Inference for nonlinear dynamical systems

Quantifying suicide contagion at population scale

The spread of suicidal behavior among individuals is often described as a contagion; however, rigorous modeling of suicide as a dynamic, contagious process is minimal. Here, we develop and validate a model-inference system depicting suicide ideation and death and use it to quantify the contagion processes in the US associated with two prominent celebrity suicide events: Robin Williams during 2014 and Kate Spade and Anthony Bourdain, which occurred 3 days apart during 2018. We show that both events produced large transient increases of suicide contagion contact rates, i.e., the spread of suicidal thought and behavior, and a period of elevated suicidal ideation in the general population. Our modeling approach provides a framework for quantifying suicidal contagion and better understanding, preventing, and containing its spread.

Inference on spatiotemporal dynamics for coupled biological populations

Mathematical models in ecology and epidemiology must be consistent with observed data in order to generate reliable knowledge and evidence-based policy. Metapopulation systems, which consist of a network of connected sub-populations, pose technical challenges in statistical inference owing to nonlinear, stochastic interactions. Numerical difficulties encountered in conducting inference can obstruct the core scientific questions concerning the link between the mathematical models and the data. Recently, an algorithm has been proposed that enables computationally tractable likelihood-based inference for high-dimensional partially observed stochastic dynamic models of metapopulation systems. We use this algorithm to build a statistically principled data analysis workflow for metapopulation systems. Via a case study of COVID-19, we show how this workflow addresses the limitations of previous approaches. The COVID-19 pandemic provides a situation where mathematical models and their policy implications are widely visible, and we revisit an influential metapopulation model used to inform basic epidemiological understanding early in the pandemic. Our methods support self-critical data analysis, enabling us to identify and address model weaknesses, leading to a new model with substantially improved statistical fit and parameter identifiability. Our results suggest that the lockdown initiated on 23 January 2020 in China was more effective than previously thought.

Parameter estimation in a whole-brain network model of epilepsy: Comparison of parallel global optimization solvers

The Virtual Epileptic Patient (VEP) refers to a computer-based representation of a patient with epilepsy that combines personalized anatomical data with dynamical models of abnormal brain activities. It is capable of generating spatio-temporal seizure patterns that resemble those recorded with invasive methods such as stereoelectro EEG data, allowing for the evaluation of clinical hypotheses before planning surgery. This study highlights the effectiveness of calibrating VEP models using a global optimization approach. The approach utilizes SaCeSS, a cooperative metaheuristic algorithm capable of parallel computation, to yield high-quality solutions without requiring excessive computational time. Through extensive benchmarking on synthetic data, our proposal successfully solved a set of different configurations of VEP models, demonstrating better scalability and superior performance against other parallel solvers. These results were further enhanced using a Bayesian optimization framework for hyperparameter tuning, with significant gains in terms of both accuracy and computational cost. Additionally, we added a scalable uncertainty quantification phase after model calibration, and used it to assess the variability in estimated parameters across different problems. Overall, this study has the potential to improve the estimation of pathological brain areas in drug-resistant epilepsy, thereby to inform the clinical decision-making process.

Ordinal Outcome State-Space Models for Intensive Longitudinal Data

Intensive longitudinal (IL) data are increasingly prevalent in psychological science, coinciding with technological advancements that make it simple to deploy study designs such as daily diary and ecological momentary assessments. IL data are characterized by a rapid rate of data collection (1+ collections per day), over a period of time, allowing for the capture of the dynamics that underlie psychological and behavioral processes. One powerful framework for analyzing IL data is state-space modeling, where observed variables are considered measurements for underlying states (i.e., latent variables) that change together over time. However, state-space modeling has typically relied on continuous measurements, whereas psychological data often come in the form of ordinal measurements such as Likert scale items. In this manuscript, we develop a general estimation approach for state-space models with ordinal measurements, specifically focusing on a graded response model for Likert scale items. We evaluate the performance of our model and estimator against that of the commonly used "linear approximation" model, which treats ordinal measurements as though they are continuous. We find that our model resulted in unbiased estimates of the state dynamics, while the linear approximation resulted in strongly biased estimates of the state dynamics. Finally, we develop an approximate standard error, termed slice standard errors and show that these approximate standard errors are more liberal than true standard errors (i.e., smaller) at a consistent bias.

Using neural ordinary differential equations to predict complex ecological dynamics from population density data

Simple models have been used to describe ecological processes for over a century. However, the complexity of ecological systems makes simple models subject to modelling bias due to simplifying assumptions or unaccounted factors, limiting their predictive power. Neural ordinary differential equations (NODEs) have surged as a machine-learning algorithm that preserves the dynamic nature of the data (Chen et al. 2018 Adv. Neural Inf. Process. Syst.). Although preserving the dynamics in the data is an advantage, the question of how NODEs perform as a forecasting tool of ecological communities is unanswered. Here, we explore this question using simulated time series of competing species in a time-varying environment. We find that NODEs provide more precise forecasts than autoregressive integrated moving average (ARIMA) models. We also find that untuned NODEs have a similar forecasting accuracy to untuned long-short term memory neural networks and both are outperformed in accuracy and precision by empirical dynamical modelling . However, we also find NODEs generally outperform all other methods when evaluating with the interval score, which evaluates precision and accuracy in terms of prediction intervals rather than pointwise accuracy. We also discuss ways to improve the forecasting performance of NODEs. The power of a forecasting tool such as NODEs is that it can provide insights into population dynamics and should thus broaden the approaches to studying time series of ecological communities.

Parameter estimation and forecasting with quantified uncertainty for ordinary differential equation models using QuantDiffForecast: A MATLAB toolbox and tutorial

Mathematical models based on systems of ordinary differential equations (ODEs) are frequently applied in various scientific fields to assess hypotheses, estimate key model parameters, and generate predictions about the system's state. To support their application, we present a comprehensive, easy-to-use, and flexible MATLAB toolbox, QuantDiffForecast, and associated tutorial to estimate parameters and generate short-term forecasts with quantified uncertainty from dynamical models based on systems of ODEs. We provide software ( https://github.com/gchowell/paramEstimation_forecasting_ODEmodels/) and detailed guidance on estimating parameters and forecasting time-series trajectories that are characterized using ODEs with quantified uncertainty through a parametric bootstrapping approach. It includes functions that allow the user to infer model parameters and assess forecasting performance for different ODE models specified by the user, using different estimation methods and error structures in the data. The tutorial is intended for a diverse audience, including students training in dynamic systems, and will be broadly applicable to estimate parameters and generate forecasts from models based on ODEs. The functions included in the toolbox are illustrated using epidemic models with varying levels of complexity applied to data from the 1918 influenza pandemic in San Francisco. A tutorial video that demonstrates the functionality of the toolbox is included.

