Difference between revisions of "Epidemics:ferguson2008"

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(Transmission level)
(Prior level)
 
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<br/><br/>
 
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An example of so called SIR (Susceptible, Infectious, Recovered) model<ref name="cauchemez2004">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</ref>.
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== Cauchemez et al Model<ref name="cauchemez2004">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</ref> ==
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An example of so called SIR (Susceptible, Infectious, Recovered) model.
  
 
Joint probability of observed (''Y''), unobserved variables (<math>\nu,\,\psi</math>), and parameters (<math>\theta</math>) is given by:
 
Joint probability of observed (''Y''), unobserved variables (<math>\nu,\,\psi</math>), and parameters (<math>\theta</math>) is given by:
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<br/> where:
 
<br/> where:
 
<br/><br/>
 
<br/><br/>
<math>p\big(\theta\big)</math>, <math>\leftarrow</math> '''prior level''' (prior distribution of the parameters of the model).
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<math>p\big(\theta\big)</math>, <math>\leftarrow</math> '''prior level''' (prior distributions of the parameters of the model).
 
<br/>
 
<br/>
 
<math>p(\nu,\psi|\theta)=\prod_{ih}^{n_{ih}}p(\nu^{ih},\psi^{ih}|\theta)</math>, <math>\leftarrow</math> '''transmission level'''
 
<math>p(\nu,\psi|\theta)=\prod_{ih}^{n_{ih}}p(\nu^{ih},\psi^{ih}|\theta)</math>, <math>\leftarrow</math> '''transmission level'''
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<math>Z_{i}^{ih}</math> - the 1st day of clinical influenza of ''i''th individual in ''ih''th household
 
<math>Z_{i}^{ih}</math> - the 1st day of clinical influenza of ''i''th individual in ''ih''th household
 
<br/><br/>
 
<br/><br/>
<math>\nu_{i}^{ih},\psi_{i}^{ih} (\nu_{i}^{ih}<\psi_{i}^{ih})</math> - unobserved variables corresponding to the start end the end of the infectious period for ''i''th individual of ''ih''th household
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<math>\nu_{i}^{ih},\psi_{i}^{ih} (\nu_{i}^{ih}<\psi_{i}^{ih})</math> - unobserved variables corresponding to the start and the end of the infectious period for ''i''th individual of ''ih''th household
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=== Transmission level ===
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=== Observation level - parameters <math>\nu,\,\psi</math> ===
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<math>
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p(Y|\nu,\psi)=\left\{\begin{array}{ll}
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1, & \textrm{if\;for\;all\;infected\;}i\,(i\in I): \nu_i\in [Z_i-3,Z_i] \land \nu_i < \psi_i\\
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0, & \textrm{otherwise}
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\end{array} \right.
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</math>
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This level ensured that the unobserved data, <math>\nu,\,\psi</math>, agreed with observed data, ''Y''.
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=== 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'':
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<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.
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<math>I(t)=\{i\in I,\nu_i<t\le \psi_i\}</math> - the group of infectives just before time ''t''.
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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>.
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<br/>
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With the above, conditional on the date of the first infection <math>\nu_1</math>, we have (for the household):
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<br/>
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<math>
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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]
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</math>
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<br/>
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where ''I-{1}'' denotes infectives without the first infected
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=== Prior level ===
  
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>
 
  
=== Observation level ===
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That is prior distributions of all parameters, <math>\theta=(\mu, \sigma, \alpha, \beta, \varepsilon, \eta)</math>
  
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Gamma distribution with mean <math>\mu=3</math> and standard deviation <math>\sigma=2</math> was used for the distribution of the duration times of infectious periods. Exponential distribution for  <math>\alpha</math> and <math>\beta</math> were used
  
=== Prior level ===
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== References ==
  
That is prior distribution of parameters, <math>\theta=(\mu, \sigma, \alpha, \beta, \varepsilon, \eta)</math>
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<references/>

Latest revision as of 15:06, 25 November 2008

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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),

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

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