## The Annals of Probability

### Fluctuations of the front in a one-dimensional model for the spread of an infection

#### Abstract

We study the following microscopic model of infection or epidemic reaction: red and blue particles perform independent nearest-neighbor continuous-time symmetric random walks on the integer lattice $\mathbb{Z}$ with jump rates $D_{R}$ for red particles and $D_{B}$ for blue particles, the interaction rule being that blue particles turn red upon contact with a red particle. The initial condition consists of i.i.d. Poisson particle numbers at each site, with particles at the left of the origin being red, while particles at the right of the origin are blue. We are interested in the dynamics of the front, defined as the rightmost position of a red particle. For the case $D_{R}=D_{B}$, Kesten and Sidoravicius established that the front moves ballistically, and more precisely that it satisfies a law of large numbers. Their proof is based on a multi-scale renormalization technique, combined with approximate sub-additivity arguments. In this paper, we build a renewal structure for the front propagation process, and as a corollary we obtain a central limit theorem for the front when $D_{R}=D_{B}$. Moreover, this result can be extended to the case where $D_{R}>D_{B}$, up to modifying the dynamics so that blue particles turn red upon contact with a site that has previously been occupied by a red particle. Our approach extends the renewal structure approach developed by Comets, Quastel and Ramírez for the so-called frog model, which corresponds to the $D_{B}=0$ case.

#### Article information

Source
Ann. Probab., Volume 44, Number 4 (2016), 2770-2816.

Dates
Revised: April 2015
First available in Project Euclid: 2 August 2016

https://projecteuclid.org/euclid.aop/1470139154

Digital Object Identifier
doi:10.1214/15-AOP1034

Mathematical Reviews number (MathSciNet)
MR3531680

Zentralblatt MATH identifier
1356.60160

#### Citation

Bérard, Jean; Ramírez, Alejandro. Fluctuations of the front in a one-dimensional model for the spread of an infection. Ann. Probab. 44 (2016), no. 4, 2770--2816. doi:10.1214/15-AOP1034. https://projecteuclid.org/euclid.aop/1470139154

#### References

• [1] Alves, O. S. M., Machado, F. P. and Popov, S. Yu. (2002). The shape theorem for the frog model. Ann. Appl. Probab. 12 533–546.
• [2] Avena, L., dos Santos, R. S. and Völlering, F. (2013). Transient random walk in symmetric exclusion: Limit theorems and an Einstein relation. ALEA Lat. Am. J. Probab. Math. Stat. 10 693–709.
• [3] Bérard, J. and Ramírez, A. F. (2010). Large deviations of the front in a one-dimensional model of $X+Y\to 2X$. Ann. Probab. 38 955–1018.
• [4] Bérard, J. and Ramírez, A. F. (2012). Fluctuations of the front in a one-dimensional model for the spread of an infection. Preprint. Available at arXiv:1210.6781.
• [5] Comets, F., Quastel, J. and Ramírez, A. F. (2009). Fluctuations of the front in a one dimensional model of $X+Y\to2X$. Trans. Amer. Math. Soc. 361 6165–6189.
• [6] Dembo, A. and Zeitouni, O. (2010). Large Deviations Techniques and Applications. Stochastic Modelling and Applied Probability 38. Springer, Berlin.
• [7] den Hollander, F., dos Santos, R. and Sidoravicius, V. (2013). Law of large numbers for non-elliptic random walks in dynamic random environments. Stochastic Process. Appl. 123 156–190.
• [8] Feller, W. (1966). An Introduction to Probability Theory and Its Applications. Vol. II. Wiley, New York.
• [9] Hilário, M., den Hollander, F., Sidoravicius, V., Soares dos Santos, R. and Teixeira, A. (2014). Random Walk on Random Walks. Preprint. Available at arXiv:1401.4498.
• [10] Kesten, H. and Sidoravicius, V. (2005). The spread of a rumor or infection in a moving population. Ann. Probab. 33 2402–2462.
• [11] Kesten, H. and Sidoravicius, V. (2008). A problem in one-dimensional diffusion-limited aggregation (DLA) and positive recurrence of Markov chains. Ann. Probab. 36 1838–1879.
• [12] Kesten, H. and Sidoravicius, V. (2008). A shape theorem for the spread of an infection. Ann. of Math. (2) 167 701–766.
• [13] Kumar, N. and Tripathy, G. (2010). Velocity of front propagation in the epidemic model $A+B\rightarrow 2A$. Eur. Phys. J. B 78 201–205.
• [14] Mai, D., Sokolov, I. M., Kuzovkov, V. N. and Blumen, A. (1997). Front form and velocity in a one-dimensional autocatalytic $A+B\to 2A$ reaction. Phys. Rev. E 56 4130–4134.
• [15] Mai, J., Sokolov, I. M. and Blumen, A. (1996). Front propagation and local ordering in one-dimensional irreversible autocatalytic reactions. Phys. Rev. Lett. 77 4462–4465.
• [16] Mai, J., Sokolov, I. M. and Blumen, A. (2000). Front propagation in one-dimensional autocatalytic reactions: The breakdown of the classical picture at small particle concentrations. Phys. Rev. E 62 141–145.
• [17] McDiarmid, C. (1998). Concentration. In Probabilistic Methods for Algorithmic Discrete Mathematics. Algorithms Combin. 16 195–248. Springer, Berlin.
• [18] Panja, D. (2004). Effects of fluctuations on propagating fronts. Phys. Rep. 393 87–174.
• [19] Ramírez, A. F. and Sidoravicius, V. (2004). Asymptotic behavior of a stochastic combustion growth process. J. Eur. Math. Soc. (JEMS) 6 293–334.