Sensitivity Analysis for Calculating the Risk of Viral Infection

Alessandro Cammarata; Giuliano Cammarata

doi:10.5391/IJFIS.2023.23.4.448

Original Article

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International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(4): 448-464

Published online December 25, 2023

https://doi.org/10.5391/IJFIS.2023.23.4.448

Sensitivity Analysis for Calculating the Risk of Viral Infection

Alessandro Cammarata¹ and Giuliano Cammarata²

¹Department of Civil Engineering and Architecture, University of Catania, Catania, Italy
²Department of Electrical Electronic and Computer Engineering, University of Catania, Catania, Italy

Correspondence to :
Alessandro Cammarata (alessandro.cammarata@unict.it)

Received: August 9, 2021; Revised: November 22, 2021; Accepted: December 22, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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Abstract
1. Introduction
2. Methodology
3. Verification of Sensitivity Analysis with Case Studies
4. Conclusion for the Sensitivity Analysis
Conflict of Interest
Figures
Tables
References
Biographies

Computational relationships of infection risk depend on some parameters that are not always precisely defined but are subject to uncertainties due to measurement difficulties and the estimation of hourly production of infection quanta. Calculation results, consequently, cannot be predictive but are subject to epistemic variability. Several recent studies for severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), infection cases have demonstrated the significant influence of initial parameter uncertainties in the inverse calculation for determining the quanta/h that resulted in the infections. In the first case, the researchers found such high variability in some of the calculation parameters that the final calculation results of infection quanta varied over an extensive range. Thus, the conviction that the methods of calculating the probability of infection are reliable and, above all, deterministic does not justify the actual results that show an evident uncertainty also derived from the mathematical definitions of the various calculation methods. In this paper, a simple method for resolving epistemic indeterminacies is presented. The method employs the concepts of sensitivity analysis by fuzzy arithmetic applied to epistemic computational parameters. In addition, analytical developments for some well-known and widely used contagion risk prediction laws are provided.

Keywords: SARS-CoV-2, Risk of contagion, Sensitivity analysis, Fuzzy arithmetic

1. Introduction

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Abstract
1. Introduction
2. Methodology
3. Verification of Sensitivity Analysis with Case Studies
4. Conclusion for the Sensitivity Analysis
Conflict of Interest
Figures
Tables
References
Biographies

As the coronavirus disease 2019 (COVID-19) pandemic continues with a new and more contagious delta variant, numerous studies to calculate the risk of contagion by long-distance airborne transmission are being conducted. Unfortunately, models based on methodologies usually used for epidemiological analyses, such as the dose-effect method, flanked classic calculation models.

There is particular attention to determining, by an inverse method, the hourly production of quanta of infection. The latter is a quantity derived from the studies of Wells [1–3], partly physical and partly epidemiological, which characterizes the ability of a pathogen to infect susceptible subjects, i.e., not yet infected, present within an environment.

The analysis of the 2020 scientific publications on severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infections has shown great indeterminacies in the prediction of quanta rates and all the other parameters that become part of the calculation models, such as the number of hourly changes of external fresh air or the breathing rate.

They also included as calculation parameters the deposition losses of ground droplets and the inactivation losses due to the loss of viability in time of the virus present in the aerosol.

All the corrective parameters are homogeneous at an equivalent number of hourly changes. The set of biological parameters can increase the number of fundamental air changes, here denoted with N_r, by 0.6 to 2.3, to significant, and often unrealistic, values for calculating the probability of infection and are considered in the Fisk and Nazaroff correction [4–6].

Airborne contagion from any pathogen depends on several factors and circumstances such as

1. The spatial distribution of the pathogen.
2. The likelihood of the pathogen entering the lower parts of the respiratory tract.
3. The probability of generating contagion.
4. Calculation of how many infection rates.
5. Methods of calculating the risk of infection.

These have a decisive weight in the assessment of the risk of infection and the correct evaluation of the quanta of disease, i.e., the reverse problem.

In this paper, a method of sensitivity analysis by fuzzy arithmetic applied to epistemic computational parameters is presented. The proposed sensitivity analysis can be applied to one or more parameters simultaneously by determining all consequences of parametric variations on the probability of contagion in a single calculation. The method can be employed both for direct analysis (given the initial parameters we determine the likelihood of contagion) and inverse (given the initial parameters and the frequencies of contagion, we choose the quanta of infection, quanta/h). Furthermore, the mechanical and plant design method can be used to analyze the consequences of plant parameters (such as the number of air changes) to minimize the risk of airborne infection. For example, controlled mechanical ventilation or other systems can be employed to reduce the hourly production of quanta (see Fisk and Nazaroff corrections, through high-efficiency filters, UV-C lamps, ionizing systems such as cold plasma).

Using the fuzzy arithmetic, three curves of the probability of contagion are directly obtained: a central curve corresponding to the nominal value (crisp) of the parameters and the other two corresponding to the variations in time more and less than the central curve. The higher curves are defined as pessimistic propensity because they increase the risk of contagion concerning the central curve. Conversely, the lower curves have an optimistic propensity.

Therefore, the spread between the two extreme curves identifies an area of variability between the minimum and maximum values of the probabilities of contagion as time changes. Within these spreads, the most probable value of the risk of contagion can be found. The wider the area determined by the spread, the greater the indetermination of the probability of real contagion value. Therefore, the sensitivity analysis determines a change of mentality in dealing with the problem of estimating the likelihood of airborne infection by any pathogen and establishes new rules for risk assessment and acceptance.

2. Methodology

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Abstract
1. Introduction
2. Methodology
3. Verification of Sensitivity Analysis with Case Studies
4. Conclusion for the Sensitivity Analysis
Conflict of Interest
Figures
Tables
References
Biographies

To solve the problem of the variability of the calculation parameters, a fast but effective method based on the simplified fuzzy set is proposed, that is, the fuzzy arithmetic that reduces the complexity of the formulation of the modified calculation equations.

The fuzzy arithmetic sets and the most usual algebraic operations used within the computational equations of contagion risk are employed. For all fuzzy sets, at each time, the following values are defined: the central value (crisp), the left value, given by the difference between the central value and the variability of the chosen parameter, and the right value, given by the sum between the central value and the variability of the parameter. This simplification assumes the assumption of a triangular type of the membership function, which considerably reduces the definition of fuzzy sets [7]. By substituting each value of the fuzzy sets into the computation equations, respecting the rules of fuzzy arithmetic, we obtain three new relations referring to the three values of the fuzzy sets. Plotting together the three curves, a spread between the lower curve, defined as optimistic because it relates to lower risk values than the central curve (crisp) is observed. The upper curve is pessimistic because it relates to higher values than the higher the central curve (crisp).

This widening of the probability curves determines an area of indetermination of the calculated probability that visually illustrates the effects of the variability of the calculation parameters. To verify the proposed solution, we will analyze two example cases cited in the literature among the most characteristic of the COVID-19 pandemic. The approach with fuzzy arithmetic allows to obtain directly, with a single calculation, the results obtained experimentally with more complex elaborations.

2.1 Determining the Risk of Infection

The determination of the quanta emission rate (ERq; quanta/h) [8], for any method of calculating the risk of infection is rather complex. Moreover, in some case studies reported in the recent literature [9, 10] the results were scattered.

For the Skagit valley chorale [11], Buonanno et al. [8] utilized the dose-effect method proposed to lead to a dispersion of ERq from 550 to 1, 510 quanta/h with a mean value of 970 quanta/h. Thus, all traditional calculation methods are valid for statistical prediction but not as deterministic methods.

