Causal effect identification using causal graphs is a fundamental challenge in causal inference. While extensive research has been conducted in this area, most existing methods assume the availability of fully specified directed acyclic graphs or acyclic directed mixed graphs. However, in complex domains such as medicine and epidemiology, complete causal knowledge is often unavailable, and only partial information about the system is accessible. This paper focuses on causal effect identification within partially specified causal graphs, with particular emphasis on cluster-directed mixed graphs (C-DMGs) which can represent many different acyclic directed mixed graphs (ADMGs). These graphs provide a higher-level representation of causal relationships by grouping variables into clusters, offering a more practical approach for handling complex systems. Unlike fully specified ADMGs, C-DMGs can contain cycles, which complicate their analysis and interpretation. Furthermore, their cluster-based nature introduces new challenges, as it gives rise to two distinct types of causal effects: macro causal effects and micro causal effects, each with different properties. In this work, we focus on macro causal effects, which describe the effects of entire clusters on other clusters. We establish that the do-calculus is both sound and complete for identifying these effects in C-DMGs over ADMGs when the cluster sizes are either unknown or of size greater than one. Additionally, we provide a graphical characterization of non-identifiability for macro causal effects in these graphs.
Stochastic Petri Nets (SPNs) are an increasingly popular tool of choice for modeling discrete-event dynamics in areas such as epidemiology and systems biology, yet their parameter estimation remains challenging in general and in particular when transition rates depend on external covariates and explicit likelihoods are unavailable. We introduce a neural-surrogate (neural-network--based approximation of the posterior distribution) framework that predicts the coefficients of known covariate-dependent rate functions directly from noisy, partially observed token trajectories. Our model employs a lightweight 1D Convolutional Residual Network trained end-to-end on Gillespie-simulated SPN realizations, learning to invert system dynamics under realistic conditions of event dropout. During inference, Monte Carlo dropout provides calibrated uncertainty bounds together with point estimates. On synthetic SPNs with 20% missing events, our surrogate recovers rate-function coefficients with an RMSE = 0.108 and substantially runs faster than traditional Bayesian approaches. These results demonstrate that data-driven, likelihood-free surrogates can enable accurate, robust, and real-time parameter recovery in complex, partially observed discrete-event systems.
Wastewater-based epidemiology (WBE) is a fast emerging method for passively monitoring diseases in a population. By measuring the concentrations of pathogenic materials in wastewater, WBE negates demographic biases in clinical testing and healthcare demand, and may act as a leading indicator of disease incidence. For a WBE system to be effective, it should detect the presence of a new pathogen of concern early enough and with enough precision that it can still be localised and contained. In this study, then, we show how multiple wastewater sensors can be strategically placed across a wastewater system, to detect the presence of disease faster than if sampling was done at the wastewater treatment plant only. Our approach generalises to any tree-like network and takes into account the structure of the network and how the population is distributed over it. We show how placing sensors further upstream from the treatment plant improves detection sensitivity and can inform how an outbreak is evolving in different geographical regions. However, this improvement diminishes once individual-level shedding is modelled as highly dispersed. With overdispersed shedding, we show using real COVID-19 cases in Scotland that broad trends in disease incidence (i.e., whether the epidemic is in growth or decline) can still be reasonably estimated from the wastewater signal once incidence exceeds about 5 infections per day.
