Dynamic assessment of the risk of airborne viral infection

Abstract This paper applies the Rudnick and Milton method through the dynamic evaluation of the probability of airborne contagion, redefining all parameters and variables in discretized form. To adapt the calculation of the risk of contagion to real needs, scenarios are used to define the presence of people, infected subjects, the hourly production of the quanta of infection, and the calculation of the concentration of CO2 produced by exhalation in the air. Three case studies are discussed: a school, an office, a commercial activity. Complex scenarios include environmental sanitization, a variable number of people, and the possibility of simulating work shifts. The dynamic evaluation of the quanta of infection is also estimated, not foreseen by the Rudnick and Milton model, and involves updating the average values of the equivalent fraction of the indoor air with an improvement in the accuracy of the calculation due to the reduction of improper peaks of the stationary variables.


| INTRODUC TI ON
W.F. Wells was the first to hypothesize the mechanism of viral transmission by air through the emission from the mouth and nose of tiny organic droplets coming from the esophagus. 1 These droplets are emitted by various mechanisms such as breathing, speaking, coughing, and sneezing. Droplet emission rates vary greatly from 2 to 4 (m/s) for breathing to 60-80 (m/s) and more for sneezing. Besides, the distances traveled can vary from a few meters to ten meters for coughing. The droplets, with dimensions of 10 (μm), evaporate 1 in a few seconds forming the droplets nuclei with dimensions of a few μm, very light and such as to form an aerosol that can remain suspended in the air for a few hours. An update on droplet evaporation rate was done by Xie et al in 2007. 2 The infection caused by the droplets emitted from the mouth and/or nose is direct, affecting the respiratory tract of the affected subjects and penetrating inside the bronchi and lungs (apart from other forms of contagion of other internal organs), Beggs et al., 2004. 3 According to these recent studies, 4 the particles emitted by expiratory activity have diameters ranging from 0.8 to 5.5 μm. Studies on this mechanism have been carried out. 5,6 The lighter particles, up to 5 (μm), affect the deeper regions of the respiratory tract up to the deep alveolar region, with the possibility of activating the contagion, while particles with a diameter >5 (μm) are trapped in the upper area of the respiratory track with lower probability of contagion. 7 All this suggests that the probability of contagion associated with particles with a diameter greater than 5 (μm) must be carefully evaluated. This last scarce possibility of contagion led virologists, until the beginning of July 2020, to consider airborne viral infection to be negligible. The importance of transmission of the infection by air has been highlighted 8 for the flu but similar results also apply to the SARS-CoV-2 virus. They also emphasize how good ventilation can reduce the epidemic, as can the vaccination of about half the people present. The close contagion is said to be of short distance 9 and is significant, on average, within 2 (m) from the infected subject. Several intervention measures for the close-range contagion are available, e.g.
• use of individual protection masks; • interpersonal distance from 1 to 2 m, depending on the case; • handwashing with disinfectants to avoid the indirect contagion of droplets that fall by gravity on horizontal surfaces.
There has been much discussion on the effects of aerosol, which remain suspended in the air inside closed environments. 10 Recently, 239 scientists from around the world reiterated the possibility of infection from SARS-CoV-2 via aerosols, and the World Health Organization confirmed this hypothesis in July 2020. 11 The contagion through aerosols acts at beyond the two meters typical of direct contagion, due to the circulation induced by the air inside closed environments. 12 The aerosol is of no importance outdoors because it is immediately dispersed by air currents and diluted in a huge mass of air.
The infection risk due to airborne contagion can be evaluated by using the Wells-Riley equation, 13 with the concept of quanta emission and the dilution effects of viral loads due to ventilation in the rooms. Almost all models, proposed for calculating the probability of individual contagion, follow this approach, albeit with significant improvements. In this article, we want to present a calculation method that, using the basic idea of Rudnik and Milton 14 of spreading the infection by re-breathing the air inside the rooms by the susceptible people present. The air exhaled by the infected person contains viral loads which, in the re-breathing phase of healthy subjects, can cause airborne contagion. Matthew J. Evans, 15 carried out a risk analysis of airborne contagion arguing that normal return to work with the current COVID 19 epidemic is not possible until more than 1 in 1000 subjects is asymptomatic. Furthermore, he assumes that the assessment of the risk of airborne infection based only on the symptomatic infected is less than 1/1000 of the real one. The original method of Rudnick and Milton 14 is here completed with the calculation of the variable hourly distribution of the quanta of infection, as done with the method of Gammaitoni-Nucci in 1997, 16 and enhanced using the discretized dynamic calculation and using profiles to adapt the calculation to real situations and needs. 2. Gammaitoni-Nucci method. 16 The more recent method proposed by Rudnick-Milton 14 has had some applications recorded in the literature, 18 and some improvement proposa. 19 All these three methods express the probability of individual contagion in the mathematical form of the Poisson curve for the statistical distribution of the contagion:

