Special Issue "Lie Symmetries at Work in Biology and Medicine"

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Biology and Symmetry".

Deadline for manuscript submissions: 17 December 2020.

Special Issue Editor

Assoc. Prof. Dr. Maria Clara Nucci
Website
Guest Editor
Department of Mathematics, University of Perugia, 06123 Perugia, Italy
Interests: Lie symmetries in classical and quantum mechanics; Lie symmetries in biomathematics; magnetohydrodynamics; bits and pieces of the history of mathematics

Special Issue Information

Dear Colleagues,

Let us ponder those systems of differential equations that are proposed as mathematical models in Life Sciences. Numerical analysis is the most commonly used approach, although arbitrary parameters are in the way, and one has to guess which numbers to replace them with. Indeed, a crunching number approach. One may also look at the asymptotic behavior of solutions. However, to infinity and beyond may dismiss what happens in finite time. Therefore, the next best thing to do is search for Lie symmetries of those systems. They may be trivial, and they may be hidden, but, it is symmetries that allow for both a qualitative and even better quantitative analysis of a model, before going to infinity. This Special Issue invites contributions to show that Lie symmetries are indeed at work in biology and medicine.

Leading papers:
V. Torrisi and M.C. Nucci, "Application of Lie group analysis to a mathematical model which describes HIV transmission" in "The Geometrical Study of Differential Equations" (J.A. Leslie and T.P. Hobart, Eds.), A.M.S., Providence (2001) pp. 11-20.
M.C. Nucci, "Using Lie symmetries in epidemiology", Electron. J. Diff. Eqns. Conference 12, pp. 87-101 (2004).
M. Edwards and M.C. Nucci, "Application of Lie group analysis to a core group model for sexually transmitted diseases", J. Nonlinear Math. Phys. 13, pp. 211-230 (2006).
M.C. Nucci and K.M. Tamizhmani, "Lagrangians for biological models", J. Nonlinear Math. Phys. 19, 1250021 (2012).
M.C. Nucci and G. Sanchini, "Symmetries, Lagrangians and Conservation Laws of an Easter Island Population Model", Symmetry 7, pp. 1613-1632 (2015).

Assoc. Prof. Dr. Maria Clara Nucci
Guest Editor

Manuscript Submission Information

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Keywords

  • Lie symmetries
  • Noether symmetries
  • Differential equations in Biology and Medicine

Published Papers (3 papers)

