The outcome of a general spatial epidemic on the line

1983 ◽  
Vol 20 (4) ◽  
pp. 715-727 ◽  
Author(s):  
M. J. Faddy
Keyword(s):  
Equilibrium Solution ◽  
Numerical Results ◽  
Quasi Equilibrium ◽  
Deterministic Case ◽  
Spatial Equilibrium ◽  
Threshold Behaviour ◽  
Stochastic Epidemic ◽  
Stochastic Case ◽  
Spatial Epidemic

The general (non-spatial) stochastic epidemic is extended to allow infective individuals to move forward through a system of spatially connected locations · ··, L1, L2, · ·· (on the line) each containing susceptible individuals and the outcome of the epidemic in each of these locations is then considered. In the deterministic case, a (spatial) equilibrium solution and threshold behaviour are discussed. In the stochastic case, a (spatial) quasi-equilibrium behaviour (conditional on sufficient numbers of infectives present) is discussed; numerical results suggest some correspondence between this stochastic quasi-equilibrium and the deterministic equilibrium.

The outcome of a general spatial epidemic on the line

1983 ◽  
Vol 20 (04) ◽  
pp. 715-727 ◽  
Author(s):  
M. J. Faddy
Keyword(s):  
Equilibrium Solution ◽  
Numerical Results ◽  
Quasi Equilibrium ◽  
Deterministic Case ◽  
Spatial Equilibrium ◽  
Threshold Behaviour ◽  
Stochastic Epidemic ◽  
Stochastic Case ◽  
Spatial Epidemic

The general (non-spatial) stochastic epidemic is extended to allow infective individuals to move forward through a system of spatially connected locations · ··, L 1, L 2, · ·· (on the line) each containing susceptible individuals and the outcome of the epidemic in each of these locations is then considered. In the deterministic case, a (spatial) equilibrium solution and threshold behaviour are discussed. In the stochastic case, a (spatial) quasi-equilibrium behaviour (conditional on sufficient numbers of infectives present) is discussed; numerical results suggest some correspondence between this stochastic quasi-equilibrium and the deterministic equilibrium.

Bifurcations Induced in a Bistable Oscillator via Joint Noises and Time Delay

2016 ◽  
Vol 26 (06) ◽  
pp. 1650102 ◽  
Author(s):  
Jin Fu ◽  
Zhongkui Sun ◽  
Yuzhu Xiao ◽  
Wei Xu
Keyword(s):  
Time Delay ◽  
Qualitative Behavior ◽  
Deterministic Case ◽  
Van Der Pol ◽  
Stochastic Case ◽  
Stationary Probability Density Function ◽  
Theoretical Analyses ◽  
Probability Density Function Pdf ◽  
Good Agreement ◽  
Qualitative Changes

In this paper, noise-induced and delay-induced bifurcations in a bistable Duffing–van der Pol (DVP) oscillator under time delay and joint noises are discussed theoretically and numerically. Based on the qualitative changes of the plane phase, delay-induced bifurcations are investigated in the deterministic case. However, in the stochastic case, the response of the system is a stochastic non-Markovian process owing to the existence of noise and time delay. Then, methods have been employed to derive the stationary probability density function (PDF) of the amplitude of the response. Accordingly, stochastic P-bifurcations can be observed with the variations in the qualitative behavior of the stationary PDF for amplitude. Furthermore, results from both theoretical analyses and numerical simulations best demonstrate the appearance of noise-induced and delay-induced bifurcations, which are in good agreement.

