Linking Time-Varying Symptomatology and Intensity of Infectiousness to Patterns of Norovirus Transmission

Background Norovirus (NoV) transmission may be impacted by changes in symptom intensity. Sudden onset of vomiting, which may cause an initial period of hyper-infectiousness, often marks the beginning of symptoms. This is often followed by: a 1–3 day period of milder symptoms, environmental contamination following vomiting, and post-symptomatic shedding that may result in transmission at progressively lower rates. Existing models have not included time-varying infectiousness, though representing these features could add utility to models of NoV transmission. Methods We address this by comparing the fit of three models (Models 1–3) of NoV infection to household transmission data from a 2009 point-source outbreak of GII.12 norovirus in North Carolina. Model 1 is an SEIR compartmental model, modified to allow Gamma-distributed sojourn times in the latent and infectious classes, where symptomatic cases are uniformly infectious over time. Model 2 assumes infectiousness decays exponentially as a function of time since onset, while Model 3 is discontinuous, with a spike concentrating 50% of transmissibility at onset. We use Bayesian data augmentation techniques to estimate transmission parameters for each model, and compare their goodness of fit using qualitative and quantitative model comparison. We also assess the robustness of our findings to asymptomatic infections. Results We find that Model 3 (initial spike in shedding) best explains the household transmission data, using both quantitative and qualitative model comparisons. We also show that these results are robust to the presence of asymptomatic infections. Conclusions Explicitly representing explosive NoV infectiousness at onset should be considered when developing models and interventions to interrupt and prevent outbreaks of norovirus in the community. The methods presented here are generally applicable to the transmission of pathogens that exhibit large variation in transmissibility over an infection.


! 2!
First, for symptomatic cases: And then for asymptomatic cases: Finally, we account for the contribution of non-cases to the likelihood: The product of the contributions of all cases and non-cases in a household h is the augmented data likelihood, ! ! ! , ! ! !), for that household: Because we assume that the transmission process in each household following exposure at the point source is independent, we can then calculate the sampling probability for the entire dataset as ! !, !! !) = ! ! ! , ! ! !)

S2. Markov-Chain Monte Carlo (MCMC) Sampling Algorithm
In this section, we outline the MCMC algorithm used to sample the joint posterior distribution of event times and transmission parameters.

A. Adjusting Infection Times
Because our household transmission data are reported in terms of incidence, exact infection times for non-index cases are unobserved. To sample these missing data, we use a Gibbs sampling [2] step in which a case's infection time is sampled directly from the joint distribution of susceptible period durations and incubation period durations, conditional on the time of illness onset.
To do this, we first calculate the probability of 1) infection on each day prior to the onset of symptoms, ! ! (where ! ! ∈ ! ! , the set of all other symptom onset times in the household), and 2) the probability of an incubation period duration equal to ! ! − !: We then normalize this distribution of potential infection times by conditioning on the total probability that the infection time occurred on the interval [0, ! ! -1], i.e. that it occurred before the onset of symptoms: And the new infection time of the case is then sampled from ! ! . For more detail on this step, see Appendix A of [1].

B. Adjusting Recovery Times
To sample the unobserved recovery times in Model 1, we use a Metropolis- where ! ! is the rate parameter for the recovery period distribution on the current step, t, of the sampling algorithm and ! ! ! is its shape parameter. Because new times are sampled independently of each other, the proposal ratio, !, for this move reduces to:

C. Reversible-Jump MCMC Moves for Asymptomatic Infections
To explore the role of asymptomatic infections in these household outbreaks, we use a pair of reversible-jump MCMC moves to insert and remove asymptomatic infections from a household outbreak. Because these infections are completely unobserved, we need to sample both the infection and onset times for such infections. We assume that the incubation periods for asymptomatic infections follow the same distribution as symptomatic ones. We also use a Metropolis-Hastings (MH) move, similar to the one in S2, section B above, to adjust infection and onset times for asymptomatic infections. As with the symptomatic cases, the sampling probability of the latent period for asymptomatic cases is included in the augmented data likelihood. Finally, we sample the duration of the latent period from its density, ! ! !,! ! , as in S2.B, above. The proposal probability of this step is then Move 2: Remove Asymptomatic Infection. When removing an asymptomatic infection, we sample an individual at random from the population of individuals who have an asymptomatic infection, A, and remove the infection. Consequently, the proposal probability for this step is ! ! !! .
Using the derivation of the proposal probabilities for Steps 1 and 2, we can easily calculate the proposal ratio, !, for each step. For Move 1: For Move 2:

S3. Model Comparison with Bayes Factors.
To compute Bayes factors for the comparison of Models 2 & 3, we employed an MCMC sampling step to switch between models of time-varying infectiousness. For example, if we want to switch from Model 2 to Model 3, we set the parameters ! ! , ! ! !of Model 2 to zero with probability = 1. We then propose new values for the parameters of Model 3 ! ! , ! ! , from the proposal distributions ! ! ! and !(! ! ). So, the proposal ratio for a move from Model 2 to Model 3 is: We can then compute the acceptance probability as discussed above. The index of the accepted model is then recorded, and Bayes factors comparing Models 2 & 3 are calculated using the ratios of the marginal posterior densities for each model.  [3]. The value for the infectivity of the point-source event, ! !" appears to be very sensitive to the inclusion of asymptomatic infections. This is likely because these infections are not anchored to an observed onset time, allowing the model to place them at the time the individual dined at the point-source.

Simulated Data
To verify that the fitting procedures used for models both with and without asymptomatic infectious are accurate, we simulated outbreaks using fitted parameter values for Models 2 & 3 from Tables 2, 3 & 4 in the main text, and re-fit the model to these simulated data. We then repeated this procedure for each level of asymptomatic prevalence from 0-40%. When simulating outbreaks, we retain the size of each household in the analysis, as well as whether each individual dined at the point source. For a full simulated data analysis of Model 1, see [3].