Abstract This paper studies a time-switching advection-diffusion system modelling the competition between Aedes albopictus and Aedes aegypti mosquitoes in heterogeneous environments. The switching mechanism is induced by periodic releases of sterile Ae. albopictus mosquitoes, which are active only during their sexual lifespan within each release period. By defining a minimal release amount and four critical release period thresholds, we establish the periodic dynamics of the system, providing new insights into optimal control strategies of mosquitoes. Specifically, the trivial steady state is globally asymptotically stable if sterile releases are sufficiently frequent and abundant, which ensures the eradication of both Aedes species. For less frequent sterile releases, we prove the global asymptotic stability of the two semi-trivial periodic solutions and demonstrate the existence of a coexisting periodic solution, indicating cases where mosquito control fails. Numerical simulations are presented to validate our theoretical findings.

Abstract We extend a classical model of continuous opinion formation to explicitly include an age-structured population. We begin by considering a stochastic differential equation model which incorporates ageing dynamics and birth/death processes, in a bounded confidence type opinion formation model. We then derive and analyse the corresponding mean field partial differential equation and compare the complex dynamics on the microscopic and macroscopic levels using numerical simulations. We rigorously prove the existence of stationary states in the mean field model, but also demonstrate that these stationary states are not necessarily unique. Finally, we establish connections between this and other existing models in various scenarios.

Abstract In recent years, there has been significant interest in the effect of different types of adversarial perturbations in data classification problems. Many of these models incorporate the adversarial power, which is an important parameter with an associated trade-off between accuracy and robustness. This work considers a general framework for adversarially perturbed classification problems, in a large data or population-level limit. In such a regime, we demonstrate that as adversarial strength goes to zero that optimal classifiers converge to the Bayes classifier in the Hausdorff distance. This significantly strengthens previous results, which generally focus on $L^1$ -type convergence. The main argument relies upon direct geometric comparisons and is inspired by techniques from geometric measure theory.

Abstract In this paper, we study the distribution of the temperature within a body where the heat is transported only by radiation. Specifically, we consider the situation where both emission-absorption and scattering processes take place. We study the initial-boundary value problem given by the coupling of the radiative transfer equation with the energy balance equation on a convex domain $ \Omega \subset {\mathbb{R}}^3$ in the diffusion approximation regime, that is, when the mean free path of the photons tends to zero. Using the method of matched asymptotic expansions, we will derive the limit initial-boundary value problems for all different possible scaling limit regimes, and we will classify them as equilibrium or non-equilibrium diffusion approximation. Moreover, we will observe the formation of boundary and initial layers for which suitable equations are obtained. We will consider both stationary and time-dependent problems as well as different situations in which the light is assumed to propagate either instantaneously or with finite speed.

Abstract Seasonal changes and cyclical human activities (such as periodic fishing bans, Wolbachia-based mosquito population control, and school term breaks) have significant impacts on population dynamics. We propose a general switching dynamical model to describe these periodic changes. The existence, uniqueness and stability of positive periodic solutions are thoroughly investigated. The results are stated in terms of an introduced threshold value. To demonstrate their practicability, the obtained results are applied to two biological situations.

Abstract In this paper, we consider an optimal distributed control problem for a reaction-diffusion-based SIR epidemic model with human behavioural effects. We develop a model wherein non-pharmaceutical intervention methods are implemented, but a portion of the population does not comply with them, and this non-compliance affects the spread of the disease. Drawing from social contagion theory, our model allows for the spread of non-compliance parallel to the spread of the disease. The quantities of interest for control are the reduction in infection rate among the compliant population, the rate of spread of non-compliance and the rate at which non-compliant individuals become compliant after, e.g., receiving more or better information about the underlying disease. We prove the existence of global-in-time solutions for fixed controls and study the regularity properties of the resulting control-to-state map. The existence of optimal control is then established in an abstract framework for a fairly general class of objective functions. Necessary first–order optimality conditions are obtained via a Lagrangian-based stationarity system. We conclude with a discussion regarding minimisation of the size of infected and non-compliant populations and present simulations with various parameters values to demonstrate the behaviour of the model.

Abstract Machine learning has exhibited substantial success in the field of natural language processing (NLP). For example, large language models have empirically proven to be capable of producing text of high complexity and cohesion. However, at the same time, they are prone to inaccuracies and hallucinations. As these systems are increasingly integrated into real-world applications, ensuring their safety and reliability becomes a primary concern. There are safety critical contexts where such models must be robust to variability or attack and give guarantees over their output. Computer vision had pioneered the use of formal verification of neural networks for such scenarios and developed common verification standards and pipelines, leveraging precise formal reasoning about geometric properties of data manifolds. In contrast, NLP verification methods have only recently appeared in the literature. While presenting sophisticated algorithms in their own right, these papers have not yet crystallised into a common methodology. They are often light on the pragmatical issues of NLP verification, and the area remains fragmented. In this paper, we attempt to distil and evaluate general components of an NLP verification pipeline that emerges from the progress in the field to date. Our contributions are twofold. First, we propose a general methodology to analyse the effect of the embedding gap – a problem that refers to the discrepancy between verification of geometric subspaces, and the semantic meaning of sentences which the geometric subspaces are supposed to represent. We propose a number of practical NLP methods that can help to quantify the effects of the embedding gap. Second, we give a general method for training and verification of neural networks that leverages a more precise geometric estimation of semantic similarity of sentences in the embedding space and helps to overcome the effects of the embedding gap in practice.

Abstract The article considers systems of interacting particles on networks with adaptively coupled dynamics. Such processes appear frequently in natural processes and applications. Relying on the notion of graph convergence, we prove that for large systems the dynamics can be approximated by the corresponding continuum limit. Well-posedness of the latter is also established.

Abstract In this paper, we study the existence of travelling wave solutions and the spreading speed for the solutions of an age-structured epidemic model with nonlocal diffusion. Our proofs make use of the comparison principles both to construct suitable sub/super-solutions and to prove the regularity of travelling wave solutions.

Abstract We study the Cauchy problem on the real line for the nonlocal Fisher-KPP equation in one spatial dimension,\begin{equation*} u_t = D u_{xx} + u(1-\phi *u), \end{equation*}where$\phi *u$is a spatial convolution with the top hat kernel,$\phi (y) \equiv H\left (\frac{1}{4}-y^2\right )$. After observing that the problem is globally well-posed, we demonstrate that positive, spatially periodic solutions bifurcate from the spatially uniform steady state solution$u=1$as the diffusivity,$D$, decreases through$\Delta _1 \approx 0.00297$(the exact value is determined in Section 3). We explicitly construct these spatially periodic solutions as uniformly valid asymptotic approximations for$D \ll 1$, over one wavelength, via the method of matched asymptotic expansions. These consist, at leading order, of regularly spaced, compactly supported regions with width of$O(1)$where$u=O(1)$, separated by regions where$u$is exponentially small at leading order as$D \to 0^+$. From numerical solutions, we find that for$D \geq \Delta _1$, permanent form travelling waves, with minimum wavespeed,$2 \sqrt{D}$, are generated, whilst for$0 \lt D \lt \Delta _1$, the wavefronts generated separate the regions where$u=0$from a region where a steady periodic solution is created via a distinct periodic shedding mechanism acting immediately to the rear of the advancing front, with this mechanism becoming more pronounced with decreasing$D$. The structure of these transitional travelling wave forms is examined in some detail.