Evaluation of infection risk for SARS-CoV-2 transmission on university campuses

Abstract

In 2021, COVID-19 has been widely spread worldwide. Few studies focused on infection risk in the different confined spaces on university campuses. However, obtaining the quanta value of SARS-CoV-2 using the conventional method is not identical. Therefore, in our study, the estimation method of the quanta value was improved via statistically analyzing the viral load of the infectors and fitting the droplet number concentration of different particle sizes. Moreover, the infection risk and efficacy of typical engineering control measures in confined spaces such as dormitories, classrooms, gyms, libraries, and refectories were evaluated using the improved Wells-Riley equation. The results demonstrated that: (1) the quanta value range of SARS-CoV-2 was 20.49 ∼ 454.87 quanta/h, which was confirmed by the existing literature; (2) the infection risk in dormitories and classrooms was 100% (exposure time was eight hours) and 5% (exposure time was 1.5 h), respectively, while basic reproduction numbers were both 2.8; (3) The combined control measures mainly based on engineering measures such as ventilation, high efficiency particle air (HEPA) filter, ultraviolet germicidal irradiation (UVGI), partially based on surgical masks were recommended. The findings could provide suggestions for universities to scientifically formulate intervention measures and self-protection means for students.

Highlight

• The estimation method of the quanta generation rate was improved.

• Statistically analyzed the distribution of viral load based on published data.

• The droplet number of different particle sizes was determined.

• The quanta generation rate was found to be substantially influenced by the respiratory intensity and activity level.

• Infection risks in the dormitories and classrooms were the highest.

Introduction

By June 2021, the COVID-19 pandemic has caused 1.7 hundred million infections and more than 3.5 million deaths (Jin et al. 2020; JHU 2021). In response to the outbreak, over 133 US colleges and universities closed the campus temporarily and bore a burden to implement a reopen plan (CDC 2021a). Particularly, university students seem to be uncontrollable due to the free activity. In 2019, the number of Chinese university students exceeded 30 million (NBS 2019). The Chinese government has issued the Technical Plan for the Prevention and Control of COVID-19 Epidemics in Colleges and Universities in Autumn and Winter. The students have returned to their campuses in the 2020–2021 academic year (MOE 2020). How should they protect themselves after returning to school? Which confined space has the greater infection risk? These questions must be answered to prevent the SARS-CoV-2 from spreading among colleges and universities.

A university campus can be regarded as a small community, including functional buildings, such as libraries, classrooms, dining halls, dormitories, gyms, and offices (Xu, Gang, and Peng 2006). The lifestyles, social activities, and contact behavior of university students are significantly different from those of K-12 school students. Undergraduates may spend more time in dormitories and classrooms, while postgraduates are primarily in laboratories, offices, or dormitories. Moreover, the activity level and respiratory intensity in different functional buildings are inconsistent, which causes differences in SARS-CoV-2 transmission. Consequently, it is necessary to research the infection risk of different confined spaces on university campuses separately.

Knowing the SARS-CoV-2 transmission route is the basis for studying infection risks and formulating epidemic prevention measures. The WHO identified four modes existed: contact and droplet transmission, airborne transmission, and fomite transmission (WHO 2020a, 2020b). The American Society of Heating, Refrigerating and Air-Conditioning Engineers (ASHRAE) and the Federation of European Heating and Air-Conditioning Associations (REHVA) respectively emphasized the possibility of long-range transmission and fecal aerosol transmission (Schoen 2020; REHVA 2020). The COVID-19 Diagnosis and Treatment Plan (Trial Eighth Edition) issued by the Chinese National Health Commission figured out that droplets and contact transmission were the major transmission route (NHC. 2020). Currently, the epidemic prevention measures of Chinese universities contain access control, social distancing, temperature testing. However, those measures do not distinguish between the different confined spaces. Moreover, some universities no longer maintain closed-off management attributable to normalized epidemic prevention. Therefore, exploring the transmission risk based on different physical buildings is crucial.

The Wells-Riley equation is most commonly used for assessing the infection risk of respiratory diseases (To and Chao 2010). Wells proposed the quanta concept in 1955 (Wells 1955), and (Riley, Murphy, and Riley 1978) improved this concept to form the Wells-Riley equation. So far, applications of the Wells-Riley equation in confined spaces, such as medical buildings (Furuya 2013), schools (Gao et al. 2012), prisons (Mushayabasa 2013), and public transport (Yan et al. 2017), have been reported. Regarding the infection risk of respiratory disease in schools, (Riley, Murphy, and Riley 1978) first adopted the Wells-Riley equation to research a measles outbreak in a US suburban elementary school. (Taylor et al. 2016) predicted an annual tuberculosis infection rate of 18.58% in a South African rural school. (Hella et al. 2017) discovered that the annual tuberculosis transmission risk in a 50-person class of a Tanzanian school was 4.02% at an average CO₂ concentration of 655 ppm. (Gao et al. 2012) conducted statistical analysis of schools at different levels in Hong Kong and explored the impact of different infection prevention precautions, including ventilation, on reducing influenza infectors. Moreover, systematic studies of respiratory disease control strategies that integrated schools as a community or part of the city have been proposed (Gao, Wei, Cowling, et al. 2016; Gao, Wei, Lei, et al. 2016; Zhang et al. 2018). Research on campus respiratory disease aimed at primary and secondary schools and university campuses still needs more attention.

