Difference between revisions of "Epidemics:ferguson2008"
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(→Transmission level - parameters \alpha,\,\varepsilon,\,\beta,\,(\eta)) |
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This level ensured that the unobserved data, <math>\nu,\,\psi</math>, agreed with observed data, ''Y''. | This level ensured that the unobserved data, <math>\nu,\,\psi</math>, agreed with observed data, ''Y''. | ||
| − | === Transmission level - parameters <math>\alpha,\,\varepsilon,\,\beta,\,(\eta)</math> === | + | === Transmission level - parameters <math>\alpha,\,\varepsilon,\,\beta,\,(\eta),\,\mu,\,\sigma</math> === |
The instantaneous risk of infection for an individual at time ''t'' in household of size ''n'': | The instantaneous risk of infection for an individual at time ''t'' in household of size ''n'': | ||
| Line 47: | Line 47: | ||
<br/> | <br/> | ||
<br/>where <math>\alpha, \varepsilon</math> - instantaneous risk of infection from the community and within household, respectively. | <br/>where <math>\alpha, \varepsilon</math> - instantaneous risk of infection from the community and within household, respectively. | ||
| + | <math>I(t)=\{i\in I,\nu_i<t\le \psi_i\}</math> - the group of infectives just before time ''t''. | ||
| − | The duration of infectious period for ''i''th infective <math>d_i=\psi_i-\nu_i</math> is taken from gamma distribution with mean <math>\mu_i</math> and standard deviation <math>\sigma_i</math> | + | The duration of infectious period for ''i''th infective <math>d_i=\,\psi_i-\nu_i</math> is taken from the gamma distribution with mean <math>\mu_i</math> and standard deviation <math>\sigma_i</math>. |
| + | <br/> | ||
| + | With the above, conditional on the date of the first infection <math>\nu_1</math>, we have (for the household): | ||
| + | <br/> | ||
| + | <math> | ||
| + | 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] | ||
| + | </math> | ||
| + | <br/> | ||
| + | where ''I-{1}'' denotes infectives without the first infected | ||
=== Prior level === | === Prior level === | ||
Revision as of 12:56, 21 November 2008
| Virtual society | Virus spread | Literature | Version 
|
|---|---|---|---|
| Polish virtual society epidemics model | |||
| Guinea pigs epidemics model | |||
| Virus genetic evolution epidemics model | |||
Contents
[hide]Cauchemez et al Model[1]
An example of so called SIR (Susceptible, Infectious, Recovered) model.
Joint probability of observed (Y), unobserved variables (
), and parameters (
) is given by:
,
where:
, 
prior level (prior distributions of the parameters of the model).
, transmission level
, observation level
Y - indicator function: 
(for ith, (
) individual of ihth household (of size 
) on jth day (
)), if clinical influenza was observed, 
otherwise. 
- all observations from household ihth, Y - observations from all households.
- group of individuals at ihth household with at least 1 day of clinical influenza, 
- remaining members of the ihth household.
- the 1st day of clinical influenza of ith individual in ihth household
- unobserved variables corresponding to the start and the end of the infectious period for ith individual of ihth household
Observation level - parameters
![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.](/images/math/8/4/1/84177b6b9d5f65471dd2cf3d05c5bb89.png)
This level ensured that the unobserved data, , agreed with observed data, Y.
Transmission level - parameters 
The instantaneous risk of infection for an individual at time t in household of size n:
,
where 
- instantaneous risk of infection from the community and within household, respectively.
- the group of infectives just before time t.
The duration of infectious period for ith infective 
is taken from the gamma distribution with mean 
and standard deviation 
.
With the above, conditional on the date of the first infection 
, 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]](/images/math/0/7/8/078bfa5057da059d963a2770eb8eb196.png)
where I-{1} denotes infectives without the first infected
Prior level
That is prior distributions of all parameters, 
References
- Jump up ↑ 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
