Density evolution in stochastic dynamical systems with memory: A universal algorithm

Stochastic dynamical systems with memory are usually modeled using stochastic functional differential equations. Quantifying the probability density evolution in these systems remains an open problem with strong practical applications. However, due to a lack of efficient methods for computing the probability density of stochastic functional differential equations in their general form, the application of these systems are severely restricted. We address this challenge by presenting a universal approach for computing the evolution of probability density in a broad class of stochastic dynamical systems with memory. The proposed approach approximates the stochastic functional equation via a discrete model derived from the Euler scheme and recursively estimates its probability density by computing that of the discretized counterpart. The method is deterministic and computationally efficient. To validate and demonstrate its effectiveness, we apply it to compute both transient and long-term probability density evolution for some typical climate models.

Dynamics of a stochastic delay differential model for COVID-19 infection with asymptomatic infected and interacting people: Case study in the UAE

Public health science is increasingly focusing on understanding how COVID-19 spreads among humans. For the dynamics of COVID-19, we propose a stochastic epidemic model, with time-delays, Susceptible-Infected-Asymptomatic-Quarantined-Recovered (SIAQR). One global positive solution exists with probability one in the model. As a threshold condition of persistence and existence of an ergodic stationary distribution, we deduce a generalized stochastic thresholdR 0 s < R 0 . To estimate the percentages of people who must be vaccinated to achieve herd immunity, least-squares approaches were used to estimateR 0 from real observations in the UAE. Our results suggest that whenR 0 > 1 , a proportion max ( 1 - 1 / R 0 ) of the population needs to be immunized/vaccinated during the pandemic wave. Numerical simulations show that the proposed stochastic delay differential model is consistent with the physical sensitivity and fluctuation of the real observations.

Analysis of a stochastic HBV infection model with delayed immune response

Considering the environmental factors and uncertainties, we propose, in this paper, a higher-order stochastically perturbed delay differential model for the dynamics of hepatitis B virus (HBV) infection with immune system. Existence and uniqueness of an ergodic stationary distribution of positive solution to the system are investigated, where the solution fluctuates around the endemic equilibrium of the deterministic model and leads to the stochastic persistence of the disease. Under some conditions, infection-free can be obtained in which the disease dies out exponentially with probability one. Some numerical simulations, by using Milstein's scheme, are carried out to show the effectiveness of the obtained results. The intensity of white noise plays an important role in the treatment of infectious diseases.

Stochastic modelling of PTEN regulation in brain tumors: A model for glioblastoma multiforme

This work is the outcome of the partnership between the mathematical group of Department DISBEF and the biochemical group of Department DISB of the University of Urbino "Carlo Bo" in order to better understand some crucial aspects of brain cancer oncogenesis. Throughout our collaboration we discovered that biochemists are mainly attracted to the instantaneous behaviour of the whole cell, while mathematicians are mostly interested in the evolution along time of small and different parts of it. This collaboration has thus been very challenging. Starting from [23,24,25], we introduce a competitive stochastic model for post-transcriptional regulation of PTEN, including interactions with the miRNA and concurrent genes. Our model also covers protein formation and the backward mechanism going from the protein back to the miRNA. The numerical simulations show that the model reproduces the expected dynamics of normal glial cells. Moreover, the introduction of translational and transcriptional delays offers some interesting insights for the PTEN low expression as observed in brain tumor cells.

Exact dynamics of stochastic linear delayed systems: application to spatiotemporal coordination of comoving agents

Delayed dynamics result from finite transmission speeds of a signal in the form of energy, mass, or information. In stochastic systems the resulting lagged dynamics challenge our understanding due to the rich behavioral repertoire encompassing monotonic, oscillatory, and unstable evolution. Despite the vast literature, quantifying this rich behavior is limited by a lack of explicit analytic studies of high-dimensional stochastic delay systems. Here we fill this gap for systems governed by a linear Langevin equation of any number of delays and spatial dimensions with additive Gaussian noise. By exploiting Laplace transforms we are able to derive an exact time-dependent analytic solution of the Langevin equation. By using characteristic functionals we are able to construct the full time dependence of the multivariate probability distribution of the stochastic process as a function of the delayed and nondelayed random variables. As an application we consider interactions in animal collective movement that go beyond the traditional assumption of instantaneous alignment. We propose models for coordinated maneuvers of comoving agents applicable to recent empirical findings in pigeons and bats whereby individuals copy the heading of their neighbors with some delay. We highlight possible strategies that individual pairs may adopt to reduce the variance in their velocity difference and/or in their spatial separation. We also show that a minimum in the variance of the spatial separation at long times can be achieved with certain ratios of measurement to reaction delay.

ON HALANAY-TYPE ANALYSIS OF EXPONENTIAL STABILITY FOR THE θ-MARUYAMA METHOD FOR STOCHASTIC DELAY DIFFERENTIAL EQUATIONS

Using an approach that has its origins in work of Halanay, we consider stability in mean square of numerical solutions obtained from the θ-Maruyama discretization of a test stochastic delay differential equation [Formula: see text] interpreted in the Itô sense, where W(t) denotes a Wiener process. We focus on demonstrating that we may use techniques advanced in a recent report by Baker and Buckwar to obtain criteria for asymptotic and exponential stability, in mean square, for the solutions of the recurrence [Formula: see text]

The Pathwise Convergence of Approximation Schemes for Stochastic Differential Equations

The authors of this paper study approximation methods for stochastic differential equations, and point out a simple relation between the order of convergence in the pth mean and the order of convergence in the pathwise sense: Convergence in the pth mean of order α for all p ≥ 1 implies pathwise convergence of order α – ε for arbitrary ε > 0. The authors then apply this result to several one-step and multi-step approximation schemes for stochastic differential equations and stochastic delay differential equations. In addition, they give some numerical examples.

A biologically motivated signal transmission approach based on stochastic delay differential equation

Based on the stochastic delay differential equation (SDDE) modeling of neural networks, we propose an effective signal transmission approach along the neurons in such a network. Utilizing the linear relationship between the delay time and the variance of the SDDE system output, the transmitting side encodes a message as a modulation of the delay time and the receiving end decodes the message by tracking the delay time, which is equivalent to estimating the variance of the received signal. This signal transmission approach turns out to follow the principle of the spread spectrum technique used in wireless and wireline wideband communications but in the analog domain rather than digital. We hope the proposed method might help to explain some activities in biological systems. The idea can further be extended to engineering applications. The error performance of the communication scheme is also evaluated here.

Mean-square stability of a stochastic model for bacteriophage infection with time delays

We consider the stability properties of the positive equilibrium of a stochastic model for bacteriophage infection with discrete time delay. Conditions for mean-square stability of the trivial solution of the linearized system around the equilibrium are given by the construction of suitable Lyapunov functionals. The numerical simulations of the strong solutions of the arising stochastic delay differential system suggest that, even for the original non-linear model, the longer the incubation time the more the phage and bacteria populations can coexist on a stable equilibrium in a noisy environment for very long time.