At the other end of the spectrum, the infection risk is closely related to the indoor air environment of university campus buildings. (Shen et al. 2021) proposed a systematic approach to evaluating the effectiveness of IAQ strategies for reducing the SARS-CoV-2 airborne infection risk. (Liu, Liu, and Liu 2020) surveyed the physical symptoms of 399 students in Sichuan Province through the questionnaire. The results showed that the physical discomfort incidence was 34.85% (mild 26.26%; moderate 8.59%). (Zhao et al. 2020) employed the improved SEIR model to simulate the COVID-19 epidemiology in a US university with 38000 participants, which provided advice on reopening the campus. Various studies explored the relationship between the indoor environmental improvement measures and the infection risk reduction, and the results showed that enhanced ventilation played an essential role in the reduction of influenza and tuberculosis (Nardell et al. 1991; Fennelly and Nardell 1998; Escombe et al. 2007; Li et al. 2007). Although ASHRAE and REHVA guidelines recommend the integration of UVGI of the ventilation system to inactivate SARS-COV-2, and multiple studies have also underlined the contributions of ventilation to fight SARS-COV-2 in different scenarios (Zhang 2020; Dai and Zhao 2020; Rothamer et al. 2021; Blocken et al. 2021; Stabile et al. 2021; Buonanno, Stabile, et al. 2020), how indoor environment improvement helps reduce the transmission risk of SARS-COV-2 should be clarified furthermore.

The scope of this paper is to evaluate the infection risk of SARS-CoV-2 transmission in different university confined spaces. Five confined spaces and specific exposure scenarios of a dormitory, classroom, library, gymnasium, and dining hall were considered. The viral load distribution and droplet concentration fitted curve were obtained based on the peer-reviewed literature. The quanta generation rates were estimated using the forward calculation method according to viral load conservation. The infection risk was quantified using the improved Wells-Riley equation. The efficacy of infection prevention precautions in different exposure scenarios was discussed. The results could provide suggestions for the students’ self-protection and in formulating intervention measures for universities.

Methodology

Prospective objects

Confined spaces

We selected a university campus in China for the research. Considering the daily routines, the learning needs, and the breathing intensity of students, we constructed five specific exposure scenarios: the dormitory, classroom, library, gym, and dining hall. Complex buildings such as teaching buildings, libraries, and gyms usually install central - air conditioning systems, and rooms of different sizes can independently regulate the air supply volume (Xu, Gang, and Peng 2006; Li and Luo 2011; Liu Hua and De Ying 2017). Therefore, we simply studied the common confined spaces of different buildings. Among them, a medium-sized classroom in the teaching building with a volume of 296 m³ and a large space study room in the library with a height of 16 m was selected, respectively. For the gym, the competition hall (MOHURD and GASC 2003) with a floor height of 19.5 m was considered. The air change rates referred to Design Code for Heating Ventilation and Air Conditioning of Civil Buildings (GB50736-2012) (MOHURD 2012). In particular, the dormitory utilized natural ventilation with an air change rate of 0.5 ACH, while other scenarios all adopted mechanical ventilation. At present, almost all Chinese universities strictly adopt control measures such as temperature testing and social distancing. Hence, the number of infectors in each scenario was 1. Moreover, we assumed that a student who was playing basketball was the infector in the gym. The detailed design parameters were shown in Table 1.

Table 1. Descriptions of the confined spaces.

Scenario	Dormitory	Classroom	Gym	Library	Dining hall
Floor area (m^2)	20	80	8000	300	1050
Height (m)	3	3.7	19.5	16	4.8
Volume (m^3)	60	296	156000	4800	5040
Ventilation rate (ACH)	0.5 (NV)	6.08 (MV)	4.5 (MV)	2 (MV)	3 (MV)
Number of infectors	1

Exposed scenarios

The transmission of respiratory diseases starts with self-respiratory activity, and the respiratory intensity may cause substantial differences in the disease infectiousness from the source. Since determining the activity level and breathing intensity in different confined spaces is impossible, we designed and classified the exposure scenarios in a statistical sense, as shown in Table 2. We considered the infector activities in the dormitory, classroom, gym library was talking and sleeping, studying and talking, playing basketball, studying and whispering, respectively. The breathing rate was estimated based on the preset activity type (Adams 1993; Buonanno, Stabile, et al. 2020). While the flow and individual behavior of susceptible people in the dining hall were highly random. The stay time of susceptible people was significantly lower than that of other scenarios. We assumed that “no talking” was strictly implemented, so the pulmonary rate was 1.38 m³/h. The susceptible number was equal to the designed seat number.