AN ADAPTIVE EBCA MODEL PROBING THE PROBLEM OF RIDING AGAINST THE TRAFFIC FLOW

2011 ◽  
Vol 22 (02) ◽  
pp. 191-208 ◽  
Author(s):  
TAO WANG ◽  
JUN-FENG WANG
Keyword(s):  
Phase Transition ◽  
Average Density ◽  
Deterministic Case ◽  
Average Flow Rate ◽  
Traffic System ◽  
The Road ◽  
Wrong Direction ◽  
Stochastic Case ◽  
The Right ◽  
Good Agreement

Despite the rule that cyclists must ride on the right half of the road is written into the state vehicle code, the phenomenon of riding against the bicycle flow is still serious. To investigate the effect of bicycles going in the wrong direction, a Bi-Directional Adaptive EBCA model is developed in this paper. The phase transition F-J as well as the phase transition F-S-F are suggested by observing the spatial-temporal pattern. The deterministic case that the linear relationship between the average flow rate and the bicycle number disappears when the average density exceeds a particular value is shown. Under the stochastic case, the impacts of the avoiding probability Ps and the returning probability Pr on the traffic system are analyzed. The results of the simulation are in good agreement with the realistic bicycle flow.

The threshold behaviour of epidemic models

1983 ◽  
Vol 20 (2) ◽  
pp. 227-241 ◽  
Author(s):  
Frank Ball
Keyword(s):  
Almost Sure Convergence ◽  
Death Process ◽  
Time Interval ◽  
Birth And Death Process ◽  
Initial Number ◽  
Finite Time Interval ◽  
Threshold Behaviour ◽  
Total Size ◽  
Stochastic Epidemic ◽  
General Stochastic

We provide a method of constructing a sequence of general stochastic epidemics, indexed by the initial number of susceptibles N, from a time-homogeneous birth-and-death process. The construction is used to show strong convergence of the general stochastic epidemic to a birth-and-death process, over any finite time interval [0, t], and almost sure convergence of the total size of the general stochastic epidemic to that of a birth-and-death process. The latter result furnishes us with a new proof of the threshold theorem of Williams (1971). These methods are quite general and in the remainder of the paper we develop similar results for a wide variety of epidemics, including chain-binomial, host-vector and geographical spread models.

Global dynamics of deterministic and stochastic epidemic systems with nonmonotone incidence rate

2018 ◽  
Vol 11 (08) ◽  
pp. 1850101 ◽  
Author(s):  
Tao Feng ◽  
Zhipeng Qiu
Keyword(s):  
Epidemic Model ◽  
Reproduction Number ◽  
Intervention Strategy ◽  
Sufficient Conditions ◽  
Global Dynamics ◽  
Stochastic Noise ◽  
Global Stability Analysis ◽  
Stochastic Epidemic ◽  
Stochastic Case ◽  
Deterministic Framework

This paper is devoted to studying the dynamics of a susceptible-infective-latent-infective (SILI) epidemic model that is subject to the combined effects of environmental noise and intervention strategy. We extend the classical SILI epidemic model from a deterministic framework to a stochastic one. For the deterministic case, the global stability analysis of the solution is carried out in terms of the basic reproduction number. For the stochastic case, sufficient conditions for the extinction of diseases are obtained. Then, the existence of stationary distribution and asymptotic behavior of the solution are further studied to illustrate the cycling phenomena of recurrent diseases. Numerical simulations are conducted to verify these analytical results. It is shown that both stochastic noise and intervention strategy contribute to the control of diseases.

Dynamic and stochastic propagation of the Brenier optimal mass transport

2019 ◽  
Vol 30 (6) ◽  
pp. 1264-1299
Author(s):  
ALISTAIR BARTON ◽  
NASSIF GHOUSSOUB
Keyword(s):  
Phase Space ◽  
Mean Field ◽  
Mean Field Games ◽  
Deterministic Case ◽  
Variational Solutions ◽  
Stochastic Case ◽  
Image Position ◽  
Fixed End ◽  
Stochastic Setting ◽  
Jacobi Equations

