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2 results

  • Epidemic processes on digraphs of random mappings

  • Part of
  • Jerzy Jaworski
  • Journal:
    Journal of Applied Probability / Volume 36 / Issue 3 / September 1999
    Published online by Cambridge University Press:
    14 July 2016, pp. 780-798
    Print publication:
    September 1999

  • A random mapping (Tn;q) of a finite set V, V = {1,2,…,n}, into itself assigns independently to each i ∊ V its unique image j ∊ V with probability q if i = j and with probability P = (1-q)/(n−1) if i ≠ j. Three versions of epidemic processes on a random digraph GT representing (Tn;q) are studied. The exact probability distributions of the total number of infected elements as well as the threshold functions for these epidemic processes are determined.

  • Asymptotic final-size distribution for some chain-binomial processes

  • Gianpaolo Scalia-Tomba
  • Journal:
    Advances in Applied Probability / Volume 17 / Issue 3 / September 1985
    Published online by Cambridge University Press:
    01 July 2016, pp. 477-495
    Print publication:
    September 1985

  • The classical Reed-Frost process is generalized by allowing infection probabilities to depend on current epidemic size. Such a process can be imbedded in a simple Markov process derived from i.i.d. waiting times. The final size of the epidemic has the same distribution as the time for the first crossing of a certain linear barrier of the imbedding process. The asymptotic distribution of the final size can be derived from some weak convergence results for the imbedding process. The existence of a distribution determining set of harmonic functions for these chain-binomial processes is also established.