Table 2. Detailed information of the exposure scenarios.

Scenarios	Dormitory	Classroom	Gym	Library	Dining hall
Exposed subject	Sleep or talking	Study or talking	Playing basketball	Study or whisper	Eating
Pulmonary ventilation rate (m^3/h) $(p)$	0.49	1.38	3.3	0.49	1.38
Total susceptible population $(N)$	4	61	3511	600	200
Total exposure time $(t)$	8 h	1.5 h	2 h	4 h	0.5 h

Estimation of quanta generation rate

Estimation method and steps

We employed the Wells-Riley equation for quantifying the infection risk of the exposure scenarios. However, a significant problem with applications of the Wells-Riley equation is the selection of the quanta value, which was the only statistically based parameter that implied virus infectivity (To and Chao 2010). Table 3 summarizes the quanta values of common respiratory viruses other than SARS-CoV-2. In Table 3, LN presents the lognormal distribution. Scholars usually conducted backward calculations from an outbreak case or directly estimated it based on the previous studies to obtain the appropriate quanta value. Theoretically, except for the calculation parameters in the Wells-Riley equation, the quanta value depends on many reasons. No uniform determining standard has been acknowledged. The COVID-19 pandemic was still raging. (Dai and Zhao 2020) studied the quanta value by fitting the values of MERS, influenza TB, SARS. The results did not distinguish different exposure scenarios. Based on the assumption that the viral load was conserved, (Buonanno, Stabile, et al. 2020; Buonanno, Morawska, et al. 2020) forward calculated the estimated quanta generation rate under different activity levels.

Table 3. The quanta values of common airborne pathogen.

Airborne pathogen	Scenario	Quanta value (quanta/h)	Data resource
Influenza	School, train	LN (66.91,1.53)	(Liao, Chang, and Liang 2005)
		3.5	(Gao et al. 2012)
	Aircraft	15 ∼ 128	(Rudnick and Milton 2003)
		28	(Gao, Li, and Leung 2009)
		LN (68.67,1.52)	(Liao, Chen, and Chang 2008)
	Hospital	515	(To and Chao 2010)
	Gym	76.18	(Andrade et al. 2018)
	All	100	(Azimi and Stephens 2013)
Rhinovirus	All	0.6 ∼ 7.8	(Rudnick and Milton 2003)
Tuberculosis	Office	12.7	(Nardell et al. 1991)
		13	(Basu and Galvani 2008)
		250	(Catanzaro 1982)
	Prison	1	(Johnstone-Robertson et al. 2011)
		1.25	(Urrego et al. 2015)
		10	(Mushayabasa 2013)
		12.7	(Hella et al. 2017)
	Community	1	(Wood et al. 2010)
	Gym	12.7	(Andrade et al. 2018)
	Hospital	1	(Noakes and Sleigh 2009)
		1.3	(Saab 1997)
		8.2	(Escombe et al. 2007)
	Market	1.25	(Hella et al. 2017)
	Café, care facilities	17.58	(Furuya, Nagamine, and Watanabe 2009; Furuya 2013)
	Cell block	30	(Cooper-Arnold et al. 1999)
	All	1.25 ∼ 60	(Beggs et al. 2003)
SARS	Hospital	LN (28.94, 2.66)	(Liao, Chen, and Chang 2008)
H1N1	Aircraft	50 ∼ 128	(Wagner, Coburn, and Blower 2009)
	Car	67	(Knibbs, Morawska, and Bell 2012)
Small pox	School	2	(Gao, Wei, Lei, et al. 2016)
Measles	School	8	(Riley, Murphy, and Riley 1978)
		93	(Riley, Murphy, and Riley 1978)
	Aircraft	LN (106.75, 1.93)	(Liao, Chen, and Chang 2008)
chickenpox	Aircraft	LN (59.07,1.99)	(Liao, Chen, and Chang 2008)

According to the viral load in exhaled droplets of different sizes, we adopted an estimation method to calculate the quanta generation rate. The expression is shown in Equation (1)(1) $$q_{\mathit{estimated}}=c_v{c}_{i}pv$$(1) (Chen et al. 2020). (1) $$q_{\mathit{estimated}}=c_v{c}_{i}pv$$(1) where: $q_{\mathit{estimated}}$ is the estimated quanta generation rate, quanta/h; $c_v$ is the viral load, RNA copies/mL; $c_i$ is the conversion factor, and we consider the worst case and set this value to 0.1 (Buonanno, Stabile, et al. 2020); $p$ is the pulmonary ventilation rate, as summarized in Table 2, m³/h; $v$ is the volume of the total droplets, m³.

