Dynamic causal modelling

Dynamic causal modelling (DCM) is the application of variational Bayes to estimate the parameters of state-space models and, crucially, the evidence for alternative models of the same data.11 It was developed to model interactions among neuronal populations and has been used subsequently in radar, medical nosology and recently epidemiology.11–15 Variational Bayes is also known as approximate Bayesian inference and is computationally more efficient than Bayesian techniques based on sampling procedures (eg, approximate Bayesian computation), which predominate in epidemiological modelling.16–18 The particular dynamic causal model used here embeds an SEIR model of immune status into a model that includes all latent factors generating data; namely, location, infection symptom and testing status. Please see the foundational paper for structural details of the model used in this paper15 and the generic (variational Laplace) scheme used to estimate model parameters and evidence.

DCM differs from conventional epidemiological modelling in that it uses mean field approximations and standard variational procedures to model the evolution of probability densities.16 This contrasts with epidemiological modelling that generally uses stochastic realisations of epidemiological dynamics to approximate probability densities with sample densities.17 19–21 One advantage of variational procedures is that they are orders of magnitude more efficient, enabling end-to-end model inversion or fitting within minutes (on a laptop) as opposed to hours or days (on a supercomputer).17 More importantly, variational procedures provide an efficient way of assessing the quality of one model relative to another, in terms of model evidence (a.k.a., marginal likelihood).22 This enables one to compare different models using Bayesian model comparison (a.k.a. structure learning) and use the best model for nowcasting, forecasting or, indeed, test competing hypotheses about viral transmission.

More generally, Bayesian model comparison plays a central role in testing hypotheses given (often sparse or noisy) data. It eschews intuitive assumptions, about whether there are sufficient data to test this or that, by evaluating the evidence for competing hypotheses or models. If there is sufficient information in the data to disambiguate between two hypotheses, the difference in log-evidence will enable one to confidently assert one model is more likely than the other. Note that this automatically ensures that the model is identifiable, in relation to the model parameters or prior assumptions in question.

Dynamic causal models can be extended to generate any kind of epidemiological data at hand: for example, the number of positive antigen tests. This requires careful consideration of how positive tests are generated, by modelling latent variables such as the bias towards testing people with or without infection or, indeed, the time-dependent capacity for testing. In short, everything that matters—in terms of the latent (hidden) causes of the data—can be installed in the model, including lockdown, self-isolation and other processes that underwrite viral transmission. Model comparison can then be used to assess whether the effect of a latent cause is needed to explain the data—by withdrawing the effect and seeing if model evidence increases or decreases. Here, we leverage the efficiency of DCM to evaluate the evidence for a series of models that are distinguished by heterogeneity or variability in the way that populations respond to an epidemic. The dynamic causal model used for the analyses below is summarised in terms of its structure (figure 1) and parameters (table 1).

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Figure 1

A location, infection, symptom and testing (LIST) model. This schematic summarises a LIST model used for the accompanying analyses. This model is formally equivalent to the model by Friston et al.56 It includes a state (isolated) to model people who have yet to be exposed to the virus or are shielding because they think they may be infectious (within their home or elsewhere). It also includes a (seronegative) state to model individuals who have pre-existing immunity, for example, via cross-reactivity38 39 or other protective host factors.41 42 This absorbing state plays the role of the recovered or removed states of Susceptible, Exposed, Infected and Removed (SEIR) models—once entered, people stay in this state for the duration of the outbreak. One can leave any of the remaining states. For example, one occupies the deceased state for a day and then moves to healthy on the following day. Similarly, one occupies the state of testing positive or negative for a day, and then moves to the untested state the following day. This enables the occupancy of various states to be quantified in terms of daily rates. The discs represent the four factors of the model, and the segments correspond to their states (ie, compartments). The green disc is the closest to a conventional (ie, SEIR) model that is embedded within three other factors. The states within any factor are mutually exclusive. In other words, every individual has to be in one state associated with four factors. The orange boxes represent the observable outputs that are generated by this model, in this instance, daily reports of positive tests and deaths. The rate of transition between states—or the dwell time within any state—rests on the model parameters that, in some instances, can be specified with fairly precise prior densities—listed in table 1.

Heterogeneity in exposure

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Figure 2

Heterogeneity of exposure, susceptibility and transmission. Upper panel: this schematic illustrates the composition of a population in terms of a proportion that is not exposed to the virus, a proportion that is not susceptible when exposed and a further proportion of susceptible people who cannot transmit the virus. These proportions are unknown but can be estimated from the data. Lower panel: this schematic illustrates the parameterisation of heterogeneity in terms of the parameters in table 1.

