Deep learning of contagion dynamics on complex networks

A simple yet effective approach for predicting disease spread using mathematically-inspired diffusion-informed neural networks

The COVID-19 outbreak has highlighted the importance of mathematical epidemic models like the Susceptible-Infected-Recovered (SIR) model, for understanding disease spread dynamics. However, enhancing their predictive accuracy complicates parameter estimation. To address this, we proposed a novel model that integrates traditional mathematical modeling with deep learning which has shown improved predicted power across diverse fields. The proposed model includes a simple artificial neural network (ANN) for regional disease incidences, and a graph convolutional neural network (GCN) to capture spread to adjacent regions. GCNs are a recent deep learning algorithm designed to learn spatial relationship from graph-structured data. We applied the model to COVID-19 incidences in Spain to evaluate its performance. It achieved a 0.9679 correlation with the test data, outperforming previous models with fewer parameters. By leveraging the efficient training methods of deep learning, the model simplifies parameter estimation while maintaining alignment with the mathematical framework to ensure interpretability. The proposed model may allow the more robust and insightful analyses by leveraging the generalization power of deep learning and theoretical foundations of the mathematical models.

A quantum mechanics-based framework for infectious disease modeling

Traditional infectious disease models often use fixed compartments to represent different states of individuals. However, these models can be limited in accurately reflecting real-world conditions of individuals. In this study, we integrate quantum mechanics into infectious disease modeling, developing a quantum mechanics-based model that effectively addresses the limitations of traditional compartmental models and introduces a novel approach to understanding disease dynamics. Firstly, we examined the individual infection process and the model's evolutionary dynamics, deriving both the disease-free equilibrium point and the model's basic reproduction number. Secondly, the proposed model is simulated on a quantum circuit. The simulation results are utilized to analyze the model's parameter sensitivity and verify its rationality. The results indicate that the model's predictions align with the general patterns of viral transmission and are capable of replicating the structural attributes of compartmental models. Finally, we apply the model to simulate the spread of COVID-19. The observed similarity between the simulated results and actual infection trends demonstrates the model's effectiveness in accurately capturing viral transmission dynamics. Comparative experiments show that the proposed model significantly improves accuracy over traditional models. By leveraging quantum mechanics, our method offers a fresh perspective in infectious disease modeling, broadening the application of quantum mechanics methodologies in understanding information propagation within the macroscopic world.

Identifying Influential Nodes Based on Evidence Theory in Complex Network

Influential node identification is an important and hot topic in the field of complex network science. Classical algorithms for identifying influential nodes are typically based on a single attribute of nodes or the simple fusion of a few attributes. However, these methods perform poorly in real networks with high complexity and diversity. To address this issue, a new method based on the Dempster-Shafer (DS) evidence theory is proposed in this paper, which improves the efficiency of identifying influential nodes through the following three aspects. Firstly, Dempster-Shafer evidence theory quantifies uncertainty through its basic belief assignment function and combines evidence from different information sources, enabling it to effectively handle uncertainty. Secondly, Dempster-Shafer evidence theory processes conflicting evidence using Dempster's rule of combination, enhancing the reliability of decision-making. Lastly, in complex networks, information may come from multiple dimensions, and the Dempster-Shafer theory can effectively integrate this multidimensional information. To verify the effectiveness of the proposed method, extensive experiments are conducted on real-world complex networks. The results show that, compared to the other algorithms, attacking the influential nodes identified by the DS method is more likely to lead to the disintegration of the network, which indicates that the DS method is more effective for identifying the key nodes in the network. To further validate the reliability of the proposed algorithm, we use the visibility graph algorithm to convert the GBP futures time series into a complex network and then rank the nodes in the network using the DS method. The results show that the top-ranked nodes correspond to the peaks and troughs of the time series, which represents the key turning points in price changes. By conducting an in-depth analysis, investors can uncover major events that influence price trends, once again confirming the effectiveness of the algorithm.

