Corrigendum: Covid 19 and beyond: a procedure for HVAC systems to address infectious aerosol illness transmission

A Corrigendum on
Covid 19 and beyond: a procedure for HVAC systems to address infectious aerosol illness transmission

by Walkinshaw DS and Horstman RH (2023) Front. Built Environ. 9:999126. doi: 10.3389/fbuil.2023.999126

In the published article, there were several errors made in the text:

Two definitions were not properly spaced.

A correction has been made to Inhalation equations, Paragraph 3. The definition sentences previously stated:

“N_T = number of occupants in the space t = occupancy time after the exposed group including the infector enter the space”

The corrected definition sentences appear below:

“N_T = number of occupants in the space

t = occupancy time after the exposed group including the infector enter the space”

The definitions of r_o and P are not provided.

A correction has been made to the Infection Equations following Equation 14.

$r_o=\left(1-\frac{P}{P}\left(1-e^{-D\left(\frac{{QD}_{50}}{{HID}_{50}}\right)}\right)\right)$

The definitions to be added after Equation 14 appear below:

$r_o=\left(1-\frac{P}{P}\left(1-e^{-D\left(\frac{{QD}_{50}}{{HID}_{50}}\right)}\right)\right)(14)$

where

$r_o$ = reproduction rate, the average number of secondary transmissions per infectious person

$P$ = disease prevalence, the fraction of the population that is shedding infectious aerosols at the rate used in calculating dose, D

Missing words, small p and wrong equation number.

A correction has been made to Designing HVAC Systems, Paragraph 1, missing phrase introducing the equation; variable p should be cap P; Equation 14 should be Equation 15. This sentence and the following equation previously stated:

… for the average infectious period of the index.

$\mathrm{p}=1/\text{NT}$

The corrected variable name and equation number appear below:

“for the average infectious period of the index, the disease prevalence at the beginning then is”

$P=1/N_T$

A variable definition missing.

A correction has been made to Designing HVAC systems, Paragraph 5. This sentence previously stated:

“…maximum reproduction number either in the setting…”

The corrected sentence appears below:

“…maximum reproduction number r_o either in the setting…”

The caption of Table 2 requires more information. The caption previously was: Table 2 Outdoor air exposure estimates”

The corrected Table 2 caption appears below:

“Table 2 Occupancy experience example input data for four settings”

There was an error in the below factor, which originally stated: “then by a factor (i.e. 1.127) to obtain.” The corrected factor appears below

“then by a factor (i.e. 0.898) to obtain.”

The below sub-section name should have been a higher level:

A correction has been made to Design of HVAC systems, Equal reproduction and Equal reproduction and local prevalence.

The equation number in text is incorrect

A correction has been made to Designing HVAC systems, Equal reproduction, Paragraph 8. This sentence previously stated:

“…using Equation 15 where n_o = 1.”

The corrected sentence appears below:

“…using Equation 14 where:

$n_o=P{N}_T$

$r_o=\frac{n}{n_o}\sim {(1-\mathrm{e}}^{-D\left(\frac{{QD}_{50}}{{HID}_{50}}\right)})\mathit{P}$

$\mathrm{D}=\frac{D_G}{N_S}\sim \frac{Q_B{Pq}_G}{\left(VE\right)Q_p}\left[t+\frac{v_o}{\left(VE\right)Q_p}\left(e^{-\left(\frac{\left(VE\right)Q_p}{v_o}\right)t}\right)-1\right]$

The equation number and lack of capitalization for p is incorrect.

A correction has been made to Designing HVAC systems Occupancy experience example. The equation number previously and variable P was:

$r_o\sim \left[1-e^{-\left(\frac{Q_{B}P{q}_{n}t}{Q_p+Q_f}\right)}\right]/\mathrm{p}(15)$

The corrected equation number and variable P appears below:

$r_o\sim \left[1-e^{-\left(\frac{Q_{B}P{q}_{n}t}{Q_p+Q_f}\right)}\right]/\mathrm{P}(16)$

The sub-sub-section title for Recirculation and filter flow was place on too high a level and should have been downsized.

