Epidemics:ferguson2008

Virtual society	Virus spread	Literature	Version
Polish virtual society epidemics model
Guinea pigs epidemics model
Virus genetic evolution epidemics model

Cauchemez et al Model

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. $I(t)=\{i\in I,\nu_i<t\le \psi_i\}$ - 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