Formal modelling and approximation-based analysis for mode-switching population dynamics

This thesis explores aspects of model specification and analysis for population dynamics which arise when modelling complex interactions and communication structures in agent or component collectives. The motivating examples come from the design of man-made systems where the optimal parametrisations for the behaviours of agents or components are not known a priori. In particular, we introduce a formal modelling framework to support the specification of control problems for collective dynamics in a high-level process algebraic language. A natural choice for the underlying semantics is to consider continuous time Markov decision processes due to their close relation to continuous time Markov chains that have traditionally been used as the mathematical model in numerous high-level modelling languages for stochastic dynamics. Although the theory of the resulting decision processes has a long history, the practical considerations, like computation time, present challenges due to the problem of state space explosion when considering large systems with complex behaviours. State space explosion problems are especially apparent in formal modelling paradigms where the specification of models usually happens at a component or an agent level in terms of a discrete set of states with defined rules for composing the specified behaviours into the dynamics of a system. Such specifications often give rise to very large models which are costly to analyse in full detail. However, when analysing models of collectives we are usually interested in the resulting macro-scale dynamics in terms of some aggregate measures. With that in mind, the second aspect of analysing collective dynamics that is considered in this thesis relates to fluid, linear noise and moment closure-based approximation methods which aim to give a good representation of the macro-scale dynamics of the models while being computationally less costly to analyse. We address a class of models where the population structure results from the assumption that components or agents can only be distinguished from each other based on the state they are in and focus on the particular cases where the population dynamics can be separated into a discrete set of modes. Our study of these models is motivated by considering information propagation via broadcast communication where the behaviour of components can change drastically when new information is received from the rest of the population. We consider existing approximation methods for resulting stochastic processes and propose a novel approach for applying these methods to models incorporating broadcast communication where each level of information available to the collective corresponds to a discrete dynamic mode. The resulting approximations combine continuous dynamics with discrete stochastic jumps and are not immediately simple to treat numerically. To that end we propose further approximations that allow for a computationally efficient analysis. Finally, we demonstrate how the formal modelling framework in conjunction with the developed approximation methods can be used for an example in policy synthesis

SiSyPHE: a python package for the simulation of systems of interacting mean-field particles with high efficiency

Over the past decades, the study of systems of particles has become an important part of many research areas, from theoretical physics to applied biology and computational mathematics. One of the main motivations in mathematical biology is the modelling of large animal societies and the emergence of complex patterns from simple behavioral rules, e.g., flocks of birds, fish schools, ant colonies, etc. In the microscopic world, particle systems are used to model a wide range of phenomena, from the collective motion of spermatozoa to the anarchical development of cancer cells. Within this perspective, there are at least three important reasons to conduct large scale computer simulations of particle systems. First, numerical experiments are essential to calibrate the models and test the influence of each parameter in a controlled environment. For instance, the renowned Vicsek model (Vicsek et al., 1995) is a minimal model of flocking, which exhibits a complex behavior, studied numerically in particular in (Chaté et al., 2008). Secondly, particle simulations are used to check the validity of macroscopic models that describe the statistical behavior of particle systems. These models are usually based on partial differential equations (PDE) derived using phenomenological considerations that are often difficult to justify mathematically (Degond et al., 2021; Degond & Motsch, 2008; Dimarco & Motsch, 2016). Finally, inspired by models in biology, there is an ever growing literature on the design of algorithms based on the simulation of artificial particle systems to solve tough optimization problems (Grassi & Pareschi, 2020; Kennedy & Eberhart, 1995; Pinnau et al., 2017; Totzeck, 2021) and to construct new more efficient Markov Chain Monte Carlo methods (Cappé et al., 2004; Clarté et al., 2021; Del Moral, 1998, 2013; Doucet et al., 2001). The simulation of systems of particles is also at the core of molecular dynamics (Leimkuhler & Matthews, 2015), although the present library is not specifically written for this purpose. The SiSyPHE library builds on recent advances in hardware and software for the efficient simulation of large scale interacting mean-field particle systems, both on the GPU and on the CPU. The versatile object-oriented Python interface of the library is designed for the simulation and comparison of new and classical many-particle models of collective dynamics in mathematics and active matter physics, enabling ambitious numerical experiments and leading to novel conjectures and results