System identifiability in a time-evolving agent-based model

Mathematical models are a valuable tool for studying and predicting the spread of infectious agents. The accuracy of model simulations and predictions invariably depends on the specification of model parameters. Estimation of these parameters is therefore extremely important; however, while some parameters can be derived from observational studies, the values of others are difficult to measure. Instead, models can be coupled with inference algorithms (i.e., data assimilation methods, or statistical filters), which fit model simulations to existing observations and estimate unobserved model state variables and parameters. Ideally, these inference algorithms should find the best fitting solution for a given model and set of observations; however, as those estimated quantities are unobserved, it is typically uncertain whether the correct parameters have been identified. Further, it is unclear what 'correct' really means for abstract parameters defined based on specific model forms. In this work, we explored the problem of non-identifiability in a stochastic system which, when overlooked, can significantly impede model prediction. We used a network, agent-based model to simulate the transmission of Methicillin-resistant staphylococcus aureus (MRSA) within hospital settings and attempted to infer key model parameters using the Ensemble Adjustment Kalman Filter, an efficient Bayesian inference algorithm. We show that even though the inference method converged and that simulations using the estimated parameters produced an agreement with observations, the true parameters are not fully identifiable. While the model-inference system can exclude a substantial area of parameter space that is unlikely to contain the true parameters, the estimated parameter range still included multiple parameter combinations that can fit observations equally well. We show that analyzing synthetic trajectories can support or contradict claims of identifiability. While we perform this on a specific model system, this approach can be generalized for a variety of stochastic representations of partially observable systems. We also suggest data manipulations intended to improve identifiability that might be applicable in many systems of interest.

Fitting a stochastic model of intensive care occupancy to noisy hospitalization time series during the COVID-19 pandemic

Intensive care occupancy is an important indicator of health care stress that has been used to guide policy decisions during the COVID-19 pandemic. Toward reliable decision-making as a pandemic progresses, estimating the rates at which patients are admitted to and discharged from hospitals and intensive care units (ICUs) is crucial. Since individual-level hospital data are rarely available to modelers in each geographic locality of interest, it is important to develop tools for inferring these rates from publicly available daily numbers of hospital and ICU beds occupied. We develop such an estimation approach based on an immigration-death process that models fluctuations of ICU occupancy. Our flexible framework allows for immigration and death rates to depend on covariates, such as hospital bed occupancy and daily SARS-CoV-2 test positivity rate, which may drive changes in hospital ICU operations. We demonstrate via simulation studies that the proposed method performs well on noisy time series data and apply our statistical framework to hospitalization data from the University of California, Irvine (UCI) Health and Orange County, California. By introducing a likelihood-based framework where immigration and death rates can vary with covariates, we find, through rigorous model selection, that hospitalization and positivity rates are crucial covariates for modeling ICU stay dynamics and validate our per-patient ICU stay estimates using anonymized patient-level UCI hospital data.

Estimation of nonlinear mixed-effects continuous-time models using the continuous-discrete extended Kalman filter

Many intensive longitudinal measurements are collected at irregularly spaced time intervals, and involve complex, possibly nonlinear and heterogeneous patterns of change. Effective modelling of such change processes requires continuous-time differential equation models that may be nonlinear and include mixed effects in the parameters. One approach of fitting such models is to define random effect variables as additional latent variables in a stochastic differential equation (SDE) model of choice, and use estimation algorithms designed for fitting SDE models, such as the continuous-discrete extended Kalman filter (CDEKF) approach implemented in the dynr R package, to estimate the random effect variables as latent variables. However, this approach's efficacy and identification constraints in handling mixed-effects SDE models have not been investigated. In the current study, we analytically inspect the identification constraints of using the CDEKF approach to fit nonlinear mixed-effects SDE models; extend a published model of emotions to a nonlinear mixed-effects SDE model as an example, and fit it to a set of irregularly spaced ecological momentary assessment data; and evaluate the feasibility of the proposed approach to fit the model through a Monte Carlo simulation study. Results show that the proposed approach produces reasonable parameter and standard error estimates when some identification constraint is met. We address the effects of sample size, process noise variance, and data spacing conditions on estimation results.

Anchoring the mean generation time in the SEIR to mitigate biases in ℜ0 estimates due to uncertainty in the distribution of the epidemiological delays

The basic reproduction number, ℜ 0 , is of paramount importance in the study of infectious disease dynamics. Primarily, ℜ 0 serves as an indicator of the transmission potential of an emerging infectious disease and the effort required to control the invading pathogen. However, its estimates from compartmental models are strongly conditioned by assumptions in the model structure, such as the distributions of the latent and infectious periods (epidemiological delays). To further complicate matters, models with dissimilar delay structures produce equivalent incidence dynamics. Following a simulation study, we reveal that the nature of such equivalency stems from a linear relationship between ℜ 0 and the mean generation time, along with adjustments to other parameters in the model. Leveraging this knowledge, we propose and successfully test an alternative parametrization of the SEIR model that produces accurate ℜ 0 estimates regardless of the distribution of the epidemiological delays, at the expense of biases in other quantities deemed of lesser importance. We further explore this approach's robustness by testing various transmissibility levels, generation times and data fidelity (overdispersion). Finally, we apply the proposed approach to data from the 1918 influenza pandemic. We anticipate that this work will mitigate biases in estimating ℜ 0 .

Malaria, climate variability, and interventions: modelling transmission dynamics

Assessment of the relative impact of climate change on malaria dynamics is a complex problem. Climate is a well-known factor that plays a crucial role in driving malaria outbreaks in epidemic transmission areas. However, its influence in endemic environments with intensive malaria control interventions is not fully understood, mainly due to the scarcity of high-quality, long-term malaria data. The demographic surveillance systems in Africa offer unique platforms for quantifying the relative effects of weather variability on the burden of malaria. Here, using a process-based stochastic transmission model, we show that in the lowlands of malaria endemic western Kenya, variations in climatic factors played a key role in driving malaria incidence during 2008-2019, despite high bed net coverage and use among the population. The model captures some of the main mechanisms of human, parasite, and vector dynamics, and opens the possibility to forecast malaria in endemic regions, taking into account the interaction between future climatic conditions and intervention scenarios.

