QSDs for one-species reaction networks

Abstract: This talk is concerned with quasi-stationary distributions (QSDs) on $\mathbb{N}_0$ for one-species stochastic reaction networks conceived as continuous time Markov chains (CTMCs). A QSD describes the long time behaviour of the CTMC before absorption. Examples arise in ecology, epidemiology, cellular biology and chemistry.

Let $(X_t)_{t\ge 0}$, $X_t\in \mathbb{N}_0$, describe the species count of a one-species reaction network that eventually is absorbed into a trapping set $A\subseteq\mathbb{N}_0$. For example, $S\to 2S$ and $S\to 0$, here $A=\{0\}$. A probability distribution $\nu$ with support on $A^c=\mathbb{N}_0\!\setminus\! A$ is a QSD, if the following holds:

$\bullet$ If the initial count $X_0$ is chosen from $\nu$, then $X_t$, $t>0$, remains distributed as $\nu$, provided the chain is not absorbed.

Formulated in math terms: $$\mathbb{P}_\nu(X_t\in B\, |\, t<\tau_A)=\nu(B),\quad B\subseteq A^c,\quad\text{for all}\quad t\ge 0,$$ where $ \tau_A=\inf\{t\ge0 \colon X_t\in A\}$ is the time until absorption, and $\mathbb{P}_{\nu}$ is the distribution of the process before absorption with initial distribution $\nu$.

We start by looking at some examples to shape the intuition. Then, we characterise all QSDs $\nu$ in terms of a finite number of generating terms $\nu(J)=(\nu(i_1),\ldots,\nu(i_d))$, $J=\{i_1,\ldots,i_d\}$ and a corresponding eigenvalue $\theta\in\R$. This is to say, $\nu(n)=\sum_{j\in J}R_j(n,\theta)\nu(j)$, where the coefficients $R_j(n,\theta)$ are given recursively in $n$. Based on this and relying on Perron-Frobenius theory for infinite matrices, we prove existence of an extremal QSD for Kingman's parameter $\theta_K>0$, provided the CTMC is ultimately absorbed. Furthermore, we show the existence of a series of polynomials of increasing degree, $ \rho_n(\theta)$, such that $\theta(n)\downarrow\theta_K$, where $\theta(n)$ is the smallest real root of $ \rho_n(\theta)$. These results mimic results for birth-death processes (seminal work by Karlin and McGregor).

We further discuss numerial means to find the generator and to determine Kingman's parameter. The results are illustrated with numerical examples from stochastic reaction network theory.

This is joint work with Chuang Xu (University of Hawaii) and Mads Chr Hansen (formerly at University of Copenhagen).

chemical biologychemical kineticsalgebraic geometrydynamical systemsprobability

Audience: researchers in the topic

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Seminar on the Mathematics of Reaction Networks

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This seminar series focuses on progress in mathematical theory for the study of reaction networks, mainly in biology and chemistry. The scope is broad and accommodates works arising from dynamical systems, stochastics, algebra, topology and beyond.

We aim at providing a common forum for sharing knowledge and encouraging discussion across subfields. In particular we aim at facilitating interactions between junior and established researchers. These considerations will be represented in the choice of invited speakers and we will strive to create an excellent, exciting and diverse schedule.

The seminar runs approximately every other week on Thursdays, at 17:00 Brussels time (observe that this webpage shows the schedule in your current time zone). Each session consists of two 25-minute talks followed by 5-minute questions. After the two talks, longer discussions will take place for those interested. To this end, we will use breakout rooms. For this to work well, you need to have the latest version of Zoom installed (version 5.3.0 or higher), and use the desktop client or mobile app (not supported on ChromeOS).

We look forward hearing about new work and meeting many of you over zoom! Many of the talks are recorded; to see the recording, from Past Talks, open details of the listed talk for a video link.

The organizers.

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