The equation number was incorrect:

A correction has been made to Occupancy experience example. The equation number previously was:

$Q_R=\frac{Q_f}{\eta }\text{\hspace{0.17em}cfm}/\text{person}$

The corrected equation number appears below:

$Q_R=\frac{Q_f}{\eta }\text{\hspace{0.17em}cfm}/\text{person}(17)$

The caption for Table 3 needed clarification. The caption for Table 3 as published was: “Determining the filter efficiency and the flow required”

Table 3

Table 3. Determining the filter efficiency and flow required.

The Table 3 caption as corrected appears below:

“Table 3 Occupancy experience example recirculation flow requirements using Merv 13 filters $\mathrm{\eta }=0.62$ to achieve a combined reproduction rate r_o of 2.5 for the four settings.”

Table 3 had too many decimal places and had to be expanded to include the metabolic and breathing rates that were used in the Occupancy experience example. The corrected Table 3 and it’s caption appear below:

The caption of Table 4 needed more information, and some figure were incorrect. The corrected table caption appears below:

Table 4

Table 4. Compilation of Figure 5 using Equation 18 and national prevalence P = 0.03.

The caption of Table 5 required clarification and some values in the Filter Eff column were incorrect. The corrected table caption appears below:

Table 5

Table 5. Compilation of Figure 6 using Equation 19 and a national prevalence P = 0.03.

A variable name was incorrect and the equation number is missing.

A correction has been made to Designing HVAC systems, Equal reproduction example,

$r_o=\left(1-e^{-D\left(\frac{{QD}_{50}}{{HID}_{50}}\right)}\right)/\mathrm{p}\sim \frac{D_G}{{1.443HID}_{50}}$

The denominator variable should be cap P and the equation number is (19). The corrected equation appears below:

$r_o=\left(1-e^{-D\left(\frac{{QD}_{50}}{{HID}_{50}}\right)}\right)/\mathrm{P}\sim \frac{D_G}{{1.443HID}_{50}}(19)$

There was an error missing equation number.

A correction has been made to Designing HVAC systems, Equal reproduction and local prevalence example

The equation as it now stands

$\left[r_o=N_S\left({1-e}^{-D\left(\frac{{QD}_{50}}{{HID}_{50}}\right)}\right)\right]$

The equation corrected

$\left[r_o=N_S\left({1-e}^{-D\left(\frac{{QD}_{50}}{{HID}_{50}}\right)}\right)\right](20)$

The caption of Table 6 needed more information, and some of the figures presented were inaccurate. The corrected table caption appears below:

Table 6

Table 6. Compilation of Figure 7 using Equation 20 and local prevalence P = 1/N_T for various settings for a reproduction rate r_o = 2.5

The caption of Figure 5, 6 and 7 required additional detail. The corrected titles appear below:

Figure 5 Equal reproduction example using Equation 18 and national prevalence P = 0.03 for ventilation designed for r_o = 2.5.

Figure 6 Equal reproduction example using the Wells-Riley approximation Equation 19 and national prevalence P = 0.03 for ventilation designed for reproduction r_o = 2.5.

Figure 7 Equal reproduction example ventilation requirements for various settings to achieve r_o = 2.5 using Equation 20, the data in Table 6 and local prevalence P = 1/N_T (Equation 15).

The references to Tables 4, 5, Figure 5 and Figure 6 were missing. A corrections have been made to

The missing sentence below has been added to Designing HVAC systems, Equal reproduction example. :

“Equation 18 is used in calculating the recirculation flow requirements shown in Table 4 and Figure 5 for the various settings.”

And a correction has been made to the section Occupancy experience example. The figure and table reference previously was:

Figure 5; Table 4

The corrected reference appears below:

Figure 6; Table 5

The paragraph heading Occupancy Experience was incorrectly located.

A correction has been made to Occupancy Experience. The heading was previously before “This procedure sets a national….” and has been move to before “During COVID-19 Linka et al.

The derivation of Equation 20 was not fully explained.

A correction has been made to missing equations to explain the derivation of Equation 20.

${\mathit{r}}_{\mathit{o}}={\mathit{N}}_{\mathit{s}}\left({\mathbf{1}-\mathit{e}}^{-\mathit{D}\left(\frac{\mathit{Q}\mathit{D}\mathbf{50}}{\mathit{H}\mathit{I}\mathit{D}\mathbf{50}}\right)}\right)$

The missing equations follow.

