The effect of avoiding known infected neighbors on the persistence of a recurring infection process. (English) Zbl 1507.60126

Summary: We study a generalization of the classical contact process (SIS epidemic model) on a directed graph \(G\). Our model is a continuous-time interacting particle system in which at every time, each vertex is either healthy or infected, and each oriented edge is either active or inactive. Infected vertices become healthy at rate 1 and pass the infection along each active outgoing edge at rate \(\lambda\). At rate \(\alpha\), healthy individuals deactivate each incoming edge from their infected neighbors, and an inactive edge becomes active again as soon as its tail vertex becomes healthy. When \(\alpha =0\), this model is the same as the classical contact process on a static graph. We study the persistence time of this epidemic model on the lattice \(\mathbb{Z}\), the \(n\)-cycle \({\mathbb{Z}_n}\), and the \(n\)-star graph. We show that on \(\mathbb{Z}\), for every \(\alpha > 0\), there is a phase transition in \(\lambda\) between almost sure extinction and positive probability of indefinite survival; on \({\mathbb{Z}_n}\) we show that there is a phase transition between poly-logarithmic and exponential survival time as the size of the graph increases. On the star graph, we show that the survival time is \({n^{\Delta +o(1)}}\) for an explicit function \(\Delta (\alpha ,\lambda)\) whenever \(\alpha > 0\) and \(\lambda > 0\). In the cases of \(\mathbb{Z}\) and \({\mathbb{Z}_n}\), our results qualitatively match what has been shown for the classical contact process, while in the case of the star graph, the classical contact process exhibits exponential survival for all \(\lambda > 0\), which is qualitatively different from our result. This model presents a challenge because, unlike the classical contact process, it has not been shown to be monotonic in the infection parameter \(\lambda\) or the initial infected set.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
05C22 Signed and weighted graphs
92D30 Epidemiology
91D30 Social networks; opinion dynamics
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[1] Noam Berger, Christian Borgs, Jennifer Chayes, and Amin Saberi, On the spread of viruses on the internet, Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms (New York), ACM, 2005, pp. 301-310. · Zbl 1297.68029
[2] Shankar Bhamidi, Danny Nam, Oanh Nguyen, and Allan Sly, Survival and extinction of epidemics on random graphs with general degrees, Ann. Probab. 49 (2021), no. 1, 244-286. · Zbl 1478.60254
[3] Van Hao Can, Metastability for the contact process on the preferential attachment graph, Internet Math. (2017), 45pp. · Zbl 1491.05165
[4] Van Hao Can and Bruno Schapira, Metastability for the contact process on the configuration model with infinite mean degree, Electron. J. Probab. 20 (2015), no. 26, 22pp. · Zbl 1327.82051
[5] Shirshendu Chatterjee and Rick Durrett, Contact processes on random graphs with power law degree distributions have critical value 0, Ann. Probab. 37 (2009), no. 6, 2332-2356. · Zbl 1205.60168
[6] Rick Durrett and Claudia Neuhauser, Epidemics with recovery in \[d=2\], Ann. Appl. Probab. 1 (1991), no. 2, 189-206. · Zbl 0733.92022
[7] Rick Durrett and Roberto H. Schonmann, Large deviations for the contact process and two dimensional percolation, Probab. Theory Related Fields 77 (1988), no. 4, 583-603. · Zbl 0621.60108
[8] Richard Durrett, Oriented percolation in two dimensions, Ann. Probab. 12 (1984), no. 4, 999-1040. · Zbl 0567.60095
[9] Eric Foxall, The SEIS model, or, the contact process with a latent stage, J. Appl. Probab. 53 (2016), no. 3, 783-801. · Zbl 1351.60102
[10] Thilo Gross and Bernd Blasius, Adaptive coevolutionary networks: a review, J. R. Soc. Interface 5 (2008), no. 20, 259-271.
[11] Thilo Gross, Carlos J. Dommar D’Lima, and Bernd Blasius, Epidemic dynamics on an adaptive network, Phys. Rev. Lett. 96 (2006), 208701.
[12] Dongchao Guo, Stojan Trajanovski, Ruud Bovenkamp, Huijuan Wang, and Piet Van Mieghem, Epidemic threshold and topological structure of susceptible-infectious-susceptible epidemics in adaptive networks, Phys. Rev. E 88 (2013), 042802.
[13] Xiangying Huang and Rick Durrett, The contact process on random graphs and galton-watson trees, ALEA Lat. Am. J. Probab. Math. Stat., 17 (2020), 159-182. · Zbl 1439.60082
[14] Emmanuel Jacob and Peter Mörters, The contact process on scale-free networks evolving by vertex updating, R. Soc. Open Sci. 4 (2017), no. 5, 14pp. · Zbl 1367.05194
[15] Karly A. Jacobsen, Mark G. Burch, Joseph H. Tien, and Grzegorz A. Rempala, The large graph limit of a stochastic epidemic model on a dynamic multilayer network, J. Biol. Dyn. 12, (2016), no 1, 746-788. · Zbl 1447.92433
[16] Yufeng Jiang, Remy Kassem, Grayson York, Matthew Junge, and Rick Durrett, SIR epidemics on evolving graphs, 1901.06568, (2019).
[17] T. M. Liggett, R. H. Schonmann, and A. M. Stacey, Domination by product measures, Ann. Probab. 25 (1997), no. 1, 71-95. · Zbl 0882.60046
[18] T.M. Liggett, Stochastic interacting systems: Contact, voter and exclusion processes, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Springer, 1999. · Zbl 0949.60006
[19] Thomas Mountford, Jean-Christophe Mourrat, Daniel Valesin, and Qiang Yao, Exponential extinction time of the contact process on finite graphs, Stochastic Process. Appl. 126 (2016), no. 7, 1974-2013. · Zbl 1346.82023
[20] Thomas Mountford, Daniel Valesin, and Qiang Yao, Metastable densities for the contact process on power law random graphs, Electron. J. Probab. 18 (2013), 36 pp. · Zbl 1281.82018
[21] Robin Pemantle, The contact process on trees, Ann. Probab. 20 (1992), no. 4, 2089-2116. · Zbl 0762.60098
[22] Daniel Remenik, The contact process in a dynamic random environment, Ann. Appl. Probab. 18 (2008), no. 6, 28pp. · Zbl 1181.60153
[23] David Sivakoff, Contact process on a graph with communities, ALEA Lat. Am. J. Probab. Math. Stat. 14 (2017), 9-31. · Zbl 1355.60123
[24] András Szabó-Solticzky, Luc Berthouze, Istvan Z Kiss, and Péter L Simon, Oscillating epidemics in a dynamic network model: stochastic and mean-field analysis, J. Math. Biol. 72 (2016), no. 5, 1153-1176. · Zbl 1333.05276
[25] Ilker Tunc, Maxim S. Shkarayev, and Leah B. Shaw, Epidemics in adaptive social networks with temporary link deactivation, J. Stat. Phys. 151 (2013), no. 1-2, 355-366. · Zbl 1329.92141
[26] Achillefs Tzioufas, Oriented percolation with density close to one, 1311.2952v4 (2014).
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