Transmission of SARS-CoV-2 before and after symptom onset: impact of nonpharmaceutical interventions in China

Nonpharmaceutical interventions, such as contact tracing and quarantine, have been the primary means of controlling the spread of SARS-CoV-2; however, it remains uncertain which interventions are most effective at reducing transmission at the population level. Using serial interval data from before and after the rollout of nonpharmaceutical interventions in China, we estimate that the relative frequency of presymptomatic transmission increased from 34% before the rollout to 71% afterward. The shift toward earlier transmission indicates a disproportionate reduction in transmission post-symptom onset. We estimate that, following the rollout of nonpharmaceutical interventions, transmission post-symptom onset was reduced by 82% whereas presymptomatic transmission decreased by only 16%. The observation that only one-third of transmission was presymptomatic at baseline, combined with the finding that NPIs reduced presymptomatic transmission by less than 20%, suggests that the overall impact of NPIs was driven in large part by reductions in transmission following symptom onset. This implies that interventions which limit opportunities for transmission in the later stages of infection, such as contact tracing and isolation, are particularly important for control of SARS-CoV-2. Interventions which specifically reduce opportunities for presymptomatic transmission, such as quarantine of asymptomatic contacts, are likely to have smaller, but non-negligible, effects on overall transmission. Supplementary Information The online version contains supplementary material available at 10.1007/s10654-021-00746-4.

Case incidence based on case pairs (red) and non-Hubei cases (blue), with both curves scaled to their respective maximums and the non-Hubei incidence curve shifted 7 days to the left.

Fig. S15
Daily estimates of based on case pair incidence data (red) and non-Hubei incidence data (blue). Dashed vertical line marks January 23, the beginning of the Wuhan lockdown and the start of NPI rollout across the country. Gray shading covers region in which is likely to be underestimated due to right-truncation.

Applying deviance information criteria (DIC) to models of presymptomatic transmission of SARS-CoV-2
A comprehensive guide to deviance information criteria (DIC) for missing data models exists in the paper by Celeux et al; here, we introduce only those definitions and concepts applicable to our models. The first part is devoted to DIC in general, while the second part shows how DIC is calculated for our models in particular.

Part I: Theory
Introduction Let ( | ) be the likelihood of observing some data , given a model with parameter(s) .
The deviance of model with parameters is defined as DIC can be readily applied to model fitting by MCMC because the posterior distribution of is approximated by the post-convergence Markov chain. As long as the likelihood function ( | ) is available in closed form, the posterior mean deviance, ( ), can be estimated by averaging ( ) over all of the steps in the chain, and ( ) simply requires an estimator of , such as the posterior mean.

Extension of DIC to missing data models
With data-augmented models, the likelihood function ( | ) is often not available in closed form, because it depends on missing data as well as the observed data . In this case, ( | ) is called as the observed likelihood, while ( , | ) is termed the complete likelihood.
The DIC for a missing data model can therefore be re-written in terms of the complete likelihood, as follows: Since the data are, by definition, missing, this quantity can not be computed directly; however, if the distribution of is known or can be approximated (e.g. using a data augmentation MCMC algorithm), it is sufficient to take the expectation of DIC with respect to : The first term in this expression can be estimated using the posterior distributions of and from a data augmentation MCMC algorithm, but the second term requires calculation of the posterior mean [ | , ] for each value of , which is inconvenient. However, this term can be reformulated to make estimation more straightforward. Recall that the formula for DIC is as follows: And recall that, if we set ℎ( ) = 1, the formula for deviance simplifies to The second term in this expression can be estimated by running a second MCMC with the parameters fixed at ( ) and taking the expectation of the log-likelihood with respect to the posterior distribution of the missing data .

Part II: Application
We now show how DIC is calculated for the models of pre-symptomatic transmission described in the Materials & Methods. Recall that the generation interval is assumed to follow a gamma distribution with shape parameter and rate parameter (incubation-independent model) or / (incubation-dependent model). The incubation periods of the infector and infectee are denoted and , respectively, while refers to the serial interval.
We used an MCMC algorithm with data augmentation to estimate the parameters and , as well as the missing data and , by fitting to the serial interval data . Thus, in the notation of Part 1, the serial interval data are the observed data , the incubation periods and comprise the missing data , and the parameters and are the model parameters .
At the end of Part 1, we arrived at the following expression for DIC for a missing data model: where is the number of iterations in the thinned converged Markov chain, with a superscript ( ) denoting the th iteration; is the number of infector-infectee pairs represented in the serial interval data set, with a subscript ( ) denoting the th pair; is the generation interval distribution (with parameters and ); and is the prior for the incubation period.
The second component of the expression can similarly be rewritten: where and are the parameter estimates (in this case, posterior means) from the original MCMC, and apostrophes indicate quantities from a secondary MCMC in which and are fixed at and , respectively.