With the announcement of regulated vaccines and an imminent programme of vaccination, there are several pressing questions about the rate and extent of vaccination required to contain and possibly suppress the spread of SARS-CoV-2 (Saad-Roy, Wagner et al. 2020)¹. Established quantitative criteria can provide guidance on the rollout and extent of requisite vaccination programs². For example, it is generally considered that a herd immunity of 67% of population will be sufficient to suppress community transmission of SARS-CoV-2. These kinds of estimates usually assume an idealised epidemiology, in which every member of the population is exchangeable, and the reproduction ratio is fixed, here, R₀ = 3. However, heterogeneity in exposure, susceptibility and transmission—coupled with fluctuating reproduction ratios—may mean that the proportion of the population requiring vaccination may be less than 67% (Fontanet and Cauchemez 2020, Gomes, Corder et al. 2020, Lourenco, Pinotti et al. 2020). Furthermore, pre-existing immunity (Le Bert, Tan et al. 2020, Ng, Faulkner et al. 2020) may further reduce the number of people who need to be vaccinated. In what follows, we use dynamic causal modelling to assimilate information from past responses to the outbreak to predict what one might expect under different vaccination scenarios in the future. This offers a slightly more nuanced treatment that considers vaccination in the context of heterogeneity, ongoing restrictions and public health measures.

The purpose of this technical note is to describe the model in question and illustrate its application to data at the time of writing (6-12-2021). Its purpose is to provide a coarse-grained way of assessing the average effectiveness of public health measures and, in particular, how they contextualise each other. The analyses that follow are therefore presented purely for illustrative purposes. We anticipate using this model to quantify the effectiveness of vaccination in a few months when appropriate data become available.

Dynamic causal modelling may play a useful role in this setting because it features a state-space model of both epidemiological and behavioural responses to infection at a population level. Furthermore, some of these responses can be characterised as interventions, enabling model fitting—based upon empirical data from the past—to inform the structure and parameters that underwrite viral spread in the future (Friston, Parr et al. 2020, Friston, Parr et al. 2020). Here, we use a dynamic causal model to generate posterior predictive densities over some key outcomes (e.g., fatality rates, confirmed cases and vaccination rates). This dynamic causal model (DCM) has been progressively optimised using Bayesian model comparison during the course of the epidemic—to ensure that it has the right balance of expressivity (i.e., model complexity) and predictive accuracy. The details of this model can be found in (Friston, Flandin et al. 2020), along with (i) the Master equation that underwrites the generative model, (ii) the prior over model parameters (and their sources) and (iii) the variational scheme used for model inversion and comparison.

For people not familiar with dynamic causal modelling (DCM), it can be regarded as an analytic version of a stochastic agent-based model, where the sample densities are replaced by a mean field approximation. This allows one to model the ensemble density dynamics analytically. In turn, this affords an efficient way of fitting the model to observed time series (i.e., data assimilation), while, at the same time, quantifying uncertainty about model parameters. Most importantly, it allows one to score each model in terms of (a variational bound on) model evidence; thereby enabling model comparison or structure learning. A model with a high evidence (a.k.a. marginal likelihood) has greater predictive validity in the sense that the likelihood of new data, under the posterior predictive density will, on average, be greater. Crucially, model evidence is a function of the data, which means the predictive validity of the model with the greatest evidence increases with the available data. We use this feature of DCM to licence its application to the scenario modelling at hand.

Our focus is on the dynamics of vaccination and interactions with other interventions; in particular, the adaptive restriction of contact rates through social distancing, lockdowns, and travel restrictions. In the current model, these mechanisms are lumped together in terms of a single probability that any member of the population will move from a low (e.g., home) to a high risk (e.g., work) location on any given day. This contact rate is itself a function of the prevalence of infection. This function models things like ‘tier systems’ and ‘circuit breakers’—as implemented as a governmental level—but, ultimately, driven by (measures of) the prevalence of infection and its fluctuations (e.g., the reproduction ratio or R-number). In brief, contact rates are a non-linear threshold function of a linear mixture of various prevalence levels (including the number of people who are currently infected and seroprevalence). Please see (Friston, Flandin et al. 2020) for details. The second important mitigating mechanism is the supported isolation of contacts; namely, reducing viral transmission by minimising the contact rates with people who are infected but have yet to become infectious. Of all the mitigating mechanisms, this is the most efficient; in the sense that it is the most specific. In other words, it targets the small proportion of the population that can transmit the virus to others. This should be contrasted with restricting contact rates through social distancing: this mechanism applies to everyone and is more costly to implement at several levels.

For example, consider heterogeneity of transmission due to overdispersion (Lloyd-Smith, Schreiber et al. 2005, Endo, Abbott et al. 2020, Gomes, Corder et al. 2020). This is sometimes cast in terms of superspreading; namely, when the few infect the many. Assume, for example, that only 20% of the population can transmit the virus to someone else. If one were able to identify superspreaders, then it would only be necessary to vaccinate—or isolate—20% of the population to eliminate community transmission. However, there is no way of knowing who is likely to infect someone else in advance. Or is there? This, on one view, is the rationale for contact tracing (critically, backward contact tracing or ‘cluster busting’). In other words, one can identify a subpopulation with an enriched probability that it contains people who are likely to infect others in the near future. This is why contact tracing and supported isolation has played a key role in containing the virus in many countries with established public health measures. One of our key questions was whether vaccination would enable efficient contact tracing in the United Kingdom by reducing the prevalence of infection.

