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A stochastic SIS epidemic model with heterogeneous contacts

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  • Economou, A.
  • Gómez-Corral, A.
  • López-García, M.

Abstract

A stochastic model for the spread of an SIS epidemic among a population consisting of N individuals, each having heterogeneous infectiousness and/or susceptibility, is considered and its behavior is analyzed under the practically relevant situation when N is small. The model is formulated as a finite time-homogeneous continuous-time Markov chain X. Based on an appropriate labeling of states, we first construct its infinitesimal rate matrix by using an iterative argument, and we then present an algorithmic procedure for computing steady-state measures, such as the number of infected individuals, the length of an outbreak, the maximum number of infectives, and the number of infections suffered by a marked individual during an outbreak. The time till the epidemic extinction is characterized as a phase-type random variable when there is no external source of infection, and its Laplace–Stieltjes transform and moments are derived in terms of a forward elimination backward substitution solution. The inverse iteration method is applied to the quasi-stationary distribution of X, which provides a good approximation of the process X at a certain time, conditional on non-extinction, after a suitable waiting time. The basic reproduction number R0 is defined here as a random variable, rather than an expected value.

Suggested Citation

  • Economou, A. & Gómez-Corral, A. & López-García, M., 2015. "A stochastic SIS epidemic model with heterogeneous contacts," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 421(C), pages 78-97.
    • Handle: RePEc:eee:phsmap:v:421:y:2015:i:c:p:78-97
    DOI: 10.1016/j.physa.2014.10.054
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    References listed on IDEAS

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    1. van Doorn, Erik A. & Pollett, Philip K., 2013. "Quasi-stationary distributions for discrete-state models," European Journal of Operational Research, Elsevier, vol. 230(1), pages 1-14.
    2. Vega-Redondo,Fernando, 2007. "Complex Social Networks," Cambridge Books, Cambridge University Press, number 9780521857406, October.
    3. Vega-Redondo,Fernando, 2007. "Complex Social Networks," Cambridge Books, Cambridge University Press, number 9780521674096, October.
    4. Farida Chamchod & Shigui Ruan, 2012. "Modeling the Spread of Methicillin-Resistant Staphylococcus aureus in Nursing Homes for Elderly," PLOS ONE, Public Library of Science, vol. 7(1), pages 1-9, January.
    5. Robert R Wilkinson & Kieran J Sharkey, 2013. "An Exact Relationship Between Invasion Probability and Endemic Prevalence for Markovian SIS Dynamics on Networks," PLOS ONE, Public Library of Science, vol. 8(7), pages 1-8, July.
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    Cited by:

    1. Amador, J. & Gómez-Corral, A., 2020. "A stochastic epidemic model with two quarantine states and limited carrying capacity for quarantine," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 544(C).
    2. Polly-Anne Jeffrey & Martín López-García & Mario Castro & Grant Lythe & Carmen Molina-París, 2020. "On Exact and Approximate Approaches for Stochastic Receptor-Ligand Competition Dynamics—An Ecological Perspective," Mathematics, MDPI, vol. 8(6), pages 1-31, June.
    3. Teng, Zhidong & Wang, Lei, 2016. "Persistence and extinction for a class of stochastic SIS epidemic models with nonlinear incidence rate," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 451(C), pages 507-518.
    4. Karen M Ong & Michael S Phillips & Charles S Peskin, 2020. "A mathematical model and inference method for bacterial colonization in hospital units applied to active surveillance data for carbapenem-resistant enterobacteriaceae," PLOS ONE, Public Library of Science, vol. 15(11), pages 1-32, November.
    5. Maria Gamboa & Maria Jesus Lopez-Herrero, 2020. "The Effect of Setting a Warning Vaccination Level on a Stochastic SIVS Model with Imperfect Vaccine," Mathematics, MDPI, vol. 8(7), pages 1-23, July.
    6. Antonio Gómez-Corral & Martín López-García & Maria Jesus Lopez-Herrero & Diana Taipe, 2020. "On First-Passage Times and Sojourn Times in Finite QBD Processes and Their Applications in Epidemics," Mathematics, MDPI, vol. 8(10), pages 1-25, October.

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