Dynamic causal modelling of COVID-19

Modeling COVID-19 disease processes by remote elicitation of causal Bayesian networks from medical experts

BACKGROUND

COVID-19 is a new multi-organ disease causing considerable worldwide morbidity and mortality. While many recognized pathophysiological mechanisms are involved, their exact causal relationships remain opaque. Better understanding is needed for predicting their progression, targeting therapeutic approaches, and improving patient outcomes. While many mathematical causal models describe COVID-19 epidemiology, none have described its pathophysiology.

METHODS

In early 2020, we began developing such causal models. The SARS-CoV-2 virus's rapid and extensive spread made this particularly difficult: no large patient datasets were publicly available; the medical literature was flooded with sometimes conflicting pre-review reports; and clinicians in many countries had little time for academic consultations. We used Bayesian network (BN) models, which provide powerful calculation tools and directed acyclic graphs (DAGs) as comprehensible causal maps. Hence, they can incorporate both expert opinion and numerical data, and produce explainable, updatable results. To obtain the DAGs, we used extensive expert elicitation (exploiting Australia's exceptionally low COVID-19 burden) in structured online sessions. Groups of clinical and other specialists were enlisted to filter, interpret and discuss the literature and develop a current consensus. We encouraged inclusion of theoretically salient latent (unobservable) variables, likely mechanisms by extrapolation from other diseases, and documented supporting literature while noting controversies. Our method was iterative and incremental: systematically refining and validating the group output using one-on-one follow-up meetings with original and new experts. 35 experts contributed 126 hours face-to-face, and could review our products.

RESULTS

We present two key models, for the initial infection of the respiratory tract and the possible progression to complications, as causal DAGs and BNs with corresponding verbal descriptions, dictionaries and sources. These are the first published causal models of COVID-19 pathophysiology.

CONCLUSIONS

Our method demonstrates an improved procedure for developing BNs via expert elicitation, which other teams can implement to model emergent complex phenomena. Our results have three anticipated applications: (i) freely disseminating updatable expert knowledge; (ii) guiding design and analysis of observational and clinical studies; (iii) developing and validating automated tools for causal reasoning and decision support. We are developing such tools for the initial diagnosis, resource management, and prognosis of COVID-19, parameterized using the ISARIC and LEOSS databases.

Multiple waves of COVID-19: a pathway model approach

A generalized pathway model, with time-dependent parameters, is applied to describe the mortality curves of the COVID-19 disease for several countries that exhibit multiple waves of infections. The pathway approach adopted here is formulated explicitly in time, in the sense that the model's growth rate for the number of deaths or infections is written as an explicit function of time, rather than in terms of the cumulative quantity itself. This allows for a direct fit of the model to daily data (new deaths or new cases) without the need of any integration. The model is applied to COVID-19 mortality curves for ten selected countries and found to be in very good agreement with the data for all cases considered. From the fitted theoretical curves for a given location, relevant epidemiological information can be extracted, such as the starting and peak dates for each successive wave. It is argued that obtaining reliable estimates for such characteristic points is important for studying the effectiveness of interventions and the possible negative impact of their relaxation, as it allows for a direct comparison of the time of adoption/relaxation of control measures with the peaks and troughs of the epidemic curve.

A 12-month projection to September 2022 of the COVID-19 epidemic in the UK using a dynamic causal model

Objectives

Predicting the future UK COVID-19 epidemic allows other countries to compare their epidemic with one unfolding without public health measures except a vaccine program.

Methods

A Dynamic Causal Model was used to estimate key model parameters of the UK epidemic, such as vaccine effectiveness and increased transmissibility of Alpha and Delta variants, the effectiveness of the vaccine program roll-out and changes in contact rates. The model predicts the future trends in infections, long-COVID, hospital admissions and deaths.

Results

Two-dose vaccination given to 66% of the UK population prevents transmission following infection by 44%, serious illness by 86% and death by 93%. Despite this, with no other public health measures used, cases will increase from 37 million to 61 million, hospital admissions from 536,000 to 684,000 and deaths from 136,000 to 142,000 over 12 months. A retrospective analysis (conducted after the original submission of this report) allowed a comparison of these predictions of morbidity and mortality with actual outcomes.

