# Epidemics:ferguson2008

## 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:

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

where:

$p\big(\theta\big)$, $\leftarrow$ 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 $\nu,\,\psi$

$p(Y|\nu,\psi)=\left\{\begin{array}{ll} 1, & \textrm{if\;for\;all\;infected\;}i\,(i\in I): \nu_i\in [Z_i-3,Z_i] \land \nu_i < \psi_i\\ 0, & \textrm{otherwise} \end{array} \right.$

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 - 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

## References

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