| E XIS TING ME THODS FOR C ALCUL ATING THE PROBAB ILIT Y OF IND IVIDUAL CONTAG I ON BY AIR
where p is the probability of contagion and μ is the Poisson factor. The factor μ includes some fundamental quantities, such as: This last quantity has an epidemiological definition and contains two fundamental information: the number of viral loads and the term of infectivity, that is the probability that the viral load starts an infection. 4,20 In practice, for the quanta (q) the definition holds: Definition 1 Q = number of quanta/unit time × term of infectivity.
Quanta are not easily calculated directly, also due to the great uncertainty of the RNA pairs, but are estimated epidemiologically from epidemic cases by replacing the calculated probability, p, by the number of infected. 21 An alternative approach to the quanta

Practical implications
• The proposed method allows for better passive protection of infections by alternating work periods with full personnel and short rest periods.
• The proposed method with sensitivity analysis allows to determine the spread of individual probabilities due to the variability of the parameters.
• The method makes it possible to determine the best operating conditions for intervening on HVAC systems or for the insertion of CMV units both in individual rooms and in larger work areas.
• The use of a method based on the number of equivalent air-changes allows to reduce energy and plant costs by meeting the requests of the major plant manufacturers. method is the dose-response method. 20 The latter requires a sufficient amount of data to build a dose-response relationship. The conceptual simplicity of the quanta is now criticized by epidemiologists because the infection depends not only on the viral load but also on the general health of the infected subject and the type of viral load absorption. The Wells-Riley relationship, 13  The first observation depends on an architectural variable, the volume of the room V, on which we can hardly intervene.
Consequently, in small rooms, it is good to stay for as little time as possible. The second observation depends on external air ventilation. This can be of two types: natural ventilation 22 and forced ventilation. 23 3 | ME THODOLOGY

| Theoretical basis of the Rudnick and Milton method
The method of Rudnick and Milton,14 wanted to deepen and modify the calculation relationships proposed up to then exclusively based on the balance of the quanta of infection and the infected in the environments. The basic idea of the new method is that the virus in circulation is conveyed through the exhalation air of the infected which, subsequently, is re-inhaled (re-breathing) by susceptible subjects in the environment. The higher this breathing activity, the greater the possibility of transmitting the viral infection by air. Other research has been published in subsequent years, 18,19,24 based on the re-breathing mechanism and the use of CO 2 as a marker. Due to its flexibility and ease of application, in the following description, only the Rudnick and Milton method will be recalled. Since the air exhaled by people inside the environment always contains, for the internal metabolism, a percentage of CO 2 , as well as a lower percentage of O 2 compared to the inhaled air, it can be hypothesized to use the CO 2 concentration to trace the breathing activity of the subjects inside the environment. The novelty of the method consists, in fact, in taking into account the people present in the environment and the amount of air breathed and re-inhaled in repeated breathing cycles. The human body inhales a flow of air that depends on the age, the activity performed, the state of health of the subject, and the functioning of the pulmonary system. In all cases, in the exhalation phase, part of the oxygen present in the inspired air is replaced by CO 2 . Using appropriate sensors, it is possible to trace the global respiratory activity by tracing the CO 2 emitted. Of course, the hypothesis is valid that within the environment there are no other sources of CO 2 other than that of expiratory origin.
Other basic assumptions for the mathematical developments are as follows: • limited room size to avoid vorticity effects and unevenness in air distribution. The simplifying hypothesis of perfect mixing is therefore valid; • constant temperature and humidity in the calculation period; • constant viral activity and survival time in the calculation period; • calculation period limited to a fraction of a day (work shifts, lesson hours, commercial activity hours).

Unlike what indicated by Rudnick and Milton in their publication,
instead of making a balance of CO 2 fractions present in the ambient air of volume V and produced by the exhalation of the subjects in the same volume V e , a balance of the CO 2 concentrations for the two volumes is provided. Therefore, in the hypothesis of well-mixed ventilation (perfect mixing), it is possible to write a balance for the CO 2 in the environment, i.e.
The advantage of this setting is to be able to correlate the concentrations, expressed in (ppm), to the ventilation air-flow rate and therefore to the whole set: building-HVAC system-people. The dynamic method proposed in this article derives from this approach.
Solving the previous equation for the ratio V e /V, we obtain: where f is the equivalent fraction of internal air 14  being V e the volume of exhaled air and p the breathing rate. To calculate the difference ΔC = C − C 0 , the steady-state expression was taken into account: it is derived that: The average value over time t is: It is important to note that both expressions (8) and (9)

