Stochastic EM algorithm for partially observed stochastic epidemics with individual heterogeneity.

We develop a stochastic epidemic model progressing over dynamic networks, where infection rates are heterogeneous and may vary with individual-level covariates. The joint dynamics are modeled as a continuous-time Markov chain such that disease transmission is constrained by the contact network structure, and network evolution is in turn influenced by individual disease statuses. To accommodate partial epidemic observations commonly seen in real-world data, we propose a stochastic EM algorithm for inference, introducing key innovations that include efficient conditional samplers for imputing missing infection and recovery times which respect the dynamic contact network. Experiments on both synthetic and real datasets demonstrate that our inference method can accurately and efficiently recover model parameters and provide valuable insight at the presence of unobserved disease episodes in epidemic data.

IDENTIFIABILITY OF INFECTION MODEL PARAMETERS EARLY IN AN EPIDEMIC.

It is known that the parameters in the deterministic and stochastic SEIR epidemic models are structurally identifiable. For example, from knowledge of the infected population time series I(t) during the entire epidemic, the parameters can be successfully estimated. In this article we observe that estimation will fail in practice if only infected case data during the early part of the epidemic (prepeak) is available. This fact can be explained using a well-known phenomenon called dynamical compensation. We use this concept to derive an unidentifiability manifold in the parameter space of SEIR that consists of parameters indistinguishable from I(t) early in the epidemic. Thus, identifiability depends on the extent of the system trajectory that is available for observation. Although the existence of the unidentifiability manifold obstructs the ability to exactly determine the parameters, we suggest that it may be useful for uncertainty quantification purposes. A variant of SEIR recently proposed for COVID-19 modeling is also analyzed, and an analogous unidentifiability surface is derived.

Estimating the basic reproduction number from noisy daily data.

In this paper, we propose an easy to implement generalized linear models (GLM) methodology for estimating the basic reproduction number, R0, a major epidemic parameter for assessing the transmissibility of an infection. Our approach rests on well known qualitative properties of the classical SIR and SEIR systems for large populations. Moreover, we assume that information at the individual network level is not available. In inference we consider non homogeneous Poisson observation processes and mainly concentrate on epidemics that spread through a completely susceptible population. Further, we examine the performance of the estimator under various scenarios of relevance in practice, like partially observed data. We perform a detailed simulation study and illustrate our approach on Covid-19 Canadian data sets. Finally, we present extensions of our methodology and discuss its merits and practical limitations, in particular the challenges in estimating R0 when mitigation measures are applied.

A non-parametric Hawkes model of the spread of Ebola in west Africa.

Recently developed methods for the non-parametric estimation of Hawkes point process models facilitate their application for describing and forecasting the spread of epidemic diseases. We use data from the 2014 Ebola outbreak in West Africa to evaluate how well a simple Hawkes point process model can forecast the spread of Ebola virus in Guinea, Sierra Leone, and Liberia. For comparison, SEIR models that fit previously to the same data are evaluated using identical metrics. To test the predictive power of each of the models, we simulate the ability to make near real-time predictions during an actual outbreak by using the first 75% of the data for estimation and the subsequent 25% of the data for evaluation. Forecasts generated from Hawkes models more accurately describe the spread of Ebola in each of the three countries investigated and result in a 38% reduction in RMSE for weekly case estimation across all countries when compared to SEIR models (total RMSE of 59.8 cases/week using SEIR compared to 37.1 for Hawkes). We demonstrate that the improved fit from Hawkes modeling cannot be attributed to overfitting and evaluate the advantages and disadvantages of Hawkes models in general for forecasting the spread of epidemic diseases.

Estimating unconfirmed COVID-19 infection cases and multiple waves of pandemic progression with consideration of testing capacity and non-pharmaceutical interventions: A dynamic spreading model.

The novel coronavirus disease 2019 (COVID-19), caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), has unique epidemiological characteristics that include presymptomatic and asymptomatic infections, resulting in a large proportion of infected cases being unconfirmed, including patients with clinical symptoms who have not been identified by screening. These unconfirmed infected individuals move and spread the virus freely, presenting difficult challenges to the control of the pandemic. To reveal the actual pandemic situation in a given region, a simple dynamic susceptible-unconfirmed-confirmed-removed (D-SUCR) model is developed taking into account the influence of unconfirmed cases, the testing capacity, the multiple waves of the pandemic, and the use of non-pharmaceutical interventions. Using this model, the total numbers of infected cases in 51 regions of the USA and 116 countries worldwide are estimated, and the results indicate that only about 40% of the true number of infections have been confirmed. In addition, it is found that if local authorities could enhance their testing capacities and implement a timely strict quarantine strategy after identifying the first infection case, the total number of infected cases could be reduced by more than 90%. Delay in implementing quarantine measures would drastically reduce their effectiveness.

