A network with tunable clustering, degree correlation and degree distribution, and an epidemic thereon
 Frank Ball,
 Tom Britton,
 David Sirl
 … show all 3 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
A random network model which allows for tunable, quite general forms of clustering, degree correlation and degree distribution is defined. The model is an extension of the configuration model, in which stubs (halfedges) are paired to form a network. Clustering is obtained by forming small completely connected subgroups, and positive (negative) degree correlation is obtained by connecting a fraction of the stubs with stubs of similar (dissimilar) degree. An SIR (Susceptible \(\rightarrow \) Infective \(\rightarrow \) Recovered) epidemic model is defined on this network. Asymptotic properties of both the network and the epidemic, as the population size tends to infinity, are derived: the degree distribution, degree correlation and clustering coefficient, as well as a reproduction number \(R_*\) , the probability of a major outbreak and the relative size of such an outbreak. The theory is illustrated by Monte Carlo simulations and numerical examples. The main findings are that (1) clustering tends to decrease the spread of disease, (2) the effect of degree correlation is appreciably greater when the disease is close to threshold than when it is well above threshold and (3) disease spread broadly increases with degree correlation \(\rho \) when \(R_*\) is just above its threshold value of one and decreases with \(\rho \) when \(R_*\) is well above one.
 Andersson H (1997) Epidemics in a population with social structures. Math Biosci 140:79–84 CrossRef
 Andersson H (1998) Limit theorems for a random graph epidemic model. Ann Appl Prob 8:1331–1349 CrossRef
 Andersson H (1999) Epidemic models and social networks. Math Sci 24(2):128–147
 Andersson H, Britton T (2000) Stochastic epidemic models and their statistical analysis. Springer lecture notes in statistics, vol 151. Springer, New York
 Badham J, Stocker R (2010) The impact of network clustering and assortativity on epidemic behaviour. Theor Pop Biol 77:71–75 CrossRef
 Ball FG (1983) The threshold behaviour of epidemic models. J Appl Prob 20:227–241 CrossRef
 Ball FG (1986) A unified approach to the distribution of total size and total area under the trajectory of the infectives in epidemic models. Adv Appl Prob 18:289–310 CrossRef
 Ball FG (2000) Susceptibility sets and the final outcome of stochastic SIR epidemic models. Research Report 00–09. Division of Statistics, School of Mathematical Sciences, University of Nottingham
 Ball FG, Lyne OD (2001) Stochastic multitype SIR epidemics among a population partitioned into households. Adv Appl Prob 33:99–123 CrossRef
 Ball FG, Mollison D, ScaliaTomba G (1997) Epidemics with two levels of mixing. Ann Appl Prob 7:46–89 CrossRef
 Ball FG, Neal P (2002) A general model for stochastic SIR epidemics with two levels of mixing. Math Biosci 180:73–102 CrossRef
 Ball FG, O’Neill PD (1999) The distribution of general final state random variables for stochastic epidemic models. J Appl Prob 36:473–491 CrossRef
 Ball FG, Sirl DJ (2012) An SIR epidemic model on a population with random network and household structure, and several types of individuals. Adv Appl Prob 44:63–86 CrossRef
 Ball FG, Sirl DJ, Trapman P (2009) Threshold behaviour and final outcome of an epidemic on a random network with household structure. Adv Appl Prob 41:765–796 CrossRef
 Ball FG, Sirl DJ, Trapman P (2010) Analysis of a stochastic SIR epidemic on a random network incorporating household structure. Math Biosci 224: 53–73 (See also erratum (2010), ibid. 225:81.)
 Ball FG, Sirl DJ, Trapman P (2012) Epidemics on random intersection graphs (submitted). arXiv:1011.4242
 Britton T, Lindholm M, Turova T (2011) A dynamic network in a dynamic population: asymptotic properties. J Appl Prob 48:1163–1178 CrossRef
 Britton T, Nordvik MK, Liljeros F (2007) Modelling sexually transmitted infections: the effect of partnership activity and number of partners on \(R_0\) . Theor Pop Biol 72:389–399 CrossRef
 Britton T, Deijfen M, Lindholm M, Lagerås AN (2008) Epidemics on random graphs with tunable clustering. J Appl Prob 45:743–756 CrossRef