Conclusion

Vaccination alone will not control the epidemic. Relaxation of mitigating public health measures carries several risks, which include overwhelming the health services, the creation of vaccine resistant variants and the economic cost of huge numbers of acute and chronic cases.

Dynamic causal modelling of immune heterogeneity

An interesting inference drawn by some COVID-19 epidemiological models is that there exists a proportion of the population who are not susceptible to infection-even at the start of the current pandemic. This paper introduces a model of the immune response to a virus. This is based upon the same sort of mean-field dynamics as used in epidemiology. However, in place of the location, clinical status, and other attributes of people in an epidemiological model, we consider the state of a virus, B and T-lymphocytes, and the antibodies they generate. Our aim is to formalise some key hypotheses as to the mechanism of resistance. We present a series of simple simulations illustrating changes to the dynamics of the immune response under these hypotheses. These include attenuated viral cell entry, pre-existing cross-reactive humoral (antibody-mediated) immunity, and enhanced T-cell dependent immunity. Finally, we illustrate the potential application of this sort of model by illustrating variational inversion (using simulated data) of this model to illustrate its use in testing hypotheses. In principle, this furnishes a fast and efficient immunological assay-based on sequential serology-that provides a (1) quantitative measure of latent immunological responses and (2) a Bayes optimal classification of the different kinds of immunological response (c.f., glucose tolerance tests used to test for insulin resistance). This may be especially useful in assessing SARS-CoV-2 vaccines.

COVID-19: Short term prediction model using daily incidence data

Background

Prediction of the dynamics of new SARS-CoV-2 infections during the current COVID-19 pandemic is critical for public health planning of efficient health care allocation and monitoring the effects of policy interventions. We describe a new approach that forecasts the number of incident cases in the near future given past occurrences using only a small number of assumptions.

Methods

Our approach to forecasting future COVID-19 cases involves 1) modeling the observed incidence cases using a Poisson distribution for the daily incidence number, and a gamma distribution for the series interval; 2) estimating the effective reproduction number assuming its value stays constant during a short time interval; and 3) drawing future incidence cases from their posterior distributions, assuming that the current transmission rate will stay the same, or change by a certain degree.

Results

We apply our method to predicting the number of new COVID-19 cases in a single state in the U.S. and for a subset of counties within the state to demonstrate the utility of this method at varying scales of prediction. Our method produces reasonably accurate results when the effective reproduction number is distributed similarly in the future as in the past. Large deviations from the predicted results can imply that a change in policy or some other factors have occurred that have dramatically altered the disease transmission over time.

Conclusion

We presented a modelling approach that we believe can be easily adopted by others, and immediately useful for local or state planning.

Second waves, social distancing, and the spread of COVID-19 across the USA

We recently described a dynamic causal model of a COVID-19 outbreak within a single region. Here, we combine several instantiations of this (epidemic) model to create a (pandemic) model of viral spread among regions. Our focus is on a second wave of new cases that may result from loss of immunity-and the exchange of people between regions-and how mortality rates can be ameliorated under different strategic responses. In particular, we consider hard or soft social distancing strategies predicated on national (Federal) or regional (State) estimates of the prevalence of infection in the population. The modelling is demonstrated using timeseries of new cases and deaths from the United States to estimate the parameters of a factorial (compartmental) epidemiological model of each State and, crucially, coupling between States. Using Bayesian model reduction, we identify the effective connectivity between States that best explains the initial phases of the outbreak in the United States. Using the ensuing posterior parameter estimates, we then evaluate the likely outcomes of different policies in terms of mortality, working days lost due to lockdown and demands upon critical care. The provisional results of this modelling suggest that social distancing and loss of immunity are the two key factors that underwrite a return to endemic equilibrium.

