Quasi-stationary Distribution Research Articles

Insights of infected Schwann cells extinction and inherited randomness in a stochastic model of leprosy

Investigating disease progression, transmission of infection and impacts of Multidrug Therapy (MDT) to inhibit demyelination in leprosy involves a certain amount of difficulty in terms of the in-built uncertain complicated and complex intracellular cell dynamical interactions. To tackle this scenario and to elucidate a more realistic, rationalistic approach of examining the infection mechanism and associated drug therapeutic interventions, we propose a four-dimensional ordinary differential equation-based model. Stochastic processes has been employed on this deterministic system by formulating the Kolmogorov forward equation introducing a transition state and the quasi-stationary distribution, exact distribution analysis have been investigated which allow us to estimate an expected time to extinction of the infected Schwann cells into the human body more prominently. Additionally, to explore the impact of uncertainty in the key intracellular factors, the stochastic system is investigated incorporating random perturbations and environmental noises in the disease dissemination, proliferation and reinfection rates. Rigorous numerical simulations validating the analytical outcomes provide us significant novel insights on the progression of leprosy and unravelling the existing major treatment complexities. Analytical experiments along with the simulations utilizing Monte-Carlo method and Euler–Maruyama scheme involving stochasticity predicts that the bacterial density is underestimated due to the recurrence of infection and suggests that maintaining a drug-efficacy rate in the range 0.6−0.8 would be substantially efficacious in eradicating leprosy.

Quasi-stationary distribution for continuous-state branching processes with competition

We study quasi-stationary distribution of the continuous-state branching process with competition introduced by Berestycki et al. (2018). This process is defined as the unique strong solution to a stochastic integral equation with jumps. An important example is the logistic branching process proposed by Lambert (2005). We establish the strong Feller property, trajectory Feller property, Lyapunov condition, weak Feller property and irreducibility, respectively. These properties together allow us to prove that if the competition is strong enough near +∞, then there is a unique quasi-stationary distribution, which attracts all initial distributions with exponential rates.

Concentration of quasi-stationary distributions for one-dimensional diffusions with applications

Concentration of quasi-stationary distributions for one-dimensional diffusions with applications

Quasi-stationary distribution of a single species model under demographic stochasticity and Allee effects

In this paper, we utilize an absorbed diffusion process to model the dynamics of a single species under the influence of demographic stochasticity and component Allee effects. The trajectories of stochastic solutions exhibits multi-scale dynamics distinct from those of the corresponding mean-field model. The primary focus is on analyzing transient dynamics before extinction, which is described by the quasi-stationary distribution. The paper provides results on the existence, uniqueness, and exponential convergence to the quasi-stationary distribution for initial distributions with compact support. Due to the singularity of the noise at 0, we employ the idea of changing variables to transfer the model into one-dimensional Kolmogorov diffusions with a drift possibly exploding at 0, to prove the main results.

Quasistationary Distribution for The Invasion Model on a Complete Bipartite Graph

The Invasion Model on the complete bipartite graph was introduced and studied by physicists as a rudimentary model for opinion dynamics on complex networks. Our main result provides the limit of the Quasistationary Distribution (QSD) for the model as one partition size tends to infinity. This limit is a highly dispersed measure. The results are in stark contrast to the Voter Model whose QSD is essentially a perturbation to absorbing states. A distinct feature not present in the Voter Model and demanding a new approach developed here is the appearance of two time scales with a non-trivial interaction. Our analysis is based on decoupling of the scales and using a Stein-type operator to identify a limit distribution. It also involves an application of a known duality with coalescing Markov chains through which asymptotic survival rates associated with the QSD are obtained.

Quasi-stationary distribution for strongly Feller Markov processes by Lyapunov functions and applications to hypoelliptic Hamiltonian systems

We establish a general result on the existence and uniqueness of a quasi-stationary distribution \mu_{\mathcal{D}} of a strongly Feller Markov process (X_{t},t\ge 0) killed when it exits a domain \mathcal{D} , under some Lyapunov function condition. Our result covers the case of hypoelliptic damped Hamiltonian systems. Our method is based on a characterization of the essential spectral radius by means of Lyapunov functions and measures of noncompactness.