First discovered by L. R. Taylor (1961, Nature), Taylor's Power Law (TPL) correlates the mean (M) population abundances and the corresponding variances (V) across a set of insect populations using a power function (V=aM^b). TPL has demonstrated its 'universality' across numerous fields of sciences, social sciences, and humanities. This universality has inspired two main prongs of exploration: one from mathematicians and statisticians, who might instinctively respond with a convergence theorem similar to the central limit theorem of the Gaussian distribution, and another from biologists, ecologists, physicists, etc., who are more interested in potential underlying ecological or organizational mechanisms. Over the past six decades, TPL studies have produced a punctuated landscape with three relatively distinct periods (1960s-1980s; 1990s-2000s, and 2010s-2020s) across the two prongs of abstract and physical worlds. Eight themes have been identified and reviewed on this landscape, including population spatial aggregation and ecological mechanisms, TPL and skewed statistical distributions, mathematical/statistical mechanisms of TPL, sample vs. population TPL, population stability, synchrony, and early warning signals for tipping points, TPL on complex networks, TPL in macrobiomes, and in microbiomes. Three future research directions including fostering reciprocal interactions between the two prongs, heterogeneity measuring, and exploration in the context of evolution. The significance of TPL research includes practically, population fluctuations captured by TPL are relevant for agriculture, forestry, fishery, wildlife-conservation, epidemiology, tumor heterogeneity, earthquakes, social inequality, stock illiquidity, financial stability, tipping point events, etc.; theoretically, TPL is one form of power laws, which are related to phase transitions, universality, scale-invariance, etc.
Metapopulation models are commonly used in ecology, evolution, and epidemiology. These models usually entail homogeneity assumptions within patches and study networks of migration between patches to generate insights into conservation of species, differentiation of populations, and persistence of infectious diseases. Here, focusing on infectious disease epidemiology, we take a complementary approach and study the effects of individual variation within patches while neglecting any form of disease transmission between patches. Consistently with previous work on single populations, we show how metapopulation models that neglect in-patch heterogeneity also underestimate basic reproduction numbers ($\mathcal{R}_{0}$) and the effort required to control or eliminate infectious diseases by uniform interventions. We then go beyond this confirmatory result and introduce a scheme to infer distributions of individual susceptibility or exposure to infection based on suitable stratifications of a population into patches. We apply the resulting metapopulation models to a simple case study of the COVID-19 pandemic.
Long-term forecasting of chaotic systems from short-term observations remains a fundamental and underexplored challenge due to the intrinsic sensitivity to initial conditions and the complex geometry of strange attractors. Existing approaches often rely on long-term training data or focus on short-term sequence correlations, struggling to maintain predictive stability and dynamical coherence over extended horizons. We propose PhyxMamba, a novel framework that integrates a Mamba-based state-space model with physics-informed principles to capture the underlying dynamics of chaotic systems. By reconstructing the attractor manifold from brief observations using time-delay embeddings, PhyxMamba extracts global dynamical features essential for accurate forecasting. Our generative training scheme enables Mamba to replicate the physical process, augmented by multi-token prediction and attractor geometry regularization for physical constraints, enhancing prediction accuracy and preserving key statistical invariants. Extensive evaluations on diverse simulated and real-world chaotic systems demonstrate that PhyxMamba delivers superior long-term forecasting and faithfully captures essential dynamical invariants from short-term data. This framework opens new avenues for reliably predicting chaotic systems under observation-scarce conditions, with broad implications across climate science, neuroscience, epidemiology, and beyond. Our code is open-source at https://github.com/tsinghua-fib-lab/PhyxMamba.
In the era of AI, neural networks have become increasingly popular for modeling, inference, and prediction, largely due to their potential for universal approximation. With the proliferation of such deep learning models, a question arises: are leaner statistical methods still relevant? To shed insight on this question, we employ the mechanistic nonlinear ordinary differential equation (ODE) inverse problem as a testbed, using physics-informed neural network (PINN) as a representative of the deep learning paradigm and manifold-constrained Gaussian process inference (MAGI) as a representative of statistically principled methods. Through case studies involving the SEIR model from epidemiology and the Lorenz model from chaotic dynamics, we demonstrate that statistical methods are far from obsolete, especially when working with sparse and noisy observations. On tasks such as parameter inference and trajectory reconstruction, statistically principled methods consistently achieve lower bias and variance, while using far fewer parameters and requiring less hyperparameter tuning. Statistical methods can also decisively outperform deep learning models on out-of-sample future prediction, where the absence of relevant data often leads overparameterized models astray. Additionally, we find that statistically principled approaches are more robust to accumulation of numerical imprecision and can represent the underlying system more faithful to the true governing ODEs.