This was modelled in terms of an effective population size that is less than the total (census) population. The effective population comprises individuals who are in contact with other infected individuals. The remainder of the population are assumed to be geographically sequestered from a regional outbreak or are shielded from it. For example, if the population of the UK was 68 million, and the effective population was 39 million, then only 57% are considered to participate in the outbreak. Of this effective population, a certain proportion are susceptible to infection.

Heterogeneity in transmission

We modelled heterogeneity in transmission with a free parameter (with a prior of one half and a prior SD of 1/16). This parameter corresponds to the proportion of susceptible people who are unlikely to transmit the virus, that is, individuals who move directly from a state of being infected to a seronegative state (as opposed to moving to a seropositive state after a period of being infectious). We associated this transition with a mild infection44 that does not entail seroconversion, for example, recovery in terms of T-cell-mediated responses.39 41 In short, the seronegative state plays the role of a seropositive state of immunity for people who never become infectious, either because they are not susceptible to infection or have a mild infection (with or without symptoms, eg, children).

Modelling heterogeneity of susceptibility in terms of susceptible and non-susceptible individuals can be read as modelling the difference between old (susceptible) and young (non-susceptible) people.24 However, in contrast to models with age-stratification, the current model does not consider different contact rates between different strata (eg, contact matrices). Instead, we finesse this limitation with location-specific contact rates—parameterised by the number of people one is exposed to in different locations. Although full stratification is straightforward to implement (please see DEM_COVID_I.m for a Matlab demonstration that can be read as pseudocode), this simplified model of heterogeneity is sufficient to make definitive inferences about the joint contribution of lockdown and population immunity to viral transmission.

The prior means of non-susceptible and non-infectious people of 50% was chosen on the basis of secondary attack rates in well-defined cohorts.34 45 A prior variance of 1/256 corresponds to a range of between 42% and 58% (99% CI). Note that these priors are just constraints (that are also used as starting estimates). The posterior estimate can be markedly different from the prior if the data are sufficiently informative.

This kind of model is sufficiently expressive to reconcile the apparent disparity between morbidity/mortality rates and low seroprevalence observed empirically31 (https://www.gov.uk/government/publications/national-covid-19-surveillance-reports/sero-surveillance-of-covid-19). We will see below that Bayesian model comparison suggests there is very strong evidence46 for all three types of heterogeneity.

Given a suitable dynamic causal model one can use standard variational techniques to fit the empirical data and estimate model parameters. Having estimated the requisite model parameters, one can then reconstitute the most likely trajectories of latent states: namely, the probability of being in different locations, states of infection, symptom and testing states. An example is provided in figure 3 using daily cases and death-by-date data from the UK from 30 January 2020 to 1 September 2020.

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Figure 3

Latent causes. This figure illustrates posterior predictions of the most likely latent states for the UK. Here, the outcomes in the upper two panels (dots) are supplemented with the underlying latent causes or expected states in the lower four panels (the first state in each factor has been omitted for clarity, ie, home, susceptible, healthy and untested). These latent or expected states generate the observable outcomes in the upper two panels. The solid lines are colour-coded and correspond to the states of the four factors in figure 2. For example, under the location factor, the probability of being found at in a high-risk location (work: blue line) rises at the onset of the outbreak, falls during lockdown and then slowly recovers during unlocking. At the same time, the proportion of the population who are not exposed to the virus (removed: yellow line) falls, as the exposed proportion increases to the effective population size. During lockdown, the probability of isolating oneself rises to about 3% during the peak of the pandemic (isolated: purple line). After about 6 weeks, lockdown starts to relax and slowly tails off, with accompanying falls in morbidity (in terms of symptoms: blue line in the symptom panel) and mortality (in terms of death rate). Note that seroprevalence (AB +ve: yellow line in the infection panel) peaks at about 50 weeks and then starts to decline. Please see software and data note for details about the model inversion and data used to prepare this figure—and figure 6 for CIs on fatality rates. (This figure can be reproduced automatically by invoking the Matlab routine DEM_COVID_COUNTRY.m).