The parameters contained in the proposed calculation reports suffer from the same basic assumptions:

• Uniform air distribution (perfect mixing).
• Temporal stability of respiratory activity (m³/h).
• The constancy of ERq during the calculation period.
• Stationary conditions achieved for ERq.
• The constancy of the number of hourly changes (/h).

Nobody performed a sensitivity analysis of the airborne risk of infection to these parameters. Therefore, in regular use, calculation predictions should be considered with great caution.

Mathematical models provide correct results only if specific input data are provided. If we assume known and constant parameters that are not constant at all, such as the ERq, the hourly changes of respiratory activity, the ventilation airflow rate (which varies from the nominal design conditions during the day), the type of phonation (speech, singing), hence we can’t expect that the models for calculating the risk of contagion provide correct values.

2.2 Methods for Calculating the Risk of Infection

Reference is made to the most widely used expressions both because they are historically best known and because they are easier to use. The procedures for sensitivity analysis will be applied to the Wells-Riley [12], Gammaitoni-Nucci [9], and Buonanno-Morawska-Stabile [8] expressions. This choice is motivated by two reasons: these are the most widely used expressions and are easy to implement. There are other more complex computational methods [6, 13], that use computational assumptions based on CO₂ production per exhalation (Rudnick and Milton’s method). The latter are not discussed in this paper.

All computational relationships are expressed in the mathematical form of the Poisson distribution

(1)P=1-e-μ,

which nicely interpolates the distribution of contagions, according to Wells’ formulation. Some fundamental quantities are included in the exponent, such as

• The airflow rate (m³/h) of breathing of the individual of ambient air, usually indicated through the letter p.
• The initial number of infected, usually one, is denoted by the letter I.
• The exposure time (h) from entry into the environment, denoted by t_esp.
• The total fresh air flow rate, Q (m³/h), calculated as the product of the number of changes per hour, N_r (/h), and the volume, V (m³), of the room.
• The number of infection quanta, (quanta/h), produced by the infected inside the environment.

2.2.1 Wells-Riley

The Wells-Riley expression [12], is the oldest and most used and is expressed as follows:

(2)P=(1-e(-I·q·p·tNr·V))·100,

where the following quantities can be easily recognized

• I number of infected, usually set equal to 1;
• q quanta rate (quanta/h);
• p breathing air flow rate (m³/h);
• N_r number of air changes (/h);
• V volume of the environment (m³);
• t calculation time (h).

2.2.2 Gammaitoni Nucci method

Gammaitoni and Nucci [9] introduced a novel computational methodology that overcomes the stationarity assumption of the Wells-Riley expression. The latter is

(3)P=(1-e-pqIVCdt+e-Cdt-1Cd2)·100.

The exponent contains the following parameters

• p lung activity (m³/h);
• q number of quanta per hour (quanta/h);
• I number of initial infected;
• V volume of the environment (m³);
• C_d dispersion coefficient, equivalent to the number of changes per hour, N_r (/h);
• t calculation time (h).

2.2.3 Buonanno-Morawska-Stabile method

The method used in [8] is different from the previous ones and is called the dose-response method [14]. Starting from the temporal distribution of quanta, determined by the Gammaitoni-Nucci expression

(4)n (t,ERq)=n0·e-IVRR·t+ERq·IIVRR·V(1-e-IVRR·t)

being IVRR the number of air changes per hour (that is, N_r), and n the quanta concentration (quanta/m³), ER_q is the emission rate (quanta/h). The dose definition of quanta of infection absorbed in time is given by

(5)Dq(ERq)=IR∫0Tn (t) dt (quanta),

where IR is the respiration rate (m³/h) and D_q is the quanta dose. The probability of contagion is finally expressed as

(6)P=(1-e-Dq)·100.

2.2.4 General considerations for infection risk calculation reports

All calculation methodologies share the following features:

1. The probability of individual contagion decreases as the volume of the environment increases.
2. The likelihood of individual infection decreases as the number of hourly changes in fresh outside air increases.
3. The likelihood of individual infection increases with increasing exposure time from the time of entry into the environment.

The first observation depends on an architectural variable, the volume of the environment V . The second observation depends on the external air ventilation. The third observation is that P depends on the exposure time.

2.3 Variability of Calculation Parameters

Variabilities in the computational parameters are present in all the expressions above reported and generally have the characteristics examined in detail below. The primary assumption underlying the sensitivity analysis proposed here is that the parameters are epistemic and not random.

2.3.1 Number of quanta/h

The number of quanta/h is generally not very precise to define and represents the most critical parameter for calculating the risk of contagion. This variability derives from biological characteristics, the activity performed by the infected person, the type of emission (breath, speech, cough, sneeze), the position of the infected person about other susceptible subjects, the physical conditions, and the age of the infected person. The action of this parameter is positive, i.e., as q increases, the probability of contagion increases and vice versa.

2.3.2 Respiration rate, p

Respiration rate is biologically defined and depends on the subject’s age, activity, metabolism, and physical conditions. Conventional values are employed, typically 0.48 m³/h for a sedentary adult. For other activities, it is necessary to know the metabolism, expressed in Met = 56 W/m², to calculate the correct value. Also, this parameter is not very precise, and its variability has a non-negligible weight on calculating the risk of infection. The action of p is positive and, therefore, as p increases, the probability of contagion P increases too and vice versa.

2.3.3 Number of air changes per hour, N_r

The number of air changes per hour, N_r is defined during the design phase for systems with mechanical ventilation but is entirely undefined in the case of natural ventilation that depends on external environmental conditions. Even in the presence of mechanical ventilation, N_r can vary according to the elaborate flow rate of air handling units (AHU), especially if they have a variable flow rate (VAV). The airflow rate in a room, served by a centralized aeraulic network, mainly depends on the flow rate determined by the fan, the pressure drops of the distribution network, the network balancing, the state of the filters of the terminals, the room regulation, the state of maintenance of the ducts and their losses.

For forced ventilation, knowledge of N_r is also uncertain. Compared to the other parameters, N_r is erroneously considered the most reliable one. The action of N_r is adverse, i.e., as N_r increases, the probability of contagion decreases and vice versa.

Controlling the number of outdoor air changes per hour for physiological changes is critical to reducing the risk of infection, as demonstrated for passive protection systems [15].

2.3.4 Influence of quanta rate reduction systems

Following the directions of Nicas and Fisk and Nazaroff [4, 16], it is possible to account for quanta rate reduction schemes by considering their effects as an equivalent increase in the number of hourly changes, N_r

(7)L=kventilation+kfilter·efilter+kUN·eUV+kdep+kinert,

where the subscripts refer to

• k_ventilation is the number of changes per hour by ventilation, N_r;
• k_filter is the equivalent number of changes per hour due to filtration;
• k_UV is the equivalent number of changes per hour due to UV radiation;
• k_dep is the equivalent number of air changes per hour due to droplets;
• k_inert is the equivalent number of hourly changes due to reduction in viral viability.

Each term may or may not be present, except the first one. The sum of all terms greatly increases the value of N_r. The previous expression can be put in the form

(8)L=Nr+FN,

where the Fisk and Nazaroff term holds

(9)FN=kfilter·efilter+kUN·eUV+kdep+kinert.

In [17], the methods for calculating each term in Eq. (9) are explained.