We introduce the SoftBart approach from Bayesian ensemble learning to estimate the relationship between multipollutant mixtures and health on chronic exposures in epidemiology research. This approach offers several key advantages over existing methods: (1) it is computationally efficient and well-suited for analyzing large datasets; (2) it is flexible in estimating various correlated nonlinear functions simultaneously; and (3) it accurately identifies active variables within highly correlated multipollutant mixtures. Through simulations, we demonstrate the method's superiority by comparing its accuracy in estimating and quantifying uncertainties for both main and interaction effects with the commonly used method, BKMR. Last, we apply the method to analyze a multipollutant dataset with 10,110 participates from the Nurses' Health Study.
In science, we are interested not only in forecasting but also in understanding how predictions are made, specifically what the interpretable underlying model looks like. Data-driven machine learning technology can significantly streamline the complex and time-consuming traditional manual process of discovering scientific laws, helping us gain insights into fundamental issues in modern science. In this work, we introduce a pre-trained symbolic foundation regressor that can effectively compress complex data with numerous interacting variables while producing interpretable physical representations. Our model has been rigorously tested on non-network symbolic regression, symbolic regression on complex networks, and the inference of network dynamics across various domains, including physics, biochemistry, ecology, and epidemiology. The results indicate a remarkable improvement in equation inference efficiency, being three times more effective than baseline approaches while maintaining accurate predictions. Furthermore, we apply our model to uncover more intuitive laws of interaction transmission from global epidemic outbreak data, achieving optimal data fitting. This model extends the application boundary of pre-trained symbolic regression models to complex networks, and we believe it provides a foundational solution for revealing the hidden mechanisms behind changes in complex phenomena, enhancing interpretability, and inspiring further scientific discoveries.
Data fusion, the process of combining observational and experimental data, can enable the identification of causal effects that would otherwise remain non-identifiable. Although identification algorithms have been developed for specific scenarios, do-calculus remains the only general-purpose tool for causal data fusion, particularly when variables are present in some data sources but not others. However, approaches based on do-calculus may encounter computational challenges as the number of variables increases and the causal graph grows in complexity. Consequently, there exists a need to reduce the size of such models while preserving the essential features. For this purpose, we propose pruning (removing unnecessary variables) and clustering (combining variables) as preprocessing operations for causal data fusion. We generalize earlier results on a single data source and derive conditions for applying pruning and clustering in the case of multiple data sources. We give sufficient conditions for inferring the identifiability or non-identifiability of a causal effect in a larger graph based on a smaller graph and show how to obtain the corresponding identifying functional for identifiable causal effects. Examples from epidemiology and social science demonstrate the use of the results.
Detecting anomalous edges in dynamic graphs is an important task in many applications over evolving triple-based data, such as social networks, transaction management, and epidemiology. A major challenge with this task is the absence of structural-temporal coupling information, which decreases the ability of the representation to distinguish anomalies from normal instances. Existing methods focus on handling independent structural and temporal features with embedding models, which ignore the deep interaction between these two types of information. In this paper, we propose a structural-temporal coupling anomaly detection architecture with a dynamic graph transformer model. Specifically, we introduce structural and temporal features from two integration levels to provide anomaly-aware graph evolutionary patterns. Then, a dynamic graph transformer enhanced by two-dimensional positional encoding is implemented to capture both discrimination and contextual consistency signals. Extensive experiments on six datasets demonstrate that our method outperforms current state-of-the-art models. Finally, a case study illustrates the strength of our method when applied to a real-world task.
The aim of this paper is to present an, admittedly somewhat subjective, bird's eye view of the mathematical theory concerning the spread of an infectious disease in a susceptible host population with static structure, culminating in a future-oriented description of various modelling challenges.
