On the asymptotic geometrical behaviour of a class of contact interaction processes with a monotone infection rate

Schürger, Klaus

doi:10.1007/bf00534880

Cited by 23 publications

(21 citation statements)

References 13 publications

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“…The reproduction rule for abnormal cells may also be modified so that the rate of infection at a healthy (normal) site is monotonic in the number of unhealthy (abnormal) neighbours, rather than proportional, as we have been assuming. The almost sure analogue of (3) for this modification is shown by Schiirger (8). Although it will not concern us in this paper, we point out that the exact nature of the norm || ||' is still unknown.…”

Section: \ a A { ) (Xea) Atrate \{Y E A C -^Y -X\\ =mentioning

confidence: 94%

“…(Theset A may be random.) Also, let T (^>8) be the hitting time of <f> for (£,l A ' 8) ) an(^ (Q A ' s) )> which one is meant will be clear from the context. T A is the hitting time of <f> for (E,t A ' s) )-When dealing with the processes started at time 0, we will frequently drop the time superscript, and write (^• 4i0) ) as (£?…”

Section: Mauby Bbamson and David Griffeathmentioning

confidence: 99%

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On the Williams-Bjerknes tumour growth model: II

Bramson

Griffeath

1980

Math. Proc. Camb. Phil. Soc.

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Williams and Bjerknes introduced in 1972 a stochastic model for the spread of cancer cells. Cells, normal and abnormal (cancerous), are situated on a planar lattice. With each cellular division, one daughter cell stays put, while the other usurps the position of a neighbour; abnormal cells reproduce at a faster rate than normal cells. We are interested in the long-term behaviour of this system. We showed in 'On the WilliamsBjerknes Tumour Growth Model I ' d ) that, provided it lives forever, the tumour will eventually contain a ball of linearly expanding radius. Here it is shown that the rate of expansion is actually linear, and that the region of infection has an asymptotic shape which is given by some (unknown) norm. To demonstrate that the tumour contains a ball of linearly expanding radius, we applied in (l) certain techniques common to the field of interacting particle systems; in particular we made use of certain auxiliary Markov chains which are imbedded in the dual processes of our model. Here different techniques are applied. We show that the first infection times of different sites are almost subadditive, and we exhibit various regularity properties of the infection times such as bounds on their moments. These properties of the Williams-Bjerknes process allow one to follow the outline prescribed by Richardson in (7), where he showed that such a process does indeed exhibit a linear growth whose shape is prescribed by a norm. IntroductionIn a 1972 paper, Williams and Bjerknes(9) proposed a simple stochastic model for the spread of cancer cells. Based on biological considerations, they restricted attention to the basal layer of an epithelium, thereby obtaining a two-dimensional setting. The cells, normal and abnormal, are situated on a planar lattice. With each cellular division, one daughter cell stays put while the other usurps the position of a neighbour. Splitting of each normal cell is assumed to occur at rate 1, whereas due to 'carcinogenic advantage', each cancer cell splits at rate K > 1.Assuming the lattice to be Z 2 , these axioms give rise to a simple continuous time Markov chain on the state space iS 0 = {finite subsets of Z 2 }. (Actually, the hexagonal lattice was preferred in (9), though the square and triangular lattice were considered as well. Our results apply to all three lattices; Z 2 is chosen largely for notational convenience.) If we let E,f denote the set of sites occupied by cancer cells at time t, given that the original cancerous population occupies A eS 0 , then the processes (£,i) t > 0 are strong Markov (see, for example, (2)). Their jump rates are given by

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Section: \ a A { ) (Xea) Atrate \{Y E A C -^Y -X\\ =mentioning

confidence: 94%

Section: Mauby Bbamson and David Griffeathmentioning

confidence: 99%

On the Williams-Bjerknes tumour growth model: II

Bramson

Griffeath

1980

Math. Proc. Camb. Phil. Soc.

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“…If θ > 1 this argument needs a substantial extra step, as the dynamics can only be successfully restarted from a fairly large set. As is the case in Example 1, it is possible that an x ∈ A t is nowhere near, say, a large fully occupied square; this renders the usual shape theorem (of [5] and [15]) false.…”

Section: Introductionmentioning

confidence: 93%

“…At first glance, the dynamics described above do not differ appreciably from those in [15] and [5]. Indeed, if θ = 1, then the standard subadditive arguments work: the main observation is that, for any time t 0 , one can get a lower bound on the original occupied set at any time t ≥ t 0 by picking an occupied point at time t 0 , throwing the other occupied points away, and restarting the dynamics.…”

Section: Introductionmentioning

confidence: 95%

Random threshold growth dynamics

Bohman

Gravner

1999

Random Struct. Alg.

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A site in Z 2 becomes occupied with a certain probability as soon as it sees at least a threshold number of already occupied sites in its neighborhood. Such randomly growing sets have the following regularity property: a large fully occupied set exists within a fixed distance (which does not increase with time) of every occupied point. This property suffices to prove convergence to an asymptotic shape.

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“…Thus the probability density (12) is a unique positive density satisfying (13). It is easy to see that this statement is equivalent to saying that the spatial birth-and-death process described above is a unique time reversible spatial birth-and-death process specified by the unit death rate and by the birth rates c(ñ(x, X)), x ∈ Λ (given a current state X), where c(·) is a positive function.…”

Section: The Models With Similar Dynamicsmentioning

confidence: 99%

On Equilibrium Distribution of a Reversible Growth Model

Shcherbakov

Yambartsev

2012

J Stat Phys

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We study a probabilistic model of interacting spins indexed by elements of a finite subset of the d-dimensional integer lattice, d ≥ 1. Conditions of time reversibility are examined. It is shown that the model equilibrium distribution converges to a limit distribution as the indexing set expands to the whole lattice. The occupied site percolation problem is solved for the limit distribution. Two models with similar dynamics are also discussed.

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On the asymptotic geometrical behaviour of a class of contact interaction processes with a monotone infection rate

Cited by 23 publications

References 13 publications

On the Williams-Bjerknes tumour growth model: II

On the Williams-Bjerknes tumour growth model: II

Random threshold growth dynamics

On Equilibrium Distribution of a Reversible Growth Model

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