The possibility of zero infection follows a Poisson distribution:

$Poisson=\frac{e^{-\lambda }{\lambda }^x}{x!}$

For zero infection, x = 0

${\lambda }^x={\lambda }^0=1$

$x!=0!=1$

Probability of zero infections

$Poisson=e^{-\lambda }$

Probability of infection for susceptible individual:

$r={1-e}^{-\lambda }$

Risk for occupant i:

$r_i={1-e}^{-{\lambda }_i}$

Total risk for all occupants:

$n={\displaystyle \sum _{i=1}^{N_s}}\left({1-e}^{-{\lambda }_i}\right)$

Total risk for all occupants assuming a perfectly mixed volume:

$n=N_s\left({1-e}^{-\lambda }\right)$

Lambda is the quantity of the statistical unit inhaled for i^th individual over time:

${\lambda }_i={\displaystyle \underset{0}{\overset{t}{\int }}}{Q}_B{C}_i\left(t\right)\partial t$

$Q_B=\text{cfh\hspace{0.17em}}breathingcouldvarywithtimeand\text{\hspace{0.17em}individual}$

$C_i\left(t\right)=\text{concentration\hspace{0.17em}}ofquantainaircouldvarywithtimeandlocationquanta/ft3$$t=\text{hours}$

${\lambda }_i=\text{quanta\hspace{0.17em}}inhaledbyoccupantiovertimeperiontatlocationi$

$C_i\left(t\right)=C_{V,i}\left(\frac{QD50}{HID50}\right)$

$C_{V,i}=\text{concentration\hspace{0.17em}}ofvirusinaircouldvarywithtimeandlocationvirus/ft3$

$QD50=\text{quanta\hspace{0.17em}}dosethecausesdiseasein50\%ofthegroup,0.693quanta:$

$\frac{n}{N_s}=0.5=\left({1-e}^{-\lambda }\right)=\left({1-e}^{-0.693}\right)$

$HID50=\text{virus\hspace{0.17em}}dosethatcausesdiseasein50\%ofthegroup$

${\lambda }_i={\displaystyle \underset{0}{\overset{t}{\int }}}{Q}_B{C}_i\left(t\right)\partial t={\displaystyle \underset{0}{\overset{t}{\int }}}{Q}_B{C}_{V,i}\left(\frac{QD50}{HID50}\right)\partial t$

${\displaystyle \underset{0}{\overset{t}{\int }}}{Q}_B{C}_{V,i}\partial t=D$

For the transient case, the virus dose is:

$D=\frac{Q_B{n}_o{q}_n}{Q_T}\left[t+\frac{1}{ACH}\left(e^{-\left(ACH\right)t}-1\right)\right]$

The reproduction number shows the number of infections from an infector, no = 1:

$D=\frac{Q_B{q}_n}{Q_T}\left[t+\frac{1}{ACH}\left(e^{-\left(ACH\right)t}-1\right)\right]$

$q_n=\frac{virus}{hr}frominfector$

$N_s=\text{number\hspace{0.17em}}ofsusceptibles$

$n=r_o=N_s\left({1-e}^{-\lambda }\right)=N_s\left({1-e}^{-D\left(\frac{QD50}{HID50}\right)}\right)$

For steady state case t > 5 air changes:

$D=\frac{Q_B{q}_n}{Q_T}\left[t+\frac{1}{ACH}\left(e^{-\left(ACH\right)t}-1\right)\right]=\frac{Q_B{q}_n}{Q_T}t$

$r_o=N_s\left({1-e}^{-\frac{Q_B{q}_n}{Q_T}t\left(\frac{QD50}{HID50}\right)}\right)$

${\mathbf{r}}_{\mathbf{o}}={\mathbf{N}}_{\mathbf{s}}\left({1-\mathbf{e}}^{-\mathbf{D}\left(\frac{\mathbf{Q}\mathbf{D}50}{\mathbf{H}\mathbf{I}\mathbf{D}50}\right)}\right)$

In terms of quanta:

$q_G=\frac{QD50q_n}{HID50}$

You get the Wells-Riley equation:

$r_o=N_s\left({1-e}^{-\frac{Q_B{q}_{G}t}{Q_T}}\right)$ (20)

Extra clarification was need in the section Occupancy experience, following “…in that setting.

The additional sentence is: “Steady state occupancies only are considered ”

Clarification was needed in the section Equal reproduction, following “and transient”

The below was added: “(leaving or entering)”.

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