Network Epidemic Modeling And Global Analysis In Terms Of Pair Approximation

The epidemic transmission among population can be seen as propagation behaviors onnetwork which obey a law along network edges. Network epidemic models in terms of pairsreflect the mutual influence between network topology and diseases spread, which are morepractical than traditional homogeneous mixing models. Epidemic models according to pairapproximation, one of network epidemic models, regard network edges with differentattributes as variables and research dynamic evolution of these variables on network. Therehave been many results about pair approximation models, but dynamical analysis for thesemodels has no been found. Therefore, paper systematically makes global stability analysisof SIS pair approximation models on regular and random networks and performs numericalsimulations, supplying and enriching findings in pair approximation models on network.Besides, with the in-depth study on epidemics, the role, stochasticity plays on infectiousdiseases modeling, becomes increasingly important. The advantage of stochastic models isthat it considers the necessary stochastic factor which deterministic models have ignored.Thus, paper, by applying Markov Process in stochastic process, builds an SIS stochasticpair approximation epidemic model with birth and death on network and deduces itsdeterministic model, then dynamic analysis and numerical simulation of the obtained modelare made. The research results not only investigate the change of dynamic network edgesbut make network modeling methods based on pair approximation diversified.Chapter1, firstly, introduces study significance on network epidemic models, statisticscharacteristics describing network topology and classical four networks. Then, it presentsthe development situation of network pair approximation epidemic models and gives theintroduction about Markov Process in stochastic process and relative fundamental theory. Finally, the development of stochastic pair approximation models is briefly depicted.Chapter2, in terms of SIS pair approximation model on regular and random network,reduces dimensionality of the model and closes obtained model according to the topologicalcharacteristics of the two networks. Then, the reproduction number can be attained and theglobally dynamical behaviors are analyzed and demonstrated by applying mathematicalmethods like Lyapunov and Dulac function. Besides, appropriate numerical simulation isperformed to verify the theoretical validity. The conclusion provides theoretical basis forpair approximation models research.Chapter3, in accordance with Markov Process in stochastic process and applyingmathematical tools such as transition probability, Q matrix, Kolmogorov equation andmoment generating function, derives an SIS stochastic pair approximation epidemic modelwith birth and death on network. Then we obtain the reproduction number and verify thatvia numerical simulation.