The Contact Process with switching
Jochen Blath1 ; Felix Hermann1 ; Michel Reitmeier1
1 Goethe-Universität Frankfurt am Main, Robert-Mayer Str. 10, 60629 Frankfurt, Germany
MathematicS In Action, Tome 12 (2023) no. 1, pp. 135-154.

In this paper, we introduce a type switching mechanism for the Contact Process. That is, we allow the individual particles/sites to switch between two (or more) types independently of one another, and the different types may exhibit specific infection and recovery dynamics. Such type switches can e.g. be motivated from biology, where “phenotypic switching” is common among micro-organisms. Our framework includes as special cases systems with switches between “active” and “dormant” states (the Contact Process with dormancy, CPD), and the Contact Process in a randomly evolving environment (CPREE) introduced by Broman (2007). The “standard” multi-type Contact Process (without type-switching) can also be recovered as a limiting case.

After constructing the process from a graphical representation, we first establish basic properties that are mostly analogous to the classical Contact Process. We then provide couplings between several variants of the system, obtaining sufficient conditions for the existence of a phase transition. Further, we investigate the effect of the switching parameters on the critical value of the system by providing rigorous bounds obtained from the coupling arguments as well as numerical and heuristic results. Finally, we investigate scaling limits for the process as the switching parameters tend to 0 (slow switching regime) resp. ∞ (fast switching regime). We conclude with a brief discussion of further model variants and questions for future research.

Publié le :
DOI : 10.5802/msia.35
Classification : 60K35, 92D25, 92D30
Mots clés : Contact Process, Coupling, Dormancy, Random environment, Switching
Jochen Blath 1 ; Felix Hermann 1 ; Michel Reitmeier 1

1 Goethe-Universität Frankfurt am Main, Robert-Mayer Str. 10, 60629 Frankfurt, Germany
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{MSIA_2023__12_1_135_0,
     author = {Jochen Blath and Felix Hermann and Michel Reitmeier},
     title = {The {Contact} {Process} with switching},
     journal = {MathematicS In Action},
     pages = {135--154},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {12},
     number = {1},
     year = {2023},
     doi = {10.5802/msia.35},
     language = {en},
     url = {https://msia.centre-mersenne.org/articles/10.5802/msia.35/}
}
TY  - JOUR
AU  - Jochen Blath
AU  - Felix Hermann
AU  - Michel Reitmeier
TI  - The Contact Process with switching
JO  - MathematicS In Action
PY  - 2023
SP  - 135
EP  - 154
VL  - 12
IS  - 1
PB  - Société de Mathématiques Appliquées et Industrielles
UR  - https://msia.centre-mersenne.org/articles/10.5802/msia.35/
DO  - 10.5802/msia.35
LA  - en
ID  - MSIA_2023__12_1_135_0
ER  - 
%0 Journal Article
%A Jochen Blath
%A Felix Hermann
%A Michel Reitmeier
%T The Contact Process with switching
%J MathematicS In Action
%D 2023
%P 135-154
%V 12
%N 1
%I Société de Mathématiques Appliquées et Industrielles
%U https://msia.centre-mersenne.org/articles/10.5802/msia.35/
%R 10.5802/msia.35
%G en
%F MSIA_2023__12_1_135_0
Jochen Blath; Felix Hermann; Michel Reitmeier. The Contact Process with switching. MathematicS In Action, Tome 12 (2023) no. 1, pp. 135-154. doi : 10.5802/msia.35. https://msia.centre-mersenne.org/articles/10.5802/msia.35/

[1] Martina Baar; Loren Coquille; Hannah Mayer; Michael Hoelzel; Meri Rogava; Thomas Tueting; Anton Bovier A stochastic model for immunotherapy of cancer, Sci. Rep., Volume 6 (2016), 24169 | DOI

[2] Nathalie Q. Balaban; Jack Merrin; Remy Chait; Lukasz Kowalik; Stanislas Leibler Bacterial persistence as a phenotypic switch, Science, Volume 305 (2004) no. 5690, pp. 1622-1625 | DOI

[3] Ten lectures on particle systems, Springer (1995), pp. 97-201 | DOI | Zbl

[4] Jochen Blath; Felix Hermann; Martin Slowik A branching process model for dormancy and seed banks in randomly fluctuating environments, J. Math. Biol., Volume 83 (2021) no. 2 | DOI | MR | Zbl

