Stochastic methods for epidemiology and biochemical reaction networks

eSMB2020 eSMB2020 Follow Wednesday at 11:15am EDT
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Hye-Won Kang and Wasiur KhudaBukhsh


Stochastic modelling is becoming increasingly popular in biological sciences. The ability to account for intrinsic fluctuations and uncertainty in experimental outcomes has been an advantage of stochastic methods. The application of stochastic tools has proven to be tremendously useful in analyzing biological data. In particular, stochastic methods have found usefulness in studying the spread of infectious diseases, in understanding the biophysics of enzyme kinetics, metabolism, immune-response mechanisms, in constructing phylogenetic trees etc. The objective of this two-part mini-symposium is to highlight some of the recent advances in the closely related areas of stochastic epidemiology and biochemical reaction networks - both at the molecular as well as the ecological scale. The first session will focus on epidemiology, while the second will discuss biochemical reaction networks in broad generality. Both sessions will cover a wide range of themes (including applications and techniques) giving a broad overview of the two fields. Specific topics include new asymptotic results/approximations, multiscale methods and statistical inference algorithms. Network-based approaches to epidemic modelling will also receive attention. With the recent Ebola and COVID-19 outbreaks in West Africa and China respectively, these methods are not only interesting from a theoretical standpoint, but also potentially important for timely public health interventions.

Ankit Gupta

ETH Zurich, Switzerland,
"The probability distribution of the reconstructed phylogenetic tree with occurrence data"
Stochastic birth-death processes are extensively used in epidemiology to model the underlying population dynamics of infected individuals. In such models the infection history of extant population naturally gives rise to a phylogenetic tree which can be used to study the evolution of the epidemiological process in the past. In this talk we study the problem of computing the probability distribution of such phylogenetic trees arising from partially sampled birth death processes. We consider observations from three distinct sampling schemes. First, individuals can be sampled and removed, through time, and included in the tree. Second, they can be occurrences which are sampled and removed through time and not included in the tree. Third, extant individuals can be sampled and included in the tree. The outcome of the process is thus composed of the reconstructed phylogenetic tree spanning all individuals sampled and included in the tree, and a timeline of occurrence events which are not placed along the tree. We derive a formula for computing the joint probability density of this outcome, which can readily be used to perform maximum likelihood or Bayesian estimation of the parameters of the birth-death model. In the context of epidemiology, our probability density enables the estimation of transmission rates through a joint analysis of epidemiological case count data and phylogenetic trees reconstructed from pathogen sequences.

Grzegorz Rempala

The Ohio State University, United States,
"Mathematical Model of a Pandemic: 2019-20 Coronavirus Analysis"
The modeling of a pandemic may be typically divided into three time phases: the early stochastic one, the mid-course deterministic one and the final, also stochastic. I will show on the example of Corona virus pandemic of 2019 how such model may be effectively used for predictions about disease dynamics applying both multiscale approximation and the idea of survival dynamical system obtained from the aggregate network model.

Hye-Won Kang

University of Maryland at Baltimore County, United States
"A stochastic model for enzyme clustering in glucose metabolism"
A sequence of metabolic enzymes tightly regulates glycolysis and gluconeogenesis. It has been hypothesized that these enzymes form multienzyme complexes and regulate glucose flux. In the previous work, it was identified that several rate-limiting enzymes form multienzyme complexes and control the direction of glucose flux between energy metabolism and building block biosynthesis. A recent study introduced a mathematical model to support this finding, in which the association of the rate-limiting enzymes into multienzyme complexes in included. However, this model did not fully account for dynamic and random movement of the enzyme clusters, as observed in the experiment. In this talk, I will introduce a stochastic model for enzyme clustering in glucose metabolism. The model will describe both the enzyme kinetics and the spatial organization of metabolic enzyme complexes. Then, I will discuss underlying model assumptions and approximation methods

Wasiur KhudaBukhsh

The Ohio State University, United States,
"Incorporating delays and non-Markovian dynamics into biochemical reaction networks"
Markov models for many biophysical systems are often found to be unrealistic because of the assumption that the interactions occur instantaneously or that the inter-reaction times follow an exponential distribution. In this talk, we consider relaxing those assumptions by incorporating delays into the system’s dynamics. We show that this modification leads to approximations by means of Partial Differential Equation (PDE) limits instead of the classical Ordinary Differential Equation (ODE) ones. Describing the dynamics by means of measure-valued processes is at the heart of such approximations. While the theory is developed for a general class of chemical reaction networks, we will also discuss some concrete examples.

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