turned text into variables when needed
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Luis Aleixo
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1 changed files with 45 additions and 45 deletions
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@ -10,19 +10,19 @@ The :mod:`cara.apps.calculator.model_generator` module is responsible to bind al
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The :py:mod:`cara.apps.calculator.report_generator` module is responsible to bind the results from the model calculations into the respective output variables presented in the CARA report.
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The :mod:`cara.models` module itself implements the core CARA methods. A useful feature of the implementation is that we can benefit from vectorization, which allows running multiple parameterization of the model at the same time.
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Unlike other similar models, some of the CARA variables are considered for a given aerosol diameter **D**,
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Unlike other similar models, some of the CARA variables are considered for a given aerosol diameter :math:`D`,
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as the behavior of the virus-laden particles in the room environment and inside the susceptible host (once inhaled) are diameter-dependent.
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These variables are identified by **(D)** in the variable name, such as the **emission rate** -- **vR(D)**, **removal rate** -- **vRR(D)**, and **concentration** -- **C(t, D)**.
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These variables are identified by :math:`D` in the variable name, such as the **emission rate** -- :math:`\mathrm{vR}(D)`, **removal rate** -- :math:`\mathrm{vRR}(D)`, and **concentration** -- :math:`C(t, D)`.
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Despite the outcome of the CARA results include the entire range of diameters, throughout the model,
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most of the variables and parameters are kept in their diameter-dependent form for any possible detailed analysis of intermediate results.
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Only the final quantities shown in output, such as the concentration and the dose, are integrated over the diameter distribution.
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This is performed thanks to a Monte-Carlo integration at the level of the dose (vD\ :sup:`total`\) which is computed over a distribution of particle diameters,
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This is performed thanks to a Monte-Carlo integration at the level of the dose (:math:`\mathrm{vD^{total}}`) which is computed over a distribution of particle diameters,
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from which the average value is then calculated -- this is equivalent to an analytical integral over diameters
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provided the sample size is large enough.
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It is important to distinguish between 1) Monte-Carlo random variables (which are vectorised independently on its diameter-dependence) and 2) numerical Monte-Carlo integration for the diameter-dependence
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Since the integral of the diameter-dependent variables are solved when computing the dose -- vD\ :sup:`total`\ -- while performing some of the intermediate calculations,
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Since the integral of the diameter-dependent variables are solved when computing the dose -- :math:`\mathrm{vD^{total}}` -- while performing some of the intermediate calculations,
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we normalize the results by *dividing* by the Monte-Carlo variables that are diameter-independent, so that they are not considered in the Monte-Carlo integration (e.g. :meth:`cara.models.ConcentrationModel.normed_integrated_concentration`).
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Expiration
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@ -30,7 +30,7 @@ Expiration
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The **Expiration** class (representing the expiration of aerosols by an infected person) has the `Particle` -- :attr:`cara.models.Expiration.particle` -- as one of its properties,
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which represents the virus-laden aerosol with a vectorised parameter: the particle `diameter` (assuming a perfect sphere).
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For a given aerosol diameter, one :class:`cara.models.Expiration` object provides the aerosol **volume - Vp(D)**, multiplied by the **mask outward efficiency - ηout(D)** to include the filteration capacity, when applicable.
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For a given aerosol diameter, one :class:`cara.models.Expiration` object provides the aerosol **volume** - :math:`V_p(D)`, multiplied by the **mask outward efficiency** - :math:`η_\mathrm{out}(D)` to include the filteration capacity, when applicable.
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The BLO model represents the distribution of diameters used in the model. It corresponds to the sum of three lognormal distributions, weighted by the **B**, **L** and **O** modes.
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The aerosol diameter distributions are given by the :meth:`cara.monte_carlo.data.BLOmodel.distribution` method.
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@ -44,52 +44,52 @@ an integration of the lognormal distributions is performed over all aerosol diam
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Under the :mod:`cara.apps.calculator.model_generator`, when it comes to generate the Expiration model, the `diameter` property is sampled through the BLO :meth:`cara.monte_carlo.data.BLOmodel.distribution` method, while the value for the **cn** is given by the :meth:`cara.monte_carlo.data.BLOmodel.integrate` method.