In Equation (1)(1) $$q_{\mathit{estimated}}=c_v{c}_{i}pv$$(1) , Only $c_v$ and $v$ were unknown. Hence, the following steps were made:

1. We statistically analyzed the distribution of viral load based on the published peer-reviewed literature.

2. We fitted the particle size distribution of respiratory particles in different scenarios.

3. We calculated the quanta generation rate using the Monte-Carlo method. Detailed information of $c_v$ and $v$ was illustrated in Section “Viral load” and Section “Respiratory particles”, respectively.

Viral load

How to ascertain the viral load in the exhaled droplets was an unresolved problem. Theoretically, viruses that entered the upper respiratory tract may be released into the air during respiratory activities and cause infection. (Walsh et al. 2020) investigated the viral load in nearly 50 studies. The results showed that the viral load was almost the same between pre-symptomatic, asymptomatic, and symptomatic subjects, and the viral load of saliva was higher than that of other parts, reaching a peak within a few days after infection. As the infector viral load varied with the time after infection, we counted the viral load of throat swabs and saliva swabs reported by (Zheng et al. 2020), (Rothe et al. 2020), (To et al. 2020), (Kim et al. 2020), (Pan et al. 2020), (Wölfel et al. 2020), (Cao et al. 2020), (Yoon et al. 2020), (Dubert et al. 2020) and (Han et al. 2020). The logarithmic results are shown in Figure 1. There were significant differences in viral load reported in different studies. (Kim et al. 2020), (To et al. 2020), (Zheng et al. 2020), (Yoon et al. 2020), and (Wölfel et al. 2020) showed a median of ${10}^{4.5}\sim {10}^{6.5}$ RNA copies/mL, (Dubert et al. 2020), (Han et al. 2020), and (Rothe et al. 2020) showed a higher median of ${10}^8\sim {10}^9$ RNA copies/mL, while (Pan et al. 2020) and (Cao et al. 2020) showed a median of less than ${10}^{4.5}$ RNA copies/mL. The viral load of different infectors in the same studies was also remarkably different. Considering the viral load distribution in the above literature is a better approach to reflect the true viral load.

Fig. 1. Viral load in different specimens.

We fitted a total of 397 data points from the above literature using the Monte Carlo method and performed Kolmogorov-Smirnov (KS) goodness of fit testing. As shown in Figure 2, the results had a normal distribution with a median value of 4.87 and a standard deviation of 1.90. The P-value was greater than 0.1, indicating that the fitting results were reliable.

Fig. 2. Viral load distribution: (a) Histogram and Fitting Curve; (b) Normal Q-Q Plot.

Respiratory particles

According to Equation (1)(1) $$q_{\mathit{estimated}}=c_v{c}_{i}pv$$(1) , the estimated quanta generation rate was also influenced by the exhaled droplets. It was calculated by Equation (2)(2) $$v={\int }_0^{D_{\mathit{max}}}{N}_d\left(D\right)d{V}_d\left(D\right)$$(2) . (2) $$v={\int }_0^{D_{\mathit{max}}}{N}_d\left(D\right)d{V}_d\left(D\right)$$(2) where $N_d\left(D\right)$ is the droplet number concentration, $\mathit{part}./m^3;$ $V_d\left(D\right)$ is the volume of a single droplet, as shown in Equation (3)(3) $$V_d\left(D\right)=\frac{4}{3}\pi {\left(\frac{D}{2}\right)}^3$$(3) . $D_{\mathit{max}}$ is the maximum aerodynamic diameter of the exhaled droplet, $\mu m,$ with a value of 20. (3) $$V_d\left(D\right)=\frac{4}{3}\pi {\left(\frac{D}{2}\right)}^3$$(3)

Some scholars have carried out experimental testing and simulation studies with respect to $N_d\left(D\right)$ (Duguid 1946; Lindsley et al. 2012; Lindsley et al. 2016; Morawska et al. 2009; Chen et al. 2020; Mao et al. 2020; Alsved et al. 2020). The aforementioned literature mainly focused on the number of droplets with different particle sizes during coughing, speaking, and breathing. Furthermore, the experimental results were subject to individual differences and contingencies. Even for the same study, the results varied from person to person. (Morawska et al. 2009) experimentally measured the droplets number concentration in different respiratory activities. To reduce the contingency, we assigned different weights to the designed exposure scenarios for fitting, based on the results. For example, students studying quietly in the library, the weight is 100% whispered counting. The droplet number concentration after weight allocation and the curve after fitting are shown in Figure 3. However, it was still unknown that what kind of fitting type should be employed. Since there were only four test values, we applied cubic polynomial, and $R^2$ was 1 (Burden, Burden, and Faires 2016). The fitting formula is expressed in Equation (4)(4) $$N_d\left(D\right)=F_3{D}^3+F_2{D}^2+F_{1}D+F_0$$(4) . (4) $$N_d\left(D\right)=F_3{D}^3+F_2{D}^2+F_{1}D+F_0$$(4) where $F_3,$ $F_2,$ $F_1,$ and $F_0$ are cubic term, quadratic term, primary term, and constant term coefficient, respectively. The fitted results are also shown in Figure 3.

