The Annals of Applied Probability

Partial immunization processes

Alan Stacey

Full-text: Open access

Abstract

Partial immunization processes are generalizations of the contact process in which the susceptibility of a site to infection depends on whether or not it has been previously infected. Such processes can exhibit a phase of weak survival, in which the process survives but drifts off to infinity, even on graphs such as $\mathbb{Z}^d$, where no such phase exists for the contact process. We establish that whether or not strong survival occurs depends only on the rate at which sites are reinfected and not on the rate at which sites are infected for the first time. This confirms a prediction by Grassberger, Chaté and Rousseau. We then study the processes on homogeneous trees, where the behaviour is also related to that of the contact process whose infection rate is equal to the reinfection rate of the partial immunization process. However, the phase diagram turns out to be substantially richer than that of either the contact process on a tree or partial immunization processes on $\mathbb{Z}^d$.

Article information

Source
Ann. Appl. Probab., Volume 13, Number 2 (2003), 669-690.

Dates
First available in Project Euclid: 18 April 2003

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1050689599

Digital Object Identifier
doi:10.1214/aoap/1050689599

Mathematical Reviews number (MathSciNet)
MR1970282

Zentralblatt MATH identifier
1030.60092

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Contact process immunization tree weak survival critical value phase diagram

Citation

Stacey, Alan. Partial immunization processes. Ann. Appl. Probab. 13 (2003), no. 2, 669--690. doi:10.1214/aoap/1050689599. https://projecteuclid.org/euclid.aoap/1050689599


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References

  • [1] ATHREy A, K. B. and NEY, P. E. (1972). Branching Processes. Springer, Berlin.
  • [2] BEZUIDENHOUT, C. and GRIMMETT, G. (1990). The critical contact process dies out. Ann. Probab. 18 1462-1482.
  • [3] BEZUIDENHOUT, C. and GRIMMETT, G. (1991). Exponential decay for subcritical and percolation processes. Ann. Probab. 19 984-1009.
  • [4] BOLLOBÁS, B. (1998). Modern Graph Theory. Springer, New York.
  • [5] DURRETT, R. (1988). Lectures Notes on Particle Sy stems and Percolation. Wadsworth, Belmont, CA.
  • [6] DURRETT, R. (1995). Ten Lectures on Particle Sy stems. Lecture Notes in Math. 1608. Springer, New York.
  • [7] DURRETT, R. and SCHINAZI, R. (2000). Boundary modified contact processes. J. Theoret. Probab. 13 575-594.
  • [8] GRASSBERGER, P., CHATÉ, H. and ROUSSEAU, G. (1997). Spreading in media with long-time memory. Phy s. Rev. E 55 2488-2495.
  • [9] HARRIS, T. E. (1974). Contact interactions on a lattice. Ann. Probab. 2 969-988.
  • [10] HARRIS, T. E. (1978). Additive set-valued Markov processes and graphical methods. Ann. Probab. 6 355-378.
  • [11] KULASMAA, K. (1982). The spatial general epidemic and locally dependent random graphs. J. Appl. Probab. 19 745-758.
  • [12] LALLEY, S. (1999). Growth profile and invariant measures for the weakly supercritical contact process on a homogeneous tree. Ann. Probab. 27 206-225.
  • [13] LALLEY, S. and SELLKE, T. (1998). Limit set of a weakly supercritical contact process on a homogeneous tree. Ann. Probab. 26 644-657.
  • [14] LIGGETT, T. M. (1985). Interacting Particle Sy stems. Springer, New York.
  • [15] LIGGETT, T. M. (1996). Multiple transition points for the contact process on the binary tree. Ann. Probab. 24 1675-1710.
  • [16] LIGGETT, T. M. (1999). Stochastic Interacting Sy stems: Contact, Voter and Exclusion Processes. Springer, New York.
  • [17] Ly ONS, R. (2000). Phase transitions on nonamenable graphs. J. Math. Phy s. 41 1099-1126.
  • [18] PEMANTLE, R. (1992). The contact process on trees. Ann. Probab. 20 2089-2116.
  • [19] SALZANO, M. and SCHONMANN, R. H. (1998). A new proof that for the contact process on homogeneous trees local survival implies complete convergence. Ann. Probab. 26 1251- 1258.
  • [20] SALZANO, M. and SCHONMANN, R. H. (1999). The second lowest extremal invariant measure of the contact process II. Ann. Probab. 27 845-875.
  • [21] SCHONMANN, R. H. (1998). The triangle condition for contact processes on homogeneous trees. J. Statist. Phy s. 90 1429-1440.
  • [22] STACEY, A. M. (1996). The existence of an intermediate phase for the contact process on trees. Ann. Probab. 24 1711-1726.
  • [23] STACEY, A. M. (2001). The contact process on finite trees. Probab. Theory Related Fields 121 551-576.