The article covers the continuous McKendrick model, which is key in epidemiology and population dynamics, developed by Anderson Gray McKendrick. It describes population age structure changes via birth and death rates as continuous functions, shedding light on population dynamics through differential equations. The article delves into the equations, boundary conditions, and survival rates, offering insights into disease progression and population dynamics. It aims for an asymptotic analysis of population dynamics using previous studies and illustrative examples. The main finding is that the population nears extinction only if the expected offspring per individual ($R$) is less than 1.
Machine learning detects patterns, block chain guarantees trust and immutability, and modern causal inference identifies directional linkages, yet none alone exposes the full energetic anatomy of complex systems; the Hamiltonian Higher Order Elasticity Dynamics(2HOED) framework bridges these gaps. Grounded in classical mechanics but extended to Economics order elasticity terms, 2HOED represents economic, social, and physical systems as energy-based Hamiltonians whose position, velocity, acceleration, and jerk of elasticity jointly determine systemic power, Inertia, policy sensitivity, and marginal responses. Because the formalism is scaling free and coordinate agnostic, it transfers seamlessly from financial markets to climate science, from supply chain logistics to epidemiology, thus any discipline in which adaptation and shocks coexist. By embedding standard econometric variables inside a Hamiltonian, 2HOED enriches conventional economic analysis with rigorous diagnostics of resilience, tipping points, and feedback loops, revealing failure modes invisible to linear models. Wavelet spectra, phase space attractors, and topological persistence diagrams derived from 2HOED expose multistage policy leverage that machine learning detects only empirically and block chain secures only after the fact. For economists, physicians and other scientists, the method opens a new causal energetic channel linking biological or mechanical elasticity to macro level outcomes. Portable, interpretable, and computationally light, 2HOED turns data streams into dynamical energy maps, empowering decision makers to anticipate crises, design adaptive policies, and engineer robust systems delivering the predictive punch of AI with the explanatory clarity of physics.
Biological time can be measured in two ways: in generations and in physical time. When generation intervals differ between individuals, these two clocks diverge, which impedes our ability to relate mathematical models to real populations. In this paper we show that nevertheless, these disparate clocks become equivalent in the long run via a simple identity relating generational and physical time. This equivalence allows us to directly translate statements from mathematical models to the physical world and vice versa. As an application, we obtain a generalized Euler-Lotka equation linking the basic reproduction number $R_0$ to the growth rate, and derive several information-theoretic bounds on these quantities. We also show how the fitness of a lineage can be defined consistently in population models, with applications to microbial growth, epidemiology and population biology.
There is a plethora of highly stochastic non-linear dynamical systems in fields such as molecular biology, chemistry, epidemiology, and ecology. Yet, none of the currently available stochastic models are both accurate and computationally efficient for long-term predictions of large systems. The Linear Noise Approximation (LNA) model for biochemical reaction networks is analytically tractable, which makes it computationally efficient for simulation, analysis, and inference. However, it is only accurate for linear systems and short-time transitions. Other methods can achieve greater accuracy across a wider range of systems, including non-linear ones, but lack analytical tractability. This paper seeks to challenge the prevailing view by demonstrating that the Linear Noise Approximation can indeed capture non-linear dynamics after certain modifications. We introduce a new framework that utilises centre manifold theory allowing us to identify simple interventions to the LNA that do not significantly compromise its computational efficiency. We develop specific algorithms for systems that exhibit oscillations or bi-stability and demonstrate their accuracy and computational efficiency across multiple examples.
The study of infectious disease propagation is essential for understanding and controlling epidemics. One of the most useful tools for gaining insights into the spread of infectious diseases is mathematical modelling. In terms of mathematical epidemiology, the main models are based on compartments, such as SI, SIR, and SEIR. These models offer mathematical frameworks for representing the proliferation dynamics of various diseases, for instance flu and smallpox. In this work, we explore these models using two distinct mathematical approaches, Cellular Automata (CA) and ODEs. They are able to reproduce the spread dynamics of diseases with their own individuality. CA models incorporate the local interaction among individuals with discrete time and space, while ODEs provide a continuous and simplified view of a disease propagation in large and homogeneous populations. By comparing these two approaches, we find that the shape of the curves of all models is similar for both representations. Although, the growth rates differ between CA and ODE. One of our results is to show that the CA yields a power-law growth, while the ODE growth rate is well-represented by an exponential function. Furthermore, a substantial contribution of our work is using a hyperbolic tangent to fit the initial growth of infected individuals for all the considered models. Our results display a strong correlation between simulated data and adjusted function. We mainly address this successful result by the fact that the hyperbolic function captures both growing: the power-law (when considered the first terms of infinite sums) and combinations of exponential (when the hyperbolic function is written via exponential). Therefore, our work shows that when modelling a disease the choice of mathematical representation is crucial, in particular to model the onset of an epidemic.
