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update concentration plot

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Andre Henriques 2022-06-23 10:11:46 -04:00
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cara/docs/full_diameter_dependence.rst
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@ -57,6 +57,7 @@ Interface Focus 20210076, https://doi.org/10.1098/rsfs.2021.0076), as follows:
The later integral, which is giving the total volumetric particle emission concentration (in mL/m\ :sup:`3` \), is a example of a numerical Monte-Carlo integration over the particle diameters,
since :math:`\mathrm{vR}(D)` is a diameter-dependent quantity. :math:`E_{c, j}` is calculated using a Monte-Carlo sampling of the BLO distribution given by :math:`N_p(D)`, which contains the scaling factor :math:`cn`.
Note that :math: `D_{\mathrm{max}}` value will differe, depending on the type of exposrue (see below).
In the code, for a given Expiration, we use different methods to perform the calculations *set-by-step*:
@ -85,19 +86,21 @@ Long-range approach
Starting with the long-range concentration of virus-laden aerosols of a given size **D**, that is based on the mass balance equation between the emission and removal rates, is given by:
:math:`C(t, D)=\frac{\mathrm{vR}(D) \cdot N_{\mathrm{inf}}}{\lambda_{\mathrm{vRR}}(D) \cdot V_r}-\left (\frac{\mathrm{vR}(D) \cdot N_{\mathrm{inf}}}{\lambda_{\mathrm{vRR}}(D) \cdot V_r}-C_0(D) \right )e^{-\lambda_{\mathrm{vRR}}(D)t}` ,
:math:`C_{\mathrm{LR}}(t, D)=\frac{\mathrm{vR}(D) \cdot N_{\mathrm{inf}}}{\lambda_{\mathrm{vRR}}(D) \cdot V_r}-\left (\frac{\mathrm{vR}(D) \cdot N_{\mathrm{inf}}}{\lambda_{\mathrm{vRR}}(D) \cdot V_r}-C_0(D) \right )e^{-\lambda_{\mathrm{vRR}}(D)t}` ,
and uses this :meth:`cara.models.ConcentrationModel.concentration` method, which cumputes the long-range concentration, as a function of the exposure time and particle diamter.
The long-range concentration, integrated over the exposure time (in piecewise constant steps), :math:`C(D)`, is given by the :meth:`cara.models.ConcentrationModel.integrated_concentration`
In the :math:`C(t, D)` equation above, the **emission rate** :math:`\mathrm{vR}(D)` and **viral removal rate** :math:`\lambda_{\mathrm{vRR}}` (:meth: `infectious_virus_removal_rate`) are both diameter-dependent.
???? Hence, the concentration needs, equally, to be normalized by the diameter-dependent quantities: emission rate ??.
???? Since the emission rate is, in turn, dependent on other diameter-independent variables (:math:`\mathrm{vl}_\mathrm{in}` and :math:`\mathrm{BR}_k``), those parameters shall also be excluded when calculating the integral, the concentration method was written to be normalized by the emission rate.
In the :math:`C_{\mathrm{LR}}(t, D)` equation above, the **emission rate** :math:`\mathrm{vR}(D)` and **viral removal rate** :math:`\lambda_{\mathrm{vRR}}` (:meth: `infectious_virus_removal_rate`) are both diameter-dependent.
???? Hence, the concentration needs, equally, to be normalized by the diameter-dependent quantities: emission rate (AND VRR??).
???? Since the emission rate is, in turn, dependent on other diameter-independent variables (:math:`\mathrm{vl}_\mathrm{in}` and :math:`\mathrm{BR}_k``) (WHERE IS THIS?), those parameters shall also be excluded when calculating the integral,
the concentration method was written to be normalized by the emission rate.
In other words, we can split the concentration in two different formulations:
* Normalized concentration :meth:`cara.models.ConcentrationModel._normed_concentration` : :math:`\mathrm{C_\mathrm{normed}}(t, D)` that cumputes the concentration without including the emission rate.
???? * Concentration :meth:`cara.models.ConcentrationModel.concentration` : :math:`C(t, D) = [\mathrm{C_\mathrm{normed}}(t, D) \cdot \mathrm{vR}(D)] \cdot \mathrm{BR_k} \cdot \mathrm{vl_{in}}` , where :math:`\mathrm{vR}(D)` that cumputes the full concentration, still as a function of the particle diameter,
* Normalized concentration :meth:`cara.models.ConcentrationModel._normed_concentration` : :math:`\mathrm{C_\mathrm{LR, normed}}(t, D)` that cumputes the concentration without including the emission rate.