Testing and tracking in the UK: A dynamic causal modelling study

By equipping a previously reported dynamic causal modelling of COVID-19 with an isolation state, we were able to model the effects of self-isolation consequent on testing and tracking. Specifically, we included a quarantine or isolation state occupied by people who believe they might be infected but are asymptomatic—and could only leave if they test negative. We recovered maximum posteriori estimates of the model parameters using time series of new cases, daily deaths, and tests for the UK. These parameters were used to simulate the trajectory of the outbreak in the UK over an 18-month period. Several clear-cut conclusions emerged from these simulations. For example, under plausible (graded) relaxations of social distancing, a rebound of infections is highly unlikely. The emergence of a second wave depends almost exclusively on the rate at which we lose immunity, inherited from the first wave. There exists no testing strategy that can attenuate mortality rates, other than by deferring or delaying a second wave. A testing and tracking policy—implemented at the present time—will defer any second wave beyond a time horizon of 18 months. Crucially, this deferment is within current testing capabilities (requiring an efficacy of tracing and tracking of about 20% of asymptomatic infected cases, with 50,000 tests per day). These conclusions are based upon a dynamic causal model for which we provide some construct and face validation—using a comparative analysis of the United Kingdom and Germany, supplemented with recent serological studies.

Second waves, social distancing, and the spread of COVID-19 across the USA

We recently described a dynamic causal model of a COVID-19 outbreak within a single region. Here, we combine several instantiations of this (epidemic) model to create a (pandemic) model of viral spread among regions. Our focus is on a second wave of new cases that may result from loss of immunity-and the exchange of people between regions-and how mortality rates can be ameliorated under different strategic responses. In particular, we consider hard or soft social distancing strategies predicated on national (Federal) or regional (State) estimates of the prevalence of infection in the population. The modelling is demonstrated using timeseries of new cases and deaths from the United States to estimate the parameters of a factorial (compartmental) epidemiological model of each State and, crucially, coupling between States. Using Bayesian model reduction, we identify the effective connectivity between States that best explains the initial phases of the outbreak in the United States. Using the ensuing posterior parameter estimates, we then evaluate the likely outcomes of different policies in terms of mortality, working days lost due to lockdown and demands upon critical care. The provisional results of this modelling suggest that social distancing and loss of immunity are the two key factors that underwrite a return to endemic equilibrium.

Quantifying the shift in social contact patterns in response to non-pharmaceutical interventions

Social contact mixing plays a critical role in influencing the transmission routes of infectious diseases. Moreover, quantifying social contact mixing patterns and their variations in a rapidly evolving pandemic intervened by changing public health measures is key for retroactive evaluation and proactive assessment of the effectiveness of different age- and setting-specific interventions. Contact mixing patterns have been used to inform COVID-19 pandemic public health decision-making; but a rigorously justified methodology to identify setting-specific contact mixing patterns and their variations in a rapidly developing pandemic, which can be informed by readily available data, is in great demand and has not yet been established. Here we fill in this critical gap by developing and utilizing a novel methodology, integrating social contact patterns derived from empirical data with a disease transmission model, that enables the usage of age-stratified incidence data to infer age-specific susceptibility, daily contact mixing patterns in workplace, household, school and community settings; and transmission acquired in these settings under different physical distancing measures. We demonstrated the utility of this methodology by performing an analysis of the COVID-19 epidemic in Ontario, Canada. We quantified the age- and setting (household, workplace, community, and school)-specific mixing patterns and their evolution during the escalation of public health interventions in Ontario, Canada. We estimated a reduction in the average individual contact rate from 12.27 to 6.58 contacts per day, with an increase in household contacts, following the implementation of control measures. We also estimated increasing trends by age in both the susceptibility to infection by SARS-CoV-2 and the proportion of symptomatic individuals diagnosed. Inferring the age- and setting-specific social contact mixing and key age-stratified epidemiological parameters, in the presence of evolving control measures, is critical to inform decision- and policy-making for the current COVID-19 pandemic.