Fig. 3. Droplet concentrations fitting curve.

Calculating tool and accuracy

To characterize the uncertainty and variability, we employed Monte Carlo simulation to quantify the uncertainty with regard to the quanta generation rate based on the fitted viral load distribution and droplet number concentration curves. We used K-S statistics to optimize the goodness of fit of the distributions. We adopted Crystal Ball software (Version 2000.2, Decisioneering, Inc., Denver, CO, USA) to analyze the data and estimate the distribution parameters. The number of iterations was 10,000 to ensure the stability of the results.

The transmission model

The Wells-Riley equation

We quantified the infection risk of the five exposure scenarios based on the Wells-Riley equation. However, the Wells-Riley equation fails to reflect the change of quanta concentration over time, resulting in a slightly higher risk result (Beggs et al. 2003). Therefore, we further refined it according to the improved equation proposed by (Gammaitoni and Nucci 1997), taking into account respiration protection and various engineering measures. The following assumptions were made: (1) The quanta generation rate varied with time; (2) The incubation period of the disease was longer than the time scale of the model; (3) The droplets were distributed uniformly and instantaneously in the room. The number of susceptible individuals and the indoor quanta concentration conformed to differential equations (5)(5) $$\frac{dS}{dt}=-\frac{p(1-{\eta }_s)}{V}cS$$(5) and (6)(6) $$\frac{dc}{dt}=-rc+q_{\mathit{estimated}}I$$(6) : (5) $$\frac{dS}{dt}=-\frac{p(1-{\eta }_s)}{V}cS$$(5) (6) $$\frac{dc}{dt}=-rc+q_{\mathit{estimated}}I$$(6) where $S$ is the number of susceptible individuals, when $t=0,$ $S$ is the initial number of susceptible individuals $S(0),$ $S\left(0\right)=N-1;$ ${\eta }_s$ is the respiratory protective efficiency, %, if no respiratory protection was taken, ${\eta }_s=0;$ $V$ is the volume of confined space, m³; $c$ is the quanta concentration of confined space, quanta/m³; $r$ is the infection virus removal rate, ${\mathrm{h}}^{-1}.$

After solving the differential equations (5)(5) $$\frac{dS}{dt}=-\frac{p(1-{\eta }_s)}{V}cS$$(5) and (6)(6) $$\frac{dc}{dt}=-rc+q_{\mathit{estimated}}I$$(6) , the change of quanta concentration with time is shown in Equation (7)(7) $$c\left(t\right)=\frac{q_{\mathit{estimated}}\mathrm{I}}{rV}\left(1-\exp \left(-rt\right)\right)+c\left(0\right)\exp (-rt)$$(7) . (7) $$c\left(t\right)=\frac{q_{\mathit{estimated}}\mathrm{I}}{rV}\left(1-\exp \left(-rt\right)\right)+c\left(0\right)\exp (-rt)$$(7) where $c(0)$ is the initial quanta concentration of the confined space, quanta/m³. $r$ consists of four contributions: ventilation, particle deposition, viral inactivity, and engineering control measures such as UVGI and HEPA. Consequently, $r$ is given by Equation (8)(8) $$r=\mathit{AER}+{\lambda }_V+{\lambda }_{\mathit{Dep}}+{\lambda }_{UV}+h_r{\eta }_r$$(8) . (8) $$r=\mathit{AER}+{\lambda }_V+{\lambda }_{\mathit{Dep}}+{\lambda }_{UV}+h_r{\eta }_r$$(8) where $\mathit{AER}$ is the air exchange rate via ventilation, ${\mathrm{h}}^{-1};$ ${\lambda }_V$ is the rate coefficient of viral activity, ${\mathrm{h}}^{-1},$ with a value of 0.64 (van Doremalen et al. 2020); ${\lambda }_{\mathit{Dep}}$ is the particle deposition rate, ${\mathrm{h}}^{-1},$ with a value of 0.24 (Chatoutsidou and Lazaridis 2019); ${\lambda }_{UV}$ is the rate coefficient of inactivation by ultraviolet irradiation, ${\mathrm{h}}^{-1},$ $h_r$ is the equivalent virus removal rate of the filter, ${\mathrm{h}}^{-1},$ ${\eta }_r$ is the filtration efficiency, %.

If the infector leaves the confined space, $q_{\mathit{estimated}}=0,$ and $c(t)$ is expressed as Equation (9)(9) $$c\left(t\right)=c\left(0\right)\exp (-\mathrm{rt})$$(9) . (9) $$c\left(t\right)=c\left(0\right)\exp (-\mathrm{rt})$$(9)

The integral of quanta concentration over time was substituted into the Wells-Riley equation to obtain the infection risk of a susceptible individual from time $T_1$ to time $T_2,$ as shown in Equation (10)(10) $$R=1-\exp (-p(1-{\eta }_s){\int }_{t_1}^{t_2}c\left(t\right)dt)$$(10) . (10) $$R=1-\exp (-p(1-{\eta }_s){\int }_{t_1}^{t_2}c\left(t\right)dt)$$(10) where: $R$ is the infection risk of a susceptible individual.

