A Bayesian Approach to Quantifying the Effects of Mass Poultry Vaccination upon the Spatial and Temporal Dynamics of H5N1 in Northern Vietnam

Outbreaks of H5N1 in poultry in Vietnam continue to threaten the livelihoods of those reliant on poultry production whilst simultaneously posing a severe public health risk given the high mortality associated with human infection. Authorities have invested significant resources in order to control these outbreaks. Of particular interest is the decision, following a second wave of outbreaks, to move from a “stamping out” approach to the implementation of a nationwide mass vaccination campaign. Outbreaks which occurred around this shift in policy provide a unique opportunity to evaluate the relative effectiveness of these approaches and to help other countries make informed judgements when developing control strategies. Here we use Bayesian Markov Chain Monte Carlo (MCMC) data augmentation techniques to derive the first quantitative estimates of the impact of the vaccination campaign on the spread of outbreaks of H5N1 in northern Vietnam. We find a substantial decrease in the transmissibility of infection between communes following vaccination. This was coupled with a significant increase in the time from infection to detection of the outbreak. Using a cladistic approach we estimated that, according to the posterior mean effect of pruning the reconstructed epidemic tree, two thirds of the outbreaks in 2007 could be attributed to this decrease in the rate of reporting. The net impact of these two effects was a less intense but longer-lasting wave and, whilst not sufficient to prevent the sustained spread of outbreaks, an overall reduction in the likelihood of the transmission of infection between communes. These findings highlight the need for more effectively targeted surveillance in order to help ensure that the effective coverage achieved by mass vaccination is converted into a reduction in the likelihood of outbreaks occurring which is sufficient to control the spread of H5N1 in Vietnam.


Details of MCMC algorithm
The posterior distribution was sampled by proceeding according to the following MCMC algorithm: (1). Starting values of the kernel and time-to-report are assumed to follow a gamma distribution.
The duration between infection and report for each outbreak are repeatedly drawn from a uniform proposal distribution of between 1 and 30 days until a possible augmented starting set of infection times (i.e. every outbreak except for the first to occur has a potential source of infection) is obtained. (2). Each kernel and time to report parameter are updated using a standard, random-walk singlecomponent Metropolis-Hastings algorithm where proposed updates are drawn on a log-scale from a normal distribution centred on the existing parameter value with a variance calibrated, by assessing the ratio of proposed moves which are accepted, so as to achieve good MCMC mixing (ideally as close to the "golden" acceptance ratio of 23% [1] but a ratio of 10-50% was ajudged to represent adequate mixing).
(3). The duration between infection and report, and thus the infection time itself, of individual outbreaks are updated using a Metropolis-Hastings independence sampler [2] with the same proposal distribution as that used in step (1) to obtain a starting set. Using now to denote the set of current infection times, the current infection time of a randomly selected commune is replaced with the proposed infection time with probability where denotes the set of infection times with the proposed replacing the current infection time. As the likelihood is conditional on the existence a unique initial infection, any move resulting in more than one initial outbreak was allocated zero likelihood and rejected. In order to achieve faster convergence it is desirable to update more than one infection time during this step (updating 10% of infection times appears to be a reasonably good rule of thumb). Evaluating the entire likelihood for each of these proposed time results in the algorithm becoming computationally intensive. In view of this the sampler was designed to calculate the acceptance probability by comparing the existing and proposed likelihood in the interval , the only section of the likelihood affected by the proposed move.

Estimating the change in infectivity between waves
Estimating the proportional change in infectivity between outbreaks is an easily interpretable measure of the incremental effect of control policies. This is particularly the case when estimating the overall protective effect of a vaccination campaign. One method for estimating this statistic from the fitting procedure is to fit the model to two waves simultaneously, allowing only the kernel parameter and the infection time parameters to vary between the waves.

Comparing Risk Maps
Where risk maps are being used to estimate the effects of control policies it is necessary to standardise the risk maps by using the same denominator poultry population for each set of posterior parameter distribution.

Reconstructing the epidemic process.
When the times at which a commune is infected and remains infectious are observed the reproductive number and the expected distance over which infection was transmitted can by estimated by reconstructing the epidemic tree [3]. The probability that commune is infected by is then However, when the times of infection are not available a sufficiently large number of samples of have to be drawn at regular intervals during the MCMC. The probability that is infects is then estimated by: where is the posterior mean of estimated by calculating the mean of the obtained samples.
From this the expected distance of an infected commune from the commune which infected it can then be defined as Furthermore, the effective reproductive number of an infected commune can be calculated by summing the infection probabilities between the commune and all potential offspring communes: This provides an indication of how the reproductive number changes as the outbreak wave progresses due to changes in the level of susceptibility within the communes around outbreaks and the application of control measures. If the reproductive number remains below unity for a sufficiently long period of time the epidemic has a high probability of dying out.