The minimal quasi-stationary distribution of the absorbed M/M/∞ queue

The minimal quasi-stationary distribution of the absorbed M/M/∞ queue

Quasi-stationary distributions of one-dimensional generalized diffusions without natural boundaries

Quasi-stationary distributions of one-dimensional generalized diffusions without natural boundaries

Large deviation principle for quasi-stationary distributions and multiscale dynamics of absorbed singular diffusions

Large deviation principle for quasi-stationary distributions and multiscale dynamics of absorbed singular diffusions

Nonequilibrium phase transition of a one dimensional system reaches the absorbing state by two different ways

We study the nonequilibrium phase transitions from the absorbing phase to the active phase for the model of diseases spreading (Susceptible-Infected-Refractory-Susceptible (SIRS)) on a regular one-dimensional lattice. In this model, particles of three species (S, I, and R) on a lattice react as follows: S+I→2I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S+I\\rightarrow 2I$$\\end{document} with probability λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}, I→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I\\rightarrow R$$\\end{document} after infection time τI\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au _I$$\\end{document} and R→I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R\\rightarrow I$$\\end{document} after recovery time τR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au _R$$\\end{document}. In the case of τR>τI\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au _R>\ au _I$$\\end{document}, this model has been found to have two critical thresholds separating the active phase from absorbing phases. The first critical threshold λc1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda _{c1}$$\\end{document} corresponds to a low infection probability and the second critical threshold λc2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda _{c2}$$\\end{document} corresponds to a high infection probability. At the first critical threshold λc1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda _{c1}$$\\end{document}, our Monte Carlo simulations of this model suggest the phase transition to be of directed percolation class (DP). However, at the second critical threshold λc2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda _{c2}$$\\end{document} we observe that the system becomes so sensitive to initial values conditions which suggest the phase transition to be a discontinuous transition. We confirm this result using order parameter quasistationary probability distribution and finite-size analysis for this model at λc2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda _{c2}$$\\end{document}.

Partisan voter model: Stochastic description and noise-induced transitions.

We give a comprehensive mean-field analysis of the partisan voter model (PVM) and report analytical results for exit probabilities, fixation times, and the quasistationary distribution. In addition, and similarly to the noisy voter model, we introduce a noisy version of the PVM, named the noisy partisan voter model (NPVM), which accounts for the preferences of each agent for the two possible states, as well as for idiosyncratic spontaneous changes of state. We find that the finite-size noise-induced transition of the noisy voter model is modified in the NPVM leading to the emergence of intermediate phases that were absent in the standard version of the noisy voter model, as well as to both continuous and discontinuous transitions.

Metastable oscillations in an evolutionary game: Synchronization and control

Evolutionary games with a finite number of players may manifest transient oscillatory dynamics before reaching an absorbing state. Such oscillations could be prone to synchronization, however its details and specific properties are not known. Here, we investigate synchronization of metastable oscillations in the finite-dimensional evolutionary game “Battle of the Sexes”, as well as the possibility of controlling the properties of these oscillations by a multiplicative periodic signal. The appropriately generalized frequency and phase quantifiers demonstrate the possibility of synchronization, and the asymmetry of frequency range, induced by the finite size of population. Evolution of oscillations under strengthening periodic driving depends on whether synchronization is reached: in the synchronous regime the amplitude of oscillations grows, while their lifetime decreases, whereas in the asynchronous regime their amplitude decays to zero with the quasi-stationary distribution becoming unimodal.

The asymptotic tails of limit distributions of continuous-time Markov chains

AbstractThis paper investigates tail asymptotics of stationary distributions and quasi-stationary distributions (QSDs) of continuous-time Markov chains on subsets of the non-negative integers. Based on the so-called flux-balance equation, we establish identities for stationary measures and QSDs, which we use to derive tail asymptotics. In particular, for continuous-time Markov chains with asymptotic power law transition rates, tail asymptotics for stationary distributions and QSDs are classified into three types using three easily computable parameters: (i) super-exponential distributions, (ii) exponential-tailed distributions, and (iii) sub-exponential distributions. Our approach to establish tail asymptotics of stationary distributions is different from the classical semimartingale approach, and we do not impose ergodicity or moment bound conditions. In particular, the results also hold for explosive Markov chains, for which multiple stationary distributions may exist. Furthermore, our results on tail asymptotics of QSDs seem new. We apply our results to biochemical reaction networks, a general single-cell stochastic gene expression model, an extended class of branching processes, and stochastic population processes with bursty reproduction, none of which are birth–death processes. Our approach, together with the identities, easily extends to discrete-time Markov chains.

Asymptotics and criticality for a space-dependent branching process

We investigate a non-conservative semigroup determined by a branching process tracing the evolution of particles moving in a domain in . When a particle is killed at the boundary, a new generation of particles with mean number is born at a random point in the domain. Between branching, the particles are driven by a diffusion process with Dirichlet boundary conditions. According to the sign of , we distinguish super/sub-critical regimes and determine the exact exponential rate for the total number of particles , with depending explicitly on . We prove the Yaglom limit , where the quasi-stationary distribution ν is determined by the resolvent of the Dirichlet kernel at the point . The main application is in particle systems, where the normalization of the semigroup by its total mass gives the hydrodynamic limit of the Bak-Sneppen branching diffusions (BSBD). Since ν is the asymptotic profile under equilibrium, and the family of quasi-stationary distributions ν is indexed by , the model provides an explicit example of self-organized criticality.