Heterogeneity in exposure, susceptibility and transmission

In what follows, we use DCM to revisit some assumptions implicit in conventional epidemiological modelling. We follow the three lines of arguments rehearsed in Okell et al (ibid). The first can be summarised as: under herd immunity the cumulative mortality rate per million of the population should plateau at roughly the same level in different countries.

This is true if, and only if, the same proportion of the population can transmit the virus. In other words, a plateau to endemic equilibrium—based on the removal of susceptible people from the population—only requires people are susceptible to infection to be immune. If this proportion depends on the composition of a country’s population (ie, demography), mortality rates could differ from country to country. This can be illustrated by using models with heterogeneous population structures, of the sort summarised in figure 2. Figure 4 shows the data and ensuing predictions for 10 countries, using the format of Okell et al.2 These countries were chosen because, at time of writing, they had a well-defined first peak in conjunction with a high fatality rate, in relation to other countries.

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Figure 4

This figure reproduces the quantitative results in the Okell et al (ibid). However, here, they are based on the predictions of a dynamic causal model of the epidemic in the 10 countries with the highest mortality on 15 June 2020. The left panel shows the cumulative deaths per million as a function of time to illustrate variation over countries. The middle panel plots the log deaths per million in the 6 weeks following lockdown against the corresponding mortality before lockdown. The right panel plots the estimated cumulative deaths against estimated seroprevalence in the effective population over a period of 180 days. Note that these graphics include both empirical data (dots and circles) and predictions (coloured lines) based on the parameter estimates of the dynamic causal modelling (CIs have been omitted for clarity). The predictions of seroprevalence are, in this example, based purely on new cases and deaths. Although it would be relatively straightforward to include empirical seroprevalence data in the dynamic causal modelling, we have presented the seroprevalence predictions to illustrate the predictive validity of the model—given that the predictions are in line with empirical reports (ie, between 5% and 20%).

Heterogeneity of transmission may be particularly important here.3 This is sometimes framed in terms of overdispersion or the notion of superspreading and amplification events.4 6 37 For example, if only 20% of the population were able to develop a sufficient viral load to infect others, then protective immunity in this subpopulation would be sufficient for an innocuous endemic equilibrium.47 Furthermore, if seroconversion occurs largely in the subpopulation spreading the virus,48 a sufficient herd immunity may only require a seroprevalence of around 10% of the effective population (right panel of figure 4). In short, one might challenge the assumption that COVID-19 is spread homogeneously across the population. Indeed, heterogeneity is becoming increasingly evident in high-risk settings, and in the variation in the period of infectivity across ages.

This begs the question: is there evidence for heterogeneity in the dispersion of SARS-CoV-2? And, if so, does this heterogeneity vary from country to country? Figure 5 answers this question using Bayesian model comparison. It shows—under the models in question—there is overwhelming evidence for heterogeneity of exposure, susceptibility and transmission. And that a substantial proportion of each country’s population does not contribute to viral transmission. These proportions vary from country to country, leading to the differential mortality rates as shown in figure 4.

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Figure 5

Bayesian model comparison of dynamic causal models of the coronavirus outbreak in 10 countries with the greatest mortality rates. The upper panel shows the drop in log evidence when removing a component from the model. This reduction is known as log Bayes factor. For example, if the No build-up of herd immunity model for the UK has a log evidence of −54, then it is exp(54) times less likely than the full model that includes a build-up of immunity. By convention, a difference in log evidence of 3 is taken to indicate strong evidence in favour of the full model. The full model accommodates heterogeneity at three levels. The first considers an effective population that is exposed to infection, where the remainder are geographically isolated or shielding. Of this effective population, a certain proportion cannot be infected (eg, due to host factors such as pre-existing immunity). Of the remaining susceptible population, a certain proportion cannot transmit the virus (eg, through having a short period of viral shedding). This leaves a subpopulation who are both susceptible to infection and capable of transmission. By removing these three kinds of heterogeneity—and re-evaluating the evidence for the reduced models in relation to the full model—one can assess the evidence for the contribution of heterogeneity. Similarly, one can remove social distancing and the build-up of seropositive immunity from the model and evaluate their respective contribution. The results in the upper panel indicate very strong evidence in favour of the full model for nearly all countries. The exceptions are Brazil and Spain—where the entire population appears to participate in the outbreak—and Mexico, where social distancing is not evident. Mexico is an interesting example where removing a model parameter increases model evidence (via a reduction model complexity). Full model: this model incorporates all three forms of heterogeneity, social distancing and seropositive immunity. Exposure: heterogeneity to exposure was removed by equating the effective and total population. Susceptibility: heterogeneity of susceptibility was reduced by decreasing the prior proportion of non-susceptible people by a factor of e. Transmission: heterogeneity of transmission was reduced by decreasing the prior proportion of non-infectious people by a factor of e. Lockdown: social distancing was reduced by increasing the social distancing threshold by a factor of e² . Immunity: the effect of immunity was reduced by decreasing the period of seropositivity by a factor of e² . The lower panels provide the posterior expectations (ie, most likely value given the data) of the effective population under the full model, alongside the census population (left panel). The subsequent panels show the non-susceptible proportion of the effective population (middle panel) and non-infectious proportion of the susceptible population (right panel). CIs have been submitted for clarity but were in the order of 10%–20%.