Spatially-heterogeneous embedded stochastic SEIR models for the 2014-2016 Ebola outbreak in West Africa.

The dynamics of human infectious diseases are challenging to understand, particularly when a pathogen spreads spatially over a large region. We present a stochastic, spatially-heterogeneous model framework derived from the foundational SEIR compartmental model. These models utilize a graph structure of spatial locations, facilitating mobility via random walks while progressing through disease states, parameterized by the net probability flux between locations. The analysis is bolstered by Approximate Bayesian Computation, by which epidemiological and mobility parameter distributions are estimated, including an empirically adjusted reproductive number, while model structure proposals are compared using Bayes Factors. The utility of this novel class of models is demonstrated through application to the 2014-2016 Ebola outbreak in West Africa. The flexibility of such models, whose complexity may be adjusted as desired, and complementary methods of analysis enable the exploration of various spatial divisions and mobility schema, while maintaining the essential spatiotemporal disease dynamics.

Data-driven prediction of COVID-19 cases in Germany for decision making.

BACKGROUND: The COVID-19 pandemic has led to a high interest in mathematical models describing and predicting the diverse aspects and implications of the virus outbreak. Model results represent an important part of the information base for the decision process on different administrative levels. The Robert-Koch-Institute (RKI) initiated a project whose main goal is to predict COVID-19-specific occupation of beds in intensive care units: Steuerungs-Prognose von Intensivmedizinischen COVID-19 Kapazitäten (SPoCK). The incidence of COVID-19 cases is a crucial predictor for this occupation. METHODS: We developed a model based on ordinary differential equations for the COVID-19 spread with a time-dependent infection rate described by a spline. Furthermore, the model explicitly accounts for weekday-specific reporting and adjusts for reporting delay. The model is calibrated in a purely data-driven manner by a maximum likelihood approach. Uncertainties are evaluated using the profile likelihood method. The uncertainty about the appropriate modeling assumptions can be accounted for by including and merging results of different modelling approaches. The analysis uses data from Germany describing the COVID-19 spread from early 2020 until March 31st, 2021. RESULTS: The model is calibrated based on incident cases on a daily basis and provides daily predictions of incident COVID-19 cases for the upcoming three weeks including uncertainty estimates for Germany and its subregions. Derived quantities such as cumulative counts and 7-day incidences with corresponding uncertainties can be computed. The estimation of the time-dependent infection rate leads to an estimated reproduction factor that is oscillating around one. Data-driven estimation of the dark figure purely from incident cases is not feasible. CONCLUSIONS: We successfully implemented a procedure to forecast near future COVID-19 incidences for diverse subregions in Germany which are made available to various decision makers via an interactive web application. Results of the incidence modeling are also used as a predictor for forecasting the need of intensive care units.

Assessing the impact of adherence to Non-pharmaceutical interventions and indirect transmission on the dynamics of COVID-19: a mathematical modelling study.

Adherence to public health policies such as the non-pharmaceutical interventions implemented against COVID-19 plays a major role in reducing infections and controlling the spread of the diseases. In addition, understanding the transmission dynamics of the disease is also important in order to make and implement efficient public health policies. In this paper, we developed an SEIR-type compartmental model to assess the impact of adherence to COVID-19 non-pharmaceutical interventions and indirect transmission on the dynamics of the disease. Our model considers both direct and indirect transmission routes and stratifies the population into two groups: those that adhere to COVID-19 non-pharmaceutical interventions (NPIs) and those that do not adhere to the NPIs. We compute the control reproduction number and the final epidemic size relation for our model and study the effect of different parameters of the model on these quantities. Our results show that there is a significant benefit in adhering to the COVID-19 NPIs.

Coronavirus Disease 2019 on the Heels of Ebola Virus Disease in West Africa.