Inferring the disruption of rabies circulation in vampire bat populations using a betaherpesvirus-vectored transmissible vaccine

Transmissible vaccines are an emerging biotechnology that hold prospects to eliminate pathogens from wildlife populations. Such vaccines would genetically modify naturally occurring, nonpathogenic viruses ("viral vectors") to express pathogen antigens while retaining their capacity to transmit. The epidemiology of candidate viral vectors within the target wildlife population has been notoriously challenging to resolve but underpins the selection of effective vectors prior to major investments in vaccine development. Here, we used spatiotemporally replicated deep sequencing to parameterize competing epidemiological mechanistic models of Desmodus rotundus betaherpesvirus (DrBHV), a proposed vector for a transmissible vaccine targeting vampire bat-transmitted rabies. Using 36 strain- and location-specific time series of prevalence collected over 6 y, we found that lifelong infections with cycles of latency and reactivation, combined with a high R0 (6.9; CI: 4.39 to 7.85), are necessary to explain patterns of DrBHV infection observed in wild bats. These epidemiological properties suggest that DrBHV may be suited to vector a lifelong, self-boosting, and transmissible vaccine. Simulations showed that inoculating a single bat with a DrBHV-vectored rabies vaccine could immunize >80% of a bat population, reducing the size, frequency, and duration of rabies outbreaks by 50 to 95%. Gradual loss of infectious vaccine from vaccinated individuals is expected but can be countered by inoculating larger but practically achievable proportions of bat populations. Parameterizing epidemiological models using accessible genomic data brings transmissible vaccines one step closer to implementation.

Resolving the enigma of Iquitos and Manaus: A modeling analysis of multiple COVID-19 epidemic waves in two Amazonian cities

The two nearby Amazonian cities of Iquitos and Manaus endured explosive COVID-19 epidemics and may well have suffered the world's highest infection and death rates over 2020, the first year of the pandemic. State-of-the-art epidemiological and modeling studies estimated that the populations of both cities came close to attaining herd immunity (>70% infected) at the termination of the first wave and were thus protected. This makes it difficult to explain the more deadly second wave of COVID-19 that struck again in Manaus just months later, simultaneous with the appearance of a new P.1 variant of concern, creating a catastrophe for the unprepared population. It was suggested that the second wave was driven by reinfections, but the episode has become controversial and an enigma in the history of the pandemic. We present a data-driven model of epidemic dynamics in Iquitos, which we also use to explain and model events in Manaus. By reverse engineering the multiple epidemic waves over 2 y in these two cities, the partially observed Markov process model inferred that the first wave left Manaus with a highly susceptible and vulnerable population (≈40% infected) open to invasion by P.1, in contrast to Iquitos (≈72% infected). The model reconstructed the full epidemic outbreak dynamics from mortality data by fitting a flexible time-varying reproductive number [Formula: see text] while estimating reinfection and impulsive immune evasion. The approach is currently highly relevant given the lack of tools available to assess these factors as new SARS-CoV-2 virus variants appear with different degrees of immune evasion.

Hidden variable models reveal the effects of infection from changes in host survival

The impacts of disease on host vital rates can be demonstrated using longitudinal studies, but these studies can be expensive and logistically challenging. We examined the utility of hidden variable models to infer the individual effects of infectious disease from population-level measurements of survival when longitudinal studies are not possible. Our approach seeks to explain temporal deviations in population-level survival after introducing a disease causative agent when disease prevalence cannot be directly measured by coupling survival and epidemiological models. We tested this approach using an experimental host system (Drosophila melanogaster) with multiple distinct pathogens to validate the ability of the hidden variable model to infer per-capita disease rates. We then applied the approach to a disease outbreak in harbor seals (Phoca vituline) that had data on observed strandings but no epidemiological data. We found that our hidden variable modeling approach could successfully detect the per-capita effects of disease from monitored survival rates in both the experimental and wild populations. Our approach may prove useful for detecting epidemics from public health data in regions where standard surveillance techniques are not available and in the study of epidemics in wildlife populations, where longitudinal studies can be especially difficult to implement.

Effects of vaccination on mitigating COVID-19 outbreaks: a conceptual modeling approach

This paper is devoted to investigating the impact of vaccination on mitigating COVID-19 outbreaks. In this work, we propose a compartmental epidemic ordinary differential equation model, which extends the previous so-called SEIRD model ^[1,2,3,4] by incorporating the birth and death of the population, disease-induced mortality and waning immunity, and adding a vaccinated compartment to account for vaccination. Firstly, we perform a mathematical analysis for this model in a special case where the disease transmission is homogeneous and vaccination program is periodic in time. In particular, we define the basic reproduction number $ \mathcal{R}_0 $ for this system and establish a threshold type of result on the global dynamics in terms of $ \mathcal{R}_0 $. Secondly, we fit our model into multiple COVID-19 waves in four locations including Hong Kong, Singapore, Japan, and South Korea and then forecast the trend of COVID-19 by the end of 2022. Finally, we study the effects of vaccination again the ongoing pandemic by numerically computing the basic reproduction number $ \mathcal{R}_0 $ under different vaccination programs. Our findings indicate that the fourth dose among the high-risk group is likely needed by the end of the year.

An SEIR Model with Time-Varying Coefficients for Analyzing the SARS-CoV-2 Epidemic

In this study, we propose a time-dependent susceptible-exposed-infected-recovered (SEIR) model for the analysis of the SARS-CoV-2 epidemic outbreak in three different countries, the United States, Italy, and Iceland using public data inherent the numbers of the epidemic wave. Since several types and grades of actions were adopted by the governments, including travel restrictions, social distancing, or limitation of movement, we want to investigate how these measures can affect the epidemic curve of the infectious population. The parameters of interest for the SEIR model were estimated employing a composite likelihood approach. Moreover, standard errors have been corrected for temporal dependence. The adoption of restrictive measures results in flatten epidemic curves, and the future evolution indicated a decrease in the number of cases.