Integrating artificial intelligence with mechanistic epidemiological modeling: a scoping review of opportunities and challenges

Integrating prior epidemiological knowledge embedded within mechanistic models with the data-mining capabilities of artificial intelligence (AI) offers transformative potential for epidemiological modeling. While the fusion of AI and traditional mechanistic approaches is rapidly advancing, efforts remain fragmented. This scoping review provides a comprehensive overview of emerging integrated models applied across the spectrum of infectious diseases. Through systematic search strategies, we identified 245 eligible studies from 15,460 records. Our review highlights the practical value of integrated models, including advances in disease forecasting, model parameterization, and calibration. However, key research gaps remain. These include the need for better incorporation of realistic decision-making considerations, expanded exploration of diverse datasets, and further investigation into biological and socio-behavioral mechanisms. Addressing these gaps will unlock the synergistic potential of AI and mechanistic modeling to enhance understanding of disease dynamics and support more effective public health planning and response.

Deep learning resilience inference for complex networked systems

Resilience, the ability to maintain fundamental functionality amidst failures and errors, is crucial for complex networked systems. Most analytical approaches rely on predefined equations for node activity dynamics and simplifying assumptions on network topology, limiting their applicability to real-world systems. Here, we propose ResInf, a deep learning framework integrating transformers and graph neural networks to infer resilience directly from observational data. ResInf learns representations of node activity dynamics and network topology without simplifying assumptions, enabling accurate resilience inference and low-dimensional visualization. Experimental results show that ResInf significantly outperforms analytical methods, with an F1-score improvement of up to 41.59% over Gao-Barzel-Barabási framework and 14.32% over spectral dimension reduction. It also generalizes to unseen topologies and dynamics and maintains robust performance despite observational disturbances. Our findings suggest that ResInf addresses an important gap in resilience inference for real-world systems, offering a fresh perspective on incorporating data-driven approaches to complex network modeling.

A combined neural ODE-Bayesian optimization approach to resolve dynamics and estimate parameters for a modified SIR model with immune memory

We propose a novel hybrid approach that integrates Neural Ordinary Differential Equations (NODEs) with Bayesian optimization to address the dynamics and parameter estimation of a modified time-delay-type Susceptible-Infected-Removed (SIR) model incorporating immune memory. This approach leverages a neural network to produce continuous multi-wave infection profiles by learning from both data and the model. The time-delay component of the SIR model, expressed through a convolutional integral, results in an integro-differential equation. To resolve these dynamics, we extend the NODE framework, employing a Runge-Kutta solver, to handle the challenging convolution integral, enabling us to fit the data and learn the parameters and dynamics of the model. Additionally, through Bayesian optimization, we enhance prediction accuracy while focusing on long-term dynamics. Our model, applied to COVID-19 data from Mexico, South Africa, and South Korea, effectively learns critical time-dependent parameters and provides accurate short- and long-term predictions. This combined methodology allows for early prediction of infection peaks, offering significant lead time for public health responses.

Reconstructing networks from simple and complex contagions

Network scientists often use complex dynamic processes to describe network contagions, but tools for fitting contagion models typically assume simple dynamics. Here, we address this gap by developing a nonparametric method to reconstruct a network and dynamics from a series of node states, using a model that breaks the dichotomy between simple pairwise and complex neighborhood-based contagions. We then show that a network is more easily reconstructed when observed through the lens of complex contagions if it is dense or the dynamic saturates, and that simple contagions are better otherwise.

Predicting epidemic threshold in complex networks by graph neural network

To achieve precision in predicting an epidemic threshold in complex networks, we have developed a novel threshold graph neural network (TGNN) that takes into account both the network topology and the spreading dynamical process, which together contribute to the epidemic threshold. The proposed TGNN could effectively and accurately predict the epidemic threshold in homogeneous networks, characterized by a small variance in the degree distribution, such as Erdős-Rényi random networks. Usability has also been validated when the range of the effective spreading rate is altered. Furthermore, extensive experiments in ER networks and scale-free networks validate the adaptability of the TGNN to different network topologies without the necessity for retaining. The adaptability of the TGNN is further validated in real-world networks.

