A note on monotonicity of spatial epidemic models. (English) Zbl 1423.92239

Tzioufas, Achillefs

Braz. J. Probab. Stat. 33, No. 3, 674-684 (2019).

Summary: The epidemic process on a graph is considered for which infectious contacts occur at rate which depends on whether a susceptible is infected for the first time or not. We show that the Vasershtein coupling extends if and only if secondary infections occur at rate which is greater than that of initial ones. Nonetheless we show that, with respect to the probability of occurrence of an infinite epidemic, the said proviso may be dropped regarding the totally asymmetric process in one dimension, thus settling in the affirmative this special case of the conjecture for arbitrary graphs due to A. Stacey [Ann. Appl. Probab. 13, No. 2, 669–690 (2003; Zbl 1030.60092)].

MSC:

92D30	Epidemiology
60E15	Inequalities; stochastic orderings

Keywords:

three state contact processes; stochastic domination; attractiveness; contact process; standard spatial epidemic

Citations:

Zbl 1030.60092

References:

[1]	Durrett, R. (1981). An introduction to infinite particle systems. Stochastic Processes and Their Applications11, 109-150. · Zbl 0462.60095
[2]	Durrett, R. (1988). Lecture Notes on Particle Systems and Percolation. Belmont: Wadsworth. · Zbl 0659.60129
[3]	Durrett, R. (1995). Ten Lectures on Particle Systems. Lecture Notes in Math.1608. New York: Springer. · Zbl 0840.60088
[4]	Durrett, R. and Schinazi, R. B. (2000). Boundary modified contact processes. Journal of Theoretical Probability13, 575-594. · Zbl 0995.60090
[5]	Grassberger, P., Chate, H. and Rousseau, G. (1997). Spreading in media with long-time memory. Physical Review E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics55, 2488-2495.
[6]	Griffeath, D. (1979). Additive and Cancelative Interacting Particle Systems. Lecture Notes in Math.724. Berlin: Springer. · Zbl 0412.60095
[7]	Grimmett, G. R. (1999). Percolation, 2nd ed. Berlin: Springer. · Zbl 0926.60004
[8]	Harris, T. E. (1978). Additive set valued Markov processes and graphical methods. Annals of Probability6, 355-378. · Zbl 0378.60106
[9]	Holley, R. (1972). An ergodic theorem for interacting systems with attractive interactions. Probability Theory and Related Fields24, 325-334. · Zbl 0251.60066
[10]	Kuulasmaa, K. (1982). The spatial general epidemic and locally dependent random graphs. Journal of Applied Probability19, 745-758. · Zbl 0509.60094
[11]	Liggett, T. M. (1985). Interacting Particle Systems. New York: Springer. · Zbl 0559.60078
[12]	Liggett, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. New York: Springer. · Zbl 0949.60006
[13]	Liggett, T. M., Schonmann, R. H. and Stacey, A. M. (1997). Domination by product measures. Annals of Probability25, 71-95. · Zbl 0882.60046
[14]	Lindvall, T. (2002). Lectures on the Coupling Method. New York: Dover. · Zbl 1013.60001
[15]	Mollison, D. (1977). Spatial contact models for ecological and epidemic spread. Journal of the Royal Statistical Society, Series B, Statistical Methodology39, 283-326. · Zbl 0374.60110
[16]	Schinazi, R. B. (1999). Classical and Spatial Stochastic Processes. Boston: Birkhauser. · Zbl 0919.60002
[17]	Schonmann, R. H. (1986). The asymmetric contact process. Journal of Statistical Physics44, 505-534. · Zbl 0649.92018
[18]	Stacey, A. M. (2003). Partial immunization processes. The Annals of Applied Probability13, 669-690. · Zbl 1030.60092
[19]	Van Den Berg, J., Grimmett, G. R. and Schinazi, R. B. (1998). Dependent random graphs and spatial epidemics. The Annals of Applied Probability8, 317-336. · Zbl 0946.92028