Chapter 6

“Closed Box Model” can be Used to Optimize Space to Reduce Virus Transmission

Methods

This study uses the COVID-19 Aerosol Transmission Estimator tool developed by Jimenez and Peng (2020) during the COVID-19 global pandemic to understand the influences of different factors in the spread viruses on in public spaces. The model was adapted for various public transportation applications. The model is utilized and studied to understand the relative impacts of various factors to the spread of viral contagions in a city bus, light rail unit, and an airplane.

Multivariable regression is used to produce a summary of the influence of each input variable for three types of public transportation vehicles. Reducing the model down to one equation is a useful tool to identify a path forward toward reducing the aerosol transmission risk by quickly identifying which factors to work on. This study also looks at a detailed example of changing transit bus airflow and filtration is analyzed to illustrate an approach and metrics that can be employed to making investment decisions to improve public health. The transmission risks are compared amongst different modes of transport. The transmission risks are also compared to vehicle accident rates to understand how viral transmission risks line up against total travel risk. Finally, recommendations are made regarding the best ways to invest to improve transit vehicle safety overall.

COVID-19 Aerosol Transmission Estimator

Jimenez and Peng (2020) developed this spreadsheet-based risk calculator as an aid to understand what factors influence COVID-19 and aerosol transmission. The COVID-19 Aerosol Transmission Estimator estimates the probability of COVID-19 virus transmission in-between vehicle occupants during a single trip. It uses numerous estimations and multiplies them by several more estimations. The absolute risk level calculated may not be accurate, but the relative improvements when comparing different scenarios are directionally correct. The different input variables are independent. If the model shows that filter A has half the risk as filter B, that one-half ratio will hold even if the number of people on the bus triples. See Table 31 for an example.

Table 31: Independent nature of input variables as it applies to risk of infection.

Model Assumptions

These assumptions are inherent to the COVID-19 Aerosol Transmission Estimator and the equations it uses:

1. The model estimates the spread of the COVID-19 Omicron B variant by aerosol transmission only.
2. It does not estimate transmission by direct physical contact.
3. It does not estimate transmission by directly coughing or sneezing on other people.

(Jimenez & Peng)

The Total Loss Rate is a combination of several factors that reduce the number of viral quanta in the cabin air. The units are Hr^-1, total airflow per hour divided by vehicle volume, or ACH. Facemasks reduce quanta and risk of infection in Equations 25 and 27.

Table 32: Processes that contribute to the removal of virus particles from the vehicle cabin air and example values for a transit bus.

The (1 – e ^–TLR * ^t) term in Equation 26 equals approximately 1 in many transit situations. These are the required conditions:

1. Each passenger is on the vehicle for at least 5 minutes.
2. The transit vehicle is equipped with an HVAC unit and the evaporator fan is running.

   Or

   A significant fraction of the windows is open, and the vehicle is moving at speed. This scenario is not considered in this report.

Equation 27: Quanta dose per person

QIPP = AQC * BR * t * (1-IME * FWF)

Where:

QIPP = Quanta inhaled per person (Quanta)

AQC = Average quanta concentration (Quanta * m^-3)

BR = Breathing rate (m³ min^-1)

t = time (h)

IME = Inhalation mask efficiency (fraction)

FWF = Fraction wearing facemasks

(Jimenez & Peng)

Equation 28: Probability of infection from a given viral dose

“Wells-Riley model”

P_t = 1 - e^-D_q

Where:

P_t = The probability of Infection. (%)

D^Q = The viral dose in quanta. (fraction)

(Buonanno et al.) (Riley et al.)

Model Input Values

The Aerosol Transmission Estimator as received had examples of a subway car, classroom, supermarket, choir practice room, and stadium. It was designed to be adaptable, and new scenarios were created for Bus

Equation 29

y = a₀ + a₁x + a₂z…

Where:

y = The dependent variable

x, z = Independent variables

a₀, a₁, a_x = _{multiple regression parameters}

(Lipson)

The tools and equations to calculate the multivariable regression parameters are designed to work on data sets. In this case, there is no data per se, so a hypothetical data set was created. 120 random variables were created bounded by the ranges shown in Table 33. The net distribution has the mean at the center of the range and a uniform distribution throughout. The sample variation information is ignored, as there is no real data. The seven parameters most applicable for public transit operators were chosen for the regression and are shown in Table 34.

