Epidemics:ferguson2008

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An example of so called SIR (Susceptible, Infectious, Recovered) model[1].

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),

where:

p\big(\theta\big), \leftarrow prior level (prior distribution 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 end the end of the infectious period for ith individual of ihth household

Transmission level

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.

The duration of infectious period for ith infective d_i=\psi_i-\nu_i is taken from gamma distribution with mean \mu_i and standard deviation \sigma_i

Observation level

Prior level

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