This study utilized modeling and simulation to examine the effectiveness of current and potential future COVID-19 response interventions in the West African countries of Guinea, Liberia, and Sierra Leone. A comparison between simulations can highlight which interventions could have an effect on the pandemic in these countries. An extended compartmental model was used to run simulations incorporating multiple vaccination strategies and non-pharmaceutical interventions (NPIs). In addition to the customary categories of susceptible, exposed, infected, and recovered (SEIR) compartments, this COVID-19 model incorporated early and late disease states, isolation, treatment, and death. Lessons learned from the 2014-2016 Ebola virus disease outbreak-especially the optimization of each country's resource allocation-were incorporated in the presented models. For each country, models were calibrated to an estimated number of infections based on actual reported cases and deaths. Simulations were run to test the potential future effects of vaccination and NPIs. Multiple levels of vaccination were considered, based on announced vaccine allocation plans and notional scenarios. Increased vaccination combined with NPI mitigation strategies resulted in thousands of fewer COVID-19 infections in each country. This study demonstrates the importance of increased vaccinations. The levels of vaccination in this study would require substantial increases in vaccination supplies obtained through national purchases or international aid. While this study does not aim to develop a model that predicts the future, it can provide useful information for decision-makers in low- and middle-income nations. Such information can be used to prioritize and optimize limited available resources for targeted interventions that will have the greatest impact on COVID-19 pandemic response.

A Multi-Criteria Framework for Pandemic Response Measures.

In managing the COVID-19 pandemic, several compelling narratives seem to have played a significant role in the decision-making processes regarding which risk mitigation and management measures to implement. Many countries were to a large extent unprepared for such a situation, even though predictions about a significant probability for a pandemic to occur existed, and national governments of several countries often acted in an uncoordinated manner, which resulted in many inconsistencies in the disaster risk reduction processes. Limited evidence has also made room for strategic narratives meant to persuade the public of the chosen set of actions, even though the degree of uncertainty regarding the outcomes of these was high, further complicating the situation. In this article, we assume a normative standpoint regarding rhapsodic decision making and suggest an integrated framework for a more elaborated decision analysis under the ambiguity of how to contain the virus spread from a policy point of view, while considering epidemiologic estimations and socioeconomic factors in a multi-stakeholder-multi-criteria context based on a co-creative work process for eliciting attitudes, perceptions, as well as preferences amongst relevant stakeholder groups. The framework, applied in our paper on Romania for demonstrative purposes, is used for evaluating mitigation measures for catastrophic events such as the COVID-19 situation, to mobilize better response strategies for future scenarios related to pandemics and other hazardous events, as well as to structure the production and analysis of narratives on the current pandemic effects.

A versatile web app for identifying the drivers of COVID-19 epidemics.

BACKGROUND: No versatile web app exists that allows epidemiologists and managers around the world to comprehensively analyze the impacts of COVID-19 mitigation. The http://covid-webapp.numerusinc.com/ web app presented here fills this gap. METHODS: Our web app uses a model that explicitly identifies susceptible, contact, latent, asymptomatic, symptomatic and recovered classes of individuals, and a parallel set of response classes, subject to lower pathogen-contact rates. The user inputs a CSV file of incidence and, if of interest, mortality rate data. A default set of parameters is available that can be overwritten through input or online entry, and a user-selected subset of these can be fitted to the model using maximum-likelihood estimation (MLE). Model fitting and forecasting intervals are specifiable and changes to parameters allow counterfactual and forecasting scenarios. Confidence or credible intervals can be generated using stochastic simulations, based on MLE values, or on an inputted CSV file containing Markov chain Monte Carlo (MCMC) estimates of one or more parameters. RESULTS: We illustrate the use of our web app in extracting social distancing, social relaxation, surveillance or virulence switching functions (i.e., time varying drivers) from the incidence and mortality rates of COVID-19 epidemics in Israel, South Africa, and England. The Israeli outbreak exhibits four distinct phases: initial outbreak, social distancing, social relaxation, and a second wave mitigation phase. An MCMC projection of this latter phase suggests the Israeli epidemic will continue to produce into late November an average of around 1500 new case per day, unless the population practices social-relaxation measures at least 5-fold below the level in August, which itself is 4-fold below the level at the start of July. Our analysis of the relatively late South African outbreak that became the world's fifth largest COVID-19 epidemic in July revealed that the decline through late July and early August was characterised by a social distancing driver operating at more than twice the per-capita applicable-disease-class (pc-adc) rate of the social relaxation driver. Our analysis of the relatively early English outbreak, identified a more than 2-fold improvement in surveillance over the course of the epidemic. It also identified a pc-adc social distancing rate in early August that, though nearly four times the pc-adc social relaxation rate, appeared to barely contain a second wave that would break out if social distancing was further relaxed. CONCLUSION: Our web app provides policy makers and health officers who have no epidemiological modelling or computer coding expertise with an invaluable tool for assessing the impacts of different outbreak mitigation policies and measures. This includes an ability to generate an epidemic-suppression or curve-flattening index that measures the intensity with which behavioural responses suppress or flatten the epidemic curve in the region under consideration.