2.3.5 Sensitivity analysis for the calculation of the probability of infection

In general, sensitivity analysis allows the analysis of the trend of different measured quantities when an independent model parameter varies within a specified range. This mathematical tool finds applications in various areas of engineering and finance to study the so-called possible scenarios when the desired parameters vary. The classical sensitivity analysis is based on the calculation of the partial derivative of the function Y concerning a parameter X around a given value x₀

(10)∂Y∂X|x0.

If there are many parameters, partial derivatives are written for each parameter around its working point x_{0_i}.

(11)∑N(∂Yi∂Xi|x0i).

The solution of the system of partial differential equations is not simple and often requires considerable computational resources. In the case of the probability of contagion P, one should find the following partial derivates

(12)∂P∂q|q0;∂P∂p|p0;∂P∂Nr|Nr0.

The calculated values indicate the variation of P as the parameters vary around the nominal value. The method, although general and effective, is complex, especially when there are many parameters to analyze.

2.4 The Fuzzy Arithmetic

The fuzzy sets define a range of variability of a parameter, and a membership function is in the form of fuzzy logic. Therefore, the complexity of the fuzzy sets can be reduced using fuzzy arithmetic. First, a fuzzy variable consisting of a central value, A (crisp), and a range of variability are defined. Therefore, a triangular membership of the type depicted in Figure 1 can be introduced.

The fuzzy arithmetic set is denoted with a compact form containing the crisp value A and the variability a : (A, a).

The fuzzy set generates the variability field (A−a, A, A+a), as depicted in Figure 1. On two arithmetic sets, we can define the following fuzzy arithmetic operations [18]. The use of arithmetic fuzzy sets involves the definition of the calculation functions of the conjugate possibilities for each value, i.e. left, center, and right. Each function generates its probability curve, and we can visualize all of them while distinguishing them with a dotted line, solid line, and dashed line.

In each abacus represented in the following cases, three families of fuzzy curves for the three values of the assumed quanta/h will be considered. Each family will be characterized, from the most extensive q toward the smallest, by red, green, and blue colors.

2.4.1 Sum

The sum of the two sets defined above is given by

(13)(A,a)+(B,b)=(A+B,a+b).

So, the sum set has a crisp value given by the sum of the two crisp values A + B while the variability becomes the sum a + b.

2.4.2 Sum with a constant C

The constant C can be assumed to be a zero-variable fuzzy set, i.e., one has only the crisp value C : (C, 0). Therefore, the sum of a fuzzy set and a constant hold

(14)(A,a)+C≡(A,a)+(C,0)=(A+C,a+0)=(A+C,a).

2.4.3 Difference

The difference of the two sets defined above is given by

(15)(A,a)-(B,b)=(A-B,a-b).

So, the difference set has a crisp value given by the difference of the two crisp values A − B while. the variability becomes the difference a − b.

2.4.4 Difference with a constant C

The constant C can be assumed to be a zero-variable fuzzy set, i.e., one has only the crisp value C : (C, 0). Therefore, the difference between a fuzzy set and a constant is

(16)(A,a)-C=(A-C,a).

2.4.5 Product

The product of two arithmetic sets leads to a more complex expression

(17)(A,a)·(B,b)=(A·B+a·b,A·b+B·a).

Therefore, the central value is not only the product of the two central values A − B but to this is added the product a − b of the two variabilities and the variability is the sum of the two cross products (A − b + B − a) and therefore there is a widening of the variability (spread).

The rule for products from multiple fuzzy sets can be applied

(18)(A,a)·(B,b)·(C,c)=[(A,a)·(B,b)]·(C,c)=(A·B+a·b,A·b+B·a)·(C,c),(A,a)·(B,b)·(C,c)={[(A·B+a·b)·C+(A·b+B·a)·c],(A·B+a·b)·c+C·(A·B+a·b)}.

2.4.6 Power of a fuzzy set

For the power of a fuzzy set, the multiplication rule applies and therefore

(19)(A,a)·(A,a)=(A·A+a·a,A·a+A·a)=(A2+a2,2·A·a).

2.5 Fuzzy Arithmetic for Calculating the Risk of Airborne Contagion: Wells-Riley Method

Assume that the three fundamental parameters q, p, N_r vary and add the variability of the Fisk and Nazaroff term (FN).

The fundamental parameters can be arranged in the following fuzzy sets:

• (q, dq) the arithmetic set for quanta/h, q;
• (p, dp) the arithmetic set for the breathing rate, p;
• (N_r, dN_r) the arithmetic set for the number of hourly changes, N_r.

Therefore, three probabilities of contagion can be defined.

The expressions for the central probability, P_c, left probability, P_s, and right probability, P_d, are the following:

(20)Pc=(1-e-I·q·p·tNr·V)·100,Ps=(1-e-I·[(q·p+dp·dp)-(qdp+pdq)]·t[Nr-dNr]·V)·100,Pd=(1-e-I·[(q·p+dp·dp)+(qdp+pdq)]·t[Nr+dNr]·V)·100.

2.5.1 Case 1: Variability of q

Consider a room with dimensions of 6.0 × 7.0 × 3.1 m³, with an occupancy of 5 persons per m². We consider the nominal value of quanta rate of 10, 20, and 50 quanta/h and variability of 30%, thus (q, 0.3q). The effects of deposition, k_ep = 1.7, and inerting, k_inert = 0.4, are considered in the number of hourly changes using the Fisk and Nazaroff model. As a result, the number of hourly changes varies from 0.5 to 2.5 (/h). Figure 2 gives the results of the calculation of the probability of contagion.

Each plot shows the computational assumptions of all computational parameters and quanta/h reduction solutions. For example, in the first row, the infection probabilities appear in three colors for the three cases of N_r = 0.5, 1.0, and 2.5 (/h) versus the time of exposition. The solid line represents P_c, i.e., the standard probability P calculated without the spread of quanta/h and respiratory rate p. The upper and lower curves respectively reported in dash-dotted and dashed lines, represent P_s and P_d. Observe how P_d is higher as P increases as q + dq, and P_s decreases as P decreases with q−dq. Finally, the dotted curve above P_s represents the P_d − P_s difference in percent. The Wells-Riley method has the most conservative conditions and assumes stabilization of quanta/h production, resulting in underestimating the probability values compared to the other methods that consider a variable q.

2.5.2 Case 2: Variability of N_r

Under the same conditions as in Case 1, only the variability in the number of air changes per hour, N_r, of 25% is assumed. The corresponding fuzzy set becomes (N_r, 0.25 * N_r). Figure 3 shows the simulation results. The inversion of the two curves P_s and P_d, respectively, above and below the P_c curve can be observed. This reversal of effects is due to the negative influence on the probability of infection P, which decreases when N_r increases. Negative variability reduces the probability of contagion with values to the right of the central value. It can be noted how the three curves are close to each other due to the low value (< 0.5 /h) of N_r. In this case, the spread is modest, and the effects of variability of N_r are limited.

The three families of curves correspond to the three assumed quanta/h values.

2.5.3 Variability of q and N_r

Under the same conditions as in the previous cases, the complete case is considered, with the simultaneous variability of 30% of the quanta/h (q, 0.3*dq) and the number of air changes per hour (N_r, 0.25 * N_r). Figure 4 shows the result of the calculation.

Using the Fisk and Nazaroff model, the number of hourly changes is derived. The overall effect of the variabilities of the two parameters considers the negative contribution of N_r and the positive contribution of q.