Duality between predictability and reconstructability in complex systems

Predicting the evolution of a large system of units using its structure of interaction is a fundamental problem in complex system theory. And so is the problem of reconstructing the structure of interaction from temporal observations. Here, we find an intricate relationship between predictability and reconstructability using an information-theoretical point of view. We use the mutual information between a random graph and a stochastic process evolving on this random graph to quantify their codependence. Then, we show how the uncertainty coefficients, which are intimately related to that mutual information, quantify our ability to reconstruct a graph from an observed time series, and our ability to predict the evolution of a process from the structure of its interactions. We provide analytical calculations of the uncertainty coefficients for many different systems, including continuous deterministic systems, and describe a numerical procedure when exact calculations are intractable. Interestingly, we find that predictability and reconstructability, even though closely connected by the mutual information, can behave differently, even in a dual manner. We prove how such duality universally emerges when changing the number of steps in the process. Finally, we provide evidence that predictability-reconstruction dualities may exist in dynamical processes on real networks close to criticality.

Deep-learning reconstruction of complex dynamical networks from incomplete data

Reconstructing complex networks and predicting the dynamics are particularly challenging in real-world applications because the available information and data are incomplete. We develop a unified collaborative deep-learning framework consisting of three modules: network inference, state estimation, and dynamical learning. The complete network structure is first inferred and the states of the unobserved nodes are estimated, based on which the dynamical learning module is activated to determine the dynamical evolution rules. An alternating parameter updating strategy is deployed to improve the inference and prediction accuracy. Our framework outperforms baseline methods for synthetic and empirical networks hosting a variety of dynamical processes. A reciprocity emerges between network inference and dynamical prediction: better inference of network structure improves the accuracy of dynamical prediction, and vice versa. We demonstrate the superior performance of our framework on an influenza dataset consisting of 37 US States and a PM2.5 dataset covering 184 cities in China.

Nonlinear bias toward complex contagion in uncertain transmission settings

Current epidemics in the biological and social domains are challenging the standard assumptions of mathematical contagion models. Chief among them are the complex patterns of transmission caused by heterogeneous group sizes and infection risk varying by orders of magnitude in different settings, like indoor versus outdoor gatherings in the COVID-19 pandemic or different moderation practices in social media communities. However, quantifying these heterogeneous levels of risk is difficult, and most models typically ignore them. Here, we include these features in an epidemic model on weighted hypergraphs to capture group-specific transmission rates. We study analytically the consequences of ignoring the heterogeneous transmissibility and find an induced superlinear infection rate during the emergence of a new outbreak, even though the underlying mechanism is a simple, linear contagion. The dynamics produced at the individual and group levels are therefore more similar to complex, nonlinear contagions, thus blurring the line between simple and complex contagions in realistic settings. We support this claim by introducing a Bayesian inference framework to quantify the nonlinearity of contagion processes. We show that simple contagions on real weighted hypergraphs are systematically biased toward the superlinear regime if the heterogeneity of the weights is ignored, greatly increasing the risk of erroneous classification as complex contagions. Our results provide an important cautionary tale for the challenging task of inferring transmission mechanisms from incidence data. Yet, it also paves the way for effective models that capture complex features of epidemics through nonlinear infection rates.

A data-driven epidemic model with human mobility and vaccination protection for COVID-19 prediction

Epidemiological models allow for quantifying the dynamic characteristics of large-scale outbreaks. However, capturing detailed and accurate epidemiological information often requires consideration of multiple kinetic mechanisms and parameters. Due to the uncertainty of pandemic evolution, such as pathogen variation, host immune response and changes in mitigation strategies, the parameter evaluation and state prediction of complex epidemiological models are challenging. Here, we develop a data-driven epidemic model with a generalized SEIR mechanistic structure that includes new compartments, human mobility and vaccination protection. To address the issue of model complexity, we embed the epidemiological model dynamics into physics-informed neural networks (PINN), taking the observed series of time instances as direct input of the network to simultaneously infer unknown parameters and unobserved dynamics of the underlying model. Using actual data during the COVID-19 outbreak in Australia, Israel, and Switzerland, our model framework demonstrates satisfactory performance in multi-step ahead predictions compared to several benchmark models. Moreover, our model infers time-varying parameters such as transmission rates, hospitalization ratios, and effective reproduction numbers, as well as calculates the latent period and asymptomatic infection count, which are typically unreported in public data. Finally, we employ the proposed data-driven model to analyze the impact of different mitigation strategies on COVID-19.

