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A multi-stage infectious disease model on the complete graph

  • School of Science, Beijing Jiaotong University, Beijing 100044, China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2020.02.008
  • Received Date: 23 January 2019
  • Accepted Date: 21 May 2019
  • Rev Recd Date: 21 May 2019
  • Publish Date: 28 February 2020
    Abstract

  • Abstract

    The classical contact process is an interactive particle system model based on the complete graph Cn of n points. This is a continuous-time Markov process with state space{0，1}Cn, which explores the survival of two-stage disease spread at a certain rate on the graph. However, particles in the model may have more than two states. To this end, a multi-stage infectious disease model with a propagation rate of λn(λ>0) was considered, its future trends under long-term effects was studied. And the critical value λc(λc>0) was explored, so that when λ>λc, the infectious disease survives with a high probability within the exponential time eCn; when λ<λc, the infectious disease extincts with a high probability within the logarithmic time Clnn.

    Abstract

    The classical contact process is an interactive particle system model based on the complete graph Cn of n points. This is a continuous-time Markov process with state space{0，1}Cn, which explores the survival of two-stage disease spread at a certain rate on the graph. However, particles in the model may have more than two states. To this end, a multi-stage infectious disease model with a propagation rate of λn(λ>0) was considered, its future trends under long-term effects was studied. And the critical value λc(λc>0) was explored, so that when λ>λc, the infectious disease survives with a high probability within the exponential time eCn; when λ<λc, the infectious disease extincts with a high probability within the logarithmic time Clnn.
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    • References(7)
  • [1]
    HARRIS T E. Contact interactions on a lattice[J]. The Annals of Probability, 1974, 2(6): 969-988.
    [2]
    LIGGETT T M. Interacting Particle Systems[M]. New York: Springer, 1985.
    [3]
    LIGGETT T M. Stochastic interacting systems: Contact, voter and exclusion processes[M]. New York: Springer, 1999.
    [4]
    KRONE S. The two-stage contact process[J]. The Annals of Applied Probability, 1999, 9(2): 331-351.
    [5]
    FOXALL E. New results for the two-stage contact process[J]. Journal of Applied Probability, 2015, 52(1): 258-268.
    [6]
    XUE X F. The critical infection rate of the high-dimensional two-stage contact process[J]. Statistics and Probability Letters, 2018, 140: 115-125.
    [7]
    PETERSON J. The contact process on the complete graph with random vertex-dependent infection rates[J]. Stochastic Processes and Their Applications, 2011, 121(3): 609-629.
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    [1]
    HARRIS T E. Contact interactions on a lattice[J]. The Annals of Probability, 1974, 2(6): 969-988.
    [2]
    LIGGETT T M. Interacting Particle Systems[M]. New York: Springer, 1985.
    [3]
    LIGGETT T M. Stochastic interacting systems: Contact, voter and exclusion processes[M]. New York: Springer, 1999.
    [4]
    KRONE S. The two-stage contact process[J]. The Annals of Applied Probability, 1999, 9(2): 331-351.
    [5]
    FOXALL E. New results for the two-stage contact process[J]. Journal of Applied Probability, 2015, 52(1): 258-268.
    [6]
    XUE X F. The critical infection rate of the high-dimensional two-stage contact process[J]. Statistics and Probability Letters, 2018, 140: 115-125.
    [7]
    PETERSON J. The contact process on the complete graph with random vertex-dependent infection rates[J]. Stochastic Processes and Their Applications, 2011, 121(3): 609-629.
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