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The SEIS model, or, the contact process with a latent stage

Part of: Mathematical biology in general Markov processes

Published online by Cambridge University Press:  24 October 2016

Eric Foxall
Eric Foxall*
Affiliation:
Arizona State University
*
* Postal address: School of Mathematical and Statistical Sciences, Arizona State University, PO Box 871804, Tempe, AZ, 85287-1804, USA. Email address: eric.foxall@asu.edu
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Abstract

The susceptible→exposed→infectious→susceptible (SEIS) model is well known in mathematical epidemiology as a model of infection in which there is a latent period between the moment of infection and the onset of infectiousness. The compartment model is well studied, but the corresponding particle system has so far received no attention. For the particle system model in one spatial dimension, we give upper and lower bounds on the critical values, prove convergence of critical values in the limit of small and large latent time, and identify a limiting process to which the SEIS model converges in the limit of large latent time.

Keywords

SEIS modelcontact processinteracting particle system

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces 92B99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 3 , September 2016 , pp. 783 - 801
Copyright
Copyright © Applied Probability Trust 2016 

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References

[1] Brauer, F.,van den Driessche, P. and Wu, J.(eds) (2008).Mathematical Epidemiology.Springer,Berlin.Google Scholar
[2] Durrett, R. (1980).On the growth of one-dimensional contact processes.Ann. Prob. 8,890–907.CrossRefGoogle Scholar
[3] Durrett, R. (1991).The contact process, 1974–1989. InMathematics of Random Media,American Mathematical Society,Providence, RI. pp.1–18 Google Scholar
[4] Durrett, R. (1992).Stochastic growth models: bounds on critical values.J. Appl. Prob. 29,11–20.Google Scholar
[5] Durrett, R. (2010).Random Graph Dynamics.Cambridge University Press.Google Scholar
[6] Durrett, R. and Griffeath, D. (1983).Supercritical contact processes on Z .Ann. Prob. 11,1–15.CrossRefGoogle Scholar
[7] Foxall, E.(2015).New results for the two-stage contact process.J. Appl. Prob. 52,258–268.CrossRefGoogle Scholar
[8] Griffeath, D. (1981).The basic contact processes.Stoch. Process. Appl. 11,151–185.Google Scholar
[9] Harris, T. E. (1978).Additive set-valued Markov processes and graphical methods.Ann. Prob. 6,355–378.Google Scholar
[10] Korobeinikov, A. (2004).Lyapunov functions and global properties for SEIR and SEIS epidemic models.Math. Medicine Biol. 21,75–83.Google Scholar
[11] Krone, S. M. (1999).The two-stage contact process.Ann. Appl. Prob. 9,331–351.Google Scholar
[12] Liggett, T. M. (1985).Interacting Particle Systems.Springer,New York.Google Scholar
[13] Liggett, T. M. (1995).Improved upper bounds for the contact process critical value.Ann. Prob. 23,697–723.Google Scholar
[14] Liggett, T. M. (1999).Stochastic Interacting Systems: Contact, Voter and Exclusion Processes.Springer,Berlin.CrossRefGoogle Scholar
[15] Mountford, T.,Mourrat, J.-C.,alesin, D. and Yao, Q. (2016).Exponential extinction time of the contact process on finite graphs.Stoch. Process. Appl. 126,1974–2013.Google Scholar
[16] Norris, J. R. (1997).Markov Chains.Cambridge University Press.Google Scholar
[17] Ziezold, H. and Grillenberger, C. (1988).On the critical infection rate of the one-dimensional basic contact process: numerical results.J. Appl. Prob. 25,1–8.Google Scholar