Comparing with Figure 2 reveals a reduction of the spread of the curves. Thus, there is compensation in the increase of the probability of contagion caused by the increase by the variability of q and the decrease caused by the variability of the number of hourly changes N_r. Figure 4 reports the effects of deposition, k_dep = 1.7, and inerting k_inert = 0.4. This last observation is of particular importance when performing the inverse calculation, i.e., given the number of infected and known the number of initial susceptible persons, calculate the number of quanta/h. The high variability in the parameter L may lead to incorrect results because the variability of k_dep and k_inert reduces the calculated probability of infection.

2.6 Gammaitoni-Nucci Method

What has been done for the Wells-Riley (WR) expression also applies to the Gammaitoni-Nucci (GN) method. The fuzzy expressions for the probabilities of contagion become as follows:

(21)P=(1-e-pqIVCdt+e-Cdt-1Cd2)·100,Ps=(1-e-I·[(q·p+dp·dp)-(q·dp+p·dq)]V·(Cd-dCd)t+e-(Cd-dCd)t-1(Cd2+dCd2-2·Cd·dCd))×100,Pd=(1-e-I·[(q·p+dp·dp)+(q·dp+p·dq)]V·(Cd+dCd)t+e-(Cd+dCd)t-1(Cd2+dCd2+2·Cd·dCd))×100.

2.6.1 Case 4 for GN: Variability of q, N_r

The case of 30% variability in quanta/h, (q, 0.3 * dq) of the number of changes per hour (N_r, 0.25 * N_r) for a like-for-like comparison with previous results is shown. Figure 5 reports the results. The same coefficients as the previous method for the effects of deposition and inerting are considered.

The spread results takes into account both the variability of the parameters and the greater computational accuracy of the GN method.

The results differ from those obtained through the WR method because of the GN assumptions of nonstationary distribution of quanta/h.

2.7 Buonanno-Morawska-Stabile Method

Applying the same fuzzy arithmetic rules, the following expressions for the quanta doses of the Buonanno-Morawska-Stabile (BMS) method stand

(22)Dqc=IEmpq(e-Cct+Cct-1)/(VCc2),Dqs=IEm[(qp+dpdp)-(qdp+pdq)](e-Cst+Cst-1)/(V(Cc2+dCc2-2·CddCd)),Dqd=IEm[(qp+dpdp)+(qdp+pdq)](e-Cdt+Cdt-1)/(V(Cc2+dCc2+2CddCd)),

where

• C is the number of air changes per hour, corrected according to Fisk and Nazaroff.
• E_m is the attenuation factor of the protective mask.

In the calculation of the doses of quanta absorbed, the effects of the spread widening due to the products is considered. In this case, it happens twice: first in the product pq and then in the square of C_c. The probabilities, according to Eq. (6), are

(23)Pc=(1-e-Dqc)·100,Pd=(1-e-Dqd)·100,Pd=(1-e-Dqd)·100.

2.7.1 Case 5 for BMS: variability of q and N_r

The same previous case is repeated with the equations for the BMS method. Figure 6 shows the results. The total spread is slightly larger than the cases with the WR method. However, the results obtained are fully comparable with those obtained using the GN method.

2.7.2 Case 6 for BMS: variability of k_dep and k_inert

So far, only the fundamental parameters q and p for the sensitivity analysis are considered.

Let us analyze the influence of the Fisk and Nazaroff correction parameters, i.e., k_dep and k_inert, by repeating the previous calculation and adding a variability of 10% to these parameters: k_dep = (1.7, 0.17) and k_inert = (0.4, 0.04). The effect of the masks are not included in the results of Figure 7.

The first row of the plots shows that these two parameters increase with N_r, for example, for N_r = 0.5 from 1 to 2.6 /h to 2.65 /h and 2.935 /h for the three quanta/h values.

There are, therefore, equivalent significant increases in the number of effective air changes N_r. In addition, the FN parameters have adverse effects so that as L increases, the probability of contagion P decreases. Nevertheless, the spread in all cases is significant and comparable to that obtained in the previous cases.

3. Verification of Sensitivity Analysis with Case Studies

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Abstract
1. Introduction
2. Methodology
3. Verification of Sensitivity Analysis with Case Studies
4. Conclusion for the Sensitivity Analysis
Conflict of Interest
Figures
Tables
References
Biographies

The spread effects highlighted through the sensitivity analysis and the loss of computational accuracy due to uncertainty are employed to review two real case studies reported in the literature in 2020.

3.1 Skagit Valley Chorale Case Study

Miller et al. [11] published an article on a case of SARS-CoV-2 infection during a choral rehearsal in the Skagit Valley Chorale, near Washington, DC, that occurred on March 10, 2020.

3.1.1 The outbreak that occurred in the choir practice room

In the theater, people made hand sanitization protocols, choristers had not externalized any hugs or handshakes, and verified the limits of safe distance between singers. However, during a singing rehearsal with 61 choristers lasting 2.5 hours in the 180 m² halls with a height of 4.5 m and a total volume of 810 m³, there were 53 suspected cases of infection and two deaths. Subsequently, only 33 were confirmed as infected with COVID-19. This case produced a super spreading event, i.e., high emission of quanta of infection due to singing activity in the theater room. The researchers collected valuable data to calculate the magnitude of the quanta/h emission but encountered a long series of indeterminacies summarized in Table 1.

The values reported in Table 1 indicate a substantial variability of all calculation parameters, which raised the problem of how to solve simple and directly the inverse calculation of the number of quanta of infection that caused the epidemic. Classical calculation methods would require hundreds or thousands of iteration [19] for different combinations of the parameters with the possible dispersion of the results obtained. It was precisely this case study that gave rise to the idea of using fuzzy arithmetic to solve the problem posed directly.

There were uncertainties about HVAC (heating, ventilation, and air conditioning) plants, ventilation rates, and ambient temperatures.

The respiratory activity itself was estimated from references in scientific publications, as indicated in work.

Examination of the table indicates notable variations in the following critical computational parameters: the number of hourly changes, N_r, ranges from 0.3 to 1.0 vol/h.

Respiration rates, p, are thought to have varied between 0.22 ÷ 1.38 m³/h.

For deposition losses, we assume values (ideal but not verified) varying between 0.3 ÷ 1.0 /h.

The loss of vitality, or inactivation, of the droplets is assumed to be between hypothetical values varying between 0 ÷ 0.63 /h.

3.1.2 Quanta rate calculation

Employing the BMS, the same method used by the authors can be obtained considering the following fuzzy sets:

• quanta rate: (q, 0.4 * q);
• p (p, 0.4 * p)
• N_r (N_r, 0.2 * N_r)
• K_dep (k_dep, 0.3 * k_dep)
• K_inert (k_inert, 0.2 * k_inert)

The central values considered are, see Table 1:

• q 700, 1, 700 quanta/h;
• p 0.8 m³/h;
• N_r 0.4, 0.6, 0.8 /h;
• k_dep 0.6 /h;
• k_inert 0.4 /h

Figure 8 shows the results due to all the variability of the parameters in Table 1 in a single simulation demonstrating the computational power of fuzzy arithmetic for solving complex problems.

Two horizontal lines are shown in the individual probability plots at the values 53% and 87% for 33 and 53 infected, but only the latter case was considered compared to the original research. The spread for each quanta rate case is such that the curves overlap, resulting in a single probability area covering probability values from 0% to 98%. The values of the central curves, P_c, (at full stroke, blue for q = 700 quanta/h and green for 1, 700 quanta/h) provide

• q = 1, 700 quanta/h green color, for t = 2.5 hours the values 88.5% for 0.5 /h (central value), 86.2% for 0.6 /h (central value) and 83.7% for 0.8 vol/h (central value);
• q = 700 quanta/h color blue, for t = 2.5 hours the values 59.03 % for 0.5 /h (central value), 55.74% for 0.6 /h (central value) and 52.73% for 0.8 /h (central value).