Modelling COVID-19 vaccine breakthrough infections in highly vaccinated Israel-The effects of waning immunity and third vaccination dose

In August 2021, a major wave of the SARS-CoV-2 Delta variant erupted in the highly vaccinated population of Israel. The transmission advantage of the Delta variant enabled it to replace the Alpha variant in approximately two months. The outbreak led to an unexpectedly large proportion of breakthrough infections (BTI)-a phenomenon that received worldwide attention. Most of the Israeli population, especially those aged 60+, received their second dose of the vaccination four months before the invasion of the Delta variant. Hence, either the vaccine induced immunity dropped significantly or the Delta variant possesses immunity escaping abilities, or both. In this work, we model data obtained from the Israeli Ministry of Health, to help understand the epidemiological factors involved in the outbreak. We propose a mathematical model that captures a multitude of factors, including age structure, the time varying vaccine efficacy, time varying transmission rate, BTIs, reduced susceptibility and infectivity of vaccinated individuals, protection duration of the vaccine induced immunity, and the vaccine distribution. We fitted our model to COVID-19 cases among the vaccinated and unvaccinated, for <60 and 60+ age groups, and quantified the transmission rate, the vaccine efficacy over time and the impact of the third dose booster vaccine. The peak transmission rate of the Delta variant was found to be 2.14 times higher than that of the Alpha variant. The two-dose vaccine efficacy against infection dropped significantly from >90% to ~40% over 6 months. We further performed model simulations and quantified counterfactual scenarios examining what would happen if the booster had not been rolled out. We estimated that approximately 4.03 million infective cases (95%CI 3.19, 4.86) were prevented by vaccination overall, and 1.22 million infective cases (95%CI 0.89, 1.62) averted by the booster.

Two-Stage Bayesian Optimization for Scalable Inference in State-Space Models

State-space models (SSMs) are a rich class of dynamical models with a wide range of applications in economics, healthcare, computational biology, robotics, and more. Proper analysis, control, learning, and decision-making in dynamical systems modeled by SSMs depend on the accuracy of the inferred/learned model. Most of the existing inference techniques for SSMs are capable of dealing with very small systems, unable to be applied to most of the large-scale practical problems. Toward this, this article introduces a two-stage Bayesian optimization (BO) framework for scalable and efficient inference in SSMs. The proposed framework maps the original large parameter space to a reduced space, containing a small linear combination of the original space. This reduced space, which captures the most variability in the inference function (e.g., log likelihood or log a posteriori), is obtained by eigenvalue decomposition of the covariance of gradients of the inference function approximated by a particle filtering scheme. Then, an exponential reduction in the search space of parameters during the inference process is achieved through the proposed two-stage BO policy, where the solution of the first-stage BO policy in the reduced space specifies the search space of the second-stage BO in the original space. The proposed framework's accuracy and speed are demonstrated through several experiments, including real metagenomics data from a gut microbial community.

Combining and comparing regional SARS-CoV-2 epidemic dynamics in Italy: Bayesian meta-analysis of compartmental models and global sensitivity analysis

During autumn 2020, Italy faced a second important SARS-CoV-2 epidemic wave. We explored the time pattern of the instantaneous reproductive number, R ₀(t), and estimated the prevalence of infections by region from August to December calibrating SIRD models on COVID-19-related deaths, fixing at values from literature Infection Fatality Rate (IFR) and average infection duration. A Global Sensitivity Analysis (GSA) was performed on the regional SIRD models. Then, we used Bayesian meta-analysis and meta-regression to combine and compare the regional results and investigate their heterogeneity. The meta-analytic R ₀(t) curves were similar in the Northern and Central regions, while a less peaked curve was estimated for the South. The maximum R ₀(t) ranged from 2.15 (South) to 2.61 (North) with an increase following school reopening and a decline at the end of October. The predictive performance of the regional models, assessed through cross validation, was good, with a Mean Absolute Percentage Error of 7.2% and 10.9% when considering prediction horizons of 7 and 14 days, respectively. Average temperature, urbanization, characteristics of family medicine and healthcare system, economic dynamism, and use of public transport could partly explain the regional heterogeneity. The GSA indicated the robustness of the regional R ₀(t) curves to different assumptions on IFR. The infectious period turned out to have a key role in determining the model results, but without compromising between-region comparisons.

Comparing and linking machine learning and semi-mechanistic models for the predictability of endemic measles dynamics

Measles is one the best-documented and most-mechanistically-studied non-linear infectious disease dynamical systems. However, systematic investigation into the comparative performance of traditional mechanistic models and machine learning approaches in forecasting the transmission dynamics of this pathogen are still rare. Here, we compare one of the most widely used semi-mechanistic models for measles (TSIR) with a commonly used machine learning approach (LASSO), comparing performance and limits in predicting short to long term outbreak trajectories and seasonality for both regular and less regular measles outbreaks in England and Wales (E&W) and the United States. First, our results indicate that the proposed LASSO model can efficiently use data from multiple major cities and achieve similar short-to-medium term forecasting performance to semi-mechanistic models for E&W epidemics. Second, interestingly, the LASSO model also captures annual to biennial bifurcation of measles epidemics in E&W caused by susceptible response to the late 1940s baby boom. LASSO may also outperform TSIR for predicting less-regular dynamics such as those observed in major cities in US between 1932-45. Although both approaches capture short-term forecasts, accuracy suffers for both methods as we attempt longer-term predictions in highly irregular, post-vaccination outbreaks in E&W. Finally, we illustrate that the LASSO model can both qualitatively and quantitatively reconstruct mechanistic assumptions, notably susceptible dynamics, in the TSIR model. Our results characterize the limits of predictability of infectious disease dynamics for strongly immunizing pathogens with both mechanistic and machine learning models, and identify connections between these two approaches.

Evaluation of Effectiveness of Global COVID-19 Vaccination Campaign

To model estimated deaths averted by COVID-19 vaccines, we used state-of-the-art mathematical modeling, likelihood-based inference, and reported COVID-19 death and vaccination data. We estimated that >1.5 million deaths were averted in 12 countries. Our model can help assess effectiveness of the vaccination program, which is crucial for curbing the COVID-19 pandemic.

Multilevel simulation of hard-sphere mixtures

We present a multilevel Monte Carlo simulation method for analysing multi-scale physical systems via a hierarchy of coarse-grained representations, to obtain numerically-exact results, at the most detailed level. We apply the method to a mixture of size-asymmetric hard spheres, in the grand canonical ensemble. A three-level version of the method is compared with a previously-studied two-level version. The extra level interpolates between the full mixture and a coarse-grained description where only the large particles are present -- this is achieved by restricting the small particles to regions close to the large ones. The three-level method improves the performance of the estimator, at fixed computational cost. We analyse the asymptotic variance of the estimator, and discuss the mechanisms for the improved performance.

