The convergence of a branching Brownian motion used as a model describing the spread of an epidemic

1980 ◽  
Vol 17 (2) ◽  
pp. 301-312 ◽  
Author(s):  
Frank J. S. Wang
Keyword(s):  
Diffusion Coefficient ◽  
Brownian Motion ◽  
Unit Area ◽  
Transition Probability ◽  
Random Variable ◽  
Two Dimensional ◽  
Infection Period ◽  
Branching Brownian Motion ◽  
Almost Everywhere ◽  
Probability Rate

A spatial epidemic process where the individuals are located at positions in the Euclidean space R2 is considered. The infective individuals, with an infection period that is exponentially distributed with parameter µ, move in R2 according to a Brownian motion with a diffusion coefficient σ2. The susceptible individuals may also move. But we shall use the approximation that they remain unchanged in numbers and therefore assume that the averaged ‘density' of susceptibles per unit area is the same throughout space and time. The transition probability rate of infection of a susceptible in the infinitesimal element of area dy by an infective in dx is assumed to be a function h(x – y |) of the distance | x – y | between x and y. Then our process can be considered as a two-dimensional birth and death Brownian motion. Let be the number of infective individuals in the set D at time t and . The almost everywhere convergence of the random variables to a limit random variable W(D) is established.

The convergence of a branching Brownian motion used as a model describing the spread of an epidemic

1980 ◽  
Vol 17 (02) ◽  
pp. 301-312 ◽  
Author(s):  
Frank J. S. Wang
Keyword(s):  
Diffusion Coefficient ◽  
Brownian Motion ◽  
Unit Area ◽  
Transition Probability ◽  
Random Variable ◽  
Two Dimensional ◽  
Infection Period ◽  
Branching Brownian Motion ◽  
Almost Everywhere ◽  
Probability Rate

A spatial epidemic process where the individuals are located at positions in the Euclidean space R 2 is considered. The infective individuals, with an infection period that is exponentially distributed with parameter µ, move in R 2 according to a Brownian motion with a diffusion coefficient σ 2. The susceptible individuals may also move. But we shall use the approximation that they remain unchanged in numbers and therefore assume that the averaged ‘density' of susceptibles per unit area is the same throughout space and time. The transition probability rate of infection of a susceptible in the infinitesimal element of area dy by an infective in dx is assumed to be a function h(x – y |) of the distance | x – y | between x and y. Then our process can be considered as a two-dimensional birth and death Brownian motion. Let be the number of infective individuals in the set D at time t and . The almost everywhere convergence of the random variables to a limit random variable W(D) is established.

BROWNIAN MOTION IN A TWO-DIMENSIONAL POTENTIAL: A NEW NUMERICAL SOLUTION METHOD

2002 ◽  
Vol 16 (24) ◽  
pp. 3643-3654 ◽  
Author(s):  
L. Y. CHEN
Keyword(s):  
Brownian Motion ◽  
Saddle Point ◽  
Numerical Solution ◽  
Langevin Equation ◽  
Transition Probability ◽  
Statistical Weight ◽  
Two Dimensional ◽  
Activation Rate ◽  
A New Technique ◽  
Activation Event

Studying the Brownian motion of a particle in a two-dimensional potential with two saddle-point passages connecting two wells, we compute the activation rate of the particle from one well into the other and illustrate a new technique for obtaining numerical solution to the Langevin equation for transition probability. By virtue of a Langevin equation with negative friction, this new method directly traces the active part of an activation event, without having to simulate the long period of small fluctuations in a well between two successful events, and computes the statistical weight for each successful activation. It makes feasible for us to numerically integrate the Langevin equation for transition probability even when the activation energy barrier (i.e. the potential difference between the saddle point and the well) is much greater than thermal energy k B T where other methods fail to be tractable.

scholarly journals Random Assignment Problems on 2d Manifolds

2021 ◽  
Vol 183 (2) ◽  
Author(s):  
D. Benedetto ◽  
E. Caglioti ◽  
S. Caracciolo ◽  
M. D’Achille ◽  
G. Sicuro ◽  
...  
Keyword(s):  
Assignment Problem ◽  
Unit Area ◽  
Constant Term ◽  
Beltrami Operator ◽  
Explicit Calculation ◽  
Dimensional Manifold ◽  
Two Dimensional ◽  
Random Assignment ◽  
Assignment Problems ◽  
Theoretical Formulation

