Contact process (mathematics)

The Contact Process (on a 1–D lattice): Active sites are indicated by grey circles and inactive sites by dotted circles. Active sites can activate inactive sites to either side of them at a rate r/2 or become inactive at rate 1.

The contact process is a model of an interacting particle system. It is a continuous time Markov process with state space $\{0,1\}^S$ , where $S$ is a finite or countable graph, usually Z ${}^d$ . The process is usually interpreted as a model for the spread of an infection: if the state of the process at a given time is $\eta$ , then a site $x$ in $S$ is "infected" if $\eta(x)=1$ and healthy if $\eta(x)=0$ . Infected sites become healthy at a constant rate, while healthy sites become infected at a rate proportional to the number infected neighbors. One can generalize the state space to $\{0,\ldots, \kappa\}^S$ , such is called the multitype contact process. It represents a model when more than one type of infection is competing for space.

Dynamics

More specifically, the dynamics of the basic contact process is defined by the following transition rates: at site $x$ ,

1\rightarrow0\quad\mbox{at rate }1,

0\rightarrow1\quad\mbox{at rate }\lambda\sum_{y:y\sim x}\eta(y),

where the sum is over all the neighbors in $S$ of $x$ . This means that each site waits an exponential time with the corresponding rate, and then flips (so 0 becomes 1 and vice versa).

For each graph $S$ there exists a critical value $\lambda _{c}$ for the parameter $\lambda$ so that if $\lambda>\lambda_c$ then the 1's survive (that is, if there is at least one 1 at time zero, then at any time there are ones) with positive probability, while if $\lambda<\lambda_c$ then the process dies out. For contact process on the integer lattice, a major breakthrough came in 1990 when Bezuidenhout and Grimmett showed that the contact process also dies out at the critical value. Their proof makes use of percolation theory.

Voter model

Main article: Voter model

The voter model (usually in continuous time, but there are discrete versions as well) is a process similar to the contact process. In this process $\eta (x)$ is taken to represent a voter's attitude on a particular topic. Voters reconsider their opinions at times distributed according to independent exponential random variables (this gives a Poisson process locally – note that there are in general infinitely many voters so no global Poisson process can be used). At times of reconsideration, a voter chooses one neighbor uniformly from amongst all neighbors and takes that neighbor's opinion. One can generalize the process by allowing the picking of neighbors to be something other than uniform.

Discrete time process

In the discrete time voter model in one dimension, $\xi_t(x): \mathbb{Z} \to \{0,1\}$ represents the state of particle $x$ at time $t$ . Informally each individual is arranged on a line and can "see" other individuals that are within a radius, $r$ . If more than a certain proportion, $\theta$ of these people disagree then the individual changes her attitude, otherwise she keeps it the same. Durrett and Steif (1993) and Steif (1994) show that for large radii there is a critical value $\theta_c$ such that if $\theta > \theta_c$ most individuals never change, and for $\theta \in (1/2, \theta_c)$ in the limit most sites agree. (Both of these results assume the probability of $\xi_0(x) = 1$ is one half.)

This process has a natural generalization to more dimensions, some results for this are discussed in Durrett and Steif (1993).

Continuous time process

The continuous time process is similar in that it imagines each individual has a belief at a time and changes it based on the attitudes of its neighbors. The process is described informally by Liggett (1985, 226), "Periodically (i.e., at independent exponential times), an individual reassesses his view in a rather simple way: he chooses a 'friend' at random with certain probabilities and adopts his position." A model was constructed with this interpretation by Holley and Liggett (1975).

This process is equivalent to a process first suggested by Clifford and Sudbury (1973) where animals are in conflict over territory and are equally matched. A site is selected to be invaded by a neighbor at a given time.

References

Clifford, Peter; Aidan Sudbury (1973). "A Model for Spatial Conflict". Biometrika. 60 (3): 581–588. doi:10.1093/biomet/60.3.581.
Durrett, Richard; Jeffrey E. Steif (1993). "Fixation Results for Threshold Voter Systems". The Annals of Probability. 21 (1): 232–247. doi:10.1214/aop/1176989403.
Holley, Richard A.; Thomas M. Liggett (1975). "Ergodic Theorems for Weakly Interacting Infinite Systems and The Voter Model". The Annals of Probability. 3 (4): 643–663. doi:10.1214/aop/1176996306.
Steif, Jeffrey E. (1994). "The Threshold Voter Automaton at a Critical Point". The Annals of Probability. 22 (3): 1121–1139. doi:10.1214/aop/1176988597.
Liggett, Thomas M. (1985). Interacting Particle Systems. New York: Springer Verlag. ISBN 0-387-96069-4.
Thomas M. Liggett, "Stochastic Interacting Systems: Contact, Voter and Exclusion Processes", Springer-Verlag, 1999.
C. Bezuidenhout and G. R. Grimmett, The critical contact process dies out, Ann. Probab. 18 (1990), 1462 – 1482.

Stochastic processes

Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding

Continuous time	Bessel process Birth–death process Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Point process Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage

Both	Branching process Galves–Löcherbach model Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process Stochastic chains with memory of variable length White noise

Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph

Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model

Financial models	Black–Derman–Toy Black–Karasinski Black–Scholes Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie

Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson

Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c

Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible

Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem

Inequalities	Burkholder–Davis–Gundy Doob's martingale Kunita–Watanabe

Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract

Disciplines	Actuarial mathematics Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Statistics Stochastic analysis Time series analysis Machine learning

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