Malaria remains a significant global health challenge, with sub-Saharan Africa bearing the majority of the burden. While vector control measures such as pyrethroid-based insecticidal nets and indoor residual spraying have significantly reduced malaria incidence, the emergence of insecticide resistance in Anopheles mosquito populations threatens these gains. Resistance develops through genetic mutations under prolonged selection pressure, complicating control efforts and necessitating a deeper understanding of its evolutionary dynamics. This study introduces a novel mathematical framework to investigate the emergence and spread of insecticide resistance in mosquito populations. By modelling insecticide resistance as a continuous (quantitative) trait influenced by multiple genes, we capture its variability and evolutionary transient dynamics. We propose an age-structured mosquito population model using integro-differential equations, where the resistance trait influences life-history parameters such as mortality and reproduction. Our approach provides new insights into how resistance emerges and spreads within mosquito populations over time. We analyse the model's properties, including the existence of a unique maximal bounded semiflow, and derive conditions for the existence and stability of steady states. Through parameterization and simulations, we explore the transient and long-term dynamics of resistance evolution under different scenarios. The results offer valuable insights into the evolutionary mechanisms driving insecticide resistance and inform the design of sustainable vector control strategies.
In this study, we develop epidemic reaction-diffusion models by incorporating the dependency of the diffusion rate of susceptible individuals on new infection cases, employing both Fickian and Fokker-Planck-type diffusion laws. As the first part of a two-part series, we focus on epidemics driven by frequency-dependent incidence. We explore linear, exponential and algebraic relationships between diffusion rate of the susceptible population and new infection cases to provide deeper biological insights. Our analysis establishes the global existence of solutions and characterizes the threshold dynamics using basic reproduction numbers. We find that in quasilinear parabolic systems, the Fokker-Planck-type diffusion law tends to induce spatial segregation of susceptible and infected individuals, while the Fickian law favours spatial homogenization of susceptible individuals. Additionally, the Fokker-Planck-type model, where the diffusion rate of infected individuals depends on new recovery cases, more accurately captures the cognitive diffusion behaviour of individuals.
This work investigates the dynamics of positive classical solutions to a diffusive susceptible-exposed-infected-recovered-susceptible epidemic model with a mass-action incidence mechanism in spatially heterogeneous environments. Under minimal assumptions on the initial data, the global existence of classical solutions is established. Moreover, the eventual boundedness of these solutions is proved when either the spatial domain has dimension five or lower or the susceptible and exposed subpopulations share the same diffusion rate. Next, we define the basic reproduction number, $\mathcal{R}_0$ , and demonstrate that the disease-free equilibrium is globally stable when $\mathcal{R}_0$ is sufficiently small. However, due to the complex interaction between population movement and spatial variation in transmission rates, we find that the disease may persist even when $\mathcal{R}_0$ is slightly less than one. In such cases, we show that the system admits at least two endemic equilibrium (EE) solutions, an outcome not observed under the frequency-dependent incidence mechanism. These results highlight the significant influence of the transmission mechanism on disease dynamics. Furthermore, we examine the spatial profiles of the EE solutions when diffusion rates are small. Our analysis suggests that limiting the movement of the susceptible population can significantly reduce disease prevalence, provided that the total population remains below a specific threshold. In contrast, restricting the movement of the infected, exposed, or recovered populations alone may not eradicate the disease. Overall, our findings provide important insights into the spatial dynamics of infectious diseases and may offer guidance for developing and implementing effective containment strategies.
The doubly degenerate nutrient taxis system \begin{equation} \left \{ \begin{aligned} &u_{t}=\nabla \cdot (uv\nabla u)-\chi \nabla \cdot (u<^>{\alpha }v\nabla v)+\ell uv,&x\in \Omega ,\, t\gt 0,\\[5pt] & v_{t}=\Delta v-uv,&x\in \Omega ,\, t\gt 0,\\ \end{aligned} \right . \end{equation} is considered under zero-flux boundary conditions in a smoothly bounded domain $\Omega \subset \mathbb{R}<^>3$ where $\alpha \gt 0,\chi \gt 0$ and $\ell \gt 0$ . By developing a novel class of functional inequalities to address the challenges posed by the doubly degenerate diffusion mechanism in (0.1), it is shown that for $\alpha \in (\frac {3}{2},\frac {19}{12})$ , the associated initial-boundary value problem admits a global continuous weak solution for sufficiently regular initial data. Furthermore, in an appropriate topological setting, this solution converges to an equilibrium $(u_\infty , 0)$ as $t\rightarrow \infty$ . Notably, the limiting profile $u_{\infty }$ is non-homogeneous when the initial signal concentration $v_0$ is sufficiently small, provided the initial data $u_0$ is not identically constant.