Assuming that the infector stays in the confined space from time $t=0$ to $t=T_1,$ and $c\left(0\right)=0,$ $c(t)$ is given by Equation (11)(11) $$c(t)=\{\begin{array}{c}\frac{q_{\mathit{estimated}}I}{rV}(1-\exp (-rt))\mathrm{ }0\le t\le T_{\mathit{stay}}\\ c(T_1)\exp (-rt)T_{\mathit{stay}}\le t\end{array}$$(11) . (11) $$c(t)=\{\begin{array}{c}\frac{q_{\mathit{estimated}}I}{rV}(1-\exp (-rt))\mathrm{ }0\le t\le T_{\mathit{stay}}\\ c(T_1)\exp (-rt)T_{\mathit{stay}}\le t\end{array}$$(11)

The calculation formula of $R$ can be obtained by substituting Equation (11)(11) $$c(t)=\{\begin{array}{c}\frac{q_{\mathit{estimated}}I}{rV}(1-\exp (-rt))\mathrm{ }0\le t\le T_{\mathit{stay}}\\ c(T_1)\exp (-rt)T_{\mathit{stay}}\le t\end{array}$$(11) into Equation (10)(10) $$R=1-\exp (-p(1-{\eta }_s){\int }_{t_1}^{t_2}c\left(t\right)dt)$$(10) , leading to Equation (12)(12) $$R\begin{array}{cc}=\{\begin{array}{c}1-\hspace{0.17em}\text{exp}\hspace{0.17em}\left\{-\frac{p(1-{\eta }_s)q_{\mathit{estimated}}I}{rV}\left[t+\frac{1}{r}\exp \left(-rt\right)-\frac{1}{r}\right]\right\} 0\le t\le T_{\mathit{stay}}\\ 1-\hspace{0.17em}\text{exp}\hspace{0.17em}\left\{-p(1-{\eta }_s)\left\{\begin{array}{c}\frac{q_{\mathit{estimated}}I}{rV}\left[T_1+\frac{1}{r}\exp \left(-r{T}_1\right)-\frac{1}{r}\right]+\\ \frac{q_{\mathit{estimate}\mathrm{d}}I}{rV}\left[1-\exp \left(-r{T}_1\right)\right]\left[\frac{1}{r}\exp \left(-r{T}_1\right)-\frac{1}{r}\mathrm{e}\mathrm{xp}(-rt)\right]\end{array}\right\}\right\}\end{array}& T_{\mathit{stay}}\le t\end{array}$$(12) . (12) $$R\begin{array}{cc}=\{\begin{array}{c}1-\hspace{0.17em}\text{exp}\hspace{0.17em}\left\{-\frac{p(1-{\eta }_s)q_{\mathit{estimated}}I}{rV}\left[t+\frac{1}{r}\exp \left(-rt\right)-\frac{1}{r}\right]\right\} 0\le t\le T_{\mathit{stay}}\\ 1-\hspace{0.17em}\text{exp}\hspace{0.17em}\left\{-p(1-{\eta }_s)\left\{\begin{array}{c}\frac{q_{\mathit{estimated}}I}{rV}\left[T_1+\frac{1}{r}\exp \left(-r{T}_1\right)-\frac{1}{r}\right]+\\ \frac{q_{\mathit{estimate}\mathrm{d}}I}{rV}\left[1-\exp \left(-r{T}_1\right)\right]\left[\frac{1}{r}\exp \left(-r{T}_1\right)-\frac{1}{r}\mathrm{e}\mathrm{xp}(-rt)\right]\end{array}\right\}\right\}\end{array}& T_{\mathit{stay}}\le t\end{array}$$(12)

Basic reproduction number

The $R$ value can only embody the individual risk of infection. From an epidemiological and public risk prevention perspective, the basic reproduction number $R_0$ should also be studied. $R_0$ is a vitally important quantity when exploring the spreading of a respiratory disease, which represents the number of new cases generated directly by one infector with a specific pandemic (Al-Raeei 2021). Currently, the $R_0$ can be derived directly from a mathematical model (Zhao et al. 2020) or be estimated by building a transmission model through retrospective data (Wu, Leung, and Leung 2020). In this paper, the $R_0$ was prospective rather than retrospective. The former method was adopted (Beggs et al. 2003; Buonanno, Stabile, et al. 2020; Buonanno, Morawska, et al. 2020), as Equation (13)(13) $$R_0=S\left(0\right)R$$(13) shows. (13) $$R_0=S\left(0\right)R$$(13) where $R_0$ is the basic reproduction number.