???? * Concentration :meth:`cara.models.ConcentrationModel.concentration` : :math:`C_{\mathrm{LR}}(t, D) = [\mathrm{C_\mathrm{LR, normed}}(t, D) \cdot \mathrm{vR}(D)] \cdot \mathrm{BR_k} \cdot \mathrm{vl_{in}}`, (WHERE IS THIS?)
where :math:`\mathrm{vR}(D)` that cumputes the full concentration, still as a function of the particle diameter,
where \mathrm{vR}(D) is the result of the :meth:`cara.models._PopulationWithVirus.emission_rate_when_present` method.
Note that in order to get the total concentration value in this stage, the final result should be averaged by the particle diameters (i.e. Monte-Carlo integration over diameters, see above).
@ -114,27 +117,61 @@ The integral over the exposure times is calculated directly in the class (integr
Short-range approach
*******************
The short-range concentration close to the mouth or nose of an exposed person, may be written as:
The short-range concentration is the result of a two-stage exhaled jet model developed by *JIA W. et al.* and is expressed as:
:math:`C_{\mathrm{SR}}(t, D) = \frac{1}{S({x})} \cdot (C_{0, \mathrm{SR}}(D) - C(t, D))` .
???? :math:`C_{\mathrm{SR}}(t, D) = \frac{1}{S({x})} \cdot (C_{0, \mathrm{SR}}(D) - C_{\mathrm{LR}}(t, D))` , (TO BE CONFIMED)
It depends on the **long-range concentration** of viruses, on the **dilution factor** and on the **initial concentration** of viruses on the mouth or nose of the emitter.
As for the long-range concentration, we must normalize the short-range concentration on parameters that are diameter-dependent variables, to profit from the Monte-Carlo integration.
Besides that, one should consider that for each interaction, the expiration type may be different, therefore a new distribution of diameters should be taken into consideration.
where :math: `S(x)` is the dilution factor due to jet dynamics, as a function of the interpersonal distance *x* and :math: `C_{0, \mathrm{SR}}(D)` corresponds to the initial concetration of virions at the mouth/nose outlet during exhalation.
Note that :math: `C_{0, \mathrm{SR}}(D)` is constant over time, hence only dependent on the particle diameter distribution and
The method to calculate the concentration viruses on the mouth or nose of the emitting person, has the viral load as multiplying factor:
As mentioned in *JIA W. et al.*, the jet concentration depends on the **long-range concentration** of viruses.
Here, once again, we shall normalize the short-range concentration to the diameter-dependent quantities.
IMPORTANT NOTE: since the susceptible host is physicully closer to the infector, the emitted particles are larger in size,
hence a new distribution of diameters should be taken into consideration.
As opposed to :math: `D_{\mathrm{max}} = 30 μm` for the long-range MC integration, the short-range model will assume a :math: `D_{\mathrm{max}} = 100 μm`
:math:`C_{0, SR}=(\int_{D_{min}}^{D_{\mathrm{max = 1000μm}}} N_p(D) \cdot V_p(D)\, \mathrm{d}D) \cdot 10^{-6} \cdot \mathrm{vl_{in}}` .
During as given exposure time, multiple short-range interactions can be defined in the model.
In addition, for each individual interaction, the expiration type may be different.
In other words, in the code we have one method that returns the value of :math:`N_p(D) \cdot V_p(D)`, :meth:`cara.models.Expiration.jet_origin_concentration`. Note that similarly to the `long-range` approach, the integral over the diameters is not calculated at this stage.
The initial concetration of virions at the mouth/nose, :math: `C_{0, \mathrm{SR}}(D)` is calculated as follows:
To calculate the `long-range` concentration of viruses, :math:`C(t, D)`, we profit from the :meth:`cara.models.ConcentrationModel.long_range_normed_concentration` method, normalized by the viral load, the diameter-independent variable in the concentration.