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.
We study a model in which rational agents decide whether or not to commit a crime based on a utility calculation, influenced by a judge who sets a society-wide threshold corresponding to the likelihood of an individual being found guilty and a legislator who sets a society-wide punishment level. We study how the overall crime rate is influenced by the judge’s threshold and the legislator’s punishment level, propose an objective function for the judge and legislator to minimise, and study the optimal threshold and punishment levels for this objective. We then consider the case in which the overall society is subdivided into multiple groups with varying characteristics, introducing a constraint on fairness in treatment between the groups. We study how an optimal threshold and punishment level might be chosen under this fairness constraint, what ramifications the constraints have on outcomes for individuals, and under what circumstances the constrained optimum agrees with the unconstrained optimum.
In deep learning, interval neural networks are used to quantify the uncertainty of a pre-trained neural network. Suppose we are given a computational problem $P$ and a pre-trained neural network $\Phi _P$ that aims to solve $P$ . An interval neural network is then a pair of neural networks $(\underline {\phi }, \overline {\phi })$ , with the property that $\underline {\phi }(y) \leq \Phi _P(y) \leq \overline {\phi }(y)$ for all inputs $y$ , where the inequalities are meant componentwise. $(\underline {\phi }, \overline {\phi })$ are specifically trained to quantify the uncertainty of $\Phi _P$ , in the sense that the size of the interval $[\underline {\phi }(y),\overline {\phi }(y)]$ quantifies the uncertainty of the prediction $\Phi _P(y)$ . In this paper, we investigate the phenomenon when algorithms cannot compute interval neural networks in the setting of inverse problems. We show that in the typical setting of a linear inverse problem, the problem of constructing an optimal pair of interval neural networks is non-computable, even with the assumption that the pre-trained neural network $\Phi _P$ is an optimal solution. In other words, there exist classes of training sets $\Omega$ , such that there is no algorithm, even randomised (with probability $p \geq 1/2$ ), that computes an optimal pair of interval neural networks for each training set ${\mathcal{T}} \in \Omega$ . This phenomenon happens even when we are given a pre-trained neural network $\Phi _{{\mathcal{T}}}$ that is optimal for $\mathcal{T}$ . This phenomenon is intimately linked to instability in deep learning.
We consider steady-state diffusion in a bounded planar domain with multiple small targets on a smooth boundary. Using the method of matched asymptotic expansions, we investigate the competition of these targets for a diffusing particle and the crucial role of surface reactions on the targets. We start from the classical problem of splitting probabilities for perfectly reactive targets with Dirichlet boundary conditions and improve some earlier results. We discuss how this approach can be generalised to partially reactive targets characterised by a Robin boundary condition. In particular, we show how partial reactivity reduces the effective size of the target. In addition, we consider more intricate surface reactions modelled by mixed Steklov-Neumann or Steklov-Neumann-Dirichlet problems. We provide the first derivation of the asymptotic behaviour of the eigenvalues and eigenfunctions for these spectral problems in the small-target limit. Finally, we show how our asymptotic approach can be extended to interior targets in the bulk and to exterior problems where diffusion occurs in an unbounded planar domain outside a compact set. Direct applications of these results to diffusion-controlled reactions are discussed.
We study a reaction–advection–diffusion model of a target–offender–guardian system designed to capture interactions between urban crime and policing. Using Crandall–Rabinowitz bifurcation theory and spectral analysis, we establish rigorous conditions for both steady-state and Hopf bifurcations. These results identify critical thresholds of policing intensity at which spatially uniform equilibria lose stability, leading either to persistent heterogeneous hotspots or oscillatory crime–policing cycles. From a criminological perspective, such thresholds represent tipping points in guardian mobility: once crossed, they can lock neighbourhoods into stable clusters of criminal activity or trigger recurrent waves of hotspot formation. Numerical simulations complement the theory, exhibiting stationary patterns, periodic oscillations and chaotic dynamics. By explicitly incorporating law enforcement as a third interacting component, our framework extends classical two-equation models. It offers new tools for analysing non-linear interactions, bifurcations and pattern formation in multi-agent social systems.
