These results suggest that models including time-varying infectiousness may better capture observed person-to-person norovirus transmission dynamics than approaches assuming uniform intensity of infectiousness over time. Allowing for changes in infectiousness that reflect characteristic patterns of norovirus illness can increase our ability to explain observed outbreak patterns and re-create qualitative features of these outbreaks. In particular, Models 2 & 3 were better able than Model 1 to reproduce the proportion of household outbreaks not resulting in secondary cases. Model 3 was also able to capture the probability of recrudescence in household outbreaks, potentially because the infectiousness remaining after the burst at onset is more evenly distributed over the infectious period than in Model 2. A particular strength of an approach allowing for symptom intensity to vary with time is that the roles of waning symptomatology and post-symptomatic shedding can be explored without adding model complexity, i.e. additional infectious classes. Because it eliminates latent state variables in the infectious period, this framework also facilitates straightforward model comparison.
Quantitative comparisons between Models 2 and 3 suggest that Model 3 provides a more comprehensive picture of the outbreak data than the other models presented here. This also holds across a range of plausible asymptomatic prevalence rates. These results are qualified, however, by the relatively small size of this outbreak and should be verified against datasets with a higher density of cases. It is important to note, however, that our data actually represent 70 independent replications of the household transmission process. In addition, the use of a dataset where an individual's outcomes are directly linked to her exposure is likely to decrease error in estimation relative to approaches in which only the aggregate force of infection and population-level incidence are considered, (see e.g. 
). These findings also echo results from other modeling studies which have found that asymptomatic infection is unlikely to play a major role in person-to-person transmission during an outbreak 
Although all three models are able to explain key features of the data, the qualitative fit of model 3 is the strongest of those considered here. As compared to models 1 & 2, it is able to capture both patterns of within-household transmission as well as the probability the index case will fail to transmit to any household members. Our findings are, however, limited by two factors.
First, the fact that our dataset consists primarily of self-reported illness onset times may introduce some error with respect to the actual time of infectiousness onset. Second, to provide a contrast to models 1 & 2, in model 3 the proportion of a case's infectivity at onset is fixed at 50%. This means that our model comparison results need to be interpreted as a contrast between one in which 50% of the infectiousness occurs at onset with smooth variation thereafter to one in in which infectiousness at onset is tied smoothly to variation afterwards, rather than a general comparison between a model with a spike at onset and one in which there is no such spike.
Consequently, although our findings suggest that it is important to account for increased infectiousness at onset, they should be verified and expanded using outbreak datasets with more cases and larger contact networks. Future analysis should also address variation in infectivity profiles by age, as this is likely to influence transmission. In addition, data including laboratory testing confirming symptomatic infection and identifying asymptomatic cases is necessary to verify the robustness of these results to asymptomatic transmission.
In model 3, we also assume that the 12-hour duration of the burst of infectiousness following onset is similar to the period of vomiting reported from a cohort study of norovirus infections in the community 
. Clinical challenge studies, e.g. 
, have shown longer periods of vomiting of up to several days. But this may result from dosage with quantities of norovirus much greater than likely to be encountered in the context of a real-world outbreak. Given the critical role of vomiting in norovirus transmission 
, further study is necessary to understand the distribution of the duration and intensity of vomiting in the context of real-world outbreaks and the role of contact-limiting behavior in transmission.
The methods discussed here may be extended to include the mechanisms driving the shape of infectiousness over time. For example, for HIV and other STIs, infectiousness over time may be modeled as a function of individual-level covariates, such as changes in risk behavior. Models including time-variation in the influence that individuals have on each other might also be usefully extended to studies of the diffusion of behavioral risks for chronic illness, e.g. obesity 
, where compartmental modeling approaches may result in an awkward discretization of changes in social behavior over time.
Our results underscore the idea that public health interventions need to focus on both the acute phase of infection as well the environmental contamination and post-symptomatic infectiousness that characterize norovirus outbreaks. Onset of norovirus gastroenteritis is often abrupt, with no prodrome, so public vomiting events are common. Preventing such events from occurring may not be possible, but our results demonstrate the importance of rapidly responding to such occurrences, as well as other opportunities for transmission in the initial phase of illness. Future work should test this framework and its implications for intervention in the context of community and institutional outbreaks, where issues of sanitation are most acute.