Spatio-temporal point processes (STPPs) model discrete events distributed in time and space, with important applications in areas such as criminology, seismology, epidemiology, and social networks. Traditional models often rely on parametric kernels, limiting their ability to capture heterogeneous, nonstationary dynamics. Recent innovations integrate deep neural architectures -- either by modeling the conditional intensity function directly or by learning flexible, data-driven influence kernels, substantially broadening their expressive power. This article reviews the development of the deep influence kernel approach, which enjoys statistical explainability, since the influence kernel remains in the model to capture the spatiotemporal propagation of event influence and its impact on future events, while also possessing strong expressive power, thereby benefiting from both worlds. We explain the main components in developing deep kernel point processes, leveraging tools such as functional basis decomposition and graph neural networks to encode complex spatial or network structures, as well as estimation using both likelihood-based and likelihood-free methods, and address computational scalability for large-scale data. We also discuss the theoretical foundation of kernel identifiability. Simulated and real-data examples highlight applications to crime analysis, earthquake aftershock prediction, and sepsis prediction modeling, and we conclude by discussing promising directions for the field.
There has been interest in the interactions between infectious disease dynamics and behaviour for most of the history of mathematical epidemiology. This has included consideration of which mathematical models best capture each phenomenon, as well as their interaction, but typically in a manner that is agnostic to the exact behaviour in question. Here, we investigate interacting behaviour and disease dynamics specifically related to behaviours around testing and isolation. This epidemiological-behavioural interaction is of particular interest as, prospectively, it is well-placed to be informed by real-world data temporally monitoring test results and compliance with testing policy. To carry out our investigation we extend an existing "behaviour and disease" (BaD) model by incorporating the dynamics of symptomatic testing and isolation. We provide a dynamical systems analysis of the ordinary differential equations that define this model, providing theoretical results on its behaviour early in a new outbreak (particularly its basic reproduction number) and endemicity of the system (its steady states and associated stability criteria). We then supplement these findings with a numerical analysis to inform how temporal and cumulative outbreak metrics depend on the model parameter values for epidemic and endemic regimes. As the presented interdisciplinary modelling approach can accommodate further extensions (including, but not limited to, adding testing capacity, decay in behavioural effects and multiple pathogen variants), we hope that our work will encourage further modelling studies integrating specific measured behaviours and disease dynamics that may reduce the health and economic impacts of future epidemics.
Modeling and predicting temporal point processes (TPPs) is critical in domains such as neuroscience, epidemiology, finance, and social sciences. We introduce the Spiking Dynamic Graph Network (SDGN), a novel framework that leverages the temporal processing capabilities of spiking neural networks (SNNs) and spike-timing-dependent plasticity (STDP) to dynamically estimate underlying spatio-temporal functional graphs. Unlike existing methods that rely on predefined or static graph structures, SDGN adapts to any dataset by learning dynamic spatio-temporal dependencies directly from the event data, enhancing generalizability and robustness. While SDGN offers significant improvements over prior methods, we acknowledge its limitations in handling dense graphs and certain non-Gaussian dependencies, providing opportunities for future refinement. Our evaluations, conducted on both synthetic and real-world datasets including NYC Taxi, 911, Reddit, and Stack Overflow, demonstrate that SDGN achieves superior predictive accuracy while maintaining computational efficiency. Furthermore, we include ablation studies to highlight the contributions of its core components.