The large graph limit of a stochastic epidemic model on a dynamic multilayer network

Modelling the spread of two successive SIR epidemics on a configuration model network

We present a stochastic model for two successive SIR (Susceptible → Infectious → Recovered) epidemics in the same network structured population. Individuals infected during the first epidemic might have (partial) immunity for the second one. The first epidemic is analysed through a bond percolation model, while the second epidemic is approximated by a three-type branching process in which the types of individuals depend on their position in the percolation clusters used for the first epidemic. This branching process approximation enables us to calculate, in the large population limit and conditional upon a large outbreak in the first epidemic, a threshold parameter and the probability of a large outbreak for the second epidemic. A second branching process approximation enables us to calculate the fraction of the population that are infected by such a second large outbreak. We illustrate our results through some specific cases which have appeared previously in the literature and show that our asymptotic results give good approximations for finite populations.

How to correctly fit an SIR model to data from an SEIR model?

In epidemiology, realistic disease dynamics often require Susceptible-Exposed-Infected-Recovered (SEIR)-like models because they account for incubation periods before individuals become infectious. However, for the sake of analytical tractability, simpler Susceptible-Infected-Recovered (SIR) models are commonly used, despite their lack of biological realism. Bridging these models is crucial for accurately estimating parameters and fitting models to observed data, particularly in population-level studies of infectious diseases. This paper investigates stochastic versions of the SEIR and SIR frameworks and demonstrates that the SEIR model can be effectively approximated by a SIR model with time-dependent infection and recovery rates. The validity of this approximation is supported by the derivation of a large-population Functional Law of Large Numbers (FLLN) limit and a finite-population concentration inequality. To apply this approximation in practice, the paper introduces a parameter inference methodology based on the Dynamic Survival Analysis (DSA) survival analysis framework. This method enables the fitting of the SIR model to data simulated from the more complex SEIR dynamics, as illustrated through simulated experiments.

Necessary and sufficient conditions for exact closures of epidemic equations on configuration model networks

We prove that it is possible to obtain the exact closure of SIR pairwise epidemic equations on a configuration model network if and only if the degree distribution follows a Poisson, binomial, or negative binomial distribution. The proof relies on establishing the equivalence, for these specific degree distributions, between the closed pairwise model and a dynamical survival analysis (DSA) model that was previously shown to be exact. Specifically, we demonstrate that the DSA model is equivalent to the well-known edge-based Volz model. Using this result, we also provide reductions of the closed pairwise and Volz models to a single equation that involves only susceptibles. This equation has a useful statistical interpretation in terms of times to infection. We provide some numerical examples to illustrate our results.

Projecting COVID-19 cases and hospital burden in Ohio

As the Coronavirus 2019 disease (COVID-19) started to spread rapidly in the state of Ohio, the Ecology, Epidemiology and Population Health (EEPH) program within the Infectious Diseases Institute (IDI) at The Ohio State University (OSU) took the initiative to offer epidemic modeling and decision analytics support to the Ohio Department of Health (ODH). This paper describes the methodology used by the OSU/IDI response modeling team to predict statewide cases of new infections as well as potential hospital burden in the state. The methodology has two components: (1) A Dynamical Survival Analysis (DSA)-based statistical method to perform parameter inference, statewide prediction and uncertainty quantification. (2) A geographic component that down-projects statewide predicted counts to potential hospital burden across the state. We demonstrate the overall methodology with publicly available data. A Python implementation of the methodology is also made publicly available. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics".

Hypergraphon mean field games

We propose an approach to modeling large-scale multi-agent dynamical systems allowing interactions among more than just pairs of agents using the theory of mean field games and the notion of hypergraphons, which are obtained as limits of large hypergraphs. To the best of our knowledge, ours is the first work on mean field games on hypergraphs. Together with an extension to a multi-layer setup, we obtain limiting descriptions for large systems of non-linear, weakly interacting dynamical agents. On the theoretical side, we prove the well-foundedness of the resulting hypergraphon mean field game, showing both existence and approximate Nash properties. On the applied side, we extend numerical and learning algorithms to compute the hypergraphon mean field equilibria. To verify our approach empirically, we consider a social rumor spreading model, where we give agents intrinsic motivation to spread rumors to unaware agents, and an epidemic control problem.