A Two-Phase Feature Selection Method for Identifying Influential Spreaders of Disease Epidemics in Complex Networks

Network epidemiology plays a fundamental role in understanding the relationship between network structure and epidemic dynamics, among which identifying influential spreaders is especially important. Most previous studies aim to propose a centrality measure based on network topology to reflect the influence of spreaders, which manifest limited universality. Machine learning enhances the identification of influential spreaders by combining multiple centralities. However, several centrality measures utilized in machine learning methods, such as closeness centrality, exhibit high computational complexity when confronted with large network sizes. Here, we propose a two-phase feature selection method for identifying influential spreaders with a reduced feature dimension. Depending on the definition of influential spreaders, we obtain the optimal feature combination for different synthetic networks. Our results demonstrate that when the datasets are mildly or moderately imbalanced, for Barabasi-Albert (BA) scale-free networks, the centralities' combination with the two-hop neighborhood is fundamental, and for Erdős-Rényi (ER) random graphs, the centralities' combination with the degree centrality is essential. Meanwhile, for Watts-Strogatz (WS) small world networks, feature selection is unnecessary. We also conduct experiments on real-world networks, and the features selected display a high similarity with synthetic networks. Our method provides a new path for identifying superspreaders for the control of epidemics.

Inference on a Multi-Patch Epidemic Model with Partial Mobility, Residency, and Demography: Case of the 2020 COVID-19 Outbreak in Hermosillo, Mexico

Most studies modeling population mobility and the spread of infectious diseases, particularly those using meta-population multi-patch models, tend to focus on the theoretical properties and numerical simulation of such models. As such, there is relatively scant literature focused on numerical fit, inference, and uncertainty quantification of epidemic models with population mobility. In this research, we use three estimation techniques to solve an inverse problem and quantify its uncertainty for a human-mobility-based multi-patch epidemic model using mobile phone sensing data and confirmed COVID-19-positive cases in Hermosillo, Mexico. First, we utilize a Brownian bridge model using mobile phone GPS data to estimate the residence and mobility parameters of the epidemic model. In the second step, we estimate the optimal model epidemiological parameters by deterministically inverting the model using a Darwinian-inspired evolutionary algorithm (EA)-that is, a genetic algorithm (GA). The third part of the analysis involves performing inference and uncertainty quantification in the epidemic model using two Bayesian Monte Carlo sampling methods: t-walk and Hamiltonian Monte Carlo (HMC). The results demonstrate that the estimated model parameters and incidence adequately fit the observed daily COVID-19 incidence in Hermosillo. Moreover, the estimated parameters from the HMC method yield large credible intervals, improving their coverage for the observed and predicted daily incidences. Furthermore, we observe that the use of a multi-patch model with mobility yields improved predictions when compared to a single-patch model.

Distinguishing Simple and Complex Contagion Processes on Networks

Contagion processes on networks, including disease spreading, information diffusion, or social behaviors propagation, can be modeled as simple contagion, i.e., as a contagion process involving one connection at a time, or as complex contagion, in which multiple interactions are needed for a contagion event. Empirical data on spreading processes, however, even when available, do not easily allow us to uncover which of these underlying contagion mechanisms is at work. We propose a strategy to discriminate between these mechanisms upon the observation of a single instance of a spreading process. The strategy is based on the observation of the order in which network nodes are infected, and on its correlations with their local topology: these correlations differ between processes of simple contagion, processes involving threshold mechanisms, and processes driven by group interactions (i.e., by "higher-order" mechanisms). Our results improve our understanding of contagion processes and provide a method using only limited information to distinguish between several possible contagion mechanisms.

