the pair-approximation model under Poisson distribution smallest,i.e.the accuracy of approximation methods under Poisson distribution is highest.In chapter 3,we compare the accuracy of heterogeneous approximation method Keeling proposed and super compact pairwise method Simon and Kiss proposed on heterogeneous networks by SIS models;Firstly,the detailed derivation of the super compact pair-approximation formula of Simon and Kiss is given.Then,based on these two approximation methods,K(10)1 dimensions SIS pairwise models(H-PW and HSH-PW)and three dimensions SIS pairwise model(HSL-PW)are given.It is found by calculating the basic reproduction numbers and simulating for the three models that the super compact pair-approximation method can not only reduce the dimension but make the SIS epidemic models incorporate more network topology parameters without losing models accuracy.In chapter 4,for clustering network,we similarly compare the accuracy of approximation methods Keeling presented and super compact pairwise method Sherborne,Blyuss and Kiss presented via SIS pair-approximation model.Utilizing that two methods containing clustering coefficient,we derived three SIS pairwise models with clustering: K(10)1 dimensions SIS pairwise models with clusters(HC-PW and HSHC-PW)and three dimensions SIS pairwise model with clustering(HSLC-PW).Then,simulation verifies the rationality of the model.Finally,it is found via error analysis that super compact pair-approximation method is the most accurate.Chapter 5,gives the conclusions and the prospects.