# A shape theorem for the spread of an infection

@article{Kesten2003AST, title={A shape theorem for the spread of an infection}, author={Harry Kesten and Vladas Sidoravicius}, journal={Annals of Mathematics}, year={2003}, volume={167}, pages={701-766} }

In [KSb] we studied the following model for the spread of a rumor or infection: There is a “gas” of so-called A-particles, each of which performs a continuous time simple random walk on Z d , with jump rate DA. We assume that “just before the start” the number of A-particles at x, NA(x, 0−), has a mean μA Poisson distribution and that the NA(x, 0−) ,x ∈ Z d , are independent. In addition, there are B-particles which perform continuous time simple random walks with jump rate DB. We start with a… Expand

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#### References

SHOWING 1-10 OF 28 REFERENCES

The spread of a rumor or infection in a moving population

- Mathematics, Biology
- 2003

The principal result states that if D A = D B (so that the A- and B-particles perform the same random walk), then there exist constants 0 < C i < ∞ such that almost surely C(C 2 t) C B(t) C C (C 1 t) for all large t, where C(r) = [-r, r] d . Expand

The shape theorem for the frog model with random initial configuration

- Mathematics
- 2001

We prove a shape theorem for a growing set of simple random walks on Z d , known as frog model. The dynamics of this process is described as follows: There are active particles, which perform… Expand

Models of First-Passage Percolation

- Physics
- 2004

First-passage percolation (FPP) was introduced by Hammersley and Welsh in 1965 (see [26]) as a model of fluid flow through a randomly porous material. Envision a fluid injected into the material at a… Expand

THE FRONT VELOCITY OF THE SIMPLE EPIDEMIC

- Mathematics
- 1979

The propagation rate of the one-dimensional stochastic simple epidemic converges almost surely to a front velocity for the epidemic. Percolation For a concise description of the model, consider a… Expand

The shape theorem for the frog model

- Mathematics
- 2001

In this work we prove a shape theorem for a growing set of Simple Random Walks (SRWs), known as frog model. The dynamics of this process is described as follows: There are some active particles,… Expand

Asymptotic behavior of a stochastic combustion growth process

- Mathematics
- 2004

We study a continuous time growth process on the d-dimensional hypercubic lattice Zd , which admits a phenomenological interpretation as the combustion reaction A + B → 2A, where A represents heat… Expand

First-Passage Percolation on the Square Lattice

- Mathematics
- 1978

We consider several problems in the theory of first-passage percolation on the two-dimensional integer lattice. Our results include: (i) a mean ergodic theorem for the first-passage time from (0,0)… Expand

Threshold growth dynamics

- Physics, Mathematics
- 1993

We study the asymptotic shape of the occupied region for monotone deterministic dynamics in d-dimensional Euclidean space parametrized by a threshold θ > 0, and a Borel set N ⊂ R d with positive and… Expand

Branching Random Walk with Catalysts

- Mathematics
- 2003

Shnerb et al. (2000), (2001) studied the following system of interacting particles on $\Bbb Z^d$: There are two kinds of particles, called $A$-particles and $B$-particles. The $A$-particles perform… Expand

Asymptotic shape for the chemical distance and first-passage percolation in random environment

- Mathematics
- 2003

The aim of this paper is to generalize the well-known asymptotic shape result for first-passage percolation on $\Zd$ to first-passage percolation on a random environment given by the infinite cluster… Expand