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To summarize, the Expiration contains the distribution of the diameters as a vectorised float. Depending on different expiratory types, the contributions from each mode will be different, therefore the result in the distribution also differs from model to model.
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Emission Rate - vR(D)
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Emission Rate - :math:`\mathrm{vR}(D)`
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=====================
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The mathematical equations to calculate **vR(D)** are defined in the paper
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The mathematical equations to calculate :math:`\mathrm{vR}(D)` are defined in the paper
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(Henriques A et al, Modelling airborne transmission of SARS-CoV-2 using CARA: risk assessment for enclosed spaces.
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Interface Focus 20210076, https://doi.org/10.1098/rsfs.2021.0076), as follows:
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:math:`vR(D)_j=vl_{in} . E_{c, j}(D, f_{amp}, η_{out}(D)) . BR_k` ,
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:math:`\mathrm{vR}(D)_j= \mathrm{vl_{in}} \cdot E_{c,j}(D,f_{\mathrm{amp}},\eta_{\mathrm{out}}(D)) \cdot {\mathrm{BR}}_{\mathrm{k}}` ,
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:math:`E_{c, j}^{total}=\int_0^{D_{\mathrm{max}}} E_{c,j}(D)\, \mathrm{d}D` .
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:math:`E_{c,j}^{\mathrm{total}} = \int_0^{D_{\mathrm{max}}} E_{c,j}(D)\, \mathrm{d}D` .
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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,
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since vR(D) is a diameter-dependent quantity. :math:`E_{c, j}` is calculated using a Monte-Carlo sampling of the BLO distribution given by **Np(D)**, which contains the scaling factor **cn**.
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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`.
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In the code, for a given Expiration, we use different methods to perform the calculations *set-by-step*:
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1. Calculate the non aerosol-dependent quantities in the emission rate, which is the multiplication of the diameter-**independent** variables: :meth:`cara.models.InfectedPopulation.emission_rate_per_aerosol_when_present`. This corresponds to the :math:`vl_{in} . BR_{k}` part of the vR(D) equation.
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2. Calculate the the diameter-**dependent** variable :meth:`cara.models.InfectedPopulation.aerosols`, which is the result of :math:`E_{c,j}(D) = Np(D) . Vp(D) . (1 − ηout(D))` (in mL/(m\ :sup:`3` \.µm)).
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1. Calculate the non aerosol-dependent quantities in the emission rate, which is the multiplication of the diameter-**independent** variables: :meth:`cara.models.InfectedPopulation.emission_rate_per_aerosol_when_present`. This corresponds to the :math:`\mathrm{vl_{in}} \cdot \mathrm{BR_{k}}` part of the :math:`\mathrm{vR}(D)` equation.
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2. Calculate the diameter-**dependent** variable :meth:`cara.models.InfectedPopulation.aerosols`, which is the result of :math:`E_{c,j}(D) = N_p(D) \cdot V_p(D) \cdot (1 − η_\mathrm{out}(D))` (in mL/(m\ :sup:`3` \.µm)).
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Note that this result is not integrated over the diameters at this stage, thus the units are still *'per aerosol diameter'*.
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3. Calculate the full emission rate, which is the multiplication of the two previous methods, and corresponds to **vR(D)**: :meth:`cara.models._PopulationWithVirus.emission_rate_when_present`
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1. Calculate the full emission rate, which is the multiplication of the two previous methods, and corresponds to :math:`\mathrm{vR(D)}`: :meth:`cara.models._PopulationWithVirus.emission_rate_when_present`
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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 vD\ :sup:`total` \.
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In case one would like to have intermediate results for emission rate, perform the Monte-Carlo integration of :math:`E_{c, j}^{total}` and compute :math:`vR^{total} =vl_{in} . E_{c, j}^{total} . BR_k`
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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}}`.