Fractional model for the spread of COVID-19 subject to government intervention and public perception.

COVID-19 pandemic has impacted people all across the world. As a result, there has been a collective effort to monitor, predict, and control the spread of this disease. Among this effort is the development of mathematical models that could capture accurately the available data and simulate closely the futuristic scenarios. In this paper, a fractional-order memory-dependent model for simulating the spread of COVID-19 is proposed. In this model, the impact of governmental interventions and public perception are incorporated as part of the nonlinear time-varying transmission rate. In addition, an algorithm for approximating the optimal values of the fractional order and strength of governmental interventions is provided. This approach makes our model suitable for capturing the given data set and consequently reliable for future predictions. The model simulation is performed using the two-step generalized exponential time-differencing method and tested for data from Mainland China, Italy, Saudi Arabia and Brazil. The simulation results demonstrate that the fractional order model calibrates to the data better than its integer order counterpart. This observation is further endorsed by the calculated error metrics.

Flattening the curves: on-off lock-down strategies for COVID-19 with an application to Brazil.

The current COVID-19 pandemic is affecting different countries in different ways. The assortment of reporting techniques alongside other issues, such as underreporting and budgetary constraints, makes predicting the spread and lethality of the virus a challenging task. This work attempts to gain a better understanding of how COVID-19 will affect one of the least studied countries, namely Brazil. Currently, several Brazilian states are in a state of lock-down. However, there is political pressure for this type of measures to be lifted. This work considers the impact that such a termination would have on how the virus evolves locally. This was done by extending the SEIR model with an on / off strategy. Given the simplicity of SEIR we also attempted to gain more insight by developing a neural regressor. We chose to employ features that current clinical studies have pinpointed has having a connection to the lethality of COVID-19. We discuss how this data can be processed in order to obtain a robust assessment. SUPPLEMENTARY INFORMATION: The online version contains supplementary material available at 10.1186/s13362-020-00098-w.

A model for the spread of infectious diseases compatible with case data.

For epidemics such as COVID-19, with a significant population having asymptomatic, untested infection, model predictions are often not compatible with data reported only for the cases confirmed by laboratory tests. Additionally, most compartmental models have instantaneous recovery from infection, contrary to observation. Tuning such models with observed data to obtain the unknown infection rate is an ill-posed problem. Here, we derive from the first principle an epidemiological model with delay between the newly infected (N) and recovered (R) populations. To overcome the challenge of incompatibility between model and case data, we solve for the ratios of the observed quantities and show that log(N(t)/R(t)) should follow a straight line. This simple prediction tool is accurate in hindcasts verified using data for China and Italy. In traditional epidemiology, an epidemic wanes when much of the population is infected so that 'herd immunity' is achieved. For a highly contagious and deadly disease, herd immunity is not a feasible goal without human intervention or vaccines. Even before the availability of vaccines, the epidemic was suppressed with social measures in China and South Korea with much less than 5% of the population infected. Effects of social behaviour should be and are incorporated in our model.

Spread and dynamics of the COVID-19 epidemic in Italy: Effects of emergency containment measures.

The spread of coronavirus disease 2019 (COVID-19) in Italy prompted drastic measures for transmission containment. We examine the effects of these interventions, based on modeling of the unfolding epidemic. We test modeling options of the spatially explicit type, suggested by the wave of infections spreading from the initial foci to the rest of Italy. We estimate parameters of a metacommunity Susceptible-Exposed-Infected-Recovered (SEIR)-like transmission model that includes a network of 107 provinces connected by mobility at high resolution, and the critical contribution of presymptomatic and asymptomatic transmission. We estimate a generalized reproduction number ([Formula: see text] = 3.60 [3.49 to 3.84]), the spectral radius of a suitable next-generation matrix that measures the potential spread in the absence of containment interventions. The model includes the implementation of progressive restrictions after the first case confirmed in Italy (February 21, 2020) and runs until March 25, 2020. We account for uncertainty in epidemiological reporting, and time dependence of human mobility matrices and awareness-dependent exposure probabilities. We draw scenarios of different containment measures and their impact. Results suggest that the sequence of restrictions posed to mobility and human-to-human interactions have reduced transmission by 45% (42 to 49%). Averted hospitalizations are measured by running scenarios obtained by selectively relaxing the imposed restrictions and total about 200,000 individuals (as of March 25, 2020). Although a number of assumptions need to be reexamined, like age structure in social mixing patterns and in the distribution of mobility, hospitalization, and fatality, we conclude that verifiable evidence exists to support the planning of emergency measures.