Considering that the number of actual infected was 33, the value of 700 quanta/h is sufficiently correct.

In addition to the central value, we have spreads that range from 33% to 78%. Still, concerning the 33 infected, the analysis of the three case studies in Figure 8 shows that the most realistic ventilation conditions are with N_r = 0.6 vol/h with a variability of 0.2 vol/h bringing the range of variation from 0.48 to 0.72 air changes.

The usefulness of the sensitivity analysis is evident. In fact, through a single calculation, all the parametric variabilities have been considered obtaining indications on the spread of the probability of contagion and allowing to determine the value of the most probable quanta rate.

3.2 Guangzhou Restaurant Case Study

Guangzhou Restaurant has been the subject of two scientific publications. The first publication was July 7, 2020 [20] but previewed from April 2020, the second one published by Li et al. [10] previewed on April 22, 2020, highlights in part the errors indicated in the first publication.

3.2.1 New findings and assumptions

Initially much emphasized by virologists worldwide to demonstrate responsibility of air conditioning systems, the story tells of three families A, B, C for a total of 21 people sitting around three different tables and in line with a wall-mounted air conditioner, as shown in [6]. In addition to the three families, there were more than 68 other individuals, including employers, present in the room for a total of 89 subjects. Only 10 individuals were infected out of the 89 presents in the third-floor restaurant, all belonging to the three families A, B, and C. There is uncertainty about those infected because one believes that some contagion occurred after the restaurant was occupied. No other restaurant guests or workers were infected. The new group of researchers followed up on what the first group indicated with a thorough study to determine with certainty the causes of the contagion. The dimensions of the room were 17×8.1×3.14 = 431 m³. The tables were the large type with a diameter of 1.8 m and the small type with a 1.2 m. According to Li et al. [10], the leading cause, proceeding by excluding the other types, was to be found in the poor ventilation of outside air and the concomitance of the extractors in the windows stopped during the event. In addition, there was a poor ventilation flow rate poorly distributed over the whole room of the restaurant, making the conditions even more critical in the strip of three tables whereon had a ventilation flow rate of 0.75 L/s per person. In contrast, in the rest of the room, the flow rate was 1.04 L/s per person. Thus, the researchers concluded that long-distance transmission of infection via aerosols was not a specific cause, but the deficient ventilation was undoubtedly enhanced by fresh outside air.

3.2.2 Quanta rate calculation

Considering that the number of infected is 10, two assumptions for calculating the probability of infection were introduced

1. First, 21 commensals were seated around the three tables in the ABC area, considered separate from the rest of the room. It is considered a volume equal to 1/3 of the entire room, 144 m³.
2. Eighty-nine people in total were present in the restaurant room, including the ABC and non-ABC areas, for a total volume of 432 m³.

The corresponding infection rates are

• For zone ABC only: (10−1)/(21−1) ×100 = 45.00%;
• For the entire restaurant: (10 − 1)/(89 − 1) × 100 = 10.22%

BMS method was still used with the following fuzzy sets: quanta rate: (q, 0.3q);

• p (p, 0.05 * p)
• N_r (N_r, 0.1 * N_r)
• k_dep (k_dep, .0 * k_dep)
• k_inert (k_inert .0 * k_inert)

The values for k_dep and k_inert are null because of the limited time the three families were in the restaurant (approximately 1 hour). Furthermore, new studies excluded fomite formations on the tables [20].

3.2.3 Case a: ABC zone only

Central values based on the literature, are

• q 20, 250, 300 (quanta/h).
• p₀ 48 (m³/h).
• N_r₀ 0.56, 0.65, 0.77 (/h).
• k_dep₀ 0 (/h).
• k_inert₀ 0 (/h).
• V 144 (m³).

Figure 9 shows the calculation for the case.

The results are summarized as follows: Zone ABC: for a dwell time varying between 0.9 and 1.1 hours (see black color reference lines) and for a percentage of infected of 45%, the hourly production of quanta that corresponds well, for all numbers of hourly changes, is equal to 230 (quanta/h) with a variability of 30% and with N_r = 0.66 (/h) with a variability of 10%. Thus, the spread corresponding to q = 250 (quanta/h) can reach the value of 40% probability of contagion, considering the uncertainties of the actual infected in the restaurant. Given the results obtained by Liu et al. [10] and their study on airflows using tracer gases, one believes that the area called ABC is isolated from the rest of the room and that the calculated quanta rates are equal to 250 (quanta/h).

The high number of infected, 9, is justified by short-distance contagion from aerosols, as hypothesized by Liu et al. [10] in their publication.

3.2.4 Case b: Entire restaurant room

The hypothesis of considering the whole room, both for the above considerations on air distribution and that none of the people present in the non-ABC zone became infected appears purely academic. For heuristic purposes, the study has been carried out using the following parameters:

• q 150, 170, 200 (quanta/h).
• p₀ 48 (m³/h).
• N_r₀ 0.56, 0.65, 0.77 (/h).
• k_dep₀ 0 (/h).
• k_inert₀ 0 (/h).
• V 432 (m³)

Figure 10 shows the results of the simulation. One considers that for the whole room, the calculated quanta rates, considering the spreads and overlaps for the percentage of 10.22%, are included in the range 170–200 (quanta/h), i.e., a value practically equivalent to the case of the ABC zone only.

Given the effective separation of the ABC zone from the rest of the room, as also demonstrated by Liu et al. [10], it seems convincing to consider only the ABC zone.

4. Conclusion for the Sensitivity Analysis

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4. Conclusion for the Sensitivity Analysis
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The variability in the computational parameters used in infection risk leads to a spread in the results of infection probabilities. This spread can be modest or even broad. To obtain not deterministic values but extended to a probability surface that determines a range of possible values for a given calculation time, for example, in the first plot of Figure 11 coincides with that of Figure 10. The spreads relative to the probabilities for each value of the quanta rate considered are such that they join and overlap each other forming a single area the first abacus colored in Figure 11. The calculation hypotheses considered are reported in the titles of each abacus, and the calculation method is the BMS.

Sensitivity analysis creates a spread that effectively levels the calculation methods by reducing the effects of the results’ greater or lesser deterministic accuracy. This observation results in the lack of accurate (deterministic) reliable references that are the basis of every design decision.

Three assumptions were made

• a pessimistic hypothesis that leads to higher values of the risk of contagion for each calculation hypothesis (usually the curve indicated with P_d and dashed);
• an average hypothesis that corresponds, for each calculation hypothesis, to the nominal calculation values (usually the curve denoted by solid line P_c). The middle curve describes the classical deterministic hypothesis for each calculation method.
• an optimistic assumption that leads to lower values of the risk of infection for each calculation assumption (usually the curve indicated with P_s and dotted line).

For design purposes, it is usual to refer to the worst-case design conditions, and, therefore, the pessimistic prediction should be employed. In any case, especially with overlapping spreads, a variable probability of contagion from the lower air boundary curve to the upper curve can be considered.

However, when dealing with situations where there is a lack of data to make precise calculations, this is the most frequent situation. Therefore, it is necessary to rely on sensitivity analysis and analyze the spreads obtained.

