Graphs with specified degree distributions, simple epidemics, and local vaccination strategies

Consider a random graph, having a prespecified degree distribution F, but other than that being uniformly distributed, describing the social structure (friendship) in a large community. Suppose that one individual in the community is externally infected by an infectious disease and that the disease has its course by assuming that infected individuals infect their not yet infected friends independently with probability p. For this situation, we determine the values of R 0, the basic reproduction number, and τ0, the asymptotic final size in the case of a major outbreak. Furthermore, we examine some different local vaccination strategies, where individuals are chosen randomly and vaccinated, or friends of the selected individuals are vaccinated, prior to the introduction of the disease. For the studied vaccination strategies, we determine R v , the reproduction number, and τ v , the asymptotic final proportion infected in the case of a major outbreak, after vaccinating a fraction v.

[1]  John Scott.  What is social network analysis , 2010 .

[2]  John Scott.  Social Network Analysis , 1988 .

[3]  Paul Erdös,et al.  The Giant Component 1960-1993 , 1993, Random Struct. Algorithms.

[4]  Svante Janson,et al.  The Birth of the Giant Component , 1993, Random Struct. Algorithms.

[5]  Bruce A. Reed,et al.  A Critical Point for Random Graphs with a Given Degree Sequence , 1995, Random Struct. Algorithms.

[6]  O. Kallenberg.  Foundations of Modern Probability , 2021, Probability Theory and Stochastic Modelling.

[7]  Bruce A. Reed,et al.  The Size of the Giant Component of a Random Graph with a Given Degree Sequence , 1998, Combinatorics, Probability and Computing.

[8]  Svante Janson,et al.  Random graphs , 2000, Wiley-Interscience series in discrete mathematics and optimization.

[9]  M. Newman,et al.  Epidemics and percolation in small-world networks. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[10]  Svante Janson,et al.  Random graphs , 2000, ZOR Methods Model. Oper. Res..

[11]  H. Andersson,et al.  Stochastic Epidemic Models and Their Statistical Analysis , 2000 .

[12]  M. Newman,et al.  Random graphs with arbitrary degree distributions and their applications. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  Béla Bollobás,et al.  Random Graphs: Notation , 2001 .

[14]  Reuven Cohen,et al.  Efficient immunization strategies for computer networks and populations. , 2002, Physical review letters.

[15]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[16]  Gerry Leversha,et al.  Foundations of modern probability (2nd edn), by Olav Kallenberg. Pp. 638. £49 (hbk). 2002. ISBN 0 387 95313 2 (Springer-Verlag). , 2004, The Mathematical Gazette.

[17]  A. Martin-Löf,et al.  Generating Simple Random Graphs with Prescribed Degree Distribution , 2006, 1509.06985.

[18]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[19]  Béla Bollobás,et al.  The phase transition in inhomogeneous random graphs , 2007, Random Struct. Algorithms.

[20]  Mikko Alava,et al.  Branching Processes , 2009, Encyclopedia of Complexity and Systems Science.

[21]  Svante Janson.  The Probability That a Random Multigraph is Simple , 2009, Comb. Probab. Comput..