A simple model to estimate the transmissibility of the Beta, Delta, and Omicron variants of SARS-COV-2 in South Africa

The COVID-19 pandemic caused multiple waves of mortality in South Africa, where three genetic variants of SARS-COV-2 and their ancestral strain dominated consecutively. State-of-the-art mathematical modeling approach was used to estimate the time-varying transmissibility of SARS-COV-2 and the relative transmissibility of Beta, Delta, and Omicron variants. The transmissibility of the three variants were about 73%, 87%, and 276% higher than their preceding variants. To the best of our knowledge, our model is the first simple model that can simulate multiple mortality waves and three variants' replacements in South Africa. The transmissibility of the Omicron variant is substantially higher than that of previous variants.

Cumulative cultural evolution and mechanisms for cultural selection in wild bird songs

Cumulative cultural evolution, the accumulation of sequential changes within a single socially learned behaviour that results in improved function, is prominent in humans and has been documented in experimental studies of captive animals and managed wild populations. Here, we provide evidence that cumulative cultural evolution has occurred in the learned songs of Savannah sparrows. In a first step, "click trains" replaced "high note clusters" over a period of three decades. We use mathematical modelling to show that this replacement is consistent with the action of selection, rather than drift or frequency-dependent bias. Generations later, young birds elaborated the "click train" song form by adding more clicks. We show that the new songs with more clicks elicit stronger behavioural responses from both males and females. Therefore, we suggest that a combination of social learning, innovation, and sexual selection favoring a specific discrete trait was followed by directional sexual selection that resulted in naturally occurring cumulative cultural evolution in the songs of this wild animal population.

Inferring spatial source of disease outbreaks using maximum entropy

Mathematical modeling of disease outbreaks can infer the future trajectory of an epidemic, allowing for making more informed policy decisions. Another task is inferring the origin of a disease, which is relatively difficult with current mathematical models. Such frameworks, across varying levels of complexity, are typically sensitive to input data on epidemic parameters, case counts, and mortality rates, which are generally noisy and incomplete. To alleviate these limitations, we propose a maximum entropy framework that fits epidemiological models, provides calibrated infection origin probabilities, and is robust to noise due to a prior belief model. Maximum entropy is agnostic to the parameters or model structure used and allows for flexible use when faced with sparse data conditions and incomplete knowledge in the dynamical phase of disease-spread, providing for more reliable modeling at early stages of outbreaks. We evaluate the performance of our model by predicting future disease trajectories based on simulated epidemiological data in synthetic graph networks and the real mobility network of New York State. In addition, unlike existing approaches, we demonstrate that the method can be used to infer the origin of the outbreak with accurate confidence. Indeed, despite the prevalent belief on the feasibility of contact-tracing being limited to the initial stages of an outbreak, we report the possibility of reconstructing early disease dynamics, including the epidemic seed, at advanced stages.

Inferring the effective reproductive number from deterministic and semi-deterministic compartmental models using incidence and mobility data

The effective reproduction number (ℜt) is a theoretical indicator of the course of an infectious disease that allows policymakers to evaluate whether current or previous control efforts have been successful or whether additional interventions are necessary. This metric, however, cannot be directly observed and must be inferred from available data. One approach to obtaining such estimates is fitting compartmental models to incidence data. We can envision these dynamic models as the ensemble of structures that describe the disease's natural history and individuals' behavioural patterns. In the context of the response to the COVID-19 pandemic, the assumption of a constant transmission rate is rendered unrealistic, and it is critical to identify a mathematical formulation that accounts for changes in contact patterns. In this work, we leverage existing approaches to propose three complementary formulations that yield similar estimates for ℜt based on data from Ireland's first COVID-19 wave. We describe these Data Generating Processes (DGP) in terms of State-Space models. Two (DGP1 and DGP2) correspond to stochastic process models whose transmission rate is modelled as Brownian motion processes (Geometric and Cox-Ingersoll-Ross). These DGPs share a measurement model that accounts for incidence and transmission rates, where mobility data is assumed as a proxy of the transmission rate. We perform inference on these structures using Iterated Filtering and the Particle Filter. The final DGP (DGP3) is built from a pool of deterministic models that describe the transmission rate as information delays. We calibrate this pool of models to incidence reports using Hamiltonian Monte Carlo. By following this complementary approach, we assess the tradeoffs associated with each formulation and reflect on the benefits/risks of incorporating proxy data into the inference process. We anticipate this work will help evaluate the implications of choosing a particular formulation for the dynamics and observation of the time-varying transmission rate.

Infection fatality rate and infection attack rate of COVID-19 in South American countries

BACKGROUND

The ongoing COVID-19 pandemic hit South America badly with multiple waves. Different COVID-19 variants have been storming across the region, leading to more severe infections and deaths even in places with high vaccination coverage. This study aims to assess the spatiotemporal variability of the COVID-19 pandemic and estimate the infection fatality rate (IFR), infection attack rate (IAR) and reproduction number ([Formula: see text]) for twelve most affected South American countries.

METHODS

We fit a susceptible-exposed-infectious-recovered (SEIR)-based model with a time-varying transmission rate to the reported COVID-19 deaths for the twelve South American countries with the highest mortalities. Most of the epidemiological datasets analysed in this work are retrieved from the disease surveillance systems by the World Health Organization, Johns Hopkins Coronavirus Resource Center and Our World in Data. We investigate the COVID-19 mortalities in these countries, which could represent the situation  for the overall South American region. We employ COVID-19 dynamic model with-and-without vaccination considering time-varying flexible transmission rate to estimate IFR, IAR and [Formula: see text] of COVID-19 for the South American countries.

RESULTS

We simulate the model in each scenario under suitable parameter settings and yield biologically reasonable estimates for IFR (varies between 0.303% and 0.723%), IAR (varies between 0.03 and 0.784) and [Formula: see text] (varies between 0.7 and 2.5) for the 12 South American countries. We observe that the severity, dynamical patterns of deaths and time-varying transmission rates among the countries are highly heterogeneous. Further analysis of the model with the effect of vaccination highlights that increasing the vaccination rate could help suppress the pandemic in South America.

CONCLUSIONS

This study reveals possible reasons for the two waves of COVID-19 outbreaks in South America. We observed reductions in the transmission rate corresponding to each wave plausibly due to improvement in nonpharmaceutical interventions measures and human protective behavioral reaction to recent deaths. Thus, strategies coupling social distancing and vaccination could substantially suppress the mortality rate of COVID-19 in South America.