Table 34: Input variable chosen for transit multivariable regression.

The equations to solve for the 7 regression parameters are large and lend themselves to being solved using statistical computer tools. The equations for a 2-parameter case are shown here for simplicity purposes. Larger parameter equations are calculated in a similar fashion. First, the distance from the mean is calculated for each measurement:

$Y=y_i-\bar{y}$ _{(Equation 30)}

$X=x_i-\bar{x}$ _{(Equation 31)}

$Z=zi-\bar{z}$ _{(Equation 32)}

Where:

$\bar{x},\bar{y}and\bar{z}=$ The sample average of each input
x_i, y_i and z_i = The individual data points
X, Y, Z = The distance from the mean to each data point
(Lipson)

Then, the parameters are calculated using these equations:

$a_1=\frac{\left({\displaystyle \sum YX}\right)\left(Z^2\right)-\left({\displaystyle \sum XZ}\right)\left({\displaystyle \sum YZ}\right)}{{{\displaystyle \sum X^2{\displaystyle \sum Z^2}-\left({\displaystyle \sum XZ}\right)}}^2}$ _{(Equation 33)}

$a_2=\frac{\left({\displaystyle \sum X^2}\right)\left(YZ\right)-\left({\displaystyle \sum YX}\right)\left({\displaystyle \sum XZ}\right)}{{{\displaystyle \sum X^2{\displaystyle \sum Z^2}-\left({\displaystyle \sum XZ}\right)}}^2}$ _{(Equation 34)}

$a_0=\bar{y}-a_1\bar{x}-a_2\bar{z}$ _{(Equation 35)}
(Lipson)

To solve the regression parameters, the online tool Multiple linear Regression Calculator from Statistics Kingdom was used (Multiple Linear Regression Calculator, n.d.-b). The regression was also done by taking the natural log of each individual parameter and the output to determine if there is an improvement of the fit when it is linearized. The goodness of fit was judged by the r² (multiple correlation coefficient) variable from the following equations (again for the 2-parameter case):

$r_{y\cdot xz}^2=1-\frac{s_{y\cdot xz}^2}{s_y^2}$ _{(Equation 36)}

$r_{y\cdot xz}^2$ = The multiple correlation coefficient of y and x on z
S_y·xz = The standard error of the estimate of y on x and z
S_y = sample standard deviation of y
(Lipson)

The standard error of the estimate is computed from:

$S_{y\cdot xz}=\frac{{\displaystyle \sum y^2-a_0}{\displaystyle \sum y}-a_1{\displaystyle \sum xy}-a_2{\displaystyle \sum zy}}{n-3}$ _{(Equation 37)}

Where:
n = number of data points
n - 3 = degrees of freedom, determined by the number of parameters estimated (the n’s)
(Lipson)

Bus Blower/Filter Upgrade Cost Calculations

The in-depth example will determine which possible investments a transit authority should select for the largest decrease in viral transmission per dollar of yearly cost. The aerosol model outlined above determines the relative reduction in transmission rate. These calculations, for the yearly cost of making the respective upgrades to the bus, were outlined earlier. Table 35 shows how the yearly fuel costs were calculated for the different options (shown in right column). The example shown is for OEM forward curved blowers in the base configuration.

Table 41 shows a comparison of the relative risk multiplier for each type of vehicle.

Regression Approximation of Model

The output formula from the multivariable regression procedure described previously is shown in Eqn 38. The details of the input variables are laid out in Table 42 below.

Equation 38:

Y = 0.0002750 - 0.000000242 * (X1) - 0.000000090 * (X2) - 0.0002828*(X3) - 0.0003129 * (X4) - 0.000000958 * (X5) + 0.000009865 * (X6) + 0.000009830 * (X7)

The multiple correlation coefficient r² for Equation 38 was only 0.65, which implies the model is not well correlated to a linear model. This is not surprising because of the logarithmic relationship of the number of virus particles inhaled and the risk of infection. A procedure was then employed that took the natural log (ln) of each parameter to yield the regression fit shown in Equation 39. This improved r² to 0.95. This suggests a good correlation.

