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Conceived and designed the experiments: JZ BL AH. Performed the experiments: JZ. Analyzed the data: JZ. Contributed reagents/materials/analysis tools: SB. Wrote the paper: JZ BL AH BG.

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.

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.

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.

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.

A recent spike in the number and severity of Norovirus (NoV) outbreaks worldwide

NoV illness is characterized by multiple phases of symptomatology: an initial period of explosive vomiting characteristic of illness onset, followed by 1–3 days of less severe symptoms

Representing variable infectiousness over the course of a single infection has been shown to be important for understanding the transmission dynamics of other pathogens, such as HIV

We fit three models (Models 1–3), each with a different representation of the infectious period, to household transmission data collected subsequent to a 2009 point-source outbreak of GII.12 norovirus in North Carolina

The figure illustrates the time course of infection in the 18 households in which there was a non-index case who became ill after the onset of symptoms in the index case. Filled boxes indicate an individual who dined at the point-source and became ill. Filled circles indicate individuals who became ill and did not dine at the point-source. Hollow boxes and circles along the right margin indicate the number of individuals in the household who did and did not dine at the point source and did not become ill, respectively. The additional 52 households in the analysis with no secondary cases are not pictured.

The figure illustrates the change in infectiousness over time, for Model 1 (SEIR; top), Model 2 (Exponential decay; middle) and Model 3 (Burst; bottom).

In December 2009, more than 200 individuals were sickened by a GII.12 norovirus outbreak caused by contaminated oysters served at a North Carolina restaurant. The particular GII.12 strain implicated in this outbreak is estimated to have caused 16% of reported NoV outbreaks in the United States in 2009–10

Because this work was determined by CDC human subjects review to be under the auspices of public health response, the protocol and consent procedure were not formally reviewed by an IRB, though standard practices of verbal consent and confidentiality were followed. The data were collected as part of a phone survey, so it was not possible to obtain written consent. Respondents were assured that all survey questions were voluntary and confidential. Verbal informed consent was requested and documented on the survey instrument at the time of the interview.

Because index cases were infected at a point-source event rather than during a large community outbreak that continued throughout the period assessed by the phone survey, we are able to isolate the likely source of exposure to other members of the household. In essence, our household-level transmission data provide multiple independent realizations of the stochastic transmission process, since we observed 70 exposed households, each with a distinct index case. This allows us to examine how time-varying intensity of infectiousness impacts stochastic variability in transmission

For comparison with our alternative models, we first fit an SEIR compartmental model

Model | Parameter | Definition | Value | Source |

_{PS} | Probability of infection at point source | – | EST | |

Relative infectiousness of asymptomatics | 0.05 | See text | ||

Proportion of cases asymptomatic | [0.0, 0.4] | Atmar et al. 2006 | ||

Mean duration of latency | 1 day | See text | ||

_{S} | Shape parameter of latent period distribution | 4 | See text | |

Daily symptomatic transmission rate | – | EST | ||

Mean duration of symptomatic infectiousness | – | EST | ||

_{S} | Shape par. of infectious period duration | – | EST | |

_{2} | Total infectiousness | – | EST | |

_{2} | Mean day of infectivity profile | – | EST | |

_{3} | Total infectiousness | – | EST | |

_{3} | Mean day of post-onset infectivity | – | EST | |

Proportion of infectiousness at onset | – | See text |

We then compare the ability of each model to reproduce characteristic features of the outbreak data. This allows us to assess whether a model with time-varying infectiousness may provide a more comprehensive explanation of qualitative features of NoV transmission than standard approaches. Using a quantitative Bayesian model comparison technique, we also compare the two models of time-varying infectiousness to understand if one of these representations may better explain the data.

Because asymptomatic norovirus infection is common, accounting for 15–40% of all norovirus infections

Because the outbreak data are reported in twelve-hour intervals, we use a discrete-time model where each model step represents a 12-hour period. We designate the time

Upon infection, individuals enter the latent state (E). The mean duration of latency is estimated from the outbreak data to be 1 day. We represent the latent period with a Gamma distribution with mean

After latency, individuals progress to the symptomatic (I) or asymptomatic (A) phase of infection. We assume that all individuals in the household are equally susceptible to infection, regardless of age, sex or household configuration. Infected individuals will have a symptomatic infection with probability

After latency, individuals enter either the symptomatic infectious period (

Model 2 introduces smooth variation in infectiousness over time. In this model, the infectiousness of a case decays exponentially as a function of time elapsed since the onset of symptoms. We use a discretized exponential distribution with mean_{onset}. An exponential distribution is a natural choice to represent infectivity over the symptomatic period, because its mode is at zero

We denote

In Model 3, the infectious period consists of two phases: 1) a 12-hour burst of infectivity starting with the onset of symptoms, followed by 2) a period of declining infectivity modeled by an exponential distribution with mean

Model 3 introduces a new parameter,

We fit Models 1–3 to the outbreak data using a Bayesian data augmentation (DA) approach

When fitting parameters for the mean day of the infectivity profile for Models 2 & 3 (

To understand the contribution of fully asymptomatic infections, i.e. those that enter the asymptomatic infectious phase,

All models presented here were implemented using

We compare the quality of model fit for Models 1, 2 & 3 based on qualitative fit to the data based on descriptive statistics. We perform Bayesian model selection using Bayes factors

To make qualitative comparisons, we fix the parameters for each model at the posterior mean values estimated from the outbreak data and generate 10^{4} sample outbreaks using the household sizes and point-source exposure patterns from the outbreak data. We then compare features of these sampled outbreaks to the household outbreak data via several descriptive statistics. These include: 1) the average number of secondary cases in households with at least one secondary case, 2) the average serial interval between household cases, 3) the average time from onset in the first household case to onset in the last household case in those households with secondary cases, 4) the probability of zero observed secondary cases within the household, i.e. that the index fails to generate any secondary cases, and 5) the probability of recrudescence of a household outbreak, defined here as the probability of observing a serial interval of ≥4 days between cases in a household.