Infection fatality rate and infection attack rate of COVID-19 in South American countries

BACKGROUND

The ongoing COVID-19 pandemic hit South America badly with multiple waves. Different COVID-19 variants have been storming across the region, leading to more severe infections and deaths even in places with high vaccination coverage. This study aims to assess the spatiotemporal variability of the COVID-19 pandemic and estimate the infection fatality rate (IFR), infection attack rate (IAR) and reproduction number ([Formula: see text]) for twelve most affected South American countries.

METHODS

We fit a susceptible-exposed-infectious-recovered (SEIR)-based model with a time-varying transmission rate to the reported COVID-19 deaths for the twelve South American countries with the highest mortalities. Most of the epidemiological datasets analysed in this work are retrieved from the disease surveillance systems by the World Health Organization, Johns Hopkins Coronavirus Resource Center and Our World in Data. We investigate the COVID-19 mortalities in these countries, which could represent the situation  for the overall South American region. We employ COVID-19 dynamic model with-and-without vaccination considering time-varying flexible transmission rate to estimate IFR, IAR and [Formula: see text] of COVID-19 for the South American countries.

RESULTS

We simulate the model in each scenario under suitable parameter settings and yield biologically reasonable estimates for IFR (varies between 0.303% and 0.723%), IAR (varies between 0.03 and 0.784) and [Formula: see text] (varies between 0.7 and 2.5) for the 12 South American countries. We observe that the severity, dynamical patterns of deaths and time-varying transmission rates among the countries are highly heterogeneous. Further analysis of the model with the effect of vaccination highlights that increasing the vaccination rate could help suppress the pandemic in South America.

CONCLUSIONS

This study reveals possible reasons for the two waves of COVID-19 outbreaks in South America. We observed reductions in the transmission rate corresponding to each wave plausibly due to improvement in nonpharmaceutical interventions measures and human protective behavioral reaction to recent deaths. Thus, strategies coupling social distancing and vaccination could substantially suppress the mortality rate of COVID-19 in South America.

Two waves of COIVD-19 in Brazilian cities and vaccination impact

BACKGROUNDS

Brazil has suffered two waves of Coronavirus Disease 2019 (COVID-19). The second wave, coinciding with the spread of the Gamma variant, was more severe than the first wave. Studies have not yet reached a conclusion on some issues including the extent of reinfection, the infection fatality rate (IFR), the infection attack rate (IAR) and the effects of the vaccination campaign in Brazil, though it was reported that confirmed reinfection was at a low level.

METHODS

We modify the classical Susceptible-Exposed-Infectious-Recovered (SEIR) model with additional class for severe cases, vaccination and time-varying transmission rates. We fit the model to the severe acute respiratory infection (SARI) deaths, which is a proxy of the COVID-19 deaths, in 20 Brazilian cities with the large number of death tolls. We evaluate the vaccination effect by a contrast of "with" vaccination actual scenario and "without" vaccination in a counterfactual scenario. We evaluate the model performance when the reinfection is absent in the model.

RESULTS

In the 20 Brazilian cities, the model simulated death matched the reported deaths reasonably well. The effect of the vaccination varies across cities. The estimated median IFR is around 1.2%.

CONCLUSION

Overall, through this modeling exercise, we conclude that the effects of vaccination campaigns vary across cites and the reinfection is not crucial for the second wave. The relatively high IFR could be due to the breakdown of medical system in many cities.

Inference in epidemiological agent-based models using ensemble-based data assimilation

To represent the complex individual interactions in the dynamics of disease spread informed by data, the coupling of an epidemiological agent-based model with the ensemble Kalman filter is proposed. The statistical inference of the propagation of a disease by means of ensemble-based data assimilation systems has been studied in previous works. The models used are mostly compartmental models representing the mean field evolution through ordinary differential equations. These techniques allow to monitor the propagation of the infections from data and to estimate several parameters of epidemiological interest. However, there are many important features which are based on the individual interactions that cannot be represented in the mean field equations, such as social network and bubbles, contact tracing, isolating individuals in risk, and social network-based distancing strategies. Agent-based models can describe contact networks at an individual level, including demographic attributes such as age, neighborhood, household, workplaces, schools, entertainment places, among others. Nevertheless, these models have several unknown parameters which are thus difficult to prescribe. In this work, we propose the use of ensemble-based data assimilation techniques to calibrate an agent-based model using daily epidemiological data. This raises the challenge of having to adapt the agent populations to incorporate the information provided by the coarse-grained data. To do this, two stochastic strategies to correct the model predictions are developed. The ensemble Kalman filter with perturbed observations is used for the joint estimation of the state and some key epidemiological parameters. We conduct experiments with an agent based-model designed for COVID-19 and assess the proposed methodology on synthetic data and on COVID-19 daily reports from Ciudad Autónoma de Buenos Aires, Argentina.

Challenges in estimation, uncertainty quantification and elicitation for pandemic modelling

The estimation of parameters and model structure for informing infectious disease response has become a focal point of the recent pandemic. However, it has also highlighted a plethora of challenges remaining in the fast and robust extraction of information using data and models to help inform policy. In this paper, we identify and discuss four broad challenges in the estimation paradigm relating to infectious disease modelling, namely the Uncertainty Quantification framework, data challenges in estimation, model-based inference and prediction, and expert judgement. We also postulate priorities in estimation methodology to facilitate preparation for future pandemics.

A Theoretical Model to Investigate the Influence of Temperature, Reactions of the Population and the Government on the COVID-19 Outbreak in Turkey

OBJECTIVES

The ongoing coronavirus disease 2019 (COVID-19) pandemic, which was initially identified in December 2019 in the city of Wuhan in China, poses a major threat to worldwide health care. By August 04, 2020, there were globally 695,848 deaths (Johns Hopkins University, https://coronavirus.jhu.edu/map.html). A total of 5765 of them come from Turkey (Johns Hopkins University, https://coronavirus.jhu.edu/map.html). As a result, various governments and their respective populations have taken strong measures to control the spread of the pandemic. In this study, a model that is by construction able to describe both government actions and individual reactions in addition to the well-known exponential spread is presented. Moreover, the influence of the weather is included. This approach demonstrates a quantitative method to track these dynamic influences. This makes it possible to numerically estimate the influence that various private or state measures that were put into effect to contain the pandemic had at time t. This might serve governments across the world by allowing them to plan their actions based on quantitative data to minimize the social and economic consequences of their containment strategies.