However, since the diameter distribution is different on the `short-range` interactions, we need to perform one approximation using interpolation. The set of points we want the interpolated values are the short-range particle diameters (given by the current expiration). The set of points with a known value are the long-range particle diameters (given by the initial expiration). The set of known values are the long-range concentration values normalised by the viral load. At this point, we have a procedure to calculate :math:`C_{0, \mathrm{SR}} - C(t, D)`. Given that we already have the result of the `dilution_factor`, the result of :math:`\frac{1}{S({x})} \cdot (C_{0, \mathrm{SR}} - C(t, D))` is given by the method :meth:`cara.models.ShortRangeModel.normed_concentration`. To sum up, this method calculates the virus `short-range` exposure concentration, as a function of time. It is normalized by the viral load, and the integral over the diameters is not performed at this stage.
:math:`C_{0, \mathrm{SR}}(D) = N_p(D) \cdot V_p(D) \cdot \mathrm{vl_{in}}` \cdot 10^{-6}.
The method :meth:`cara.models.ShortRangeModel.short_range_concentration` applies the multiplication by the viral load to the result of the previous method, returning the final short-range concentration for a given time.
Note, the :math: `10^{-6}` factor corresponds to the conversion from :math: `μm^{3} /cdot cm{-3}` to :math: `mL /cdot m^{3}`
The final concentration is the sum of the `short-range` and `long-range` concentrations.
To respect the diameter-dependence, the initial concentration has to be normalized by the viral load.
Hence, in the code we have one method that returns the value of :math:`N_p(D) \cdot V_p(D) \cdot 10^{-6}`, :meth:`cara.models.Expiration.jet_origin_concentration`.
Note that similarly to the `long-range` approach, the integral over the diameters is not calculated at this stage.
For mathematical consistency, the long-range concentration parameter,:math:`C_{\mathrm{LR}}(t, D)`, **in the :class:`cara.models.ShortRangeModel` class**,
???? shall also be normalized by the viral load. The :meth:`cara.models.ConcentrationModel.long_range_normed_concentration` method, is hence, normalized by the viral load. ( NOT ShortRangeModel.long_range_normed_concentration ??)
???? (DONT UNDERSTAND THE FEW LINES BELOW)
However, since the diameter distribution is different on the `short-range` interactions, we need to perform one approximation using interpolation.
The set of points we want the interpolated values are the short-range particle diameters (given by the current expiration).
The set of points with a known value are the long-range particle diameters (given by the initial expiration).
The set of known values are the long-range concentration values normalised by the viral load.
At this point, we have a procedure to calculate :math:`C_{0, \mathrm{SR}} - C_{\mathrm{LR}}(t, D)`.
Given that we already have the result of the `dilution_factor`, the result of :math:`\frac{1}{S({x})} \cdot (C_{0, \mathrm{SR}} - C_{\mathrm{LR}}(t, D))` is given by the method :meth:`cara.models.ShortRangeModel.normed_concentration`.
To sum up, this method calculates the virus `short-range` exposure concentration, as a function of time.
It is normalized by the viral load, and the integral over the diameters is not performed at this stage.
????
Note that :meth:`cara.models.ShortRangeModel._normed_concentration` method is different from :meth:`cara.models.ConcentrationModel._normed_concentration`
Finally, the method :meth:`cara.models.ShortRangeModel.short_range_concentration` applies the multiplication of the diameter-independent quantity: the viral load -
returning the final short-range concentration value for a given time and expiration activity.
???? The final concentration is the sum of the `short-range` and `long-range` concentrations. (WHERE ? cara.models.ExposureModel.concentration ?)
Note that the diameter-dependence is kept at this stage. Since other parameters downstream in code are also diameter-dependent, the Monte-Carlo integration over the aerosol sizes is computed at the level of the dose :math:`\mathrm{vD^{total}}`.
In case one would like to have intermediate results for the short-range concentration, perform the Monte-Carlo integration of :math:`C_{0, SR}^{\mathrm{total}}` and compute
:math:`C_{0, SR}^{\mathrm{total}}=(\int_{D_{min}}^{D_{\mathrm{max = 100μm}}} N_p(D) \cdot V_p(D)\, \mathrm{d}D) \cdot 10^{-6} \cdot \mathrm{vl_{in}}` .
>>>>>>>
Dose - :math:`\mathrm{vD}`
@ -181,6 +218,8 @@ The `emission_rate_per_aerosol` is introduced because of the previous normalizat
In the end, the dose is a vectorized float used in the probability of infection formula.
Short-range approach
====================