Adaptation of a population to a changing environment in the light of quasi-stationarity

AbstractWe analyze the long-term stability of a stochastic model designed to illustrate the adaptation of a population to variation in its environment. A piecewise deterministic process modeling adaptation is coupled to a Feller logistic diffusion modeling population size. As the individual features in the population become further away from the optimal ones, the growth rate declines, making population extinction more likely. Assuming that the environment changes deterministically and steadily in a constant direction, we obtain the existence and uniqueness of the quasi-stationary distribution, the associated survival capacity, and the Q-process. Our approach also provides several exponential convergence results (in total variation for the measures). From this synthetic information, we can characterize the efficiency of internal adaptation (i.e. population turnover from mutant invasions). When the latter is lacking, there is still stability, but because of the high level of population extinction. Therefore, any characterization of internal adaptation should be based on specific features of this quasi-ergodic regime rather than the mere existence of the regime itself.

Large population limit of the spectrum of killed birth-and-death processes

We consider a general class of birth-and-death processes with state space {0,1,2,3,…} which describes the size of a population going eventually to extinction with probability one. We obtain the complete spectrum of the generator of the process killed at 0 in the large population limit, that is, we scale the process by a parameter K, and take the limit K→+∞. We assume that the differential equation dx/dt=b(x)−d(x) describing the infinite population limit (in any finite-time interval) has a repulsive fixed point at 0, and an attractive fixed point x⁎>0. We prove that, asymptotically, the spectrum is the superposition of two spectra. One is the spectrum of the generator of an Ornstein-Uhlenbeck process, which is n(b′(x⁎)−d′(x⁎)), n≥0. The other one is the spectrum of a continuous-time binary branching process conditioned on non-extinction, and is given by n(d′(0)−b′(0)), n≥1. A major difficulty is that different scales and function spaces are involved. We work at the level of the eigenfunctions that we split over different regions, and study their asymptotic dependence on K in each region. In particular, we prove that the spectral gap goes to min⁡{b′(0)−d′(0),d′(x⁎)−b′(x⁎)}. This work complements a previous work of ours in which we studied the approximation of the quasi-stationary distribution and of the mean time to extinction.

Trends to equilibrium for a nonlocal Fokker–Planck equation

We obtain equilibration rates for a one-dimensional nonlocal Fokker–Planck equation with time-dependent diffusion coefficient and drift, modeling the relaxation of a large swarm of robots, feeling each other in terms of their distance, towards the steady profile characterized by uniform spreading over a finite interval of the line. The result follows by combining entropy methods for quantifying the decay of the solution towards its quasi-stationary distribution, with the properties of the quasi-stationary profile.

On quasi-stationaries for symmetric Markov processes

This paper intends to give some mild sufficient conditions for the existence and uniqueness of quasi-stationary distributions (QSDs) for general symmetric Markov processes. Under the same conditions, it is also proved that: the unique QSD attracts exponentially all initial distributions supported in the allowed states; the considered process admits a quasi-ergodic stationary distribution (QESD); the Lp-spectral bounds of its associated semigroup are independent of 1≤p≤∞. Finally, we present three typical examples to illustrate these results.

Renewal Dynamical Approach for Non-Minimal Quasi-Stationary Distributions of One-Dimensional Diffusions

We consider quasi-stationary distributions for one-dimensional diffusions via the renewal dynamical approach. We show that convergence of the iterative renewal transform to quasi-stationary distributions is equivalent to a condition on the moment growth rate of the lifetime, which is at the same time a necessary condition for the existence of Yaglom limits.

Exponential galaxy discs as the quasi-stationary distribution in a Markov chain model simulating stellar scattering

ABSTRACT Previous models have shown that stochastic scattering of stars in a two-dimensional galaxy disc can generate a time-independent surface density distribution that is an exponential divided by radius when a constant inward scattering bias is present. Here we show, using a Markov chain model, that similar profiles result from an outward scattering bias, although the disc surface density decreases slowly with time because of a net stellar outflow. The trend towards a near-exponential surface profile is robust, as it exists even if the scattering intensity has moderate radial and time dependences, subject to some limitations on the scattering rates discussed in the text. The exponential scale length of the pseudo-equilibrium disc depends on the scattering bias, the scattering length, and the size of the disc where scattering is important.