Equation 39:

Ln(Y) = -12.46 - 0.4365 · Ln(X1) - 0.1401 · Ln(X2) - 0.2256 · Ln(X3) - 0.2800 · Ln(X4) - 0.0781 · Ln(X5) + 1.121 · Ln(X6) + 1.0461 · Ln(X7)

Equation 39 was then manipulated algebraically to put the final regression fit into a more usable format shown in Equation 40.

Equation 40

Y = 0.000004051 · X1^-0.4400 · X2^-0.1401 · X3^-0.2040 · X4^-0.2799 · X5^-0.07777 · X6^1.118 · X7^1.046

Table 42 provides the details for each input parameter and their respective units. The basic estimation parameter column is compiled from Equation 38 and the improved estimation column is from Equation 40. The final column shows the percent reduction in risk for a 10% increase in the given input parameter (negative is good). This provides a relative sensitivity value for each parameter. Note that these values

Engineering Improvements

From a practical engineering standpoint, the most likely way to make an impact on viral transmission is to modify the bus HVAC system. This section will focus on transit bus application and examine several scenarios that modify the bus HVAC to reduce transmissions rates in the most cost-effective manner. Three scenarios are considered for Bus HVAC systems with:

1. Forward curved evaporator blowers without speed control using improved filtration.
2. Replace the forward curved blowers with new ones that offer more effective airflow and improve filtration.
3. Backward curved evaporator blowers with speed control and increase the blower speeds / airflow.
4. Backward curved evaporator blowers with speed control – changing blowers speeds and improving filtration.

The COVID-19 Aerosol Transmission Estimator tool was used to calculate the transmission risks based on varying the recirculation airflow and filter efficiency based on the different scenarios examined. A basic model was developed to determine the influence of the blower speeds and HVAC system efficiency on bus fuel economy. Costs were estimated from available online resources and are outlined elsewhere.

Transit Bus Scenario 1 – Forward Curved Blowers without Speed Control: Improve Filtration

This first scenario assumes that a given rooftop transit bus has the more compact forward curved blowers installed that do not have variable speed. The issue that comes up with this approach is that the better filter will increase static pressure drop in the HVAC system. This will reduce the recirculation airflow in the bus and reduce the potential amount of air that circulates through the filter, reducing the overall filter system effectiveness. The net improvement in filtration and viral transmission will therefore be net of these offsetting parameters.

This scenario changes from a MERV 8 filter to a higher efficiency MERV 13 filter in the bus HVAC system. The MERV rating refers to a filter’s ability to capture larger particles between 0.3 and 10 microns (µm). The relative ability of the filtration level of these two ratings is shown in Table 43. The MERV 13 filters are much more effective at filtering out virus particles, which are often in the .5 - 10 Micron range.

Table 43: Filtration improvement used for scenarios 1, 3 and 4. (US EPA). The MERV 13 filters will be much better at filtering out relatively small virus particles.

This scenario uses the more effective MERV 13 in place of MERV 8 filters that increase the flow resistance and lower the airflow by 10%, as described in the relationship between filter pressure drop, IAQ, and energy consumption in rooftop HVAC units. (Zaatari et al). The bus fuel consumption will increase somewhat because the reduced airflow will lower the HVAC system’s efficiency. (Zaatari et al). The cost of the filters was estimated from the information presented elsewhere. Refer to Table 44 to understand the impact of this change scenario.

Table 44: Cost and Risk impact of changing from a MERV 8 to MERV 13 filter in a bus with forward curved, single speed blowers.

The table shows that the yearly cost of running the transit bus will go up $176 due to increased filter cost, more filter changes and lower HVAC efficiency impacting bus fuel economy. The Risk Ratio parameter is the relative risk of infection from the upgrade based on the Aerosol estimator tool. The risk is reduced by 27%. The final row shows the yearly cost of a 10% risk reduction with this approach. This is a way to judge the cost effectiveness of the different approaches.

Transit Bus Scenario 2 – Forward Curved Blowers without Speed Control: Increase Airflow by Replacing the Blowers and Improve Filtration.

This second scenario changes out the forward inclined blowers to gain back the airflow that was lost because of the filter upgrade. Increasing the airflow by 10% was also examined, as shown in Table 45. It includes the cost of installing six new blowers per bus depreciated linearly over 6 years.

Transit Bus Scenario 3 – Backward Curved Blowers with Speed Control: Increase Airflow by Speed up the Blowers.