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In case one would like to have intermediate results for emission rate, perform the Monte-Carlo integration of :math:`E_{c, j}^{\mathrm{total}}` and compute :math:`\mathrm{vR^{total}} =\mathrm{vl_{in}} \cdot E_{c, j}^{\mathrm{total}} \cdot \mathrm{BR_k}`
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Long-range approach
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===================
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Concentration - C(t, D)
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Concentration - :math:`C(t, D)`
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***********************
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Starting with the long-range concentration of virus, that depends on the **emission rate**, the concentration of viruses in aerosols of a given size **D** is:
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Starting with the long-range concentration of virus, that depends on the **emission rate**, the concentration of viruses in aerosols of a given size :math:`D` is:
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: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}` ,
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where **emission rate vR(D)** and :math:`\lambda_{\mathrm{vRR}}` **(viral removal rate)** depend on the particle diameter **D**.
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Since the emission rate is dependent on diameter-independent variables (:math:`vl_{in}` and :math:`BR_k``) that should not be included when calculating the integral, the concentration method was written to be normalized by the emission rate.
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where :math:`\mathrm{vR}(D)` **(emission rate)** and :math:`\lambda_{\mathrm{vRR}}` **(viral removal rate)** depend on the particle diameter :math:`D`.
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Since the emission rate is dependent on diameter-independent variables (:math:`\mathrm{vl_{in}}` and :math:`\mathrm{BR_k}`) that should not be included when calculating the integral, the concentration method was written to be normalized by the emission rate.
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In other words, we can split the concentration in two different formulations:
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* Normed concentration : :math:`CN(t, D)` that calculates the concentration without the multiplication by the emission rate.
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* Concentration: :math:`C(t, D) = [CN(t, D) * vR(D)] * BR_k * vl_{in}`, where :math:`vR(D)` is the result of the :meth:`cara.models.Expiration.aerosols` method, while :math:`BR_k` and :math:`vl_{in}` are the diameter-independent Monte-Carlo variables.
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* Normed concentration : :math:`\mathrm{CN}(t, D)` that calculates the concentration without the multiplication by the emission rate.
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* Concentration: :math:`C(t, D) = [\mathrm{CN}(t, D) \cdot \mathrm{vR}(D)] \cdot \mathrm{BR_k} \cdot \mathrm{vl_{in}}`, where :math:`\mathrm{vR}(D)` is the result of the :meth:`cara.models.Expiration.aerosols` method, while :math:`\mathrm{BR_k}` and :math:`\mathrm{vl_{in}}` are the diameter-independent Monte-Carlo variables.
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This way, to calculate the concentration in the model, there are different methods that consider the normalization by the emission rate:
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* **_normed_concentration**, that calculates the virus long-range exposure concentration, as function of time, and normalized by the emission rate. It corresponds to the previously mentioned :math:`CN(t, D)`.
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* **_normed_concentration**, that calculates the virus long-range exposure concentration, as function of time, and normalized by the emission rate. It corresponds to the previously mentioned :math:`\mathrm{CN}(t, D)`.
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* :meth:`cara.models.ConcentrationModel.concentration`, which calculates the virus long-range exposure concentration of viruses as function of time and diameter (:math:`C(t, D)` above). Note that in order to get the total concentration value in this stage, the final result should be averaged (this is equivalent to a Monte-Carlo integration over diameters, see above). In the calculator, the integral over the diameters is performed only when doing the concentration plot. Otherwise, it is done only at a later stage, when calculating the dose (in :class:`cara.models.ExposureModel`).
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These two methods are used to calculate the concentration at a given time. At this stage to perform the integral over the diameters the resulting value should be averaged according to the Monte-Carlo integration.
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@ -101,47 +101,47 @@ The following methods calculate the integrated concentration between any two tim
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.. Note that the integral over the diameters is performed later in the dose, with the average of the samples, since the diameters are sampled according to the distribution given by **Np(D)**. The integral over different times is calculated directly in the class (integrated methods).
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Dose - vD
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Dose - :math:`\mathrm{vD}`
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*********
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The term “dose” refers to the number of viable virions that will contribute to a potential infection.
