Phase transition for SIR model with random transition rates on complete graphs

Abstract

We are concerned with the susceptible-infective-removed (SIR) model with random transition rates on complete graphs C_n with n vertices. We assign independent and identically distributed (i.i.d.) copies of a positive random variable \xi on each vertex as the recovery rates and i.i.d. copies of a positive random variable \rho on each edge as the edge infection weights. We assume that a susceptible vertex is infected by an infective one at rate proportional to the edge weight on the edge connecting these two vertices while an infective vertex becomes removed with rate equals the recovery rate on it, then we show that the model performs the following phase transition when at t = 0 one vertex is infective and others are susceptible. There exists {{\displaystyle \lambda }}_c\rangle \mathrm{0} such that when \lambda \langle {{\displaystyle \lambda }}_c, the proportion {{\displaystyle r}}_{\infty } of vertices which have ever been infective converges to 0 weakly as n\to +\infty while when \lambda \rangle {{\displaystyle \lambda }}_c, there exist c(\lambda )\rangle \mathrm{0} and b(\lambda )\rangle \mathrm{0} such that for each n\ge \mathrm{1} with probability p\ge b(\lambda ), the proportion {{\displaystyle r}}_{\infty }\ge c(\lambda ). Furthermore, we prove that {{\displaystyle \lambda }}_c is the inverse of the production of the mean of ρ and the mean of the inverse of \xi.

Keywords

Susceptible-infective-removed (SIR) model / complete graph / phase transition / random rate

References

[1] Bertacchi D, Lanchier N, Zucca F. Contact and voter processes on the infinite percolation cluster as models of host-symbiont interactions. Ann Appl Probab, 2011, 21: 1215–1252

[2] Bramson M, Durrett R, Schonmann R H. The contact process in a random environment. Ann Probab, 1991, 19: 960–983

[3] Chatterjee S, Durrett R. Contact processes on random graphs with power law degree distributions have critical value 0. Ann Probab, 2009, 37(6): 2332–2356

[4] Chen X X, Yao Q. The complete convergence theorem holds for contact processes on open clusters of ℤd×ℤ+. J Stat Phys, 2009, 135: 651–680

[5] Ethier S N, Kurtz T G. Markov Processes: Characterization and Convergence. Hoboken: John Wiley and Sons, 1986

[6] Harris T E. Additive set-valued Markov processes and graphical methods. Ann Probab, 1978, 6: 355–378

[7] Hofstad R V D. Random graphs and complex networks. Lecture notes, 2013

[8] Kesten H. Asymptotics in high dimensions for percolation. In: Grimmett G R,Welsh D J A, eds. Disorder in Physical Systems: A Volume in Honour of John M. Hammersley on the Occasion of His 70th Birthday. Oxford: Oxford Univ Press, 1990, 219–240

[9] Liggett T M. Interacting Particle Systems. New York: Springer, 1985

[10] Liggett T M. The survival of one-dimensional contact processes in random environments. Ann Probab, 1992, 20: 696–723

[11] Liggett T M. Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. New York: Springer, 1999

[12] Pastor-Satorras R, Vespignani A. Epidemic dynamics and endemic states in complex networks. Phys Rev E, 2001, 63(6): 066117

[13] Pastor-Satorras R, Vespignani A. Epidemic spreading in scale-free networks. Phys Rev Lett, 2001, 86(14): 3200–3203

[14] Peterson J. The contact process on the complete graph with random vertex-dependent infection rates. Stochastic Process Appl, 2011, 121(3): 609–629

[15] Wang J Z, Liu Z R, Xu J. Epidemic spreading on uncorrelated heterogeneous networks with non-uniform transmission. Phys A, 2007, 382(3): 715–721

[16] Wang J Z, Qian M. Discrete stochastic modeling for epidemics in networks. J Stat Phys, 2010, 140: 1157–1166

[17] Wang J Z, Qian M, Qian H. Circular stochastic fluctuations in SIS epidemics with heterogeneous contacts among sub-populations. Theor Popul Biol, 2012, 81: 223–231

[18] Xue X F. Contact processes with random vertex weights on oriented lattices. ALEA Lat Am J Probab Math Stat, 2015, 12: 245–259

[19] Xue X F. Critical value for contact processes on clusters of oriented bond percolation. Phys A, 2016, 448: 205–215

[20] Xue X F. Critical value for the contact process with random recovery rates and edge weights on regular tree. Phys A, 2016, 462: 793–806

[21] Xue X F. Phase transition for the large-dimensional contact process with random recovery rates on open clusters. J Stat Phys, 2016, 165: 845–865

[22] Xue X F. Law of large numbers for the SIR model with random vertex weights on Erdos-Renyi graph. Phys A, 2017, 486: 434–445

[23] Xue X F. Asymptotic for critical value of the large dimensional SIR epidemic on clusters. J Theoret Probab, 2017, online: 1–23

[24] Yao Q, Chen X X. The complete convergence theorem holds for contact processes in a random environment on ℤd×ℤ+. Stochastic Process Appl, 2012, 122(9): 3066–3100

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature