Fases de la investigación

The interval from the date of onset of clinical signs to date of destruction was calculated for each infected household and plotted as a frequency histogram. A lognormal distribu- tion was fitted to these data, allowing us to estimate the distribution mean and standard deviation of this interval. For those infected households without a date of onset of signs recorded (39 of 164), an estimate was obtained by taking a random draw from a lognor- mal distribution with mean and standard deviation calculated earlier, and subtracting that value from the date of destruction which was present for all case households. Sensitivity of our results to the choice of mean and standard deviation for the lognormal distribu- tion was tested by varying each parameter within a range of biological plausible values and re-running our analyses based on these altered values. Insensitivity of our results to these changes provided reassurance of the robustness of the mean and standard deviation estimates used.

Locations of HPAI infected households with coordinates recorded were plotted using Google Earth (version 5.1.3533.1731) (Google Inc. 2010). Satellite images of outbreak locations were captured at the village level and saved as georeferencing images. These images were then loaded into a Geographic Information System, allowing us to create digital maps of household locations and village boundaries (Gibin et al. 2008). For HPAI infected households with missing coordinates (32 of 164), location details were estimated by selecting random points within the boundaries of the village of interest using thespat- statpackage in R (Baddeley & Turner 2005). The approach here varied depending on the geography of each village. For those villages where the distribution of households was

characteristic of ribbon development (i.e. households located along a major roadway), a point was chosen at random along the appropriate roadway. For villages showing a clus- tered settlement pattern (i.e. households positioned around a central location) a point was chosen at random within the defined village boundaries.

We used the space-time K-function to describe the spatio-temporal interaction of infection risk. A Bayesian approach was used to quantify the disease transmission rates of HPAI H5N1 within and between communes.

Spatio-temporal interaction

The space-time K-function K(s, t)was computed to determine the magnitude of spatio- temporal interaction of HPAI H5N1 outbreaks (Diggle et al. 1995). The increase in the number of infected households within specified spatial (s) and temporal (t) separations of an arbitrarily selected infected household,D(s, t), was calculated as the observed cumu- lative number of infected households minus the expected number of infected households if no space-time interaction was present. The proportional increase in risk attributable to space-time interaction, D0(s, t), was computed as the ratio ofD(s, t)toKS(s)KT(t)

whereKS(s)defined the K-function in space andKT(t)the K-function in time. A value

ofD0(s, t)equal to 1 indicates that the risk of HPAI H5N1 was 100% greater than that ex-

pected under the assumption that space-time interaction did not exist (Diggle et al. 1995). The significance of the observed values ofD(s, t)was estimated by performing 99 Monte Carlo permutations in which each of 164 outbreaks were re-labeled with the observed 164 event times. A total of 99 estimates ofD(s, t)was obtained and the observed sum of

D(s, t)over allsandtwas then compared with the empirical frequency distribution of the 99 estimates ofD(s, t). An extreme value of the observedD(s, t)compared with this dis- tribution provided evidence of significant space-time interaction. In this study, maximum distance and time separations of 20 kilometres and 15 days were used. These analyses were implemented in the SPLANCS package (Rowlingson & Diggle 2009) within R.

Full epidemic model

A Bayesian SIR approach was used to quantify disease transmission rates of HPAI H5N1 within and between communes. The approach described here is similar to that used in

Jewell, Kypraios, Christley & Roberts (2009), Jewell, Kypraios, Neal & Roberts (2009), and Neal & Roberts (2004) and uses a continuous time inhomogenous Poisson process to model both the time to infection and time from infection to observation of clinical signs. For disease transmission rate, the basic model assumes that the infectious pressure λ

acting on a susceptible household in communej at timetis given by:

λj(t) =α+βIj(t) +ρ

X

i6=j

wi∼jIi(t) (7.1)

This model represents an epidemic process whereby individuals are homogeneously mix- ing within communes, but heterogeneously mixing (according to a Markov random field) between communes. Ij(t) returns the number of infected households at timet in com-

munej andIi(t)the number of infected households at timetin communei. The variable

wi∼j is an indicator taking the value 1 if commune iborders commune j and zero oth-

erwise. Parameter β represents the household-to-household infection rate within a com- mune and ρ represents the household-to-household infection rate between communes. Finally,αrepresents a ‘background’ infection rate common to all individuals in the pop- ulation.

Results (not shown) from the model presented in Equation 7.1 indicated a large amount of overdispersion in the dataset at the commune level. To account for this increased variability in the spatial distribution of cases relative to the underlying susceptible popu- lation, we included a random effect termηj for thejthcommune representing a measure

of commune-level infectivity relative to the population average as estimated byβ andρ. This gives the following form for the instantaneous infectious pressure:

λj(t) = α+βIj(t)ηj +ρ X i6=j wi∼jIi(t)ηi (7.2) Where η∼logN ormal(0,1/ω) (7.3)

The parameterωallows for commune-level heterogeneity in disease transmission, noting thatexp(0)= 1, giving the required unit mean in Equation 7.2.

thatdk=Ok−Ikfor individual householdk, was Gamma distributed with a fixed shape

parameter equal to 6 days, and unknown rateγsuch that

D∼Gamma(6, γ) (7.4)

The value for the shape parameter was given such that the mean infectious period (given the prior mean for the rate parameter — see below) was equal to 6 days, in accordance with Stegeman & Bouma (2004). Prior distributions for α, β, ρ, γ, and ω were chosen as a Gamma prior because this is a convenient flexible distribution that has support (i.e. ‘allowed values’) greater than or equal to 0.

α∼Gamma(0.1,0.1) β ∼Gamma(0.1,0.1) ρ∼Gamma(1,1)

γ ∼Gamma(1,1) (7.5)

A diagram of the structure of this model, in context of the SIR (susceptible, infected, removed) framework, is shown in Figure 7.2.

Using a custom Markov chain Monte Carlo (MCMC) algorithm written in C++ (see Jew- ell, Kypraios, Neal & Roberts (2009)) for details) using partial non-centering for both

γ and ω to improve mixing (Neal & Roberts 2004), the joint posterior distribution for

α, β, ρ, γ,1/ω andD was estimated. A total of 1,000,000 samples were obtained from the posterior distribution, with the first 5000 discarded as burn-in. Convergence was as- sessed visually, and autocorrelation was found to be low after thinning every 200 iterations (Kypraios 2007). An analysis of prior sensitivity was performed, varying both the prior means and variances.

To investigate commune-level epidemic risk, and quantify model fit, the Bayesian pre- dictive distribution of the epidemic was used to study likely outbreak scenarios given the observed data (Bernardo & Smith 1994). We use established continuous time stochastic simulation methodology using the epidemic model as specified above, where each sim- ulation was run using a sample from the joint posterior density (Gillespie 1976). The resulting collection of simulation results thus represents a distribution over the possible

epidemic outcomes, given our observation of the original epidemic.

7.3

Results

Dates of onset of clinical signs of disease were recorded for 76% (125 of 164) of house- hold outbreaks. Dates of destruction were recorded for 98% (161 of 164) of outbreaks. The duration from first detection of clinical signs to destruction was available for 122 outbreaks. Median duration was 2 days (IQR 1 – 4 days). Coordinates were provided for 81% (132 of 164) of affected households.