## Modelling and inference in active systems

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Non-equilibrium systems, or active systems, span virtually all disciplines of science. While the underlying processes driving these systems out of equilibrium (may that be thermal, chemical or another type of equilibrium) may be entirely unrelated, it is often the case that the same or a similar mathematical model is capable of describing either of the systems. The particular active systems we are concerned with here are (a) the microscopic interactions of active matter in colloidal physics, and (b) the epidemiological dynamics of infectious diseases in human populations. Aside from their manifestly out-of-equilibrium character, both of these systems comprise fundamentally stochastic processes. Thermal fluctuations at energy-scales comparable to the ones of colloidal dynamics lead to Brownian motion in the former. The latter is governed by both pathogen- and data-specific noise. Remarkably, applying a functional central limit theorem, we find that to linear order both of these active systems can be described by the same process – the Ornstein-Uhlenbeck process. This is despite the fact that colloidal physics is concerned with microscopic scales, while epidemiological models operate on the scale of human populations. Using this, we apply methods from Bayesian inference to quantify the respective underlying uncertainties from stochastic trajectories alone. In the following, we briefly introduce the two parts of this thesis on active systems: Active matter and Epidemiology.

Abstract Active matter consists of mesoscopic self-propelled units, active particles, which on an individual level can metabolise energy into systematic movement. The mechanics and statistical mechanics of a suspension of active particles are determined by the traction (force per unit area) on their surfaces, arising from long-ranged hydrodynamic interactions at low Reynold's number. Here, we present a solution of the direct boundary integral equation for the traction on spherical active particles interacting with each other and their surroundings. For a single particle away from any boundaries this solution is exact, as both single- and double-layer integral operators can be simultaneously diagonalised in a basis of irreducible tensor spherical harmonics. The solution, thus, can be presented as an infinite number of linear relations between the harmonic coefficients of the traction and the velocity at the boundary of the particle. These generalise Stokes laws for the force and torque. In confinement, the dynamics of an active particle are uniquely defined by its mobility matrices, fully characterising hydrodynamic interactions due to the geometry of the system alone, and so-called propulsion tensors, arising from activity. Using Jacobi's method, we analytically derive iterative solutions for these quantities, once again starting from the direct boundary integral formulation of Stokes flow. We exemplify our results by explicitly providing the dynamics of an axisymmetric squirmer in the vicinity of a plane interface. For the computationally efficient simulation of many-body systems we develop a novel linear solver, based on a Krylov subspace method, that retains the true many-body character of hydrodynamic interactions. Finally, at non-zero temperature we consider thermal fluctuations of the suspending fluid, introducing stochasticity to the model above. We review and further develop a fast Bayesian method for parameter inference and model selection for a class of active systems that can be approximated by a stationary Ornstein-Uhlenbeck process. Such systems are said to be in a non-equilibrium steady state. A key aim of this study is to understand how one may distinguish between reversible and irreversible dynamics from information contained in stochastic trajectories alone.

Abstract Epidemiological data is beset by uncertainties about the underlying epidemiological processes, and the surveillance process through which the data is acquired. We present a Bayesian inference methodology that quantifies these uncertainties, for epidemics that are modelled by non-stationary, continuous-time Markov population processes. The efficiency of the method derives from a functional central limit theorem approximation of the likelihood, akin to van Kampen's system size expansion in physics, and valid for large well-mixed populations. The Ornstein-Uhlenbeck process arising from this approximation can once again be studied using Bayesian inference, including maximum a posteriori parameter estimates, computation of the model evidence, and the determination of parameter sensitivities via the Fisher information matrix. Our methodology is implemented in PyRoss, an open-source platform for analysis of epidemiological compartment models.

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Engineering and Physical Sciences Research Council (2089780)