Why is it difficult to accurately predict the COVID-19 epidemic?

Since the COVID-19 outbreak in Wuhan City in December of 2019, numerous model predictions on the COVID-19 epidemics in Wuhan and other parts of China have been reported. These model predictions have shown a wide range of variations. In our study, we demonstrate that nonidentifiability in model calibrations using the confirmed-case data is the main reason for such wide variations. Using the Akaike Information Criterion (AIC) for model selection, we show that an SIR model performs much better than an SEIR model in representing the information contained in the confirmed-case data. This indicates that predictions using more complex models may not be more reliable compared to using a simpler model. We present our model predictions for the COVID-19 epidemic in Wuhan after the lockdown and quarantine of the city on January 23, 2020. We also report our results of modeling the impacts of the strict quarantine measures undertaken in the city after February 7 on the time course of the epidemic, and modeling the potential of a second outbreak after the return-to-work in the city.

Fractional-Order SEIQRDP Model for Simulating the Dynamics of COVID-19 Epidemic.

Goal: Coronavirus disease (COVID-19) is a contagious disease caused by a newly discovered coronavirus, initially identified in the mainland of China, late December 2019. COVID-19 has been confirmed as a higher infectious disease that can spread quickly in a community population depending on the number of susceptible and infected cases and also depending on their movement in the community. Since January 2020, COVID-19 has reached out to many countries worldwide, and the number of daily cases remains to increase rapidly. Method: Several mathematical and statistical models have been developed to understand, track, and forecast the trend of the virus spread. Susceptible-Exposed-Infected-Quarantined-Recovered-Death-Insusceptible (SEIQRDP) model is one of the most promising epidemiological models that has been suggested for estimating the transmissibility of the COVID-19. In the present study, we propose a fractional-order SEIQRDP model to analyze the COVID-19 pandemic. In the recent decade, it has proven that many aspects in many domains can be described very successfully using fractional order differential equations. Accordingly, the Fractional-order paradigm offers a flexible, appropriate, and reliable framework for pandemic growth characterization. In fact, due to its non-locality properties, a fractional-order operator takes into consideration the variables' memory effect, and hence, it takes into account the sub-diffusion process of confirmed and recovered cases. Results-The validation of the studied fractional-order model using real COVID-19 data for different regions in China, Italy, and France show the potential of the proposed paradigm in predicting and understanding the pandemic dynamic. Conclusions: Fractional-order epidemiological models might play an important role in understanding and predicting the spread of the COVID-19, also providing relevant guidelines for controlling the pandemic.

Modeling epidemics: A primer and Numerus Model Builder implementation.

Epidemiological models are dominated by compartmental models, of which SIR formulations are the most commonly used. These formulations can be continuous or discrete (in either the state-variable values or time), deterministic or stochastic, or spatially homogeneous or heterogeneous, the latter often embracing a network formulation. Here we review the continuous and discrete deterministic and discrete stochastic formulations of the SIR dynamical systems models, and we outline how they can be easily and rapidly constructed using Numerus Model Builder, a graphically-driven coding platform. We also demonstrate how to extend these models to a metapopulation setting using NMB network and mapping tools.

Modeling and predicting the long-term effects of various strategies and objectives of varicella-zoster vaccination campaigns.

BACKGROUND: Susceptible, exposed, infected, and recovered (SEIR) models are increasingly developed and used, but their simplicity contrasts with the wide variety of scenarios before launching vaccination campaigns. METHODS: We investigated the effects of some model-building choices (targets, pace, coverage rate) on the results of SEIR models in the case of vaccination against varicella and herpes zoster. RESULTS: The analysis demonstrated the need for a progressive unvaccinated to vaccinated transition and a dynamic system-equilibrium before vaccination onset. When several doses are considered, new compartments are needed to account for vaccination histories. For varicella, the delay to reach the expected coverage rate and the pace until reaching this rate have significant impacts, especially on the short-term incidence. The impact of vaccination through herd immunity should be systematically investigated. CONCLUSIONS: Graphs help understanding the progress of instantaneous incidence; however, tables of cumulative average incidence over decades should be preferred because of higher stability.