This scenario uses another common evaporator blower configuration on transit buses: backward curved blowers with speed control. The approach here will be to increase the fan speed (and airflow) by 10% and 20% to understand to understand the reduction that is possible without adding better air filters. Increasing the air circulating through the existing filter will lower the number of viral particles in the bus. Speeding up the fans should improve the HVAC cooling potential slightly. However, since the fan laws dictate that the power consumed by a blower goes up with the cube of the speed and the volume flow is only linear, the net affect will be more power consumed from the bus engine. See the equations below.

Equation 41: First Fan Law: Volume Flow
$Q_2=Q_1\left(\frac{V_2}{V_1}\right)$

Equation 42: Third Fan Law: Power

$P_2=P_1{\left(\frac{V_2}{V_1}\right)}^3$

Where:
Q₁ and Q₂: are the volumetric flow from the fan before and after the speed change (CFM)
V₁ and V₂:are the fan velocity before and after the change (RPM)
P₁ and P₂:are the fan power consumption before and after the speed change (Watts)
(Admin)

Table 46 shows the results of the calculations. The total cost increase is quite dramatic. The cost for a 10% risk reduction is also quite poor compared to the other scenarios covered so far.

Transit Bus Scenario 4 – Backward Curved Blowers with Speed Control: Improve Filtration and Vary Airflow

This scenario assumes that a MERV 8 bus filter is switched out to a MERV 13 filter, as was done in scenarios 1 and 2. Refer to Table 47. The middle column shows the calculations for the case where the airflow is not adjusted, and the airflow falls 10% because of the increased static resistance to the fan from the filter. The next column shows what happens when the airflow is increased back up to the baseline level. The last column shows the situation when the airflow is increased 10% beyond the baseline.

Discussion: Regression Approximation of Model

As can be seen in Figure 128, the Regression Approximation is a relatively good surrogate for each of the vehicle types.

The formula (Equation 40) is more portable than the entire Aerosol Transmission Estimator tool and has the advantage that it is aligned well with public transportation vehicles. It helps decision makers understand the specific situation by providing one formula to inspect.

Engineering Improvements

An examination of which cases yielded the highest return on investment, measured by Dollars / Year for a 10% risk improvement, shows it is most cost-effective to install better filtration and let the airflow

The risk of death for one trip in a private automobile is about .000 000 75. This is based on National Safety Council (NSC) data from 2021 and assumes traveling by car 1/2 of the days during a 75-year lifespan. Taking the estimated values from the bus baseline case, this would make public transit 2.3 times deadlier per trip, crash risk plus infection risk, than private automobile travel. This risk of death from a public transit vehicle crash is 24 times lower than a private automobile on a per mile basis, based on the same NSC data. This suggests that virus transmission will remain the most dangerous part of traveling on public transit long after the COVID-19 pandemic ends.

Other Important Factors for Public Transit

Newer, more energy efficient buses often have reduced air leaking in and out of the cabin. This will decrease HVAC power requirements but increase the risk of viral transmission.

The box mixing model used does not account for virus transmission by viruses deposited on surfaces, direct physical contact, or directly coughing or sneezing on another person. This model will underestimate the risk when these factors are added in, especially in crowded vehicles.

Age is an important parameter that is not in the model. The risk of death given that you are infected with COVID-19 is heavily age dependent. Thus, it matters if the vehicle is a school bus or a retirement home bus. A senior is at over 1000 times the risk of a 10-year-old. Parents have a 20-40 times higher risk of death than their kids. Age has a much larger effect than any currently known engineering improvements. A retirement home bus could theoretically justify spending 10 times more than a typical transit bus on aerosol virus transmission risk reduction.

Equation 43:
IFR = .000537 ·^.0534A
Where:
IFR = Infection Mortality Rate
A = Age in years
(Levin et al)

Final Thoughts

The COVID-19 Aerosol Transmission Estimator is a flexible tool that can be applied to a wide variety of applications, including classrooms, offices, supermarkets, indoor choir practice, restaurants, and vehicles. It is helpful to use multivariable regression techniques to isolate equations for specific applications to understand which variables are most important. This study has shown that there is a clear sequence of upgrades that minimizes the spread of aerosol viruses in public transportation vehicles for several levels of investment. The sequence is first upgrade to a MERV13 or better filter, second increase airflow, and third increase route frequency.