Epidemics:guineapigs:model - Revision history

Magd at 14:20, 11 February 2009

2009-02-11T14:20:14Z

Magd at 12:26, 24 November 2008

2008-11-24T12:26:14Z

Tzuk at 12:23, 21 November 2008

2008-11-21T12:23:51Z

Tzuk at 12:10, 21 November 2008

2008-11-21T12:10:59Z

Tzuk at 11:34, 21 November 2008

2008-11-21T11:34:34Z

Tzuk at 11:24, 21 November 2008

2008-11-21T11:24:48Z

Tzuk at 11:22, 21 November 2008

2008-11-21T11:22:58Z

Tzuk at 20:00, 19 November 2008

2008-11-19T20:00:56Z

Tzuk at 18:53, 19 November 2008

2008-11-19T18:53:43Z

Rakowski at 13:21, 14 November 2008

2008-11-14T13:21:18Z

@@ Line 4: / Line 4: @@
 <br/><br/>
 ===  Model's assumptions ===
@@ Line 163: / Line 161: @@
 <references/>

@@ Line 4: / Line 4: @@
 <br/><br/>
 ===  Model's assumptions ===
@@ Line 24: / Line 26: @@
 {| class="rtable" align="center" width=90%
 |+ align="top" | '''Tab. 1: The average virus concentration in nasal wash of individual guinea pigs infected via air. For each individual, day 1 is the day when a measurable virus quantity first appeared in its respiratory tract'''
 |-
 ! Day
@@ Line 59: / Line 60: @@
 {| class="rtable" align="center" width=80%
 |+ align="top" | '''Tab. 2: The assumed virus concentration values for the guinea pigs infected via air (A) and pre-infected (P). For the latter we took <math>\beta(t) = -1.28 t + 10.52</math> at 20<math>^\mathrm{o}</math>C and <math>-1.23 t + 12.22</math> at 5<math>^\mathrm{o}</math>C for the descending phase'''
 |-
 ! Day
@@ Line 131: / Line 131: @@
 {| class="rtable" align="center" width=50%
 |+ align="top" | '''Tab. 3: Abstract distances between the experimental data points. The values in the right-upper corner correspond to temperature 20<math>^\mathrm{o}</math>C, and in the left-lower corner, to 5<math>^\mathrm{o}</math>C. We have taken the average ''V''-coordinate values wherever there were two experiments performed under the same conditions'''
 |-
+!
@@ Line 164: / Line 163: @@
 <references/>

@@ Line 45: / Line 45: @@
 | [[Image:flu_spread_model_zuk_rakowski_radomski_Fig1a.png |frame|left| '''Fig.1a''']] || [[Image:flu_spread_model_zuk_rakowski_radomski_Fig1b.png |frame| '''Fig.1b''' ]]
 |-
-| colspan="2" | '''The two piecewise linear approximations of the infection course. For 20<math>^\mathrm{o}</math>C 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 5<math>^\mathrm{o}</math>C 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'''
+| colspan="2" | '''Fig. 1: The two piecewise linear approximations of the infection course. For 20<math>^\mathrm{o}</math>C 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 5<math>^\mathrm{o}</math>C 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'''
 |}

@@ Line 20: / Line 20: @@
 '''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.
 {| class="rtable" align="center" width=90%
@@ Line 54: / Line 55: @@
 '''4.''' The infection course in pre-infected individuals also consists of two linear (on a logarithmic scale) phases. Concentration values for the descending phase come from the measurement<ref name="lowen2007">Lowen A C, Mubareka S, Steel J, Palese P, ''Influenza Virus Transmission Is Dependent on Relative Humidity and Temperature'', PLoS Pathogens 3(10), 2007, p151</ref> (We fitted a linear function for these values. However, they were originally of a nearly linear nature.). Due to the lack of the respective experimental data, for the ascending phase we assumed the slope to be the same as for animals infected via air. The virus concentration values for those individuals are also shown in Table 2.
 {| class="rtable" align="center" width=80%
@@ Line 125: / Line 127: @@
 Distances between points representing the experiments of Lowen et al. (2007) are collected in Table 3.
 {| class="rtable" align="center" width=50%