A functional central limit theorem for SI processes on configuration model graphs

We study a stochastic compartmental susceptible–infected (SI) epidemic process on a configuration model random graph with a given degree distribution over a finite time interval. We split the population of graph vertices into two compartments, namely, S and I, denoting susceptible and infected vertices, respectively. In addition to the sizes of these two compartments, we keep track of the counts of SI-edges (those connecting a susceptible and an infected vertex) and SS-edges (those connecting two susceptible vertices). We describe the dynamical process in terms of these counts and present a functional central limit theorem (FCLT) for them as the number of vertices in the random graph grows to infinity. The FCLT asserts that the counts, when appropriately scaled, converge weakly to a continuous Gaussian vector semimartingale process in the space of vector-valued càdlàg functions endowed with the Skorokhod topology. We discuss applications of the FCLT in percolation theory and in modelling the spread of computer viruses. We also provide simulation results illustrating the FCLT for some common degree distributions.

Projecting COVID-19 Cases and Subsequent Hospital Burden in Ohio

As the Coronavirus 2019 (COVID-19) disease started to spread rapidly in the state of Ohio, the Ecology, Epidemiology and Population Health (EEPH) program within the Infectious Diseases Institute (IDI) at the Ohio State University (OSU) took the initiative to offer epidemic modeling and decision analytics support to the Ohio Department of Health (ODH). This paper describes the methodology used by the OSU/IDI response modeling team to predict statewide cases of new infections as well as potential hospital burden in the state. The methodology has two components: 1) A Dynamic Survival Analysis (DSA)-based statistical method to perform parameter inference, statewide prediction and uncertainty quantification. 2) A geographic component that down-projects statewide predicted counts to potential hospital burden across the state. We demonstrate the overall methodology with publicly available data. A Python implementation of the methodology has been made available publicly.

Highlights

We present a novel statistical approach called Dynamic Survival Analysis (DSA) to model an epidemic curve with incomplete data. The DSA approach is advantageous over standard statistical methods primarily because it does not require prior knowledge of the size of the susceptible population, the overall prevalence of the disease, and also the shape of the epidemic curve.The principal motivation behind the study was to obtain predictions of case counts of COVID-19 and the resulting hospital burden in the state of Ohio during the early phase of the pandemic.The proposed methodology was applied to the COVID-19 incidence data in the state of Ohio to support the Ohio Department of Health (ODH) and the Ohio Hospital Association (OHA) with predictions of hospital burden in each of the Hospital Catchment Areas (HCAs) of the state.

Dynamic survival analysis for non-Markovian epidemic models

We present a new method for analysing stochastic epidemic models under minimal assumptions. The method, dubbed dynamic survival analysis (DSA), is based on a simple yet powerful observation, namely that population-level mean-field trajectories described by a system of partial differential equations may also approximate individual-level times of infection and recovery. This idea gives rise to a certain non-Markovian agent-based model and provides an agent-level likelihood function for a random sample of infection and/or recovery times. Extensive numerical analyses on both synthetic and real epidemic data from foot-and-mouth disease in the UK (2001) and COVID-19 in India (2020) show good accuracy and confirm the method's versatility in likelihood-based parameter estimation. The accompanying software package gives prospective users a practical tool for modelling, analysing and interpreting epidemic data with the help of the DSA approach.

Motif-based mean-field approximation of interacting particles on clustered networks

Interacting particles on graphs are routinely used to study magnetic behavior in physics, disease spread in epidemiology, and opinion dynamics in social sciences. The literature on mean-field approximations of such systems for large graphs typically remains limited to specific dynamics, or assumes cluster-free graphs for which standard approximations based on degrees and pairs are often reasonably accurate. Here, we propose a motif-based mean-field approximation that considers higher-order subgraph structures in large clustered graphs. Numerically, our equations agree with stochastic simulations where existing methods fail.