We compare the relative strength of evidence supporting these models using Bayes factors. Bayes factors facilitate the comparison of models with differing structure. However, the large number of unobserved recovery times in Model 1, each represented by a hidden parameter, makes meaningful comparison of Model 1 with Models 2 & 3 with Bayes factors infeasible. This is because Bayes factors naturally penalize additional model structure, effectively guaranteeing that a model with fewer latent parameters (i.e., Models 2 &3) would be preferred over one with many more. So, we limit this aspect of the analysis to the two models with varying infectiousness over the symptomatic period. The Bayes factor, K_{32}, that we use to compare the models M_{2} & M_{3} is the ratio of the posterior densities of each

Because we repeat this analysis for varying rates of asymptomatic prevalence, we denote

In order to perform this portion of the analysis, we employed a reversible-jump MCMC sampling step that allowed the sampler to jump between Models 2 and 3 within a single MCMC run. This step is similar to one employed by O'Neill & Marks

When interpreting the results of these model comparisons, it is important to note that Model 3 represents a stylized scenario in which 50% of infectiousness is concentrated at onset. This means that comparisons between Model 2 and 3 are necessarily heuristic in terms of their ability to assess the explanatory power of a model that allows excess infectivity at onset versus one that does not. So, for example, if Model 3 is preferred over Model 2, this would indicate that a model with 50% of infectiousness concentrated at onset is preferred to one in which there is an exponential decay of infectiousness beginning at onset. But it should not be taken as evidence that a model with a spike in infectiousness at onset is preferred in all cases, regardless of the height of this spike.

Model | Para meter | Model 1 | Model 2 | Model 3 | |||

Est | 95% CI | Est | 95% CI | Est | 95% CI | ||

_{PS} | 0.55 | (0.38, 0.71) | 0.55 | (0.38, 0.71) | 0.54 | (0.38, 0.71) | |

– | – | 0.13 | (0.07, 0.21) | 0.14 | (0.08, 0.22) | ||

– | – | 2.78 | (1.55, 4.87) | 3.89 | (2.20, 4.95) | ||

0.05 | (0.02, 0.10) | – | – | – | – | ||

2.98 | (1.22, 4.79) | – | – | – | – | ||

_{S} | 1.04 | (0.07, 3.63) | – | – | – | – |

Data | Model 1 | Model 2 | Model 3 | |||||

Descriptive Statistic | Value | Range | Value | 5^{th}–95^{th} Quantile | Value | 5^{th}–95^{th} Quantile | Value | 5^{th}–95^{th} Quantile |

Avg#secondary cases (in hh w/1+ cases) | 1.3 | (1.0, 4.0) | 1.2 | (1.0, 3.0) | 1.3 | (1.0, 3.0) | 1.3 | (1.0, 3.0) |

Avg household outbreak | 3.1 | (0.5, 10.0) | 2.8 | (0.5, 10.0) | 3.4 | (0.5, 11.0) | 2.9 | (0.5, 10.0) |

Avg serial interval (days) | 2.3 | (0.5, 8.0) | 2.6 | (0.5, 8.0) | 2.7 | (0.6, 9.0) | 2.3 | (0.5, 9.5) |

Probability no | 0.62 | – | 0.70 | – | 0.63 | – | 0.64 | – |

Recrudescence | 0.17 | – | 0.22 | – | 0.29 | – | 0.21 | – |

Across all three models, estimates of the probability of infection at the point source are similar, with a 55% probability of infection for non-index cases exposed to the point source (

The figure shows infectiousness as a function of time since symptom onset for the estimated values of the exponential decay model (Model 2; solid line) and burst model (Model 3; dashed line).

^{4} of simulated outbreaks from Models 1–3. The first column of ^{th} to the 95^{th} quantile of the simulated distribution of the outcome. We find that all models reproduce mean values and variability in the outbreak data for 1) the number of secondary cases, 2) duration of serial intervals between cases, and 3) average household outbreak duration. The models differ in their ability to reproduce recrudescence and the proportion of households with no secondary cases. The probability of recrudescence for Model 1 (0.22) and Model 3 (0.21) are closest to the data (0.17), while Model 2 overestimates this value by a larger margin than the other two (0.29). Simulations from Models 2 & 3 generate values for the probability of observing no secondary cases (0.63 & 0.64, respectively) that are closer to the data (0.62) than Model 1 (0.70). Overall, Model 3 generates values closest to the outbreak data along those qualitative dimensions where there are differences between the candidate models.

In our quantitative comparison of models 2 and 3, we find that in a model with only symptomatic infections, support for Model 3 is strong (

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.

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

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

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.

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