Identifying the sources of structural sensitivity in partially specified biological models

Matthew W Adamson,Andrew Yu Morozov

doi:10.1038/s41598-020-73710-z

Abstract

Biological systems are characterised by a high degree of uncertainty and complexity, which implies that exact mathematical equations to describe biological processes cannot generally be justified. Moreover, models can exhibit sensitivity to the precise formulations of their component functions—a property known as structural sensitivity. Structural sensitivity can be revealed and quantified by considering partially specified models with uncertain functions, but this goes beyond well-established, parameter-based sensitivity analysis, and currently presents a mathematical challenge. Here we build upon previous work in this direction by addressing the crucial question of identifying the processes which act as the major sources of model uncertainty and those which are less influential. To achieve this goal, we introduce two related concepts: (1) the gradient of structural sensitivity, accounting for errors made in specifying unknown functions, and (2) the partial degree of sensitivity with respect to each function, a global measure of the uncertainty due to possible variation of the given function while the others are kept fixed. We propose an iterative framework of experiments and analysis to inform a heuristic reduction of structural sensitivity in a model. To demonstrate the framework introduced, we investigate the sources of structural sensitivity in a tritrophic food chain model.

Highlights

Biological systems are characterised by a high degree of uncertainty and complexity, which implies that exact mathematical equations to describe biological processes cannot generally be justified
In this paper we address the question of sensitivity analysis of a model with respect to uncertainty in its component functions, rather than just its parameters, and we outline such an approach to sensitivity analysis where the model output we’re interested in is the stability or otherwise of an equilibrium
The consideration of uncertain functions to represent biological processes in mathematical models is necessary since the use of particular equations to represent these processes is far more restrictive than is often supposed

Summary

Introduction

Biological systems are characterised by a high degree of uncertainty and complexity, which implies that exact mathematical equations to describe biological processes cannot generally be justified. We build upon previous work in this direction by addressing the crucial question of identifying the processes which act as the major sources of model uncertainty and those which are less influential To achieve this goal, we introduce two related concepts: (1) the gradient of structural sensitivity, accounting for errors made in specifying unknown functions, and (2) the partial degree of sensitivity with respect to each function, a global measure of the uncertainty due to possible variation of the given function while the others are kept fixed. The uncertainty in any representation of biological processes by a simple ‘macroscale’ mathematical model applies to the choice of equations themselves, as well as their parameters This can be problematic since models can be sensitive to the precise functions used to represent processes such as growth rates, mortality rates or feeding rates of organisms, a property known as structural sensitivity[1,2,3], even when they are robust to variations of parameters for a given choice of functions. When the uncertain inputs are parameters, a large range of techniques for UA e xists[23]

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