METHODS

A compartmental model based on SEIR that includes the risk perception of the population by an additional differential equation and uses an implicit time-dependent transmission rate is constructed. Within this model, the transmission rate depends on temperature, population, and government actions, which in turn depend on time. The model was tested using different scenarios, with the different dynamic influences being mathematically switched on and off. In addition, the real data of infected coronavirus cases in Turkey were compared with the results of the model.

RESULTS

The mathematical study of the influence of the different parameters is presented through different scenarios. Remarkably, the last scenario is also an example of a theoretical mitigation strategy that shows its maximum in August 2020. In addition, the results of the model are compared with the real data from Turkey using conventional fitting that shows good agreement.

CONCLUSIONS

Although most countries activated their pandemic plans, significant disruptions in health-care systems occurred. The framework of this model seems to be valid for a numerical analysis of dynamic processes that occur during the COVID-19 outbreak due to weather and human reactions. As a result, the effects of the measures introduced could be better planned in advance by use of this model.

The neglected role of relative humidity in the interannual variability of urban malaria in Indian cities

The rapid pace of urbanization makes it imperative that we better understand the influence of climate forcing on urban malaria transmission. Despite extensive study of temperature effects in vector-borne infections in general, consideration of relative humidity remains limited. With process-based dynamical models informed by almost two decades of monthly surveillance data, we address the role of relative humidity in the interannual variability of epidemic malaria in two semi-arid cities of India. We show a strong and significant effect of humidity during the pre-transmission season on malaria burden in coastal Surat and more arid inland Ahmedabad. Simulations of the climate-driven transmission model with the MLE (Maximum Likelihood Estimates) of the parameters retrospectively capture the observed variability of disease incidence, and also prospectively predict that of 'out-of-fit' cases in more recent years, with high accuracy. Our findings indicate that relative humidity is a critical factor in the spread of urban malaria and potentially other vector-borne epidemics, and that climate change and lack of hydrological planning in cities might jeopardize malaria elimination efforts.

Tracing contacts to evaluate the transmission of COVID-19 from highly exposed individuals in public transportation

We investigate, through a data-driven contact tracing model, the transmission of COVID-19 inside buses during distinct phases of the pandemic in a large Brazilian city. From this microscopic approach, we recover the networks of close contacts within consecutive time windows. A longitudinal comparison is then performed by upscaling the traced contacts with the transmission computed from a mean-field compartmental model for the entire city. Our results show that the effective reproduction numbers inside the buses, [Formula: see text], and in the city, [Formula: see text], followed a compatible behavior during the first wave of the local outbreak. Moreover, by distinguishing the close contacts of healthcare workers in the buses, we discovered that their transmission, [Formula: see text], during the same period, was systematically higher than [Formula: see text]. This result reinforces the need for special public transportation policies for highly exposed groups of people.

Forecast of the COVID-19 trend in India: A simple modelling approach

By February 2021, the overall impact of the COVID-19 pandemic in India had been relatively mild in terms of total reported cases and deaths. Surprisingly, the second wave in early April becomes devastating and attracts worldwide attention. Multiple factors (e.g., Delta variants with increased transmissibility) could have driven the rapid growth of the epidemic in India and led to a large number of deaths within a short period. We aim to reconstruct the transmission rate, estimate the infection fatality rate and forecast the epidemic size. We download the reported COVID-19 mortality data in India and formulate a simple mathematical model with a flexible transmission rate. We use iterated filtering to fit our model to deaths data. We forecast the infection attack rate in a month ahead. Our model simulation matched the reported deaths well and is reasonably close to the results of the serological study. We forecast that the infection attack rate (IAR) could have reached 43% by July 24, 2021, under the current trend. Our estimated infection fatality rate is about 0.07%. Under the current trend, the IAR will likely reach a level of 43% by July 24, 2021. Our estimated infection fatality rate appears unusually low, which could be due to a low case to infection ratio reported in previous study. Our approach is readily applicable in other countries and with other types of data (e.g., excess deaths).

Reconstruction of evolving gene variants and fitness from short sequencing reads

Directed evolution can generate proteins with tailor-made activities. However, full-length genotypes, their frequencies and fitnesses are difficult to measure for evolving gene-length biomolecules using most high-throughput DNA sequencing methods, as short read lengths can lose mutation linkages in haplotypes. Here we present Evoracle, a machine learning method that accurately reconstructs full-length genotypes (R² = 0.94) and fitness using short-read data from directed evolution experiments, with substantial improvements over related methods. We validate Evoracle on phage-assisted continuous evolution (PACE) and phage-assisted non-continuous evolution (PANCE) of adenine base editors and OrthoRep evolution of drug-resistant enzymes. Evoracle retains strong performance (R² = 0.86) on data with complete linkage loss between neighboring nucleotides and large measurement noise, such as pooled Sanger sequencing data (~US$10 per timepoint), and broadens the accessibility of training machine learning models on gene variant fitnesses. Evoracle can also identify high-fitness variants, including low-frequency 'rising stars', well before they are identifiable from consensus mutations.

Identifying asymptomatic spreaders of antimicrobial-resistant pathogens in hospital settings

Antimicrobial-resistant organisms (AMROs) can colonize people without symptoms for long periods of time, during which these agents can spread unnoticed to other patients in healthcare systems. The accurate identification of asymptomatic spreaders of AMRO in hospital settings is essential for supporting the design of interventions against healthcare-associated infections (HAIs). However, this task remains challenging because of limited observations of colonization and the complicated transmission dynamics occurring within hospitals and the broader community. Here, we study the transmission of methicillin-resistant Staphylococcus aureus (MRSA), a prevalent AMRO, in 66 Swedish hospitals and healthcare facilities with inpatients using a data-driven, agent-based model informed by deidentified real-world hospitalization records. Combining the transmission model, patient-to-patient contact networks, and sparse observations of colonization, we develop and validate an individual-level inference approach that estimates the colonization probability of individual hospitalized patients. For both model-simulated and historical outbreaks, the proposed method supports the more accurate identification of asymptomatic MRSA carriers than other traditional approaches. In addition, in silica control experiments indicate that interventions targeted to inpatients with a high-colonization probability outperform heuristic strategies informed by hospitalization history and contact tracing.