First-principles studies of the structure and dynamics of biomolecules

First-principles biosimulations have become an essential tool in the study of atoms and molecules and, increasingly, in modelling complex systems as those arising in biology. With the appearance of density-functional theory, and gradient-corrected exchange-correlation functionals, the ability to obtain an accurate enough solutions to the electronic Schrödinger equation for systems containing hundreds (or even thousands) of atoms has revolutionized biophysics and biochemistry. Biological systems exhibit a far higher degree of complexity than those studied in many other fields of physics. The sizes of the systems, long time scale of processes, the effect of the environment, and the range of intermolecular interactions provide challenging problems for the application of first-principles quantum mechanical simulations to biomolecular studies. This thesis concentrates on first-principles electronic structure calculations of various biological systems and processes. The dynamics of the active center of myoglobin has been studied by means of Born-Oppenheimer molecular dynamics. Similar methodology has been used to investigate the effect of hydration of the L-alanine amino acid and to predict its actual structure in aqueous solution at finite temperature. The effect of the environment, and the actual structure of several biomolecules in water have been investigated by means of vibrational spectra calculations. Different continuum models have been employed in calculations of the vibrational absorption, vibrational dichroism, Raman and Raman optical activity spectra. The treatment of large, biological systems, such as proteins in aqueous solution, entirely by ab initio methods is extremely expensive. The thesis demonstrates various approaches to overcome size and time scale limits. The work presented here is an example of how quantum mechanical techniques can successfully be applied to biologically relevant problems in rather large and complex systems.reviewe

Data-driven modelling of biological multi-scale processes

Biological processes involve a variety of spatial and temporal scales. A holistic understanding of many biological processes therefore requires multi-scale models which capture the relevant properties on all these scales. In this manuscript we review mathematical modelling approaches used to describe the individual spatial scales and how they are integrated into holistic models. We discuss the relation between spatial and temporal scales and the implication of that on multi-scale modelling. Based upon this overview over state-of-the-art modelling approaches, we formulate key challenges in mathematical and computational modelling of biological multi-scale and multi-physics processes. In particular, we considered the availability of analysis tools for multi-scale models and model-based multi-scale data integration. We provide a compact review of methods for model-based data integration and model-based hypothesis testing. Furthermore, novel approaches and recent trends are discussed, including computation time reduction using reduced order and surrogate models, which contribute to the solution of inference problems. We conclude the manuscript by providing a few ideas for the development of tailored multi-scale inference methods.Comment: This manuscript will appear in the Journal of Coupled Systems and Multiscale Dynamics (American Scientific Publishers

On the foundations of cancer modelling: selected topics, speculations, & perspectives

This paper presents a critical review of selected topics related to the modelling of cancer onset, evolution and growth, with the aim of illustrating, to a wide applied mathematical readership, some of the novel mathematical problems in the field. This review attempts to capture, from the appropriate literature, the main issues involved in the modelling of phenomena related to cancer dynamics at all scales which characterise this highly complex system: from the molecular scale up to that of tissue. The last part of the paper discusses the challenge of developing a mathematical biological theory of tumour onset and evolution

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Socio-hydrological modelling: a review asking “why, what and how?”

Interactions between humans and the environment are occurring on a scale that has never previously been seen; the scale of human interaction with the water cycle, along with the coupling present between social and hydrological systems, means that decisions that impact water also impact people. Models are often used to assist in decision-making regarding hydrological systems, and so in order for effective decisions to be made regarding water resource management, these interactions and feedbacks should be accounted for in models used to analyse systems in which water and humans interact. This paper reviews literature surrounding aspects of socio-hydrological modelling. It begins with background information regarding the current state of socio-hydrology as a discipline, before covering reasons for modelling and potential applications. Some important concepts that underlie socio-hydrological modelling efforts are then discussed, including ways of viewing socio-hydrological systems, space and time in modelling, complexity, data and model conceptualisation. Several modelling approaches are described, the stages in their development detailed and their applicability to socio-hydrological cases discussed. Gaps in research are then highlighted to guide directions for future research. The review of literature suggests that the nature of socio-hydrological study, being interdisciplinary, focusing on complex interactions between human and natural systems, and dealing with long horizons, is such that modelling will always present a challenge; it is, however, the task of the modeller to use the wide range of tools afforded to them to overcome these challenges as much as possible. The focus in socio-hydrology is on understanding the human–water system in a holistic sense, which differs from the problem solving focus of other water management fields, and as such models in socio-hydrology should be developed with a view to gaining new insight into these dynamics. There is an essential choice that socio-hydrological modellers face in deciding between representing individual system processes or viewing the system from a more abstracted level and modelling it as such; using these different approaches has implications for model development, applicability and the insight that they are capable of giving, and so the decision regarding how to model the system requires thorough consideration of, among other things, the nature of understanding that is sought