In general, the effective population is roughly half of the census population, with some variation over countries. The non-susceptible and non-infectious proportions are roughly half of the effective and susceptible populations, respectively—varying between 45% and 65%. This variation underwrites the differences in fatality rates in figure 4. It could be argued that the estimates of the proportion of non-susceptible individuals is at odds with empirical data from contained outbreaks. For example, on the aircraft carrier Charles de Gaulle, about 70% of sailors were infected (https://en.wikipedia.org/wiki/COVID-19_pandemic_on_Charles_de_Gaulle). However, this argument overlooks the contribution of heterogeneity: there were no children on the Charles de Gaulle.

Okell et al (ibid) wrote: ‘If acquisition of herd immunity was responsible for the drop in incidence in all countries, then disease exposure, susceptibility, or severity would need to be extremely different between populations’. On the basis of the above quantitative modelling, this assumption transpires to be wrong: small variations in heterogeneity of exposure and susceptibility are sufficient to explain differences between countries. Our point here is that predicates or assumptions of this sort can be evaluated quantitatively in terms of model evidence.

Do countries that went into lockdown early experience fewer deaths in subsequent weeks?

This is the second argument made by Okell et al (ibid) for the unique role of lockdown in mitigating fatalities: however, exactly the same correlation—between cumulative deaths before and after lockdown—emerges under epidemiological models that entertain heterogeneity and herd immunity (see figure 4 (middle panel)). In short, had the authors tested the hypothesis that lockdown or herd immunity were necessary to explain the data, they would have found very strong evidence for both—and may have concluded that lockdown nuances the emergence of herd immunity (see figure 5).

Does a correlation between antibodies to SARS-CoV-2 (ie, seroprevalence) and COVID-19 mortality rates imply a similar infection fatality ratio over countries?

Conventional models generally assume this is the case.2 24 The problem with this assumption is that it precludes pre-existing immunity and loss of immunity (as in SEIR models).38 Population immunity could fall over a few months due to population flux and host factors, such as loss of neutralising antibodies.48 This is important because it means that seroprevalence could fall slowly after the first wave of infection (indeed, empirical seroprevalence is not increasing and may be decreasing in the UK (https://www.ons.gov.uk/peoplepopulationandcommunity/healthandsocialcare/conditionsanddiseases/bulletins/coronaviruscovid19infectionsurveypilot/18june2020%23antibody-data). In turn, this produces a non-linear relationship between the prevalence of antibodies and cumulative deaths at the time seroprevalence is assessed. This is illustrated by the curvilinear relationships in the right panel of figure 4 (under a loss of seroprevalence with a time constant of 3 months). If one associates infection fatality ratio (IFR) with the slope of fatality rates—as a function of seroprevalence—then the IFR changes over time. In short, the IFR changes as the epidemic progresses, as the proportion of susceptible and transmitting people falls, or those at highest risk succumb early in the pandemic. This begs the question: are quantities such as IFR fit for purpose when trying to model epidemiological dynamics?

What is the impact of different rates of loss of immunity?

So, what are the implications of heterogeneity for seroprevalence and a second wave? The Bayesian model comparisons in figure 5 speak to a mechanistic role for herd immunity in mitigating a rebound of infections. Note that this model predicts seroprevalences that are consistent with empirical community studies, without ever seeing these serological data (eg, in the UK if 11% of the effective population is seropositive and the effective population is 49% of the census population, we would expect 5.4% of people to have antibodies, which was the case at the time of analysis (https://www.gov.uk/government/publications/national-covid-19-surveillance-reports/sero-surveillance-of-covid-19).