LIVING WITH COVID-19: BALANCING COSTS AGAINST BENEFITS IN THE FACE OF THE VIRUS

This paper analyses the costs and benefits of lockdown policies in the face of COVID-19. What matters for people is the quality and length of lives and one should measure costs and benefits in terms of those things. That raises difficulties in measurement, particularly in valuing potential lives saved. We draw upon guidelines used in the UK for public health decisions, as well as other measures, which allow a comparison between health effects and other economic effects. We look at evidence on the effectiveness of past severe restrictions applied in European countries, focusing on the evidence from the UK. The paper considers policy options for the degree to which restrictions are eased. There is a need to normalise how we view COVID because its costs and risks are comparable to other health problems (such as cancer, heart problems, diabetes) where governments have made resource decisions for decades. The lockdown is a public health policy and we have valued its impact using the tools that guide health care decisions in the UK public health system. The evidence suggests that the costs of continuing severe restrictions in the UK are large relative to likely benefits so that a substantial easing in general restrictions in favour of more targeted measures is warranted.

Testing and tracking in the UK: A dynamic causal modelling study

By equipping a previously reported dynamic causal modelling of COVID-19 with an isolation state, we were able to model the effects of self-isolation consequent on testing and tracking. Specifically, we included a quarantine or isolation state occupied by people who believe they might be infected but are asymptomatic—and could only leave if they test negative. We recovered maximum posteriori estimates of the model parameters using time series of new cases, daily deaths, and tests for the UK. These parameters were used to simulate the trajectory of the outbreak in the UK over an 18-month period. Several clear-cut conclusions emerged from these simulations. For example, under plausible (graded) relaxations of social distancing, a rebound of infections is highly unlikely. The emergence of a second wave depends almost exclusively on the rate at which we lose immunity, inherited from the first wave. There exists no testing strategy that can attenuate mortality rates, other than by deferring or delaying a second wave. A testing and tracking policy—implemented at the present time—will defer any second wave beyond a time horizon of 18 months. Crucially, this deferment is within current testing capabilities (requiring an efficacy of tracing and tracking of about 20% of asymptomatic infected cases, with 50,000 tests per day). These conclusions are based upon a dynamic causal model for which we provide some construct and face validation—using a comparative analysis of the United Kingdom and Germany, supplemented with recent serological studies.

Second waves, social distancing, and the spread of COVID-19 across America

We recently described a dynamic causal model of a COVID-19 outbreak within a single region. Here, we combine several of these (epidemic) models to create a (pandemic) model of viral spread among regions. Our focus is on a second wave of new cases that may result from loss of immunity-and the exchange of people between regions-and how mortality rates can be ameliorated under different strategic responses. In particular, we consider hard or soft social distancing strategies predicated on national (Federal) or regional (State) estimates of the prevalence of infection in the population. The modelling is demonstrated using timeseries of new cases and deaths from the United States to estimate the parameters of a factorial (compartmental) epidemiological model of each State and, crucially, coupling between States. Using Bayesian model reduction, we identify the effective connectivity between States that best explains the initial phases of the outbreak in the United States. Using the ensuing posterior parameter estimates, we then evaluate the likely outcomes of different policies in terms of mortality, working days lost due to lockdown and demands upon critical care. The provisional results of this modelling suggest that social distancing and loss of immunity are the two key factors that underwrite a return to endemic equilibrium.

Dynamic causal modelling of COVID-19

This technical report describes a dynamic causal model of the spread of coronavirus through a population. The model is based upon ensemble or population dynamics that generate outcomes, like new cases and deaths over time. The purpose of this model is to quantify the uncertainty that attends predictions of relevant outcomes. By assuming suitable conditional dependencies, one can model the effects of interventions (e.g., social distancing) and differences among populations (e.g., herd immunity) to predict what might happen in different circumstances. Technically, this model leverages state-of-the-art variational (Bayesian) model inversion and comparison procedures, originally developed to characterise the responses of neuronal ensembles to perturbations. Here, this modelling is applied to epidemiological populations-to illustrate the kind of inferences that are supported and how the model per se can be optimised given timeseries data. Although the purpose of this paper is to describe a modelling protocol, the results illustrate some interesting perspectives on the current pandemic; for example, the nonlinear effects of herd immunity that speak to a self-organised mitigation process.