 Coupechoux E, Lelarge M (2012) How clustering affects epidemics in random, networks. arXiv:1202.4974
 Diekmann O (1978) Thresholds nad traveling waves for the geographical spread of infection. J Math Biol 6:109–130 CrossRef
 Diekmann O, Heesterbeek JAP (2000) Mathematical epidemiology of infectious diseases. John Wiley, Chichester
 Diekmann O, Heesterbeek JAP, Metz JAJ (1990) On the definition and computation of the basic reproduction ratio \(R_0\) in models for infectious diseases in heterogeneous populations. J Math Biol 28:365–382 CrossRef
 Diekmann O, de Jong MCM, Metz JAJ (1998) A deterministic epidemic model taking account of repeated contacts between the same individuals. J Appl Prob 35:448–462 CrossRef
 Durrett R (2006) Random graph dynamics. Cambridge University Press, Cambridge CrossRef
 Erdős P, Rényi A (1959) On random graphs. Publicationes Mathematicae 6:290–297
 Feller W (1971) An introduction to probability theory and its applications, vol II, 2nd edn. Wiley, New York
 Gleeson JP (2009) Bond percolation on a class of clustered random networks. Phys Rev E 80:036107 CrossRef
 Gleeson JP, Melnik S, Hackett A (2010) How clustering affects the bond percolation threshold in complex networks. Phys Rev E 81:066114 CrossRef
 van der Hofstad R, Litvak N (2012) Degree–degree correlations in random graphs with heavytailed degrees. arXiv:1202.3071v3
 Isham V, Kaczmarska J, Nekovee M (2011) Spread of information and infection on finite random networks. Phys Rev E 83:046128 CrossRef
 Janson S (2009) The probability that a random multigraph is simple. Combinatorics Prob Comput 18:205–225 CrossRef
 Karrer B, Newman MEJ (2010) Random graphs containing arbitrary distributions of subgraphs. Phys Rev E 82:066118 CrossRef
 Ma J, van den Driessche P, Willeboordse FH (2012) Effective degree household network disease model. J Math Biol. doi:10.1007/s0028501105029
 May RM, Anderson RM (1987) Transmission dynamics of HIV infections. Nature 326:137–142 CrossRef
 Miller JC (2009) Percolation and epidemics in random clustered networks. Phys Rev E 80:020901(R)
 Mode CJ (1971) Multitype branching processes. Theory and applications. In: Modern analytic and computational methods in science and mathematics, vol 34. Elsevier, New York
 Mollison D (1977) Spatial contact models for ecological and epidemic spread. J Roy Stat Soc B 39(3):283–326
 Molloy M, Reed B (1995) A critical point for random graphs with a given degree sequence. Rand Struct Alg 6:161–179 CrossRef
 Newman MEJ, Strogatz SH, Watts DJ (2001) Random graphs with arbitrary degree distributions and their applications. Phys Rev E 64:026118 CrossRef
 Newman MEJ (2002a) Assortative mixing in networks. Phys Rev Lett 89:208701 CrossRef
 Newman MEJ (2002b) Spread of epidemic disease on networks. Phys Rev E 66:016128 CrossRef
 Newman MEJ (2003) The structure and function of complex networks. SIAM Rev 45:167–256 CrossRef
 Newman MEJ (2009) Random graphs with clustering. Phys Rev Lett 103:058701 CrossRef
 Trapman P (2007) On analytical approaches to epidemics on networks. Theor Pop Biol 71:160–173 CrossRef
 Volz E (2004) Random networks with tunable degree distribution and clustering. Phys Rev E 70:056115 CrossRef
 Warde WD, Katti SK (1971) Infinite divisibility of discrete distributions, II. Ann Math Stat 42:1088–1090
 Watts SC, Strogatz SH (1998) Collective dynamics of ‘smallworld’ networks. Nature 393:440–442 CrossRef
 Title
 A network with tunable clustering, degree correlation and degree distribution, and an epidemic thereon
 Journal

Journal of Mathematical Biology
Volume 66, Issue 45 , pp 9791019
 Cover Date
 20130301
 DOI
 10.1007/s0028501206097
 Print ISSN
 03036812
 Online ISSN
 14321416
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Branching process
 Configuration model
 Epidemic size
 Random graph
 SIR epidemic
 Threshold behaviour
 92D30
 05C80
 60J80
 Industry Sectors
 Authors

 Frank Ball ^{(1)}
 Tom Britton ^{(2)}
 David Sirl ^{(3)}
 Author Affiliations

 1. School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK
 2. Department of Mathematics, Stockholm University, Stockholm, 106 91, Sweden
 3. Mathematics Education Centre, Loughborough University, Loughborough, Leicestershire, LE11 3TU, UK