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Cauchemez et al Model[1]

An example of so called SIR (Susceptible, Infectious, Recovered) model.

Joint probability of observed (Y), unobserved variables (\nu,\,\psi), and parameters (\theta) is given by:

p\big(Y,\nu,\psi,\theta) = p(Y|\nu,\psi)\,p(\nu,\psi|\theta)\,p(\theta),


p\big(\theta\big), \leftarrow prior level (prior distributions of the parameters of the model).
p(\nu,\psi|\theta)=\prod_{ih}^{n_{ih}}p(\nu^{ih},\psi^{ih}|\theta), \leftarrow transmission level

p(Y|\nu,\psi)=\prod_{i}^{n_{ih}}p(Y^{ih}|\nu^{ih},\psi^{ih}), \leftarrow observation level

Y - indicator function: Y_{ij}^{ih}=1 (for ith, (i=1,2,..,n_{ih}) individual of ihth household (of size n_{ih}) on jth day (j=0,1,..,14)), if clinical influenza was observed, Y_{ij}^{ih}=0 otherwise. Y^{ih} - all observations from household ihth, Y - observations from all households.

I^{ih} - group of individuals at ihth household with at least 1 day of clinical influenza, S^{ih} - remaining members of the ihth household.

Z_{i}^{ih} - the 1st day of clinical influenza of ith individual in ihth household

\nu_{i}^{ih},\psi_{i}^{ih} (\nu_{i}^{ih}<\psi_{i}^{ih}) - unobserved variables corresponding to the start and the end of the infectious period for ith individual of ihth household

Observation level - parameters \nu,\,\psi

1, & \textrm{if\;for\;all\;infected\;}i\,(i\in I): \nu_i\in [Z_i-3,Z_i] \land \nu_i < \psi_i\\
0, & \textrm{otherwise}
\end{array} \right.

This level ensured that the unobserved data, \nu,\,\psi, agreed with observed data, Y.

Transmission level - parameters \alpha,\,\varepsilon,\,\beta,\,(\eta),\,\mu,\,\sigma

The instantaneous risk of infection for an individual at time t in household of size n:

\lambda(t) = \alpha + \varepsilon \sum_{i\in I(t)} \beta_i / n,

where \alpha, \varepsilon - instantaneous risk of infection from the community and within household, respectively. I(t)=\{i\in I,\nu_i<t\le \psi_i\} - the group of infectives just before time t.

The duration of infectious period for ith infective d_i=\,\psi_i-\nu_i is taken from the gamma distribution with mean \mu_i and standard deviation \sigma_i.
With the above, conditional on the date of the first infection \nu_1, we have (for the household):

p(\nu,\psi|\theta)\,=\,\prod_{i\in I}\,d_{\mu_i,\sigma_i}(\psi_i-\nu_i)\prod_{i\in I-\{1\}}\,\lambda_i(\nu_i)\exp[-\int_{\nu_1}^{\nu_i}\lambda_i(t)dt]\,\prod_{s\in S}\exp[-\int_{\nu_1}^{15}\lambda_s(t)dt]
where I-{1} denotes infectives without the first infected

Prior level

That is prior distributions of all parameters, \theta=(\mu, \sigma, \alpha, \beta, \varepsilon, \eta)

Gamma distribution with mean \mu=3 and standard deviation \sigma=2 was used for the distribution of the duration times of infectious periods. Exponential distribution for \alpha and \beta were used


  1. CAUCHEMEZ S, Carrat F, Viboud C, Valleron A J, Boelle P Y, A Bayesian MCMC approach to study transmission of influenza: application to household longitudinal data, Stat. Med., 23, (2004), p3469