Difference between revisions of "Epidemics:guineapigs:model"
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| − | + | |+ align="top" | '''The two piecewise linear approximations of the infection course. For <math>20</math> the best fit is <math>\beta(t) = 2.23 t - 0.61</math> for the growing part and <math>-1.63 t + 13.62</math> for the falling; for <math>5</math> it is <math>2.03 t - 0.01</math> and <math>-1.27 t + 12.13</math> respectively. We assume <math>\beta = 0</math> if the amount of virus is undetectable or zero''' | |
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[[Image:flu_spread_model_zuk_rakowski_radomski_Fig1a.png |frame|left| '''Fig.1a''']] | [[Image:flu_spread_model_zuk_rakowski_radomski_Fig1a.png |frame|left| '''Fig.1a''']] | ||
Revision as of 13:19, 14 November 2008
Model's assumptions
We have formulated a discrete stochastic model based on the probability of infection. To estimate this probability, we apply a simple generalized linear model (GLM), with 
as the link function. This is a standard approach that has been successfully adopted in various related contexts for modeling influenza[1][2]. The model starts from day zero, when all non pre-infected animals are healthy. For each consecutive day, we calculate the probabilities that each individual, among those not yet infected, will be infected on that day, and then we randomly check whether any of the possible infections actually happens.
The infection history is subsequently updated, and the algorithm moves to the next day.
We assume the four following conditions:
1. The probability that the ith individual will be infected on a particular day (provided that it had not been infected previously) is
where 
is a measure of virus concentration in the respiratory tract of the jth individual (decimal logarithm of the number of PFU in one ml of its nasal wash). The summation goes over individuals placed on the same and on adjacent shelves. Spatial coefficients 
are related to the probabilities that an aerosol drop, shed by the jth animal, will get to the ith guinea pig's cage. The coefficient 
reflects the dependence of the virus transmission rate on the air temperature (T) and relative humidity (H).
2. The course of influenza (virus concentration in nasal wash on the days subsequent to infection) depends only on the ambient temperature. Therefore, for each individual that became infected via air during the course of each particular experiment, the concentrations were assigned values reflecting the average case for that temperature. Table 1 shows the mean concentrations measured on particular days after infection, separately for each temperature. Functions comprising two linear pieces appear to be a reasonable guess, as shown in Figure 1.
| Day | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|
Concentration, 
|
1.52 | 4.52 | 5.63 | 6.89 | 6.06 | 3.26 | 1.88 | - | - |
Concentration, 
|
1.91 | 4.40 | 5.72 | 7.02 | 6.01 | 4.54 | 3.52 | 1.77 | - |
| Std. deviation square | 0.40 | 0.59 | 1.26 | 0.32 | 2.38 | 1.44 | 3.18 | 2.34 | - |
 Fig.1b |
the best fit is 
for the growing part and 
for the falling; for 
it is 
and 
respectively. We assume 
if the amount of virus is undetectable or zero
This may be interpreted as a two-phase infection course: first, the virus concentration grows exponentially (the scale is logarithmic), and subsequently (mostly as a result of the immune system activation) it exponentially falls. The transition between phase 1 and 2 is always sharp.
References
- ↑ FERGUSON N M, Cummings D, Cauchemez S, Fraser C, Riley S, Meeyai A, Iamsirithaworn S, and Burke D, Strategies for containing an emerging influenza pandemic in Southeast Asia, Nature, 437 (2005), p209
- ↑ STEGEMAN A, Bouma A, Elbers A R W, de Jong M C M, Nodelijk G, de Klerk F, Koch G, van Boven M,Avian Influenza A Virus (H7N7) Epidemic in The Netherlands in 2003: Course of the Epidemic and Effectiveness of Control Measures, J. Inf. Dis. 190, 2004
