A model for infection spread on graphs

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A model for infection spread on graphs

• A. J. Ganesh
• University of Bristol
• Joint work with Moez Draief (Imperial College)

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Simple epidemic model

• Undirected graph G(V,E)
• Individuals perform independent random walks on
  the graph
• Unit mean residence time at each node
• Equiprobable to go to each neighbour
• Infection can pass between individuals only when
  they are at the same site
• Pairwise infection rate ?

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Question

• If there is initially a single susceptible and a
  single infectious agent, how long does it take
  for the susceptible to become infected?
• Detailed results for complete graphs (Datta and
  Dorlas, 2004)
• What about other graphs?

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Motivation

• Many infectious diseases have limited range
• Hotspots responsible for spread
• e.g., cattle markets for foot and mouth
• transport hubs for SARS?
• Relevant to some non-biological epidemics as well
• e.g., worms/viruses spread by Bluetooth devices

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Applicability of model

• Sparsely populated network
• fluctuations matter
• pairwise interactions a good approximation
• Can use differential equation (population
  dynamics) models for densely populated networks

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Random walks on graphs

• G(V,E) connected, undirected
• Random walk Xt
• continuous time Markov process on V
• Transition rate qxy 1/degree(x) if (x,y) ? E
• Equilibrium distribution
• ?(v) degree(v)/D,
• D sum of degrees

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Coincidence time

• T(t) total coincidence time of two independent
  random walkers on G, up to time t
• Random variable, distribution depends on initial
  conditions
• Infection probability is 1?exp(??T(t))
• How to estimate these quantities?

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General results

• If random walkers start in stationary
  distribution, then
• ET(t) ?v?V ?(v)2 t
• Infection probability bounded by
• exp(???v?V ?(v)2 t)

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Large graph asymptotics

• Can we say something about how the coincidence
  time and infection probability behave on large
  graphs?
• Consider a sequence of graphs from some model
• Study scaling behaviour in the limit

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Graph model (1)

• Erdos-Renyi graph G(n,p)
• n nodes
• Edge between each pair of nodes present with
  probability p, independent of all other edges
• Advantage simplicity
• Disadvantage not realistic in many practical
  settings

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Power law graphs

• Power law graph with exponent ?
• number of vertices with degree k is proportional
  to k?? .
• Differs from classical graph models (like
  Erdos-Renyi model)
• number of vertices with degree k decays
  exponentially in k
• Many real-world networks exhibit power laws
  (e.g., hub and spoke networks)

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Power law random graph model (Chung and Lu)

• Random graph with expected degrees w1,,wn
• edge (i,j) present w.p. wi wj / ?k wk
• independent of all other edges
• Particular choice wi m(1i/i0)-1/(?-1)
• m maximal expected degree
• i0 parameter which can be expressed in terms of
  maximal and average expected degrees

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Results Homogenous graphs

• Regular graphs and Erdos-Renyi random graphs
• ET(t) t/n
• Mean time to infection scales like n

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Results power law graphs

• nET(t) / t ??
• constant, if ?gt3
• constant log(m), if ?3
• (m/d)(3??) , if 2lt?lt3
• Implication mean infection time much smaller
  than n if ?lt3
• ?lt3 corresponds to unbounded variance of node
  degree distribution

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Idea of proof

• Define
• D1 ?v degree(v)
• D2 ?v degree(v)(degree(v)?1)
• Compute expectations of these random variables
• Show that they concentrate around their expected
  values with high probability
• ?v ?(v)2 (D1D2) / (D1)2

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Open problems

• Lower bounds for infection probability
• Arbitrary initial conditions
• Other epidemic models (SIR, SIS)
• Densely populated networks