@@ Line 44: / Line 44: @@
 | [[Image:flu_spread_model_zuk_rakowski_radomski_Fig1a.png |frame|left| '''Fig.1a''']] || [[Image:flu_spread_model_zuk_rakowski_radomski_Fig1b.png |frame| '''Fig.1b''' ]]
 |-
-| colspan="2" | '''The two piecewise linear approximations of the infection course. For <math>20^\mathrm{o}</math>C 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^\mathrm{o}</math>C 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'''
+| colspan="2" | '''The two piecewise linear approximations of the infection course. For 20<math>^\mathrm{o}</math>C 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 5<math>^\mathrm{o}</math>C 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'''
 |}
@@ Line 56: / Line 56: @@
 {| class="rtable" align="center" width=80%
-|+ align="top" | '''Tab. 2: The assumed virus concentration values for the guinea pigs infected via air (A) and pre-infected (P). For the latter we took <math>\beta(t) = -1.28 t + 10.52</math> at <math>20^\mathrm{o}</math>C and <math>-1.23 t + 12.22</math> at <math>5^\mathrm{o}</math>C for the descending phase'''
+|+ align="top" | '''Tab. 2: The assumed virus concentration values for the guinea pigs infected via air (A) and pre-infected (P). For the latter we took <math>\beta(t) = -1.28 t + 10.52</math> at 20<math>^\mathrm{o}</math>C and <math>-1.23 t + 12.22</math> at 5<math>^\mathrm{o}</math>C for the descending phase'''
 <!--colspan="11" | -->
 |-
@@ Line 62: / Line 62: @@
 | 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10
 |-
-! A, <math>20^\mathrm{o}</math>C
+! A, 20<math>^\mathrm{o}</math>C
 | 0.00 || 1.62 || 3.85 || 6.08 || 7.08 || 5.45 || 3.82 || 2.18 || 0.55 || 0.00
 |-
-! A, <math>5^\mathrm{o}</math>C
+! A, 5<math>^\mathrm{o}</math>C
 | 0.00 || 2.02 || 4.05 || 6.08 || 7.06 || 5.79 || 4.52 || 3.25 || 1.98 || 0.72
 |-
-! P, <math>20^\mathrm{o}</math>C
+! P, 20<math>^\mathrm{o}</math>C
 | 5.07 || 7.30 || 6.70 || 5.42 || 4.14 || 2.87 || 1.60 || 0.32 || 0.00 || 0.00
 |-
-! P, <math>5^\mathrm{o}</math>C
+! P, 5<math>^\mathrm{o}</math>C
 | 4.97 || 7.00 || 8.54 || 7.32 || 6.10 || 4.87 || 3.64 || 2.42 || 1.20 || 0.00
 |}
@@ Line 127: / Line 127: @@
 {| class="rtable" align="center" width=50%
-|+ align="top" | '''Tab. 3: Abstract distances between the experimental data points. The values in the right-upper corner correspond to temperature <math>20^\mathrm{o}</math>C, and in the left-lower corner, to <math>5^\mathrm{o}</math>C. We have taken the average ''V''-coordinate values wherever there were two experiments performed under the same conditions'''
+|+ align="top" | '''Tab. 3: Abstract distances between the experimental data points. The values in the right-upper corner correspond to temperature 20<math>^\mathrm{o}</math>C, and in the left-lower corner, to 5<math>^\mathrm{o}</math>C. We have taken the average ''V''-coordinate values wherever there were two experiments performed under the same conditions'''
 <!--colspan="6" | -->
 |-

@@ Line 3: / Line 3: @@
 <br/><br/>
 <br/><br/>
 ===  Model's assumptions ===
@@ Line 73: / Line 74: @@
 | 4.97 || 7.00 || 8.54 || 7.32 || 6.10 || 4.87 || 3.64 || 2.42 || 1.20 || 0.00
 |}
 === Propagation of infection ===
@@ Line 101: / Line 103: @@
 We have already assumed that the virus concentration is always the same for all pre-infected individuals. The part of <math>S_\tau</math> that comes from them is then equal to this concentration (multiplied by <math>1 - \alpha_1</math> if the respective individual is located on the very top or bottom shelf).
 === An infection space ===

← Older revision		Revision as of 11:22, 21 November 2008
Line 149:		Line 149:

	=== Extraction of unknown parameters ===		=== Extraction of unknown parameters ===
		+
		+	Using the few experimental points in <math>V</math>, it is necessary to determine the unknown parameters <math>\alpha_1</math>, <math>\alpha_2</math>, and <math>\gamma(T,H)</math>. This is done by using the iterative algorithm described in section ''Propagation'' to generate a number of points corresponding to various reasonable parameter sets. Each of these sets will then be classified, separately for each temperature, as an approximation of parameter values for one particular humidity level <math>H</math> corresponding to the available experimental data, namely, the one that is represented by the experimental point of least distance to the point considered. After that, a search is performed for the most relevant values of <math>\alpha_1</math>, and <math>\alpha_2</math>, that cover the maximal number of experimental cases for both temperatures and with the smallest possible distances. The last step involves determination of <math>\gamma</math> values for each <math>T</math> and <math>H</math> separately.
		+
		+	For the simulations here we set 17 irregularly spaced checkpoints for <math>\alpha_1</math> and <math>\alpha_2</math> from 0 to 0.5, and 62 checkpoints for <math>\gamma</math> from 0.001 to 0.5. They are distributed more densely for lower parameter values.