A large spread also means considerable uncertainty that makes all values within the range possible, as illustrated in the second plot in Figure 11.

The spreads relative to the probabilities for each value of the quanta rate considered are such that they join and overlap each other forming a single area (colored in Figure 11).

The results obtained through sensitivity analysis are interesting because they show the presence of the spread of the curves of the probability of contagion as the calculation parameters vary around their nominal value. However, they initially create disconcertment because they lose the absolute, deterministic reference for the various types of calculation. However, the awareness of having a widespread also changes the way to approach both the problem of probabilistic calculation itself and the way to find the best plant solutions, when possible, to reduce the risk of contagion.

Having a large spread means that the indeterminacy due to the variability of the parameters is also significant. Therefore, one must accept a variable risk in an area (a segment for a given time value) that, on the one hand, moves the upper limit upward from the nominal curve and, on the other hand, lowers it below the nominal curve. Reducing the spread is necessary to reduce or eliminate, if possible, uncertainties in the calculation parameters by using a more careful analysis of the data. For plant variability, such as N_r, it is necessary to solve situations compromised by plant age, lack of maintenance, or deterioration of the units.

Sensitivity analysis proves to be particularly useful for inverse calculations, i.e., for determining quanta rates in the presence of real-world cases for which uncertain and highly variable data are available, as was demonstrated for the Skagit Valley Chorale case and the Guangzhou Restaurant.

The sensibility analysis also concurs to carry out simulations of the type “What if” on several possible hypotheses of calculation between some independent events.

Conflict of Interest

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4. Conclusion for the Sensitivity Analysis
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No potential conflict of interest relevant to this article was reported.

Figures

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Fig. 1.

Fuzzy set arithmetic.

Fig. 2.

Sensitivity analysis applied to the Wells-Riley relationship with variable q.

Fig. 3.

Sensitivity analysis applied to the Wells-Riley relationship with variable N_r.

Fig. 4.

Sensitivity analysis applied to Wells-Riley relationship with q and p and N_r variables.

Fig. 5.

Sensitivity analysis for Gammaitoni-Nucci with q and p and N_r variables.

Fig. 6.

Sensitivity analysis applied to the Buonanno-Morawska-Stabile relationship with q and p and N_r variables.

Fig. 7.

Sensitivity analysis applied to the relationship of Rudnick and Milton (verification) with variable k_dep and k_inert.

Fig. 8.

Simulation of the Skagit Valley Chorale with the BMS method.

Fig. 9.

Calculations of the probability of infection for Case a using the BMS method.

Fig. 10.

Calculations of the probability of infection for Case b using the BMS method.

Fig. 11.

Variability of the probabilities of individual risk the minimum P_s and maximum P_d.

Tables

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4. Conclusion for the Sensitivity Analysis
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Table. 1.

Table 1. Calculation values for the Skagit Valley Chorale simulations.

Parameter	Value(s)	Distribution
Probability of infection, p (%)	53–87	Uniform
Volumetric breathing rate, Qb (m³/h)	0.65–1.38	Uniform
Loss rate due to ventilation, v (/h)	0.3–1.0	Uniform
Loss rate due to deposition onto surfaces, k_dep (/h)	0.3–1.5	Uniform
Loss rate due to virus inactivation, k_inert (/h)	0–0.63	Uniform
Volume of rehearsal hall, V (m³)	810	Constant
Duration of rehearsal, t (h)	2.5	Constant

References

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Abstract
1. Introduction
2. Methodology
3. Verification of Sensitivity Analysis with Case Studies
4. Conclusion for the Sensitivity Analysis
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Wells, WF (1934). On air-born infection. Study II. Droplet and droplet nuclei. American Journal of Hygiene. 20, 611-618.
Riley, RL, Mills, CC, O’Grad, F, Sultan, LU, Wittstadt, F, and Shivpuri, DN (1962). Infectiousness of air from a tuberculosis ward. Ultraviolet irradiation of infected air: comparative infectiousness of different patients. American Review of Respiratory Disease. 85, 511-525.
Wells, WF (1955). Airborne Contagion and Air Hygiene: An Ecological Study of Droplet Infections. Cambridge, MA: Harvard University Press
Nicas, M, Nazaroff, WW, and Hubbard, A (2005). Toward understanding the risk of secondary airborne infection: emission of respirable pathogens. Journal of Occupational and Environmental Hygiene. 2, 143-154. https://doi.org/10.1080/15459620590918466
Lai, AC, and Nazaroff, WW (2020). Modeling indoor particle deposition from turbulent flow onto smooth surfaces. Journal of Aerosol Science. 31, 463-476. https://doi.org/10.1016/S0021-8502(99)00536-4
Cammarata, A, and Cammarata, G (2021). Dynamic assessment of the risk of airborne viral infection. Indoor Air. 31, 1759-1775. https://doi.org/10.1111/ina.12862
Anile, AM, Cammarata, G, and Greco, AM . Methodology of fuzzy set theory for the elaborations of environmental impact assessments., Proceedings of Methodology of Fuzzy Set Theory for Environmental Impact Computations Conference, 1993, Accademia Gioenia di Scienze Naturali, Catania.
Buonanno, G, Stabile, L, and Morawska, L (2020). Quantitative assessment of the risk of airborne transmission of SARS-CoV-2 infection: prospective and retrospective applications. Environment International. 145, article no 106112. https://doi.org/10.1016/j.envint.2020.106112
Gammaitoni, L, and Nucci, MC (1997). Using a mathematical model to evaluate the efficacy of TB control measures. Emerging Infectious Diseases. 3, 335-342. https://doi.org/10.3201%2Feid0303.970310
Li, Y, Qian, H, Hang, J, Chen, X, Hong, L, and Liang, P. (2020) . Evidence for probable aerosol transmission of SARS-CoV-2 in a poorly ventilated restaurant. [Online]. Available: https://doi.org/10.1101/2020.04.16.20067728
Miller, SL, Nazaroff, WW, Jimenez, JL, Jose, L, Boerstra, A, Buonanno, G, and Dancer, SJ (2021). Transmission of SARS-CoV-2 by inhalation of respiratory aerosol in the Skagit Valley Chorale superspreading event. Indoor Air. 31, 314-323. https://doi.org/10.1111/ina.12751
Riley, EC, Murphy, G, and Riley, RL (1978). Airborne spread of measles in a suburban elementary school. American Journal of Epidemiology. 107, 421-432. https://doi.org/10.1093/oxfordjournals.aje.a112560
Rudnick, SN, and Milton, DK (2003). Risk of indoor airborne infection transmission estimated rom carbon dioxide concentration. Indoor Air. 13, 237-245. https://doi.org/10.1034/j.1600-0668.2003.00189.x
Watanabe, T, Bartrand, TA, Weir, MH, Omura, T, and Haas, CN (2010). Development of a dose-response model for SARS coronavirus. Risk Analysis. 30, 1129-1138. https://doi.org/10.1111/j.1539-6924.2010.01427.x
Cammarata, A, and Cammarata, G (2021). Passive systems for reducing the risk of airborne infection in buildings with reduced ventilation. AiCARR Journal. 70, 60-69. http://www.aicarrjournal.org/cont/journalsarticles/876/volume-december-sistemi-passiviriduzione-rischio.asp
Fisk, WJ (2000). Health and productivity gains from better indoor environments and their relationship with building energy efficiency. Annual Review of Energy and the Environment. 25, 537-566. https://doi.org/10.1146/annurev.energy.25.1.537
Stephens, B. (2013) . HVAC filtration and the Wells-Riley approach to assessing risks of infectious airborne diseases. [Online]. Available: https://www.built-envi.com/publications/nafa_iit_wellsriley-FINAL.pdf
Cammarata, G, Lupo, T, and Russo, M (2006). Plant analysis in fuzzy logic: example of FMEA for a refrigeration plant. La Termotecnica.
Beggs, CB, Shepherd, SJ, and Kerr, KG (2010). Potential for airborne transmission of infection in the waiting areas of healthcare premises: stochastic analysis using a Monte Carlo model. BMC Infectious Diseases. 10. article no 247. https://doi.org/10.1186/1471-2334-10-247
Lu, J, Gu, J, Li, K, Xu, C, Su, W, and Lai, Z (2020). COVID-19 outbreak associated with air conditioning in restaurant, Guangzhou, China, 2020. Emerging Infectious Diseases. 26, 1628-1631. https://doi.org/10.3201/eid2607.200764