To address this question, we simulated different vaccination rates with and without enhanced contact tracing to quantify their interaction. This was a simulated under context-sensitive restrictions on contact rates that best explain population behaviour to date.

Dynamic causal modelling

To model the impact of vaccination, we introduced a new rate parameter determining the probability of moving from a susceptible (non-infected) state to a state of seronegative immunity, when visiting hospital (or an equivalent location). Vaccination was restricted to clinical settings because, in the model, these are frequented by clinically susceptible people and healthcare workers, who we presumed would be the first people to receive the vaccine. In the model, vaccination is taken to be hundred percent effective in precluding infection and thereby transmission. A single effective vaccination will therefore generate multiple recorded vaccinations, depending upon the number of vaccinations required for any given person, and their average effectiveness. We hope to estimate vaccine effectiveness, when the number of vaccinations administered over the next few months becomes available (using priors from the appropriate Phase 3 clinical trials).

In this DCM, there are two kinds of protective (humoral) immunity labelled seropositive and seronegative. The loss of seropositivity is modelled as a movement from seropositive to seronegative, with an estimated time constant of about 200 days. One then loses protective immunity by moving from a seronegative state to a susceptible state with a longer time constant (here, 2048 days)³. We modelled vaccination in this way because seroconversion induced by viral infection drives reduced contact rates—and we did not want to conflate vaccination induced seroconversion with seroprevalence due to infection. Furthermore, we anticipate that surveillance studies of seropositivity will deal separately with people who have and have not been vaccinated. Note that the seronegative state does not imply the absence of antibodies, rather the absence of antibodies following infection.

One can regard the seronegative state as an enduring state of immunity that does not entail seroconversion due to infection: i.e., an immune state due to pre-existing immunity (Le Bert, Tan et al. 2020, Ng, Faulkner et al. 2020), mediated by T-cells (Grifoni, Weiskopf et al. 2020, Le Bert, Tan et al. 2020, Seo, Giles et al. 2020) or vaccination (Fontanet and Cauchemez 2020, Houlihan, Vora et al. 2020, Saad-Roy, Wagner et al. 2020). This construction allows for the loss of detectable IgG antibodies (seropositivity) over months, while modelling a slower loss of protective (seronegative) immunity over years. Effective vaccination is assumed to induce an immunity of the second kind.

As previously, contact tracing is modelled with a similar rate parameter determining the probability of going into self-isolation when infected but pre-symptomatic. This models people’s response when advised by a family member, public health worker or contact tracer that they may be infected. The efficiency of contact tracing corresponds to this rate parameter. Its estimated level is about 3 to 4% based upon our past responses (see also ⁴). An efficient contact tracing is considered to be about 24%. This is based upon an idealised ability to detect 30% of new cases every day and support 80% of their contacts in self-isolation.

The parameter estimates above were based upon Bayesian model inversion using 12 sources of empirical data, shown in Figure 1. These include daily confirmed cases by specimen date, deaths with a positive PCR test within 28 days by date of death, hospital admissions, patients requiring mechanical ventilation, certified deaths, symptoms, mobility data provided by the Department of Transport, and so on. Collectively, these time series afford multiple constraints on the posterior probability of the unknown model parameters that are, universally, nonnegative rate or time constants⁵. The form and parameters of this model have been optimised using data from the first and secondary waves. This is important because assimilating data from the secondary wave mandated the inclusion of monotonic and seasonal fluctuations that, in principle, lend the model a greater predictive validity—in terms of long-term forecasting over months and years⁶. Equipped with these posterior expectations, one can roll out the probability distribution over various states into the future, under different sets of parameters.

To model different levels of vaccination, we increased the vaccination rate from 0% to 100% in 20% steps. By construction, a 100% vaccination rate (when visiting a hospital or healthcare centre) corresponds to a peak vaccination rate of about 200,000 per day, dropping to 50,000 per day after a month or two. If sustained over a year, this would ensure that about 28 million people would have received a course of vaccination by 2022. This (maximal) vaccination rate is comfortably within current public health capabilities. For example, current reports suggest that the NHS intends to use football stadiums, town halls and conference buildings in England to inoculate at least 2,000 people per centre each day. These new facilities will supplement the 1,560 community-based vaccination centres run by general practitioners, that can dispense between 200 to 500 vaccinations a day.

We did not consider higher rates of vaccination because, in this model, being vaccinated confers 100% seronegative immunity. To the extent that vaccination is only partially effective, the vaccination rates need to be adjusted accordingly. For example, if vaccination is only 50% effective, then the actual vaccination rates should be doubled. Similarly, if vaccinations are to be dispensed twice, the number of vaccinations should be doubled. Modelling vaccination as an effective vaccination means the efficacy of the vaccine does not influence the requisite herd immunity or number of people who need to receive an effective vaccination.