@@ Line 21: / Line 21: @@
 {| class="rtable" align="center" width=90%
-|+ align="top" | '''tab.1: The average virus concentration in nasal wash of individual guinea pigs infected via air. For each individual, day 1 is the day when a measurable virus quantity first appeared in its respiratory tract'''
+|+ align="top" | '''Tab. 1: The average virus concentration in nasal wash of individual guinea pigs infected via air. For each individual, day 1 is the day when a measurable virus quantity first appeared in its respiratory tract'''
 <!--colspan="10" | -->
 |-
@@ Line 27: / Line 27: @@
 | 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9
 |-
-! Concentration, <math>T = 20^o</math>
+! Concentration, <math>T = 20^\mathrm{o}</math>C
-| 1.52 || 4.52 || 5.63 || 6.89 || 6.06 || 3.26 || 1.88 || - || -
+| 1.52 || 4.52 || 5.63 || 6.89 || 6.06 || 3.26 || 1.88 || &#8212; || &#8212;
 |-
-! Concentration, <math>T = 5^o</math>
+! Concentration, <math>T = 5^\mathrm{o}</math>C
-| 1.91 || 4.40 || 5.72 || 7.02 || 6.01 || 4.54 || 3.52 || 1.77 || -
+| 1.91 || 4.40 || 5.72 || 7.02 || 6.01 || 4.54 || 3.52 || 1.77 || &#8212;
 |-
 ! Std. deviation square
-| 0.40 || 0.59 || 1.26 || 0.32 || 2.38 || 1.44 || 3.18 || 2.34 || -
+| 0.40 || 0.59 || 1.26 || 0.32 || 2.38 || 1.44 || 3.18 || 2.34 || &#8212;
 |-
 |}
@@ Line 43: / Line 43: @@
 | [[Image:flu_spread_model_zuk_rakowski_radomski_Fig1a.png |frame|left| '''Fig.1a''']] || [[Image:flu_spread_model_zuk_rakowski_radomski_Fig1b.png |frame| '''Fig.1b''' ]]
 |-
-| colspan="2" | '''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'''
+| colspan="2" | '''The two piecewise linear approximations of the infection course. For <math>20^\mathrm{o}</math>C 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^\mathrm{o}</math>C 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'''
 |}
@@ Line 54: / Line 54: @@
 '''4.''' The infection course in pre-infected individuals also consists of two linear (on a logarithmic scale) phases. Concentration values for the descending phase come from the measurement<ref name="lowen2007">Lowen A C, Mubareka S, Steel J, Palese P, ''Influenza Virus Transmission Is Dependent on Relative Humidity and Temperature'', PLoS Pathogens 3(10), 2007, p151</ref> (We fitted a linear function for these values. However, they were originally of a nearly linear nature.). Due to the lack of the respective experimental data, for the ascending phase we assumed the slope to be the same as for animals infected via air. The virus concentration values for those individuals are also shown in Table 2.
 === Propagation of infection ===
@@ Line 84: / Line 103: @@
 === An infection space ===
 === Extraction of unknown parameters ===

@@ Line 13: / Line 13: @@
 '''1.''' The probability that the ''i''th individual will be infected on a particular day (provided that it had not been infected previously) is
+: <math> p_i = 1 - \exp\left(-\gamma(T,H)\sum_j \alpha_{ij}\beta_j\right) </math>
-<math>p_i = 1 - \exp\left(-\gamma(T,H)\sum_j \alpha_{ij}\beta_j\right) </math>
+: (1)
 where <math>\beta_j</math> is a measure of virus concentration in the respiratory tract of the ''j''th 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 <math>\alpha_{ij}</math> are related to the probabilities that an aerosol drop, shed by the ''j''th animal, will get to the ''i''th guinea pig's cage. The coefficient <math>\gamma(T,H)</math> reflects the dependence of the virus transmission rate on the air temperature (''T'') and relative humidity (''H'').
@@ Line 49: / Line 48: @@
 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.
 '''3.''' A detectable virus presence in the respiratory tract begins 2 days after the infection. Such an assumption is drawn from the fact that the virus was not detectable in nasal wash earlier than the third day of experiment. A lack of infections during the first day, when the virus concentrations in pre-infected individuals is supposed to be high, is of little probability. This is consistent with the major biological and clinical experience, which shows that the influenza proper begins rapidly after 2-3 days of a rather asymptomatic and non-infectious incubation period<ref name="carrat2008">Carrat F, Vergu E, Ferguson N M, Lemaitre M, Cauchemez S, Leach S, Valleron A J, ''Time Lines of Infection and Disease in Human Influenza: A Review of Volunteer Challenge Studies'', Am. J. Epid. 167(7), 2008</ref><ref name="collier93">Collier L, Oxford J, ''Human virology'', Oxford University Press, 1993</ref>.
@@ Line 58: / Line 56: @@
 === Propagation of infection ===
 === An infection space ===