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Mathematical Modelling

A key interest of the institute is the development of mathematical models. In Systems Biology these models are used for the analysis and interpretation of the huge volume and diversity of contemporary biological data. As well as providing biological insight, modelling can make predictions about the most useful direction for future experimental research. Our particular strengths are in the analysis of regulatory and signalling networks, paramater estimation and experimental optimisation. In Epidemiology, models are also used as a tool for the analysis of case report data and to understand the impact of differing heterogeneities (e.g. climatic forcing, risk-structure); however the predictions made by epidemiological models have a more direct use, helping to advise policy and planning, and making real-time forecasts for unfolding outbreaks.

Current major research themes in mathematical modelling include:

Dynamics and function of the NF-kB signalling system

NF-kB is found throughout the animal kingdom and is a first responder to a range of cellular stresses, such as cytokines, free radicals, radiation, heavy metals and pathogens. When activated by extracellular signals, it enters the nucleus where it plays a key role in regulating genes involved in the immune and inflammatory responses. Incorrect regulation of the NF-kB signalling network has been linked to cancer, inflammatory and autoimmune diseases. Questions addressed by this project include how do the complex feedback loops control NF-κB dynamics and downstream gene expression, and how do cells achieve appropriate cell fate decisions in response to time-varying signals? This involves the development of deterministic and stochastic models and new data analysis tools to interpret and direct experimental strategy.

Perturbation theory and sensitivity analysis


There is a rapidly increasing number of complex, high dimensional deterministic models in the Zeeman Institute and our work has developed software tools for analysing such models of regulatory and signalling systems based on local analysis of parameter values using the extensive and powerful perturbation theory for differential equations. Our tools can be used for the analysis, adjustment, optimisation and design of models with large numbers of parameters and variables. This allows one to probe model dynamics and to understand behaviour under parameter changes, which can mimic perturbations to rates, pulse experiments, and the creation of specific mutations such as gene knock-outs or knock-downs. Our software also allows one to combine multiple variants of a model (i.e. a model with multiple experimental conditions) in order to determine the value of new experiments and to optimise the amount of information coming from each. More information can be found here, and on our software downloads page.

ODE models of infection dynamics

SIR_eqnThe challenges of model development should not be overlooked; it requires a deep understanding of the infection’s natural history and the ablilty to translate this into a novel mathematical formulation that captures the main biological processes. While basic model templates exist (for example the simple SIR equations), in general each problem requires a bespoke model and offers a number of unique challenges. In particular, there are often choices to be made about which elements of biological behaviour and population-level heterogeneity are necessary to generate realistic predictions.

Models however are only as good as the parameters that underpin them. We therefore put considerable emphasis on matching models to available data sources, often requiring sophisticated Bayesian parameter inference and addressing questions of model fit. This inference is often complicated by the sampling errors and biases together with the large amounts of missing data – for example it is often only the worst cases of disease (illness) that get reported, whereas models operate by capturing the transmission of all infections.

Simulation models for infection
Simulation models operate by including a wide range of factors that could not (easily) be incorporated within the traditional ODE (differential equation) formulation. These models often include elements of spatial structure, complex host-level heterogeneity and capture the stochastic (chance) elements of the transmission of infection. As such simulations attempt to accurately reflect the complexity of the real world; however a fundamental belief within the Zeeman Institute is that these models must be based upon reliable information -- parameters should be inferred from epidemiological and other data sources as much as possible. Therefore while these models are often complex, in general they rely on a limited number of parameters that can be found by matching models to data.

Examples of simulation models include: the pioneering work on foot-and-mouth disease in the UK and USA; large-scale predictions of bovine TB in the UK and methods of control; the dynamics of avian influenza (H5N1) in the poultry flocks of S.E. Asia; the spread of influenza (and other pathogens) through the commuter and travel patterns in the UK; and the prediction of spatial sources of Leigonella outbreaks from case reports.

Bespoke modelling methodologies

Often a compromise is needed between simple ODE models and full-scale simulations....