| Dynamic discretized version of the Rudnick and Milton method
Some authors 19  where the discretized parameters are gathered in vectors and expressed in boldface. In this way, it is possible to describe each calculation parameter and each variable through a sequence of numerical values independent of analytic relationships and freely modifiable as required. Notice that, in transient conditions, the probability p is evaluated considering averaged variables. The time is also discretized following the scenario. This makes it possible to obtain the desired time sequence, for example, to take into account double work shifts or to consider sanitizing the rooms during the work break to eliminate CO 2 accumulations (a total change of ambient air) and of quanta of infection from the previous work-shift ( Figure 1). Figure 2 shows an example of a vectorized scenario in graphic form.
Starting from left, the following quantities are plotted: • N p number of people present as time changes. In this case, the 17 people are present in two shifts of 4 h each with a break of 1.5 h; • I the initial numbers of infected people present during the calculation, also in vectorized form. In the case in question, there is 1 infected (constant) at the beginning of each shift, having assumed that the room was sanitized during the break;  used to calculate the probability of contagion. The spatial concentration of the quanta 21 is given by the expression: used to calculate the probability of contagion. The spatial concentration of the quanta 21 is given by the expression: The average concentration is then obtained from Eq. (18) by integrating and dividing by the time interval Δt: The average value q of the production of quanta of infection for the whole volume V, calculated in the period of time Δt, is then: The mask can also be taken into account using the term f mask , i.e. the reduction factor of the quanta flow due to the mask, as suggested by Gammaitoni and Nucci 16 : The action of the mask is directly applied to the hourly production of quanta coming from the mouth or nose, as indicated in Eq.
The value of vit , also referred to as inactivation, depends on the humidity of the air and varies from 0.5 to 1.2 (1/h). Some researchers ignore this, believing that the epidemiological definition mechanism of quanta automatically takes them into account. The hourly distribution of the quanta also follows the same changes as the distribution of the CO 2 concentration, as illustrated in Figure 4. into Eq. (12). After all, the action of these sanitization mechanisms is precisely to reduce the hourly production of quanta. In Figure 5, the hourly trend of q as a result of sanitization (assuming L as the modified number of hourly changes), and without sanitization (assuming

| B ENEFITS OF DYNAMI C C ALCUL ATI ON WITH THE MOD IFIED RUDNICK AND MILLER ME THOD. THE DYNAMI C C ALCUL ATI ON FOR A CL A SS ROOM
The dynamic calculation brings some benefits in the higher calcu- Since the factor f m depends on the ratio Δ∕C a also f m will have a transient that follows that of the concentrations, as illustrated previously and again reported in Figure 6, both for single and double work shift with sanitization, where f m has been scaled to fit in the same figure.
It follows that the dynamic calculation carried out using Eq. (12) takes into account the variability of f m in the initial transitory period which will be greater the smaller the number of hourly changes. In i.e. considering the steady-state value of the transients considered above for concentrations. The comparison between variable ΔC in dynamic and steady-state is given in Figure S2 of the supplementary information.
The stationary limit value is always significantly higher than the variable one. By imposing a stationary ΔC 0 , a stationary factor f is obtained given by the ratio Δ∕C a while in the dynamic regime f varies instantly and the average value f mm must be calculated. Figure 7 compares the two cases. Once again, the stationary method overestimates the calculation parameters and therefore the probability of individual contagion risk.  Figure 8 shows the case of a classroom for high school with the usage profiles 29 reported in the first row for N r = 0.5 (1/h). The second row reports the probability of individual contagion for a production of 50 (quanta/h) with variable temporal distribution, the reproductivity number, R 0 , also called the probability of global contagion given by the product (N p − 1)P, where N p = 17, and finally, the hourly trend of the quanta of infection. 4 It is observed that there are very high individual contagion probabilities. These are significantly reduced for N r = 5 (1/h), as illustrated in Figure 9. It can be observed, once again, how this hourly distribution reaches the regime condition much faster when the number of hourly changes is higher.

| DYNAMIC ANALYS IS DISCRE TIZED WITH A VARIAB LE N UMB ER OF PEOPLE
The discretized dynamic method allows for considering a variable number of people N p and therefore for assessing the risk of contagion due to the breathing and re-breathing of the people present in the environment. From a formal point of view, everything remains unchanged in the previously written expressions, just consider that N p is no longer constant and varies over the period considered. but not the probability of contagion P which is independent of N p as remarked in Section 3.