Results

Estimated quanta generation rate

The estimated quanta generation rate of all the exposure scenarios followed a lognormal distribution. Figure 4 illustrates the natural logarithmic results and distribution curves. The infector (athlete) in the gym had a high activity level and respiratory intensity. Therefore, the quanta generation of the gym was the highest, with a median value of 454.87 $(e^{6.12})$ quanta/h. The median value of the classroom was 194.4 ($e^{5.27}$) quanta/h. Although it was assumed that the droplet number concentration exhaled by the infector in the dormitory and classroom was the same, the inhalation rate of the infector in the classroom was higher. Therefore, the median value of the dormitory was only 83.10 ($e^{4.42}$) quanta/h. The inhalation rate and droplet concentration in the library and dining hall were lower than those in the gym, and the median values were 20.49 ($e^{3.02}$) quanta/h and 65.37 ($e^{4.18}$) quanta/h, respectively.

Fig. 4. Natural logarithm of the quanta generation rate under exposure scenarios.

Infection risk of the exposure scenarios

Quanta concentration and$$infection risk

Figure 5 presents the quanta concentration and infection risk variations with exposure time under different exposure scenarios. For comparison, Figure 5 also shows the results for the quanta condition lasting up to 24 h. When the infector left the confined space, the quanta concentration sharply dropped to zero. In the dormitory, it took about two hours for the quanta concentration to decline to zero, while in other scenarios, it was less than one hour. Accordingly, the infection risk was almost no longer increased. The dormitory volume was only 60 m³, and the ventilation rate was only 0.5 ACH. Therefore, the quanta concentration reached a stable peak of 1.0 quanta/m³ after about 3.5 h, and no significant increase was observed. Due to the high ventilation rate, the quanta concentration of the other four scenarios reached a stable peak value within one hour, and they were much smaller than that of the dormitory. The infection risk in the dormitory was also the highest. When a susceptible individual was exposed for four hours and eight hours, the infection risk was about 75% and 100%, respectively. The classroom was of a middle size (296 m³) but with a high ventilation rate (6.08 ACH), so the infection risk was about 5.0% at the end of a class period (1.5 h). The infection risk was less than 0.4% when the student watched a match in the gym for two hours. In the library, the risk still did not exceed 0.15%. Even after 30 minutes of exposure in the dining hall, the risk was only 0.3%. The dominant reasons were that the gym, library, and dining hall had larger volumes and higher ventilation rates. Consequently, the infection risk of the dormitory and the classroom was the highest.

Fig. 5. The variations of $\mathit{c}(\mathit{t})$ and $\mathit{R}:$ (a) Dormitory; (b) Classroom; (c) Gym; (d) Library; (e) Dining hall.

Basic reproduction number

Figure 6 shows the variations of $R_0$ with exposure time under different scenarios. Due to many spectators, although the individual infection risk in the gym was low, $R_0$ still increased rapidly and reached 1.3 after two hours. Although four roommates were always in the dormitory and the people mobility characteristic was not considered, the infection risk in the dormitory was much higher than that of the other confined spaces. Thus, the $R_0$ was as high as 2.8 if the exposure time reached eight hours. As for the classroom, the increased rate of $R_0$ was the highest in all confined spaces. After 1.5 h exposure, $R_0$ was 2.8. Due to the low quanta generation rate and high ventilation rate in the dining hall and library, the $R_0$ after designed exposure time was 0.06 and 0.76, respectively. The above analysis demonstrated that the transmission rate of different confined spaces had apparent differences. If the $R_0$ value could be controlled below a particular threshold value, it means the infected population will decrease gradually. Moreover, most retrospective studies set 1 as the threshold value (Buonanno, Morawska, et al. 2020). Only the dormitory and classroom were below one during the designed exposure time.

Fig. 6. Estimated ${\mathit{R}}_0$ with exposure time for the exposure scenarios.

Impact of engineering control measures

Enhance ventilation

The nature of the Wells-Riley equation reveals that the infection risk decreases with the ventilation rate. However, enhanced ventilation may also cause more energy consumption. Although the risk control cost and energy consumption were not the focus of our study, clarifying the relationship between infection risk and ventilation rate is still exceptionally significant for preventing SARS-CoV-2 transmission. In Section “Infection risk of the exposure scenarios”, we pointed out that the dormitory and classroom were the critical places for risk control on university campuses. Figure 7 illustrates the changes in the dormitory and classroom infection risk with the ventilation rate at different exposure times. Concerning the dormitory, when the exposure time was one hour, the effect of enhanced ventilation was not noticeable. Even if the ventilation rate increased from 30 m³/h to 240 m³/h (AER increased from 0.5 ACH to 4 ACH), the infection risk was reduced by less than 10%. The shorter the exposure, the lower the risk. When the exposure time increased to eight hours, the difference could reach 40%. Furthermore, when the exposure time exceeded four hours, the infection risk tended to decrease linearly. As the exposure time increased for the classroom, a slight increase in the ventilation rate was more effective in reducing the infection risk. For example, if the ventilation rate increased from 150 m³/h to 1000 m³/h (T = 1.5 h), the risk reduction value was 7%. Although enhanced ventilation can reduce the risks, there were significant differences in the effects for different confined spaces.