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The receiving dose, which is inhaled by the exposed host, in infectious virions per unit diameter, is calculated by first integrating the viral concentration profile (for a given particle diameter) over the exposure time and multiplying by a scaling factor to determine the proportion of virions which are infectious:
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:math:`\mathrm{vD}(D)=\int_{t1}^{t2}C(t, D)\;dt \cdot f_{\mathrm{inf}} \cdot \mathrm{BR}_{\mathrm{k}} \cdot f_{\mathrm{dep}}(D) \cdot (1-\eta_{\mathrm{in}})` .
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:math:`\mathrm{vD}(D)=\int_{t1}^{t2}C(t, D)\;\mathrm{d}t \cdot f_{\mathrm{inf}} \cdot \mathrm{BR}_{\mathrm{k}} \cdot f_{\mathrm{dep}}(D) \cdot (1-\eta_{\mathrm{in}})` .
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Given that the calculation is diameter-dependent, to calculate the dose in the model, the code contains different methods that consider the parameters that are dependent on the aerosol size, **D**.
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The total dose results from the sum of all the doses accumulated for each particle size is
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:math:`\mathrm{vD^{total}} = \int_0^{D_{\mathrm{max}}} \mathrm{vD}(D) \, \mathrm{d}D` .
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This calculation is computed using a Monte-Carlo integration. As previously described, many different parameters samples are generated using the probability distribution from the **Np(D)** equation.
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This calculation is computed using a Monte-Carlo integration. As previously described, many different parameters samples are generated using the probability distribution from the :math:`N_p(D)` equation.
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The dose for each of them is then computed, and their **average** value over all samples represents a good approximation of the total dose, provided that the number of samples is large enough.
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Regarding the first parameter, i.e. the concentration integrated over the time, the respective method is the :meth:`cara.models.ExposureModel._long_range_normed_exposure_between_bounds`, which calculates the long-range exposure (concentration) between two bounds (time1 and time2), normalized by the emission rate of the infected population.
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This method filters out the given bounds considering the breaks through the day (i.e. the time intervals during which there is no exposition to the virus) and calls :meth:`cara.models.ConcentrationModel.normed_integrated_concentration` that gets the integrated long-range concentration of viruses in the air between any two times.
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It corresponds to the :math:`\int_{t1}^{t2}C(t, D)\;dt` integral, normalized by the emission rate of the infected population.
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It corresponds to the :math:`\int_{t1}^{t2}C(t, D)\;\mathrm{d}t` integral, normalized by the emission rate of the infected population.
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After the calculations of the integrated concentration over the time, in order to calculate the final dose, we have to compute the remaining factors in the above equation.
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Note that the Monte-Carlo integration is performed at this stage, where all the parameters that are diameter-dependent are grouped together to calculate the final average.
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In other words, in the code the procedure is the following:
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:math:`\mathrm{vD_{normed}} = (\int_{t1}^{t2}C(t, D)\;dt \cdot V_{aerosol}(D, mask) \cdot f_{\mathrm{dep}}(D)).mean()` .
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:math:`\mathrm{vD_{normed}} = (\int_{t1}^{t2}C(t, D)\;\mathrm{d}t \cdot V_{\mathrm{aerosol}}(D, \mathrm{mask}) \cdot f_{\mathrm{dep}}(D)) \cdot \mathrm{mean()}` .
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The aerosol volume :math:`V_{aerosol}` is introduced because the integrated concentration over the time was previously normalized by the emission rate.
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The aerosol volume :math:`V_{\mathrm{aerosol}}` is introduced because the integrated concentration over the time was previously normalized by the emission rate.
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Here, to calculate the integral over the diameters we also need to consider the diameter-dependent variables that are on the emission rate, represented by the aerosol volume which depends on the diameter and on the mask type:
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:math:`V_{aerosol}(D, mask) = \mathrm{cn} \cdot Vp(D) \cdot (1 − ηout(D))` .
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:math:`V_{\mathrm{aerosol}}(D, \mathrm{mask}) = \mathrm{cn} \cdot V_p(D) \cdot (1 − \mathrm{ηout}(D))` .