Similar to how Hopf–Lax–Oleinik-type formula yield variational solutions for Hamilton–Jacobi equations on Euclidean space, optimal mass transportations can sometimes provide variational formulations for solutions of certain mean-field games. We investigate here the particular case of transports that maximize and minimize the following ‘ballistic’ cost functional on phase space TM, which propagates Brenier’s transport along a Lagrangian L, $$b_T(v, x):=\inf\left\{\langle v, \gamma (0)\rangle +\int_0^TL(t, \gamma (t), {\dot \gamma}(t))\, dt; \gamma \in C^1([0, T], M); \gamma(T)=x\right\}\!,$$ where $M = \mathbb{R}^d$, and T >0. We also consider the stochastic counterpart: \begin{align*} \underline{B}_T^s(\mu,\nu):=\inf\left\{\mathbb{E}\left[\langle V,X_0\rangle +\int_0^T L(t, X,\beta(t,X))\,dt\right]\!; X\in \mathcal{A}, V\sim\mu,X_T\sim \nu\right\}\!, \end{align*} where $\mathcal{A}$ is the set of stochastic processes satisfying dX = βX (t, X) dt + dWt, for some drift βX (t, X), and where Wt is σ(Xs: 0 ≤ s ≤ t)-Brownian motion. Both cases lead to Lax–Oleinik-type formulas on Wasserstein space that relate optimal ballistic transports to those associated with dynamic fixed-end transports studied by Bernard–Buffoni and Fathi–Figalli in the deterministic case, and by Mikami–Thieullen in the stochastic setting. While inf-convolution easily covers cost minimizing transports, this is not the case for total cost maximizing transports, which actually are sup-inf problems. However, in the case where the Lagrangian L is jointly convex on phase space, Bolza-type dualities – well known in the deterministic case but novel in the stochastic case – transform sup-inf problems to sup–sup settings. We also write Eulerian formulations and point to links with the theory of mean-field games.

scholarly journals Derivation of kernel of dynamic support vector machines: Stochastic and deterministic data process

2016 ◽  
Author(s):  
Masamichi Sato
Keyword(s):  
Gaussian Kernel ◽  
Support Vector ◽  
Deterministic Case ◽  
Bellman Equations ◽  
Data Process ◽  
Vector Machines ◽  
Stochastic Case ◽  
Geometric Context ◽  
Dynamic Support ◽  
Analytic Derivation

We give an analytic derivation of kernel of dynamic support vector machine (DSVM). We show them for the cases of the data processes with stochastic and deterministic changes. We derive the kernels by solving Bellman equations. For the stochastic case, Gaussian kernel is naturally derived. For the deterministic case, the kernel is derived in the form of traveling wave. We also give comments from physical viewpoints in the context of information geometry. Physical comments include the equivalence principle in information geometric context and the relation to AdS/CFT correspondence.

A Low‐Cost Spatial Equilibrium Solution Procedure

1972 ◽  
Vol 54 (2) ◽  
pp. 373-374 ◽  
Author(s):  
Foo‐Shiung Ho ◽  
Richard A. King
Keyword(s):  
Equilibrium Solution ◽  
Low Cost ◽  
Solution Procedure ◽  
Spatial Equilibrium

scholarly journals Compact schemes for Korteweg-de Vries equation

2017 ◽  
Vol 21 (4) ◽  
pp. 1797-1806
Author(s):  
Xiu-Ling Yin ◽  
Cheng-Jian Zhang ◽  
Jing-Jing Zhang ◽  
Yan-Qin Liu
Keyword(s):  
Vries Equation ◽  
Wave Profile ◽  
Numerical Dispersion ◽  
Compact Schemes ◽  
Deterministic Case ◽  
Order Accuracy ◽  
Small Noise ◽  
De Vries ◽  
Stochastic Case ◽  
Order Scheme

This paper proposes one family of compact schemes for Korteweg-de Vries equation. In the deterministic case, the schemes are convergent with fourth-order accuracy both in space and in time. Moreover, the schemes are stable. The numerical dispersion relation is analyzed. We compare the schemes with one second-order scheme. The numerical examples test the effect of the schemes. In the stochastic case, we simulate the wave profile and three discrete dynamical quantities for Korteweg-de Vries equation with small noise. The white noise has stochastic influence on the profile and dynamical quantities of the solution. If the size of noise increases, the perturbation on the profile and dynamical quantities will increase accordingly.

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