Wasiur KhudaBukhsh

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  Incorporating age and delay into models for biophysical systems
  Wasiur KhudaBukhsh, Wasiur Rahman Kuda Bukhsh
  In many biological systems, chemical reactions or changes in a physical state are assumed to occur instantaneously. For describing the dynamics of those systems, Markov models that require exponentially distributed inter-event times have been used widely. However, some biophysical processes such as gene transcription and translation are known to have a significant gap between the initiation and the completion of the processes, which renders the usual assumption of exponential distribution untenable. In this talk, we consider relaxing this assumption by incorporating age-dependent random time delays into the system dynamics. We do so by constructing a measure-valued Markov process on a more abstract state space, which allows us to keep track of the "ages" of molecules participating in a chemical reaction. We study the large-volume limit of such age-structured systems. We show that, when appropriately scaled, the stochastic system can be approximated by a system of Partial Differential Equations (PDEs) in the large-volume limit, as opposed to Ordinary Differential Equations (ODEs) in the classical theory. We show how the limiting PDE system can be used for the purpose of further model reductions and for devising efficient simulation algorithms.
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  Survival Dynamical Systems: individual-level survival analysis from population-level epidemic models
  Wasiur KhudaBukhsh
  Motivated by the classical Susceptible-Infected-Recovered (SIR) epidemic models proposed by Kermack and Mckendrick, we consider a class of stochastic compartmental dynamical systems with a notion of partial ordering among the compartments. We call such systems uni-directional Mass Transfer Models (MTMs). We show that there is a natural way of interpreting a uni-directional MTM as a Survival Dynamical System (SDS) that is described in terms of survival functions instead of population counts. This SDS interpretation allows us to employ tools from survival analysis to address various issues with data collection and statistical inference of unidirectional MTMs. We use the SIR model as a running example to illustrate the ideas. We also discuss several possible generalizations of the method.
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  Seminar: Wasiur KhudaBukhsh - Multi-Scale Dynamics of Stochastic Biological Systems Through the Lens of Survival Dynamical Systems (SDS)
  Wasiur KhudaBukhsh
  Stochastic methods are becoming increasingly popular in biological sciences because of their ability to account for intrinsic fluctuations and uncertainty in occurrences or (experimental) outcomes. In this talk, we shall particularly focus on stochastic methods to analyse the multi-scale aspect of many biological systems. We use the term “scale” in a broad sense. It could refer to the size of the population, the abundance/volume, the time scale, the granularity or the amount of aggregation in the data collection procedure etc. We shall consider several examples from epidemiology, biochemical reaction networks to illustrate this multi-scale feature.
  
  Given a biological system that operates over multiple scales, it is often not clear how to describe the dynamics in one scale in terms of that in another scale. Survival Dynamical System (SDS) is a way to bridge this apparent gap. An SDS, derived from the macro-scale dynamics, describes probability distributions (via survival functions) of individual trajectories at a micro scale. This SDS interpretation not only intertwines multiple scales but also allows us to employ tools from survival analysis to address various issues with data collection and statistical inference. In particular, this approach yields a new statistical inference method, which is computationally advantageous.
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  Approximate lumpability for Markovian agent-based models using local symmetries
  Wasiur KhudaBukhsh

  We study a Markovian agent-based model (MABM). Each agent is endowed with a local state that changes over time as the agent interacts with its neighbours. The neighbourhood structure is given by a graph. 

  
  

  Examples of such systems include epidemic processes, percolation, peer-to-peer systems etc. In a recent paper [Simon et al. 2011], the automorphisms of the underlying graph were used to generate a lumpable partition of the joint state space ensuring Markovianness of the lumped process for binary dynamics. However, many large random graphs tend to become asymmetric rendering the automorphism-based lumping approach ineffective as a tool of model reduction. In order to mitigate this problem, we propose a lumping method based on a notion of local symmetry, which compares only local neighbourhoods of vertices. Since local symmetry only ensures approximate lumpability, we quantify the approximation error by means of Kullback-Leibler divergence rate between the original Markov chain and a lifted Markov chain. We prove the approximation error decreases monotonically. The connections to fibrations of graphs are also discussed. (Joint work with Arnab Auddy, Yann Disser and Heinz Koeppl)