Difference between revisions of "Epidemics:ferguson2008"

Revision as of 13:02, 20 November 2008

Virtual society	Virus spread	Literature	Version
Polish virtual society epidemics model
Guinea pigs epidemics model
Virus genetic evolution epidemics 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 distribution of parameters).
$p(\nu,\psi|\theta)=\prod_{ih}^{n_{ih}}p(\nu^{ih},\psi^{ih}|\theta)$ , $\leftarrow$ 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

@@ Line 3: / Line 3: @@
 <br/><br/>
 <br/><br/>
 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:
 <br/><br/>
-<math>p(Y,\nu,\psi,\theta) = p(Y|\nu,\psi)p(\nu,\psi|\theta)p(\theta)</math>, where
+<math>p\big(Y,\nu,\psi,\theta) = p(Y|\nu,\psi)\,p(\nu,\psi|\theta)\,p(\theta)</math>,
+<br/>
+<br/> where:
 <br/><br/>
-<math>p(\theta)</math>, <math>\leftarrow</math> '''prior level''' (prior distribution of parameters).
+<math>p\big(\theta\big)</math>, <math>\leftarrow</math> '''prior level''' (prior distribution of parameters).
 <br/>
 <math>p(\nu,\psi|\theta)=\prod_{ih}^{n_{ih}}p(\nu^{ih},\psi^{ih}|\theta)</math>, <math>\leftarrow</math> '''transmission level'''
-<br/>
+<br/><br/>
 <math>p(Y|\nu,\psi)=\prod_{i}^{n_{ih}}p(Y^{ih}|\nu^{ih},\psi^{ih})</math>, <math>\leftarrow</math> '''observation level'''
 <br/>
-''Y'' - indicator function: <math>Y_{ij}^{ih}=1</math> if, for ''i''th, (<math>i=1,2,..,n_{ih}</math>) individual of ''ih''th household (of size <math>n_{ih}</math>) on ''j''th day (<math>j=0,1,..,14</math>), clinical influenza was observed,  <math>Y_{ij}^{ih}=0</math> otherwise.
+''Y'' - indicator function: <math>Y_{ij}^{ih}=1</math> (for ''i''th, (<math>i=1,2,..,n_{ih}</math>) individual of ''ih''th household (of size <math>n_{ih}</math>) on ''j''th day (<math>j=0,1,..,14</math>)), if clinical influenza was observed,  <math>Y_{ij}^{ih}=0</math> otherwise. <math>Y^{ih}</math> - all observations from household ''ih''th, ''Y'' - observations from all households.
-<br/>
+<br/><br/>
 <math>I^{ih}</math> - group of individuals at ''ih''th household with at least 1 day of clinical influenza,  <math>S^{ih}</math> - remaining members of the ''ih''th household.
-<br/>
+<br/><br/>
 <math>Z_{i}^{ih}</math> - the 1st day of clinical influenza of ''i''th individual in ''ih''th household
-<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

Difference between revisions of "Epidemics:ferguson2008"

Revision as of 13:02, 20 November 2008

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