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Research

Open AccessArticle
Finding and Breaking Lie Symmetries: Implications for Structural Identifiability and Observability in Biological Modelling
Symmetry 2020, 12(3), 469; https://doi.org/10.3390/sym12030469 - 16 Mar 2020
Cited by 1
Abstract
A dynamic model is structurally identifiable (respectively, observable) if it is theoretically possible to infer its unknown parameters (respectively, states) by observing its output over time. The two properties, structural identifiability and observability, are completely determined by the model equations. Their analysis is [...] Read more.
A dynamic model is structurally identifiable (respectively, observable) if it is theoretically possible to infer its unknown parameters (respectively, states) by observing its output over time. The two properties, structural identifiability and observability, are completely determined by the model equations. Their analysis is of interest for modellers because it informs about the possibility of gaining insight into a model’s unmeasured variables. Here we cast the problem of analysing structural identifiability and observability as that of finding Lie symmetries. We build on previous results that showed that structural unidentifiability amounts to the existence of Lie symmetries. We consider nonlinear models described by ordinary differential equations and restrict ourselves to rational functions. We revisit a method for finding symmetries by transforming rational expressions into linear systems. We extend the method by enabling it to provide symmetry-breaking transformations, which allows for a semi-automatic model reformulation that renders a non-observable model observable. We provide a MATLAB implementation of the methodology as part of the STRIKE-GOLDD toolbox for observability and identifiability analysis. We illustrate the use of the methodology in the context of biological modelling by applying it to a set of problems taken from the literature. Full article
(This article belongs to the Special Issue Lie Symmetries at Work in Biology and Medicine)
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Open AccessArticle
A Mathematical Model for Transport in Poroelastic Materials with Variable Volume:Derivation, Lie Symmetry Analysis, and Examples
Symmetry 2020, 12(3), 396; https://doi.org/10.3390/sym12030396 - 04 Mar 2020
Abstract
Fluid and solute transport in poroelastic media is studied. Mathematical modeling of such transport is a complicated problem because of the volume change of the specimen due to swelling or shrinking and the transport processes are nonlinearly linked. The tensorial character of the [...] Read more.
Fluid and solute transport in poroelastic media is studied. Mathematical modeling of such transport is a complicated problem because of the volume change of the specimen due to swelling or shrinking and the transport processes are nonlinearly linked. The tensorial character of the variables adds also substantial complication in both theoretical and experimental investigations. The one-dimensional version of the theory is less complex and may serve as an approximation in some problems, and therefore, a one-dimensional (in space) model of fluid and solute transport through a poroelastic medium with variable volume is developed and analyzed. In order to obtain analytical results, the Lie symmetry method is applied. It is shown that the governing equations of the model admit a non-trivial Lie symmetry, which is used for construction of exact solutions. Some examples of the solutions are discussed in detail. Full article
(This article belongs to the Special Issue Lie Symmetries at Work in Biology and Medicine)
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Open AccessFeature PaperArticle
Nonclassical Symmetry Solutions for Non-Autonomous Reaction-Diffusion Equations
Symmetry 2019, 11(2), 208; https://doi.org/10.3390/sym11020208 - 12 Feb 2019
Cited by 4
Abstract
The behaviour of many systems in chemistry, combustion and biology can be described using nonlinear reaction diffusion equations. Here, we use nonclassical symmetry techniques to analyse a class of nonlinear reaction diffusion equations, where both the diffusion coefficient and the coefficient of the [...] Read more.
The behaviour of many systems in chemistry, combustion and biology can be described using nonlinear reaction diffusion equations. Here, we use nonclassical symmetry techniques to analyse a class of nonlinear reaction diffusion equations, where both the diffusion coefficient and the coefficient of the reaction term are spatially dependent. We construct new exact group invariant solutions for several forms of the spatial dependence, and the relevance of some of the solutions to population dynamics modelling is discussed. Full article
(This article belongs to the Special Issue Lie Symmetries at Work in Biology and Medicine)
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Planned Papers

The below list represents only planned manuscripts. Some of these manuscripts have not been received by the Editorial Office yet. Papers submitted to MDPI journals are subject to peer-review.

Title: The symmetries of Life
Author: Mario Rasetti
Affiliation: ISI Foundation – Torino, New York
Abstract: Aim of this paper is to argue that in the discussion over the possible existence of a self-consistent scheme for the construction of a virtual ‘machine’ (term which has here a significance analogous to that of the Turing machine, i.e., a formal device which manipulates and evolves ‘states’), able to perform all what living matter – as distinguished from inert matter – can do and inanimate matter cannot, in a formal setting consistent exclusively with the quantum laws, symmetry must play a crucial role. Such symmetry is not simply Lie but bears on more complex, possibly infinite-dimensional algebraic structures. The related  theoretical construct - a conceptual framework representing and providing the operational tools of a “Life Machine”, mimicking the features of Life - implies that living matter (and - on equal footing - also intelligence, be it natural or artificial) should be represented by a generalized quantum automaton theory, endowed with a fundamental symmetry. Out of such symmetry, which has deep topological roots, emerges the capacity of the 'objects' of a living world made of quantum entities, to perform complex tasks, e.g., of replicating themselves. The field theoretical automaton that acts as virtual 'life machine' is endowed with the rules of a von Neumann constructor, but quantum and topological & field-theoretical in nature, and its dynamics is constrained by the fundamental symmetry.

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