An Edge-Based Model of SEIR Epidemics on Static Random Networks

Studies have been done using networks to represent the spread of infectious diseases in populations. For diseases with exposed individuals corresponding to a latent period, an SEIR model is formulated using an edge-based approach described by a probability generating function. The basic reproduction number is computed using the next generation matrix method and the final size of the epidemic is derived analytically. The SEIR model in this study is used to investigate the stochasticity of the SEIR dynamics. The stochastic simulations are performed applying continuous-time Gillespie's algorithm given Poisson and power law with exponential cut-off degree distributions. The resulting predictions of the SEIR model given the initial conditions match well with the stochastic simulations, validating the accuracy of the SEIR model. We varied the contribution of the disease parameters and the average degree of the network in order to investigate their effects on the spread of disease. We verified that the infection and the recovery rates show significant effects on the dynamics of the disease transmission. While the exposed rate delays the spread of the disease, increasing it towards infinity would lead to almost the same dynamics as that of an SIR case. A network with high average degree results to an early and higher peak of the epidemic compared to a network with low average degree. The results in this paper can be used as an alternative way of explaining the spread of disease and it provides implications on the control strategies applied to mitigate the disease transmission.

Survival dynamical systems: individual-level survival analysis from population-level epidemic models

In this paper, we show that solutions to ordinary differential equations describing the large-population limits of Markovian stochastic epidemic models can be interpreted as survival or cumulative hazard functions when analysing data on individuals sampled from the population. We refer to the individual-level survival and hazard functions derived from population-level equations as a survival dynamical system (SDS). To illustrate how population-level dynamics imply probability laws for individual-level infection and recovery times that can be used for statistical inference, we show numerical examples based on synthetic data. In these examples, we show that an SDS analysis compares favourably with a complete-data maximum-likelihood analysis. Finally, we use the SDS approach to analyse data from a 2009 influenza A(H1N1) outbreak at Washington State University.

A stochastic SIR network epidemic model with preventive dropping of edges

A Markovian Susceptible [Formula: see text] Infectious [Formula: see text] Recovered (SIR) model is considered for the spread of an epidemic on a configuration model network, in which susceptible individuals may take preventive measures by dropping edges to infectious neighbours. An effective degree formulation of the model is used in conjunction with the theory of density dependent population processes to obtain a law of large numbers and a functional central limit theorem for the epidemic as the population size [Formula: see text], assuming that the degrees of individuals are bounded. A central limit theorem is conjectured for the final size of the epidemic. The results are obtained for both the Molloy-Reed (in which the degrees of individuals are deterministic) and Newman-Strogatz-Watts (in which the degrees of individuals are independent and identically distributed) versions of the configuration model. The two versions yield the same limiting deterministic model but the asymptotic variances in the central limit theorems are greater in the Newman-Strogatz-Watts version. The basic reproduction number [Formula: see text] and the process of susceptible individuals in the limiting deterministic model, for the model with dropping of edges, are the same as for a corresponding SIR model without dropping of edges but an increased recovery rate, though, when [Formula: see text], the probability of a major outbreak is greater in the model with dropping of edges. The results are specialised to the model without dropping of edges to yield conjectured central limit theorems for the final size of Markovian SIR epidemics on configuration-model networks, and for the size of the giant components of those networks. The theory is illustrated by numerical studies, which demonstrate that the asymptotic approximations are good, even for moderate N.

Edge-Based Compartmental Modelling of an SIR Epidemic on a Dual-Layer Static-Dynamic Multiplex Network with Tunable Clustering

The duration, type and structure of connections between individuals in real-world populations play a crucial role in how diseases invade and spread. Here, we incorporate the aforementioned heterogeneities into a model by considering a dual-layer static–dynamic multiplex network. The static network layer affords tunable clustering and describes an individual’s permanent community structure. The dynamic network layer describes the transient connections an individual makes with members of the wider population by imposing constant edge rewiring. We follow the edge-based compartmental modelling approach to derive equations describing the evolution of a susceptible–infected–recovered epidemic spreading through this multiplex network of individuals. We derive the basic reproduction number, measuring the expected number of new infectious cases caused by a single infectious individual in an otherwise susceptible population. We validate model equations by showing convergence to pre-existing edge-based compartmental model equations in limiting cases and by comparison with stochastically simulated epidemics. We explore the effects of altering model parameters and multiplex network attributes on resultant epidemic dynamics. We validate the basic reproduction number by plotting its value against associated final epidemic sizes measured from simulation and predicted by model equations for a number of set-ups. Further, we explore the effect of varying individual model parameters on the basic reproduction number. We conclude with a discussion of the significance and interpretation of the model and its relation to existing research literature. We highlight intrinsic limitations and potential extensions of the present model and outline future research considerations, both experimental and theoretical.