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The :math:`\mathrm{cn}` factor, which represents the total number of aerosols emitted, is introduced here as a scaling factor, as otherwise the Monte-Carlo integral would be normalized to 1 as the probability distribution.
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Finally we multiply the result by all the remaining diameter-independent variables:
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:math:`\mathrm{vD^{total}} = \mathrm{vD_{normed}} \cdot f_{inf} \cdot \mathrm{BR}_{k} \cdot (1 - η_{in}) \cdot \mathrm{vR_{ND}}` ,
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:math:`\mathrm{vD^{total}} = \mathrm{vD_{normed}} \cdot f_{\mathrm{inf}} \cdot \mathrm{BR_{k}} \cdot (1 - η_{\mathrm{in}}) \cdot \mathrm{vR_{ND}}` ,
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with :math:`\mathrm{vR_{ND}} =` `emission_rate_per_aerosol` :math:`= vl_{in} \cdot \mathrm{BR}_{k}` .
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with :math:`\mathrm{vR_{ND}} =` `emission_rate_per_aerosol` :math:`= \mathrm{vl_{in}} \cdot \mathrm{BR_{k}}` .
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The `emission_rate_per_aerosol` is introduced because of the previous normalization by the emission rate, except for the diameter-dependent variable :math:`V_{aerosol}` which was already in :math:`\mathrm{vD_{normed}}`. So one should multiply by the missing parameters :math:`vl_{in}` and :math:`BR_{k}` (see :meth:`cara.models.InfectedPopulation.emission_rate_per_aerosol_when_present`).
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The `emission_rate_per_aerosol` is introduced because of the previous normalization by the emission rate, except for the diameter-dependent variable :math:`V_{\mathrm{aerosol}}` which was already in :math:`\mathrm{vD_{normed}}`. So one should multiply by the missing parameters :math:`\mathrm{vl_{in}}` and :math:`\mathrm{BR_{k}}` (see :meth:`cara.models.InfectedPopulation.emission_rate_per_aerosol_when_present`).
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In the end, the dose is a vectorized float used in the probability of infection formula.
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@ -152,18 +152,18 @@ The short-range data class models a close-range interaction **concentration** an
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Its properties are the **expiration** definition, the **activity type**, the **presence time**, and the **interpersonal distance** between any two individuals.
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When generating a full model, the short-range class is defined with a new **Expiration** distribution, given that the **min** and **max** diameters for the short-range interations are different from those used in the long-range concentration (the idea is that very large particles should not be considered in the long-range case as they fall rapidly on the floor, while they must be in for the short-range case).
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To calculate the short-range concentration, we first need to calculate what is the **concentration at the jet origin**, that depends on the diameter **D**. Very similar to what we did with the **emission rate**, we need to calculate the scaling factor from the probability distribution, **Np(D)**, as well as the **volume** for those diameters.
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To calculate the short-range concentration, we first need to calculate what is the **concentration at the jet origin**, that depends on the diameter :math:`D`. Very similar to what we did with the **emission rate**, we need to calculate the scaling factor from the probability distribution, :math:`N_p(D)`, as well as the **volume** for those diameters.
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In the code, :meth:`cara.models.Expiration.jet_origin_concentration` computes the same quatity as :meth:`cara.models.Expiration.aerosols`, except for the mask inclusion. As previously mentioned, it is normalized by the **viral load**, which is a diameter-independent property.
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When calculating the dose, we get the concentration normalized by the **viral load** and **breathing rate**, and without the **dilution factor**, since these parameters are Monte-Carlo variables that do not depend on the diameter.
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Concentration - C(t, D)
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Concentration - :math:`C(t, D)`
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***********************
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The short-range concentration close to the mouth or nose of an exposed person, may be written as:
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:math:`C_{SR}(t, D) = \frac{1}{S({x})}*(C_{0, SR}(D) - C(t, D))` .
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:math:`C_{\mathrm{SR}}(t, D) = \frac{1}{S({x})} \cdot (C_{0, \mathrm{SR}}(D) - C(t, D))` .
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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.
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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.