[5] Jochen Blath; András Tóbiás Invasion and fixation of microbial dormancy traits under competitive pressure, Stochastic Processes Appl., Volume 130 (2020) no. 12, pp. 7363-7395 | DOI | MR | Zbl

[6] Jochen Blath; András Tóbiás Microbial virus epidemics in the presence of contact-mediated host dormancy, ESAIM, Probab. Stat., Volume 27 (2023), pp. 174-220 | DOI | MR | Zbl

[7] Maury Bramson; Rick Durrett; Roberto H. Schonmann The contact process in a random environment, Ann. Probab., Volume 19 (1991) no. 3, pp. 960-983 | DOI | MR | Zbl

[8] Erik I. Broman Stochastic domination for a hidden Markov chain with applications to the Contact Process in randomly evolving environment, Ann. Probab., Volume 35 (2007) no. 6, pp. 2263-2293 | DOI | MR | Zbl

[9] Clément Dombry; Christian Mazza; Vincent. Bansaye Phenotypic diversity and population growth in a fluctuating environment, Adv. Appl. Probab., Volume 43 (2011) no. 2, pp. 375-398 | DOI | MR | Zbl

[10] Hiroko Endo; Masahiro Inoue Dormancy in cancer, Cancer Sci., Volume 110 (2019) no. 2, pp. 474-480 | DOI

[11] Robert A. Fisher; Bridget Gollan; Sophie Helaine Persistent bacterial infections and persister cells, Nat. Rev. Microbiol., Volume 15 (2017) no. 8, pp. 453-464 | DOI

[12] Simone Floreani; Cristian Giardinà; Frank den Hollander; Shubhamoy Nandan; Frank Redig Switching interacting particle systems: scaling limits, uphill diffusion and boundary layer, J. Stat. Phys., Volume 186 (2022) no. 3, 33, 45 pages | DOI | MR | Zbl

[13] Theodore E. Harris Contact interactions on a lattice, Ann. Probab., Volume 2 (1974) no. 6, pp. 969-988 | DOI | MR | Zbl

[14] Simon A. Jackson; Peter C. Fineran Bacterial dormancy curbs phage epidemics, Nature, Volume 570 (2019), pp. 173-174 | DOI

[15] Abel Klein Extinction of contact and percolation processes in a random environment, Ann. Probab., Volume 22 (1994) no. 3, pp. 1227-1251 | MR | Zbl

[16] Orkide Koyuncum; Margaret MacGibeny; Lynn Enquist Latent versus productive infection: the alpha herpesvirus switch, Future Virol., Volume 13 (2018), pp. 431-443 | DOI

[17] Jay T. Lennon; Frank den Hollander; Maite Wilke-Berenguer; Jochen Blath Principles of seed banks and the emergence of complexity from dormancy, Nat. Commun., Volume 12 (2021) no. 1 | DOI

[18] Jay T. Lennon; Stuart E. Jones Microbial seed banks: the ecological and evolutionary implications of dormancy, Nat. Rev. Microbiol., Volume 9 (2011), pp. 119-130 | DOI

[19] Kim Lewis Persister cells, Ann. Rev. Microbiol., Volume 64 (2010), pp. 357-372 | DOI

[20] Thomas M. Liggett Interacting particle systems, Grundlehren der Mathematischen Wissenschaften, 276, Springer, 1985, xv+488 pages | DOI | MR

[21] Thomas M. Liggett Improved upper bounds for the contact process critical value, Ann. Probab., Volume 23 (1995) no. 2, pp. 697-723 (Accessed 2022-05-05) | MR | Zbl

[22] Thomas M. Liggett Stochastic interacting systems: contact, voter and exclusion processes, Grundlehren der Mathematischen Wissenschaften, 324, Springer, 1999, xii+332 pages | DOI | MR

[23] Claudia Neuhauser Ergodic theorems for the multitype contact process, Probab. Theory Relat. Fields, Volume 91 (1992) no. 3-4, pp. 467-506 | DOI | MR | Zbl

[24] Daniel Remenik The contact process in a dynamic random environment, Ann. Appl. Probab., Volume 18 (2008) no. 6, pp. 2392-2420 | DOI | MR | Zbl

[25] Jeffrey E. Steif; Marcus Warfheimer The critical contact process in a randomly evolving environment dies out, ALEA, Lat. Am. J. Probab. Math. Stat., Volume 4 (2008), pp. 337-357 | MR | Zbl

Cité par Sources :