Predictive validity of this sort generally increases with model evidence. This follows from the fact that the log-evidence is accuracy minus complexity. In other words, models with the greatest evidence afford an accurate account of the data at hand, in the simplest way possible. Unlike the Akaike and widely used Bayesian information criteria (AIC and BIC), the variational bounds on log-evidence used in DCM evaluate complexity explicitly.22 Models with the greatest evidence have the greatest predictive validity because they do not overfit the data. An example of the posterior predictions afforded by the current model is provided in figures 3 and 6 that indicate the timing and amplitude of a second wave in the UK. Figure 6 focuses on fatality rates under a couple of different scenarios; namely, under a rapid loss of antibody-mediated immunity and under an accelerated FTTIS programme.

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Figure 6

Expected death rates as a function of time for a heterogeneous UK population. The trajectories (lines) and accompanying 90% Bayesian CIs (shaded areas) correspond to posterior predictions with a loss of immunity over 32 (blue) and 6 (orange) months. This subsumes loss of protective immunity due to population fluxes and the immunological status of individuals. The 6-month scenario can be considered a worst-case prediction, under which we would expect a second peak of about 100 deaths per day. Under the more optimistic prior of enduring immunity, fatality rates would be substantially lower than witnessed during the first peak (31 vs 856 per day, respectively, 95% CI 24 to 37). The green line reproduces the predictions after increasing the efficacy of find, test, trace, isolate and support to 25% (from its posterior estimate of about 1%). FTTIS, Find-Test-Trace-Isolate-Support.

These posterior predictions suggest that there may be a mild surge of fatalities over the autumn, peaking at about 100 per day. This second wave could be eliminated completely with an increase in the efficacy of contact tracing (FTTIS)—modelled as the probability of self-isolating, given one is infected but asymptomatic. It can be seen that even with a relatively low efficacy of 25%, elimination is possible by November, with convergence to zero fatality rates. Please see Friston et al 49 for a more comprehensive analysis. Note that in a month or two, death rates should disambiguate between these scenarios.

The CIs in figure 6 may appear rather tight. This reflects two issues. First, the well-known overconfidence problem with variational inference—discussed in Friston et al ¹⁵ and MacKay.⁵⁰ Second, these posterior predictive densities are based on the entire timeseries, under a dynamic causal model that constrains the functional form of the trajectories. Put simply, this means that uncertainty about the future can be reduced substantially by data from the past.

Glossary of terms

Dynamic causal modelling: the application of variational Bayes to estimate the unknown parameters of state-space models and assess the evidence for alternative models of the same data.

See http://www.scholarpedia.org/article/Dynamic_causal_modeling.

Variational Bayes (a.k.a., approximate Bayesian inference): a generic Bayesian procedure for fitting and evaluating generative models of data by optimising a variational bound on model evidence. See https://en.wikipedia.org/wiki/Variational_Bayesian_methods.

Model evidence (a.k.a., marginal likelihood): the probability of observing some data under a particular model. It is called the marginal likelihood because its evaluation entails marginalising (ie, integrating) out dependencies on model parameters. Technically, model evidence is accuracy minus complexity, where accuracy is the expected log likelihood of some data and complexity is the divergence between posterior and prior densities over model parameters.

Variational bound (a.k.a., variational free energy): known as an evidence lower bound (ELBO) in machine learning because it is always less than the logarithm of model evidence. In brief, variational free energy converts an intractable marginalisation problem—faced by sampling procedures—into a tractable optimisation problem. This optimisation furnishes the posterior density over model parameters and ensures the ELBO approximates the log evidence for a model: https://en.wikipedia.org/wiki/Variational_Bayesian_methods. Crucially, the variational bound includes an explicit estimate of model complexity, in contrast to the AIC and BIC.22

Bayesian model comparison: a procedure to compare different models of the same data in terms of model evidence. The marginal likelihood ratio of two models is known as a Bayes factor: https://en.wikipedia.org/wiki/Bayes_factor.

Agent-based simulation models: an alternative to equation-based models, usually used to simulate scenarios that are richer than models based on population dynamics. Agent-based models simulate lots of individuals to create a sample distribution over outcomes. Evaluating the marginal likelihood from the ensuing sample distributions is extremely difficult, even with state-of-the-art estimators such as the harmonic mean: see https://radfordneal.wordpress.com/2008/08/17/the-harmonic-mean-of-the-likelihood-worst-monte-carlo-method-ever/.