Quantifying the impact of COVID-19 control measures using a Bayesian model of physical distancing

Abstract

Extensive non-pharmaceutical and physical distancing measures are currently the primary interventions against coronavirus disease 2019 (COVID-19) worldwide. It is therefore urgent to estimate the impact such measures are having. We introduce a Bayesian epidemiological model in which a proportion of individuals are willing and able to participate in distancing, with the timing of distancing measures informed by survey data on attitudes to distancing and COVID-19. We fit our model to reported COVID-19 cases in British Columbia (BC), Canada, and five other jurisdictions, using an observation model that accounts for both underestimation and the delay between symptom onset and reporting. We estimated the impact that physical distancing (social distancing) has had on the contact rate and examined the projected impact of relaxing distancing measures. We found that, as of April 11 2020, distancing had a strong impact in BC, consistent with declines in reported cases and in hospitalization and intensive care unit numbers; individuals practising physical distancing experienced approximately 0.22 (0.11-0.34 90% CI [credible interval]) of their normal contact rate. The threshold above which prevalence was expected to grow was 0.55. We define the "contact ratio" to be the ratio of the estimated contact rate to the threshold rate at which cases are expected to grow; we estimated this contact ratio to be 0.40 (0.19-0.60) in BC. We developed an R package 'covidseir' to make our model available, and used it to quantify the impact of distancing in five additional jurisdictions. As of May 7, 2020, we estimated that New Zealand was well below its threshold value (contact ratio of 0.22 [0.11-0.34]), New York (0.60 [0.43-0.74]), Washington (0.84 [0.79-0.90]) and Florida (0.86 [0.76-0.96]) were progressively closer to theirs yet still below, but California (1.15 [1.07-1.23]) was above its threshold overall, with cases still rising. Accordingly, we found that BC, New Zealand, and New York may have had more room to relax distancing measures than the other jurisdictions, though this would need to be done cautiously and with total case volumes in mind. Our projections indicate that intermittent distancing measures-if sufficiently strong and robustly followed-could control COVID-19 transmission. This approach provides a useful tool for jurisdictions to monitor and assess current levels of distancing relative to their threshold, which will continue to be essential through subsequent waves of this pandemic.

Fig 1. Information regarding COVID-19 in British Columbia, Canada.

(A) Key physical distancing measures implemented in response to COVID-19. Schools closed for an annual two-week break on March 14 and then were declared indefinitely closed on March 17. (B) Time from symptom onset to reporting from case-specific data as of April 11, 2020 and (C) reported cases per day. The dashed line in panel B represents the line above which cases, by definition, have not been reported yet. Boxes indicate interquartile range and median values. (D) Hospitalization and ICU (Intensive Care Unit) census counts. All data are from the BC Centre for Disease Control [10].

Fig 2. Schematic of the epidemiological model.

Compartments are: susceptible to the virus (S); exposed (E₁); exposed, pre-symptomatic, and infectious (E₂); symptomatic and infectious (I); quarantined (Q); and recovered or deceased (R). Recovered individuals are assumed to be immune. The model includes analogous variables for individuals practising physical distancing: S_d, E_1d, E_2d, I_d, Q_d, and R_d. Solid arrows represent flow of individuals between compartments at rates indicated by the mathematical terms (see Supplement for full definitions). Dashed lines show which compartments contribute to new infections. An individual in some compartment X can begin distancing and move to the corresponding compartment X_d at rate u_d. The reverse transition occurs at rate u_r. The model quickly settles on a fraction e = u_d/(u_d + u_r) participating in distancing, and dynamics depend on this fraction, rather than on the rates u_d and u_r.

Fig 3

(A) Observed and estimated case counts, (C) estimated prevalence, and posterior estimates for (B) R_0b and (D) fraction of normal contacts (f₂) among those distancing. These projections do not account for introduced cases from other jurisdictions and they assume that distancing measures remain in place. The fraction of normal contacts is the model’s portion of contacts that remain among those who are engaged in physical distancing. In panel A, the blue line represents the posterior mean and the shaded ribbons represent 50% and 90% credible intervals on new observations. Dots and black lines represent the reported data. Grey region indicates the projection. In panel C, lines represent example draws from the posterior. In panels B and D, priors are shown in grey and posteriors in blue. In panel D, the dashed vertical line denotes the threshold above which an exponential increase in prevalence is expected (see Figure J in S1 Text). Note: Model prevalence depends on assumptions about underestimation, incubation period, and the duration of infection, none of which we can estimate well from these data (Figure M in S1 Text). Much higher values of the prevalence are consistent with our data.

Fig 4. Scenarios of relaxing distancing measures.

Distancing measures are relaxed to (A) 60% (A) and (B) 80% levels of normal contacts and exponential growth is observed at moderate and rapid rates. (C, D) Two scenarios of cycling between physical distancing levels. Here, the percentage of normal contacts alternates between 80% (dark-grey shading) and 22% (light-grey shading) at (C) 3-week and (D) 4-week intervals. Reducing contacts to 22% of normal is approximately the level estimated by our model (Fig 3). Note the lag between changes in physical distancing and reported case counts. Figure description is otherwise the same as for Fig 3A and 3C.

Fig 5

Observed and estimated case counts for (A) New York, (B) Florida, (C) Washington, (D) California, and (E) New Zealand. Solid curves represent the posterior means and shaded ribbons represent 50% and 90% credible intervals of estimated counts. Dots and black lines represent the reported data. Inset histograms show the posterior distributions of the fraction of normal contacts (f₂), with the vertical lines denoting the threshold above which an exponential increase in prevalence is expected (as in Fig 3D). (F) Reduction in movement from Google mobility transit-station data [21] colour-coded for each jurisdiction. Thin lines are raw data; thick lines are smoothed values from a generalized additive model. See Table B and Supplemental Methods in S1 Text for details on the regional modelling parameters and initialization.