Evaluating the effects of more rapid detection and culling
Once the epidemic process has been reconstructed, the effect of detecting and removing an outbreak times. This is then repeated 100000 times at regular intervals during the MCMC, randomly sampling a different each time.

Sensitivity analyses
Here we assess the robustness of our main qualitative findings to the key assumption made during the model formulation:

i. 24 hour duration between report and removal
Thus far our analysis is based upon the assumption that outbreaks in all three of the waves are removed according to the guidelines distributed to veterinary paraprofessionals in the field [4] and that a combination of movement restrictions, quarantine measures and the immediate culling of flocks within which infection has been detected combine to ensure the outbreak is removed from the wave within 24 hours of the outbreak being reported. In reality, it is likely that it takes a longer time to achieve effective control in some, or even all cases. We assess the sensitivity of our results to this uncertainty by repeating the analysis with different report to cull durations. As expected, we found that as the assumed report to cull duration increased our estimates of the infection to report distribution decreased. However, as Fig. S4 illustrates, if the duration between report and removal remains constant across all three waves, our estimates still suggest that outbreaks were not detected as rapidly during the 2007 wave.
Whilst the estimate of the reduction in per-capita infectivity following vaccination (obtained by following the methodology detailed in section 5) is incrementally muted as the duration between reporting and removal is increased across the three waves (arising from the disproportionate increase in the estimated infectious period of the two earlier waves relative to that in 2007 (Fig. S4)), these estimates are still consistent with a reduction in infectivity and remain statistically significant until the report-to-removal delay reaches approximately ten days (Table S2).
Another plausible scenario we tested was that, as a possible consequence of it being a more complicated intervention to implement or the time needed for vaccine-induced immunity to be acquired [5], outbreaks in 2007 took a longer time to be effectively controlled by ring vaccination in comparison to the mass culling campaigns during the 04/05 and 2005 waves. However, we found that even if outbreaks took over a week longer to control in 2007, the estimated mean detection time remained greater than during the previous two waves (Fig. S4), whereas the reduction in infectivity became more pronounced (Table S2).  ii.

Constant infectivity profile
As an alternative to the assumption that the communes remain infectious at a constant intensity throughout the duration of the infectious period, we explored a second scenario where we assumed infectiousness increased monotonically until the outbreak was detected, increasing rapidly during the early stages of infection but saturating quickly. For this we used the cumulative distribution function of an exponential distribution, with parameter calibrated so as to level off by the 10th day of the outbreak: .
Following the reporting of an outbreak the infectivity of the commune was then assumed to decay exponentially for the following week, with a 24 hour "half-life".
Fitting this infectivity curve to the outbreak data we once again found that the model estimated that, for the average duration of infection of the earlier waves, infectivity was lower following vaccination but that this was offset by a longer infectious period (Fig. S5). iii.

Impact of unobserved outbreaks
Applying the fitting procedure to a wave where some outbreaks remain unobserved and unreported throughout the duration of infection would result in connecting outbreaks which are further apart in both time and space. As a result, especially at a high level of unreported outbreaks, the fitting procedure is likely to overestimate the commune-level infectious period. As this factor is of particular concern following a vaccination campaign where "silent spread" may be an issue [6,7], we assessed whether it could explain the observed differences in the estimates of the infectious period and infectivity following vaccine.
We simulated an outbreak wave and then randomly selected unobserved outbreaks. Having selected a set percentage of unobserved outbreaks, we fitted the model to the remaining infections. We found that, as expected, the estimate of infectious period increases as the proportion of detected outbreaks decreases. However, as the wave is increasingly "thinned out" in this way, the fitting procedure increasingly estimates transmission over longer distances. This produces a kernel which underestimates short range and overestimates long range transmissibility (Fig. S5) Day of outbreak is the explanation for the reduction in the estimate of inter-commune transmissibility between waves whilst the notion of an increase in the level unobserved outbreaks does of course support the conclusion that there has been a decrease in overall detection capacity. It should be noted, however, that this assumes that the ability to detect outbreaks is uniform throughout Vietnam and throughout the wave of outbreaks and also that the infectious period of an outbreak would not affect, or be affected by, whether or not the outbreak was reported. In reality it is highly probable that this will not be the case. For example, short-lived outbreaks may be less likely to be detected and unreported outbreaks may remain infectious for a longer length of time than would have been the case had it been detected.