AbstractWe consider the assignment problem between two sets of N random points on a smooth, two-dimensional manifold $$\Omega $$ Ω of unit area. It is known that the average cost scales as $$E_{\Omega }(N)\sim {1}/{2\pi }\ln N$$ E Ω ( N ) ∼ 1 / 2 π ln N with a correction that is at most of order $$\sqrt{\ln N\ln \ln N}$$ ln N ln ln N . In this paper, we show that, within the linearization approximation of the field-theoretical formulation of the problem, the first $$\Omega $$ Ω -dependent correction is on the constant term, and can be exactly computed from the spectrum of the Laplace–Beltrami operator on $$\Omega $$ Ω . We perform the explicit calculation of this constant for various families of surfaces, and compare our predictions with extensive numerics.

scholarly journals Critical branching Brownian motion with absorption: Particle configurations

2015 ◽  
Vol 51 (4) ◽  
pp. 1215-1250 ◽  
Author(s):  
Julien Berestycki ◽  
Nathanaël Berestycki ◽  
Jason Schweinsberg
Keyword(s):  
Brownian Motion ◽  
Branching Brownian Motion

scholarly journals An Application of the Backbone Decomposition to Supercritical Super-Brownian Motion with a Barrier

2012 ◽  
Vol 49 (03) ◽  
pp. 671-684
Author(s):  
A. E. Kyprianou ◽  
A. Murillo-Salas ◽  
J. L. Pérez
Keyword(s):  
Brownian Motion ◽  
Wave Equation ◽  
Relative Ease ◽  
Branching Brownian Motion ◽  
Analytical Properties ◽  
Super Brownian Motion ◽  
Exit Measure ◽  
The Right

We analyse the behaviour of supercritical super-Brownian motion with a barrier through the pathwise backbone embedding of Berestycki, Kyprianou and Murillo-Salas (2011). In particular, by considering existing results for branching Brownian motion due to Harris and Kyprianou (2006) and Maillard (2011), we obtain, with relative ease, conclusions regarding the growth in the right-most point in the support, analytical properties of the associated one-sided Fisher-Kolmogorov-Petrovskii-Piscounov wave equation, as well as the distribution of mass on the exit measure associated with the barrier.

scholarly journals ONSHORE-OFFSHORE SEDIMENT MOVEMENT ON A BEACH

1978 ◽  
Vol 1 (16) ◽  
pp. 87 ◽  
Author(s):  
P. Nielsen ◽  
I.A. Svensen ◽  
C. Staub
Keyword(s):  
Experimental Data ◽  
Diffusion Coefficient ◽  
Oscillatory Flow ◽  
Sediment Flux ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Present Formulation ◽  
Exact Analytical Solutions ◽  
Net Flux ◽  
Loose Sediments

A theoretical model is developed for the movement of loose sediments in oscillatory flow with or without a net current. In the present formulation the model is two-dimensional, but may readily be extended to three dimensions. It is assumed that all movement of sediments occurs in suspension, and exact analytical solutions are given for the time variation of the concentration profile, the instantaneous sediment flux and the net flux of sediment over a wave period. The model requires as empirical input a diffusion coefficient e and pick-up function p(t), for which experimental data are presented. Two examples are discussed in detail, illustrating important aspects of the onshore-offshore sediment motion.

scholarly journals Stochastic Comparative Statics in Markov Decision Processes

2021 ◽  
Author(s):  
Bar Light
Keyword(s):  
Markov Decision Processes ◽  
Dynamic Pricing ◽  
Optimization Problems ◽  
Transition Probability ◽  
Random Variable ◽  
Comparative Statics ◽  
Decision Processes ◽  
Optimal Decision ◽  
Initial State ◽  
Markov Decision

In multiperiod stochastic optimization problems, the future optimal decision is a random variable whose distribution depends on the parameters of the optimization problem. I analyze how the expected value of this random variable changes as a function of the dynamic optimization parameters in the context of Markov decision processes. I call this analysis stochastic comparative statics. I derive both comparative statics results and stochastic comparative statics results showing how the current and future optimal decisions change in response to changes in the single-period payoff function, the discount factor, the initial state of the system, and the transition probability function. I apply my results to various models from the economics and operations research literature, including investment theory, dynamic pricing models, controlled random walks, and comparisons of stationary distributions.

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