Inferring origin-destination distribution of agent transfer in a complex network using deep gated recurrent units

Predicting the origin-destination (OD) probability distribution of agent transfer is an important problem for managing complex systems. However, prediction accuracy of associated statistical estimators suffer from underdetermination. While specific techniques have been proposed to overcome this deficiency, there still lacks a general approach. Here, we propose a deep neural network framework with gated recurrent units (DNNGRU) to address this gap. Our DNNGRU is network-free, as it is trained by supervised learning with time-series data on the volume of agents passing through edges. We use it to investigate how network topologies affect OD prediction accuracy, where performance enhancement is observed to depend on the degree of overlap between paths taken by different ODs. By comparing against methods that give exact results, we demonstrate the near-optimal performance of our DNNGRU, which we found to consistently outperform existing methods and alternative neural network architectures, under diverse data generation scenarios.

Predicting COVID-19 positivity and hospitalization with multi-scale graph neural networks

The pandemic of COVID-19 is undoubtedly one of the biggest challenges for modern healthcare. In order to analyze the spatio-temporal aspects of the spread of COVID-19, technology has helped us to track, identify and store information regarding positivity and hospitalization, across different levels of municipal entities. In this work, we present a method for predicting the number of positive and hospitalized cases via a novel multi-scale graph neural network, integrating information from fine-scale geographical zones of a few thousand inhabitants. By leveraging population mobility data and other features, the model utilizes message passing to model interaction between areas. Our proposed model manages to outperform baselines and deep learning models, presenting low errors in both prediction tasks. We specifically point out the importance of our contribution in predicting hospitalization since hospitals became critical infrastructure during the pandemic. To the best of our knowledge, this is the first work to exploit high-resolution spatio-temporal data in a multi-scale manner, incorporating additional knowledge, such as vaccination rates and population mobility data. We believe that our method may improve future estimations of positivity and hospitalization, which is crucial for healthcare planning.

The structure inference of flocking systems based on the trajectories

The interaction between the swarm individuals affects the dynamic behavior of the swarm, but it is difficult to obtain directly from outside observation. Therefore, the problem we focus on is inferring the structure of the interactions in the swarm from the individual behavior trajectories. Similar inference problems that existed in network science are named network reconstruction or network inference. It is a fundamental problem pervading research on complex systems. In this paper, a new method, called Motion Trajectory Similarity, is developed for inferring direct interactions from the motion state of individuals in the swarm. It constructs correlations by combining the similarity of the motion trajectories of each cross section of the time series, in which individuals with highly similar motion states are more likely to interact with each other. Experiments on the flocking systems demonstrate that our method can produce a reliable interaction inference and outperform traditional network inference methods. It can withstand a high level of noise and time delay introduced into flocking models, as well as parameter variation in the flocking system, to achieve robust reconstruction. The proposed method provides a new perspective for inferring the interaction structure of a swarm, which helps us to explore the mechanisms of collective movement in swarms and paves the way for developing the flocking models that can be quantified and predicted.

A graph convolutional network for predicting COVID-19 dynamics in 190 regions/countries

Introduction:

Coronavirus disease (COVID-19) rapidly spread from Wuhan, China to other parts of China and other regions/countries around the world, resulting in a pandemic due to large populations moving through the massive transport hubs connecting all regions of China via railways and a major international airport. COVID-19 will remain a threat until safe and effective vaccines and antiviral drugs have been developed, distributed, and administered on a global scale. Thus, there is urgent need to establish effective implementation of preemptive non-pharmaceutical interventions for appropriate prevention and control strategies, and predicting future COVID-19 cases is required to monitor and control the issue.

Methods

This study attempts to utilize a three-layer graph convolutional network (GCN) model to predict future COVID-19 cases in 190 regions and countries using COVID-19 case data, commercial flight route data, and digital maps of public transportation in terms of transnational human mobility. We compared the performance of the proposed GCN model to a multilayer perceptron (MLP) model on a dataset of COVID-19 cases (excluding the graph representation). The prediction performance of the models was evaluated using the mean squared error.

Results

Our results demonstrate that the proposed GCN model can achieve better graph utilization and performance compared to the baseline in terms of both prediction accuracy and stability.

Discussion

The proposed GCN model is a useful means to predict COVID-19 cases at regional and national levels. Such predictions can be used to facilitate public health solutions in public health responses to the COVID-19 pandemic using deep learning and data pooling. In addition, the proposed GCN model may help public health policymakers in decision making in terms of epidemic prevention and control strategies.