Efficient Bayesian inference of fully stochastic epidemiological models with applications to COVID-19

Epidemiological forecasts are beset by uncertainties about the underlying epidemiological processes, and the surveillance process through which data are acquired. We present a Bayesian inference methodology that quantifies these uncertainties, for epidemics that are modelled by (possibly) non-stationary, continuous-time, Markov population processes. The efficiency of the method derives from a functional central limit theorem approximation of the likelihood, valid for large populations. We demonstrate the methodology by analysing the early stages of the COVID-19 pandemic in the UK, based on age-structured data for the number of deaths. This includes maximum a posteriori estimates, Markov chain Monte Carlo sampling of the posterior, computation of the model evidence, and the determination of parameter sensitivities via the Fisher information matrix. Our methodology is implemented in PyRoss, an open-source platform for analysis of epidemiological compartment models.

A Spatiotemporal Tool to Project Hospital Critical Care Capacity and Mortality From COVID-19 in US Counties

Objectives. To create a tool to rapidly determine where pandemic demand for critical care overwhelms county-level surge capacity and to compare public health and medical responses.Methods. In March 2020, COVID-19 cases requiring critical care were estimated using an adaptive metapopulation SEIR (susceptible‒exposed‒infectious‒recovered) model for all 3142 US counties for future 21-day and 42-day periods from April 2, 2020, to May 13, 2020, in 4 reactive patterns of contact reduction-0%, 20%, 30%, and 40%-and 4 surge response scenarios-very low, low, medium, and high.Results. In areas with increased demand, surge response measures could avert 104 120 additional deaths-55% through high clearance of critical care beds and 45% through measures such as greater ventilator access. The percentages of lives saved from high levels of contact reduction were 1.9 to 4.2 times greater than high levels of hospital surge response. Differences in projected versus actual COVID-19 demands were reasonably small over time.Conclusions. Nonpharmaceutical public health interventions had greater impact in minimizing preventable deaths during the pandemic than did hospital critical care surge response. Ready-to-go spatiotemporal supply and demand data visualization and analytics tools should be advanced for future preparedness and all-hazards disaster response.

Malaria trends in Ethiopian highlands track the 2000 'slowdown' in global warming

A counterargument to the importance of climate change for malaria transmission has been that regions where an effect of warmer temperatures is expected, have experienced a marked decrease in seasonal epidemic size since the turn of the new century. This decline has been observed in the densely populated highlands of East Africa at the center of the earlier debate on causes of the pronounced increase in epidemic size from the 1970s to the 1990s. The turnaround of the incidence trend around 2000 is documented here with an extensive temporal record for malaria cases for both Plasmodium falciparum and Plasmodium vivax in an Ethiopian highland. With statistical analyses and a process-based transmission model, we show that this decline was driven by the transient slowdown in global warming and associated changes in climate variability, especially ENSO. Decadal changes in temperature and concurrent climate variability facilitated rather than opposed the effect of interventions.

Fitting stochastic predator-prey models using both population density and kill rate data

Most mechanistic predator-prey modelling has involved either parameterization from process rate data or inverse modelling. Here, we take a median road: we aim at identifying the potential benefits of combining datasets, when both population growth and predation processes are viewed as stochastic. We fit a discrete-time, stochastic predator-prey model of the Leslie type to simulated time series of densities and kill rate data. Our model has both environmental stochasticity in the growth rates and interaction stochasticity, i.e., a stochastic functional response. We examine what the kill rate data brings to the quality of the estimates, and whether estimation is possible (for various time series lengths) solely with time series of population counts or biomass data. Both Bayesian and frequentist estimation are performed, providing multiple ways to check model identifiability. The Fisher Information Matrix suggests that models with and without kill rate data are all identifiable, although correlations remain between parameters that belong to the same functional form. However, our results show that if the attractor is a fixed point in the absence of stochasticity, identifying parameters in practice requires kill rate data as a complement to the time series of population densities, due to the relatively flat likelihood. Only noisy limit cycle attractors can be identified directly from population count data (as in inverse modelling), although even in this case, adding kill rate data - including in small amounts - can make the estimates much more precise. Overall, we show that under process stochasticity in interaction rates, interaction data might be essential to obtain identifiable dynamical models for multiple species. These results may extend to other biotic interactions than predation, for which similar models combining interaction rates and population counts could be developed.

Statistical finite elements for misspecified models

We present a statistical finite element method for nonlinear, time-dependent phenomena, illustrated in the context of nonlinear internal waves (solitons). We take a Bayesian approach and leverage the finite element method to cast the statistical problem as a nonlinear Gaussian state-space model, updating the solution, in receipt of data, in a filtering framework. The method is applicable to problems across science and engineering for which finite element methods are appropriate. The Korteweg-de Vries equation for solitons is presented because it reflects the necessary complexity while being suitably familiar and succinct for pedagogical purposes. We present two algorithms to implement this method, based on the extended and ensemble Kalman filters, and demonstrate effectiveness with a simulation study and a case study with experimental data. The generality of our approach is demonstrated in SI Appendix, where we present examples from additional nonlinear, time-dependent partial differential equations (Burgers equation, Kuramoto-Sivashinsky equation).

Differential evolution to estimate the parameters of a SEIAR model with dynamic social distancing

This chapter describes the application of a recent compartment-based epidemiological model, namely the SEIAR (Susceptible-Exposed-Infectious-Asymptomatic-Recovered), to estimate the spreading of the coronavirus COVID-19 in Italy and in some of its regions. The model is here extended through the definition of a time-dependent dynamic social distancing (DSD) function, thus introducing the SEIAR–DSD model. To profitably use the SEIAR–DSD model, the most suitable values of its parameters must be found. This is performed through differential evolution, a heuristic optimization technique. This allows approximately evaluating, for each of the above-mentioned scenarios, the daily number of infectious individuals, the day(s) in which this number will reach its maximum, the corresponding value, and the future evolution of the spread.

Estimating epidemic coupling between populations from the time to invasion

Identifying the mechanisms by which diseases spread among populations is important for understanding and forecasting patterns of epidemics and pandemics. Estimating transmission coupling among populations is challenging because transmission events are difficult to observe in practice, and connectivity among populations is often obscured by local disease dynamics. We consider the common situation in which an epidemic is seeded in one population and later spreads to a second population. We present a method for estimating transmission coupling between the two populations, assuming they can be modelled as susceptible-infected-removed (SIR) systems. We show that the strength of coupling between the two populations can be estimated from the time taken for the disease to invade the second population. Confidence in the estimate is low if only a single invasion event has been observed, but is substantially improved if numerous independent invasion events are observed. Our analysis of this simplest, idealized scenario represents a first step toward developing and verifying methods for estimating epidemic coupling among populations in an ever-more-connected global human population.