| Dynamic calculation for office
We want to examine the case of an office of 10 × 20 (m 2 ), with 8 internal employees always present in two shifts of 4 h each and a break of 1 h. The sanitization of the air but not of the quanta is provided, that is, it is assumed that the infected person may be present both in the first and second shift. A modified version of the calculation program is used which simultaneously analyzes three air exchange conditions, 0.5, 1, and 2 (1/h), and three hourly productions of quanta: 40, 70, and 100 (q/h). The user profile is shown in Figure 12 with the usual information on the profile of people, infected people, concentrations, and the fraction of breathing air. Figure 13 shows

| INVER S E PROB LEM: DE TERMINATION OF THE QUANTA OF INFEC TI ON . SUPER-S PRE AD ING E VENT IN THE S K AG IT VALLE Y CHOR ALE
The possibility of multiple comparisons with multiple values of the hourly production of quanta, q, and for multiple values of air changes, The authors, based on the findings made and the facts ascertained, summarize the calculation parameters in the following Table 1.
This is an anomalous case that occurred during some choral re- The building appeared poorly served from a plant engineering point of view. There was a 20° (C) hot air generator whose operation it is unknown whether it was continuous or discontinuous. In any case, it was clear to the researchers who studied the case that the air ventilation was lacking, see Table 1. Many calculation parameters are imprecise and cannot be correctly determined afterward. Moreover, the bibliography indicated for each of them is rather vague and imprecise. Therefore, the simulations that can be done must take into account a significant variability of some fundamental parameters, as reported in Table 1. The Authors used the Gammaitoni-Nucci expression 16 for the calculation of the spatial concentration of quanta, (quanta/m 3 ), and the hourly production, (quanta/h), as described in Section 3.2. Here, we want to perform an inverse calculation for the determination of the hourly production of quanta using the data available in the original publication. 21 In the first calculation phase, the CO 2 concentration produced by the 61 present in the theater hall is determined to determine the difference ΔC required by the calculation method. We do not know the ventilation airflow or the layout of the HVAC system used. It seems that there is a heating system with a heat generator that sends hot air into the theater to maintain a nominal temperature of 20°, although some of those present said they felt cold. The operation of the system, therefore, appears without a thermostat. We have no information on the external airflow or whether there is internal recirculation. The estimated number of air changes varies from N r = 0.3 to N r = 1 (1/h), see Table 1.
The same researchers considered these values low, urging the need for effective controlled mechanical ventilation. Furthermore, the ventilation flow rate corresponding to the indicated range also entails a lack of control on ΔC which, as will be seen in the following calculations, will always be high. The researchers reported the lack of a CO 2 detector for air quality control. Choral activity is equivalent to speaking aloud and therefore droplet emission is quite high. In the aftermath, using data available in the literature, the researchers suggest a volumetric respiration rate ranging between 0.65 and 1.38 (m 3 /h). For comparison, it should be borne in mind that breathing in normal conditions (light office work activity) corresponds to a flow rate of 0.48 (m 2 /h). The volumetric composition of inhaled and exhaled air is shown in Table 2:

Quanta of infection
q=50 (q/h) q=70 (q/h) Therefore, the exhaled CO 2 flow rate will be equal to 4% of the breathed airflow. Based on the activity carried out by the singers and the high breathing rate, a CO 2 production of 0.4 (L/h) was considered. The value of the re-breathing fraction is equal to C a = 47,500 (ppm). The verification criteria already presented previously will be followed to identify the hourly production of quanta. It must be said that the variability of some parameters does not allow the direct and univocal calculation of q but it is necessary to carry out several iterations. With the verification criterion the presence of any filters is not taken into account, the difference ΔC = C − C 0 is calculated as a function of the ventilation flow-rate (in the range indicated in Table 1) Table 1. The variability of q appears considerable, as reported by the researchers in their publication. The parameters adopted for the calculation have averaged values in the intervals indicated in the article 21 and are reported in Table 3: The results for the scenario are shown in Figures 14 and 15 where a 2.5-h guideline is also shown. It is observed that the value of  dynamic method and the analogs with the static method, we have the following Table 4.
Summarizing, Table 5 reports the variability of quanta/h, expressed in terms of minimum, maximum, and mean value, for the two types of calculations of Table 4.
The average value of quanta production for the dynamic calculation is 738 (q/h) while for the static calculation it is 1088 (q/h). The results obtained in Ref. [21] are compared with the values calculated using the proposed method, as indicated in the Table 6; There is a substantial agreement for the minimum values of the

| CON CLUS IONS
The proposed discretized dynamic method offers considerable advantages over the traditional methods in use. These can be summarized as follows:

PE E R R E V I E W
The peer review history for this article is available at https://publo ns.com/publo n/10.1111/ina.12862.

DATA AVA I L A B I L I T Y S TAT E M E N T
The data that support the findings of this study are available from the corresponding author upon reasonable request.