Biographies

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1. Introduction
2. Methodology
3. Verification of Sensitivity Analysis with Case Studies
4. Conclusion for the Sensitivity Analysis
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Alessandro Cammarata received the degree in Mechanical Engineering from the University of Catania, Italy. He is an associate professor of Applied Mechanics in the University of Catania, produced more than 60 publications. It has interest in multibody systems, reduced order models of flexible systems, and robotics.

Giuliano Cammarata received the degree in Nuclear Engineering with honors in 1969 from the University of Palermo, Italy. Full professor of Technical Physics at the University of Catania, he is author of more than 200 publications on national and international journals. He has participated in research in the energy, thermomechanical and fluid dynamic fields. He has developed mathematical models for thermo fluid dynamics, applied thermodynamics, civil and industrial plant engineering, and applied acoustics. He is the author of other books in the energy sector.

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(4): 448-464

Published online December 25, 2023 https://doi.org/10.5391/IJFIS.2023.23.4.448

Sensitivity Analysis for Calculating the Risk of Viral Infection

Alessandro Cammarata¹ and Giuliano Cammarata²

¹Department of Civil Engineering and Architecture, University of Catania, Catania, Italy
²Department of Electrical Electronic and Computer Engineering, University of Catania, Catania, Italy

Correspondence to:Alessandro Cammarata (alessandro.cammarata@unict.it)

Received: August 9, 2021; Revised: November 22, 2021; Accepted: December 22, 2023

Abstract

Keywords: SARS-CoV-2, Risk of contagion, Sensitivity analysis, Fuzzy arithmetic

1. Introduction

They also included as calculation parameters the deposition losses of ground droplets and the inactivation losses due to the loss of viability in time of the virus present in the aerosol.

Airborne contagion from any pathogen depends on several factors and circumstances such as

1. The spatial distribution of the pathogen.
2. The likelihood of the pathogen entering the lower parts of the respiratory tract.
3. The probability of generating contagion.
4. Calculation of how many infection rates.
5. Methods of calculating the risk of infection.

These have a decisive weight in the assessment of the risk of infection and the correct evaluation of the quanta of disease, i.e., the reverse problem.

2. Methodology

2.1 Determining the Risk of Infection

The parameters contained in the proposed calculation reports suffer from the same basic assumptions:

• Uniform air distribution (perfect mixing).
• Temporal stability of respiratory activity (m³/h).
• The constancy of ERq during the calculation period.
• Stationary conditions achieved for ERq.
• The constancy of the number of hourly changes (/h).

Nobody performed a sensitivity analysis of the airborne risk of infection to these parameters. Therefore, in regular use, calculation predictions should be considered with great caution.

2.2 Methods for Calculating the Risk of Infection

All computational relationships are expressed in the mathematical form of the Poisson distribution

(1)

P = 1 - e^{- μ},

which nicely interpolates the distribution of contagions, according to Wells’ formulation. Some fundamental quantities are included in the exponent, such as

• The airflow rate (m³/h) of breathing of the individual of ambient air, usually indicated through the letter p.
• The initial number of infected, usually one, is denoted by the letter I.
• The exposure time (h) from entry into the environment, denoted by t_esp.
• The total fresh air flow rate, Q (m³/h), calculated as the product of the number of changes per hour, N_r (/h), and the volume, V (m³), of the room.
• The number of infection quanta, (quanta/h), produced by the infected inside the environment.

2.2.1 Wells-Riley

The Wells-Riley expression [12], is the oldest and most used and is expressed as follows:

(2)

P = (1 - e^{(- \frac{I \cdot q \cdot p \cdot t}{N_{r} \cdot V})}) \cdot 100,

where the following quantities can be easily recognized

• I number of infected, usually set equal to 1;
• q quanta rate (quanta/h);
• p breathing air flow rate (m³/h);
• N_r number of air changes (/h);
• V volume of the environment (m³);
• t calculation time (h).

2.2.2 Gammaitoni Nucci method

Gammaitoni and Nucci [9] introduced a novel computational methodology that overcomes the stationarity assumption of the Wells-Riley expression. The latter is

(3)

P = (1 - e^{- \frac{p q I}{V} \frac{C_{d} t + e^{- C_{d} t} - 1}{C_{d}^{2}}}) \cdot 100.

The exponent contains the following parameters

• p lung activity (m³/h);
• q number of quanta per hour (quanta/h);
• I number of initial infected;
• V volume of the environment (m³);
• C_d dispersion coefficient, equivalent to the number of changes per hour, N_r (/h);
• t calculation time (h).

2.2.3 Buonanno-Morawska-Stabile method

(4)

n (t, E R_{q}) = n_{0} \cdot e^{- I V R R \cdot t} + \frac{E R_{q} \cdot I}{I V R R \cdot V} (1 - e^{- I V R R \cdot t})

(5)

D_{q} (E R_{q}) = I R \int_{0}^{T} n (t) d t (quanta),

where IR is the respiration rate (m³/h) and D_q is the quanta dose. The probability of contagion is finally expressed as

(6)

P = (1 - e^{- D_{q}}) \cdot 100.

2.2.4 General considerations for infection risk calculation reports

All calculation methodologies share the following features:

1. The probability of individual contagion decreases as the volume of the environment increases.
2. The likelihood of individual infection decreases as the number of hourly changes in fresh outside air increases.
3. The likelihood of individual infection increases with increasing exposure time from the time of entry into the environment.

2.3 Variability of Calculation Parameters

2.3.1 Number of quanta/h

2.3.2 Respiration rate, p

2.3.3 Number of air changes per hour, N_r

Controlling the number of outdoor air changes per hour for physiological changes is critical to reducing the risk of infection, as demonstrated for passive protection systems [15].

2.3.4 Influence of quanta rate reduction systems

(7)

L = k_{v e n t i l a t i o n} + k_{f i l t e r} \cdot e_{f i l t e r} + k_{U N} \cdot e_{U V} + k_{d e p} + k_{i n e r t},

where the subscripts refer to

• k_ventilation is the number of changes per hour by ventilation, N_r;
• k_filter is the equivalent number of changes per hour due to filtration;
• k_UV is the equivalent number of changes per hour due to UV radiation;
• k_dep is the equivalent number of air changes per hour due to droplets;
• k_inert is the equivalent number of hourly changes due to reduction in viral viability.

Each term may or may not be present, except the first one. The sum of all terms greatly increases the value of N_r. The previous expression can be put in the form

(8)

L = N_{r} + F N,

where the Fisk and Nazaroff term holds

(9)

F N = k_{f i l t e r} \cdot e_{f i l t e r} + k_{U N} \cdot e_{U V} + k_{d e p} + k_{i n e r t} .