We study the dynamics of a delayed predator–prey system with Holling type II functional response, focusing on the interplay between time delay and carrying capacity. Using local and global Hopf bifurcation theory, we establish the existence of sequences of bifurcations as the delay parameter varies and prove that the connected components of global Hopf branches are nested under suitable conditions. A novel contribution is the demonstration that the classical limit cycle of the non-delayed system belongs to a connected component of the global Hopf bifurcation in Fuller’s space. Our analysis combines rigorous functional differential equation theory with continuation methods to characterize the structure and boundedness of bifurcation branches. We further demonstrate that delays can induce oscillatory coexistence at lower carrying capacities than in the corresponding ordinary differential equation model, yielding counterintuitive biological insights. The results contribute to the broader theory of global bifurcations in delay differential equations while providing new perspectives on nonlinear population dynamics.
In this paper, we study the relation between the property of detailed balance and the ability of discriminating between different ligands for a class of stochastic models of kinetic proofreading. We prove the existence of a critical amount of lack of detailed balance that the kinetic proofreading models must have in order to have strong specificity for a value of the binding energy $\sigma$ . We also prove that the fact that a kinetic proofreading model has a lack of detailed balance that is larger than the critical one does not necessarily yield strong discrimination properties. Indeed, there exist different sets of chemical rates, leading to the same amount of lack of detailed balance, that have strong discrimination property in some cases and not in others.
We introduce and analyse a variant of the Becker-D & ouml;ring equations that models the growth of clusters through the gain or loss of monomers. Motivated by enzymatic reactions in biology, this model incorporates irreversible fragmentation and monomers injection. We establish the well-posedness of the equations under suitable conditions on the kinetic rates. Then, as in the Becker-D & ouml;ring equations, we distinguish two cases for the long-time behaviour of our solution; however, the distinction is made from the constant rate injection of monomers. While under strong fragmentation rate the system may exhibit infinite steady states, we prove that for low injection rate and moderate fragmentation, the solution converges locally exponentially fast to the steady state. Finally, we present an efficient scheme that preserves the asymptotic and allows fast computation by sub-sampling the clusters.
In this paper, we consider the time-dependent Born-Oppenheimer approximation (BOA) of a classical quantum molecule involving a possibly large number of nuclei and electrons, described by a Schr\"odinger equation. In the spirit of Born and Oppenheimer's original idea we study quantitatively the approximation of the molecular evolution. We obtain an iterable approximation of the molecular evolution to arbitrary order and we derive an effective equation for the reduced dynamics involving the nuclei equivalent to the original Schr\"odinger equation and containing no electron variables. We estimate the coefficients of the new equation and find tractable approximations for the molecular dynamics going beyond the one corresponding to the original Born and Oppenheimer approximation.
Biochemical reaction networks (RNs) are widely applied across scientific disciplines to model complex dynamic systems. We investigate the diffusion approximation of RNs with mass-action kinetics, focusing on the identifiability of the stochastic differential equations associated to the reaction network. We derive conditions under which the law of the diffusion approximation is identifiable and provide theorems for verifying identifiability in practice. Notably, our results show that some RNs have non-identifiable reaction rates, even when the law of the corresponding stochastic process is completely known. Moreover, we show that RNs with distinct graphical structures can generate the same diffusion law under specific choices of reaction rates. Finally, we compare our framework with identifiability results in the deterministic ordinary differential equation setting and the discrete continuous-time Markov chain models for RNs.
This paper concerns the isentropic compressible Navier-Stokes equations in a three-dimensional (3D) bounded domain with slip boundary conditions and vacuum. It is shown that the classical solutions to the initial-boundary-value problem of this system with large initial energy and vacuum exist globally in time and have an exponential decay rate, which is decreasing with respect to the adiabatic exponent $\gamma \gt 1$ provided that the fluid is nearly isothermal (namely, the adiabatic exponent is close enough to 1). This constitutes an extension of the celebrated result for the one-dimensional Cauchy problem of the isentropic Euler equations that has been established in 1973 by Nishida and Smoller (Comm. Pure Appl. Math. 26 (1973), 183-200). In addition, it is also shown that the gradient of the density will grow unboundedly with an exponential rate when the initial vacuum appears (even at a point). In contrast to previous related works, where either small initial energy is required or boundary effects are absent, this establishes the first result on the global existence and exponential growth of large-energy solutions with vacuum to the 3D isentropic compressible Navier-Stokes equations with slip boundary conditions.