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@ -171,23 +171,23 @@ Besides that, one should consider that for each interaction, the expiration type
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The method to calculate the concentration viruses on the mouth or nose of the emitting person, has the viral load as multiplying factor:
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: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 vl_{in}` .
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: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}}` .
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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.
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To calculate the `long-range` concentration of viruses, `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.
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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, 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, 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.
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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.
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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.
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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.
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The final concentration is the sum of the `short-range` and `long-range` concentrations.
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Dose - vD
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Dose - :math:`vD`
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*********
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In theory, the `short-range` dose is defined as follows:
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:math:`\mathrm{vD}(D)= \mathrm{vD^{long-range}}(D) + \sum\limits_{i=1}^{n} \int_{t1}^{t2}C_{SR}(t, D)\;dt \cdot f_{\mathrm{inf}} \cdot \mathrm{BR}_{\mathrm{k}} \cdot f_{\mathrm{dep}}(D) \cdot (1-\eta_{\mathrm{in}})` ,
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:math:`\mathrm{vD}(D)= \mathrm{vD^{long-range}}(D) + \sum\limits_{i=1}^{n} \int_{t1}^{t2}C_{\mathrm{SR}}(t, D)\;\mathrm{d}t \cdot f_{\mathrm{inf}} \cdot \mathrm{BR}_{\mathrm{k}} \cdot f_{\mathrm{dep}}(D) \cdot (1-\eta_{\mathrm{in}})` ,
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where :math:`\mathrm{vD^{long-range}}(D)` is the long-range, diameter-dependent dose computed previously, and
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@ -199,17 +199,17 @@ The method :meth:`cara.models.ShortRangeModel.normed_jet_exposure_between_bounds
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Once we have the integral of the concentration normalized by the **viral load**, we multiply by the remaining diameter-dependent properties to perform the integral over the particle diameters, including the **fraction deposited** computed with an evaporation factor of `1` (as the aerosols do not have time to evaporate during a short-range interaction):
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:math:`\int_{0}^{D_{max}}C_{SR}(t, D) \cdot f_{dep}(D) \;dD` .
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:math:`\int_{0}^{D_{max}}C_{\mathrm{SR}}(t, D) \cdot f_{\mathrm{dep}}(D) \;\mathrm{d}D` .
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Note that in the code we perform the subtraction between the concentration at the jet origin and the `long-range` concentration of viruses in two steps when we calculate the dose, since the contribution of the diameter-dependent variable :math:`f_{dep}` has to be multiplied separately in substractions:
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Note that in the code we perform the subtraction between the concentration at the jet origin and the `long-range` concentration of viruses in two steps when we calculate the dose, since the contribution of the diameter-dependent variable :math:`f_{\mathrm{dep}}` has to be multiplied separately in substractions:
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`integral_over_diameters =` :math:`((C_{0, SR} \cdot f_{dep}) - (C(t, D) \cdot f_{dep})).mean()` .
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`integral_over_diameters =` :math:`((C_{0, \mathrm{SR}} \cdot f_{\mathrm{dep}}) - (C(t, D) \cdot f_{\mathrm{dep}})) \cdot \mathrm{mean()}` .
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To perform the integral, we calculate the average since it is a good approximation of the **vD** total, provided that the number of samples is large enough.
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To perform the integral, we calculate the average since it is a good approximation of the :math:`\mathrm{vD}` total, provided that the number of samples is large enough.
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Then, we add the contribution to the result of the diameter-independent vectorized properties, which are the **dilution factor**, **viral load**, **fraction of infectious virus** and **breathing rate**:
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`vD = integral_over_diameters . exhalation_rate . inhalation_rate / dilution` :math:`\cdot f_{inf} \cdot vl_{in} \cdot (1 - η_{in})` .
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`vD = integral_over_diameters . exhalation_rate . inhalation_rate / dilution` :math:`\cdot f_{\mathrm{inf}} \cdot \mathrm{vl_{in}} \cdot (1 - η_{\mathrm{in}})` .
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Note that the multiplication over the `exhalation_rate` is done at each `short-range` interaction since the `Activity` type may be different for different interactions.
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Reference in a new issue