In [17], the methods for calculating each term in Eq. (9) are explained.

2.3.5 Sensitivity analysis for the calculation of the probability of infection

(10)

{\frac{\partial Y}{\partial X} |}_{x_{0}} .

If there are many parameters, partial derivatives are written for each parameter around its working point x_{0_i}.

(11)

\sum_{N} ({\frac{\partial Y_{i}}{\partial X_{i}} |}_{x_{0 i}}) .

(12)

{\frac{\partial P}{\partial q} |}_{q_{0}}; {\frac{\partial P}{\partial p} |}_{p_{0}}; {\frac{\partial P}{\partial N r} |}_{N r_{0}} .

2.4 The Fuzzy Arithmetic

The fuzzy arithmetic set is denoted with a compact form containing the crisp value A and the variability a : (A, a).

2.4.1 Sum

The sum of the two sets defined above is given by

(13)

(A, a) + (B, b) = (A + B, a + b) .

So, the sum set has a crisp value given by the sum of the two crisp values A + B while the variability becomes the sum a + b.

2.4.2 Sum with a constant C

The constant C can be assumed to be a zero-variable fuzzy set, i.e., one has only the crisp value C : (C, 0). Therefore, the sum of a fuzzy set and a constant hold

(14)

\begin{array}{l} (A, a) + C \equiv (A, a) + (C, 0) = (A + C, a + 0) \\ = (A + C, a) . \end{array}

2.4.3 Difference

The difference of the two sets defined above is given by

(15)

(A, a) - (B, b) = (A - B, a - b) .

So, the difference set has a crisp value given by the difference of the two crisp values A − B while. the variability becomes the difference a − b.

2.4.4 Difference with a constant C

The constant C can be assumed to be a zero-variable fuzzy set, i.e., one has only the crisp value C : (C, 0). Therefore, the difference between a fuzzy set and a constant is

(16)

(A, a) - C = (A - C, a) .

2.4.5 Product

The product of two arithmetic sets leads to a more complex expression

(17)

(A, a) \cdot (B, b) = (A \cdot B + a \cdot b, A \cdot b + B \cdot a) .

The rule for products from multiple fuzzy sets can be applied

(18)

\begin{array}{l} (A, a) \cdot (B, b) \cdot (C, c) \\ = [(A, a) \cdot (B, b)] \cdot (C, c) \\ = (A \cdot B + a \cdot b, A \cdot b + B \cdot a) \cdot (C, c), \\ (A, a) \cdot (B, b) \cdot (C, c) \\ = {[(A \cdot B + a \cdot b) \cdot C + (A \cdot b + B \cdot a) \cdot c], \\ (A \cdot B + a \cdot b) \cdot c + C \cdot (A \cdot B + a \cdot b)} . \end{array}

2.4.6 Power of a fuzzy set

For the power of a fuzzy set, the multiplication rule applies and therefore

(19)

\begin{array}{l} (A, a) \cdot (A, a) = (A \cdot A + a \cdot a, A \cdot a + A \cdot a) \\ = (A^{2} + a^{2}, 2 \cdot A \cdot a) . \end{array}

2.5 Fuzzy Arithmetic for Calculating the Risk of Airborne Contagion: Wells-Riley Method

Assume that the three fundamental parameters q, p, N_r vary and add the variability of the Fisk and Nazaroff term (FN).

The fundamental parameters can be arranged in the following fuzzy sets:

• (q, dq) the arithmetic set for quanta/h, q;
• (p, dp) the arithmetic set for the breathing rate, p;
• (N_r, dN_r) the arithmetic set for the number of hourly changes, N_r.

Therefore, three probabilities of contagion can be defined.

The expressions for the central probability, P_c, left probability, P_s, and right probability, P_d, are the following:

(20)

\begin{array}{l} P_{c} = (1 - e^{- \frac{I \cdot q \cdot p \cdot t}{N_{r} \cdot V}}) \cdot 100, \\ P_{s} = (1 - e^{- \frac{I \cdot [(q \cdot p + d p \cdot d p) - (q d p + p d q)] \cdot t}{[N_{r} - d N r] \cdot V}}) \cdot 100, \\ P_{d} = (1 - e^{- \frac{I \cdot [(q \cdot p + d p \cdot d p) + (q d p + p d q)] \cdot t}{[N_{r} + d N r] \cdot V}}) \cdot 100. \end{array}

2.5.1 Case 1: Variability of q

2.5.2 Case 2: Variability of N_r

The three families of curves correspond to the three assumed quanta/h values.

2.5.3 Variability of q and N_r

2.6 Gammaitoni-Nucci Method

What has been done for the Wells-Riley (WR) expression also applies to the Gammaitoni-Nucci (GN) method. The fuzzy expressions for the probabilities of contagion become as follows:

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\begin{array}{l} P = (1 - e^{- \frac{p q I}{V} \frac{C_{d} t + e^{- C_{d} t} - 1}{C_{d}^{2}}}) \cdot 100, \\ P_{s} = (1 - e^{- \frac{I \cdot [(q \cdot p + d p \cdot d p) - (q \cdot d p + p \cdot d q)]}{V} \cdot \frac{(C_{d} - d C_{d}) t + e^{- (C_{d} - d C_{d}) t} - 1}{(C_{d}^{2} + d C_{d}^{2} - 2 \cdot C_{d} \cdot d C_{d})}}) \times 100, \\ P_{d} = (1 - e^{- \frac{I \cdot [(q \cdot p + d p \cdot d p) + (q \cdot d p + p \cdot d q)]}{V} \cdot \frac{(C_{d} + d C_{d}) t + e^{- (C_{d} + d C_{d}) t} - 1}{(C_{d}^{2} + d C_{d}^{2} + 2 \cdot C_{d} \cdot d C_{d})}}) \times 100. \end{array}

2.6.1 Case 4 for GN: Variability of q, N_r

The spread results takes into account both the variability of the parameters and the greater computational accuracy of the GN method.

The results differ from those obtained through the WR method because of the GN assumptions of nonstationary distribution of quanta/h.

2.7 Buonanno-Morawska-Stabile Method

Applying the same fuzzy arithmetic rules, the following expressions for the quanta doses of the Buonanno-Morawska-Stabile (BMS) method stand

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\begin{array}{l} D_{q c} = I E_{m} p q (e^{- C_{c} t} + C_{c} t - 1) / (V C_{c}^{2}), \\ D_{q s} = I E_{m} [(q p + d p d p) - (q d p + p d q)] (e^{- C_{s} t} + C_{s} t - 1) / (V (C_{c}^{2} + d C_{c}^{2} - 2 \cdot C_{d} d C_{d})), \\ D_{q d} = I E_{m} [(q p + d p d p) + (q d p + p d q)] (e^{- C_{d} t} + C_{d} t - 1) / (V (C_{c}^{2} + d C_{c}^{2} + 2 C_{d} d C_{d})), \end{array}

where

• C is the number of air changes per hour, corrected according to Fisk and Nazaroff.
• E_m is the attenuation factor of the protective mask.

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\begin{array}{l} P_{c} = (1 - e^{- D_{q c}}) \cdot 100, \\ P_{d} = (1 - e^{- D_{q d}}) \cdot 100, \\ P_{d} = (1 - e^{- D_{q d}}) \cdot 100. \end{array}