We describe a new method to reconstruct the permittivity distribution, of an object to image, from the remotely measured electromagnetic field. We propose to use the remote fields measured before and after injecting locally in the medium plasmonic nanoparticles. Such a technique is known in the framework of imaging using contrast agents where, in optical imaging, the nanoparticles play the role of these contrast agents. The plasmonic nanoparticles are known to enjoy resonant effects, as enhancing the applied incident field, while excited at certain particular frequencies called plasmonic resonances. These resonant frequencies encode the values of the unknown permittivity at the location of the injected nanoparticles. The imaging methods we propose mainly use this resonant effect. We show that the imaging functional build up from contrasting the fields before and after injecting the nanoparticles, measured at one single back-scattered direction, and in an explicit band of incident frequencies, reaches its maximum values, in terms of the incident frequency, precisely at the mentioned plasmonic resonances. Such a behaviour allows us to recover these plasmonic resonances from which we recover the point-wise values of the permittivity distribution. In this work, we describe the method and provide the mathematical justification of this resonant effect and its use for the optical inversion using plasmonic nanoparticles as contrast agents.
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.
In deep learning (DL), the instability phenomenon is widespread and well documented, and the most commonly used measure of stability is the Lipschitz constant. While a small Lipchitz constant is traditionally viewed as guarantying stability, it does not capture the instability phenomenon in DL for classification well. The reason is that a classification function – which is the target function to be approximated – is necessarily discontinuous, thus having an ‘infinite’ Lipchitz constant. As a result, the classical approach will deem every classification function unstable, yet basic classification functions a la ‘is there a cat in the image?’ will typically be locally very ‘flat’ – and thus locally stable – except at the decision boundary. The lack of an appropriate measure of stability hinders a rigorous theory for stability in DL, and consequently, there are no proper approximation theoretic results that can guarantee the existence of stable networks for classification functions. In this paper, we introduce a novel stability measure $\mathcal{S}(f)$ , for any classification function $f$ , appropriate to study the stability of classification functions and their approximations. We further prove two approximation theorems: First, for any $\epsilon \gt 0$ and any classification function $f$ on a compact set, there is a neural network (NN) $\psi$ , such that $\psi - f \neq 0$ only on a set of measure $\lt \epsilon$ ; moreover, $\mathcal{S}(\psi ) \geq \mathcal{S}(f) - \epsilon$ (as accurate and stable as $f$ up to $\epsilon$ ). Second, for any classification function $f$ and $\epsilon \gt 0$ , there exists a NN $\psi$ such that $\psi = f$ on the set of points that are at least $\epsilon$ away from the decision boundary.
The unprecedented success of deep learning (DL) makes it unchallenged when it comes to classification problems. However, it is well established that the current DL methodology produces universally unstable neural networks (NNs). The instability problem has caused an enormous research effort -- with a vast literature on so-called adversarial attacks -- yet there has been no solution to the problem. Our paper addresses why there has been no solution to the problem, as we prove the following mathematical paradox: any training procedure based on training neural networks for classification problems with a fixed architecture will yield neural networks that are either inaccurate or unstable (if accurate) -- despite the provable existence of both accurate and stable neural networks for the same classification problems. The key is that the stable and accurate neural networks must have variable dimensions depending on the input, in particular, variable dimensions is a necessary condition for stability. Our result points towards the paradox that accurate and stable neural networks exist, however, modern algorithms do not compute them. This yields the question: if the existence of neural networks with desirable properties can be proven, can one also find algorithms that compute them? There are cases in mathematics where provable existence implies computability, but will this be the case for neural networks? The contrary is true, as we demonstrate how neural networks can provably exist as approximate minimisers to standard optimisation problems with standard cost functions, however, no randomised algorithm can compute them with probability better than 1/2.