Fig. 7. The variations of estimated infection risk with ventilation rate: (a) Dormitory; (b) Classroom.

Susceptible flow control

As we mentioned before, the dining hall is a unique confined space with a high crow flow rate. In this section, we explored the relationship between reducing the transmission risk and the susceptible number. Figure 8 shows the variation of the $R_0$ with the susceptible number and the stay time (exposure time). Similarly, the nature of the Wells-Riley equation determines that the $R_0$ will increase with the susceptible number and the stay time. If taking one as the threshold value, even if the number of people always present was expanded to 1500, $R_0$ was only 0.28 (T = 20 minutes). If the stay time reached 60 minutes, the susceptible numbers should be controlled at 1050, with $R_0$ precisely equal to 1. If the stay time was allowed to be two hours, the susceptible number should be controlled to 450 to ensure $R_0<1.$

Fig. 8. ${\mathit{R}}_0$ with the different susceptible number and stay time.

Combined engineering measures

In addition to general ventilation (GV), frequently used engineering control measures consist of respiratory protection, ultraviolet germicidal irradiation, and high efficiency particle air (Zhang 2020; Feng, Wei, et al. 2021). In this section, the efficacy of specific combined engineering measures is discussed. Table 4 shows the calculation parameters of the surgical mask (M), UVGI, and HEPA. In Table 3, LN presents the lognormal distribution. In some university courses, students cannot wear any masks. Therefore, the following six combined measures were analyzed: (1) GV + M; (2) GV + M + UVGI; (3) GV + UVGI; (4) GV + HEPA; (5) GV + M + HEPA; (6) GV + M + UVGI + HEPA.

Table 4. Parameters of the control measures used in the infection risk model.

Parameters	Value	Symbol/Unit	Data resource
Surgical mask	N (45, 3.1)	${\eta }_s\mathrm{ }(\mathrm{\%})$	(Gammaitoni and Nucci 1997)
Dental mask	61	${\eta }_s\mathrm{ }(\mathrm{\%})$	(Weber et al. 1993)
Homemade mask	70	${\eta }_s\mathrm{ }(\mathrm{\%})$	(Davies et al. 2013)
N95 mask	99	${\eta }_s\mathrm{ }(\mathrm{\%})$	(Lee, Grinshpun, and Reponen 2008)
UVGI	N (12.0, 1.3)	${h_u\mathrm{ }(h}^{-1})$	(Xu et al. 2005)
HEPA filtration	12	$h_r\mathrm{ }(h^{-1})$	(CDC 2003; Liao, Lin, and Cheng 2013)

Figure 9 reveals the variations of $R$ and $R_0$ with exposure time for different combined measures. For comparison, the GV or NV (only dormitory) was also given.

Fig. 9. The variations of estimated $\mathit{R}$ and ${\mathit{R}}_0$ under combined engineering measures: (a) Dormitory;(b) Classroom; (c) Gym; (d) Library; (e) Dining hall.

The efficacy was not evident in the dormitory when adopting GV or GV + M due to the air change rate being only two $h^{-1}.$ However, if the UVGI or HEPA were used, the infection risk would significantly decrease by 65%. For all scenarios except for the dormitory, the order of average risk was almost the same: GV $>$ GV + M $>$ GV + M + GVUI $>$ GV + HEPA $\approx$ GV + UVGI $>$ GV + M+HEPA $>$ GV + M+UVGI + HEPA. Nevertheless, there were slight differences. The infection risk of GV + M and GV + UVGI was 35% and 28%, respectively (T = 24 h) for the classroom. While in the gym, library, and dining hall, the other measures were substantially better than the GV or GV + M. Furthermore, the efficacy of the GV + M+UVGI + HEPA was very close to the GV + M+HEPA, especially for the dining hall. The main reason was that the initial infection risk was low. Moreover, the $R_0$ value should not be ignored. For classrooms and gym, it could reach 30 and 17 respectively. To sum up, only in the infection risk reduction perspective, the combined measures were effective.

It is necessary to illustrate further the efficacy of wearing a surgical mask on infection risk reduction. Figure 10 presents the variations of estimated $R$ and $R_0$ under typical masks with different respiratory efficiencies in the classroom. For comparison, the circumstance of the only NV with an air change rate of 0.5 $h^{-1}$ was also given. Compared with NV, the infection risk was significantly reduced by 55% when only adopting GV (T = 8h). When the exposure time was below 1.5 h, the infection risk increased approximately linearly. Meanwhile, with every 10% decrease in respiratory protection efficiency, the reduction in infection risk was approximately unchanged. If the N95 mask was adopted, the infection risk was low. Even if the exposure time was 24 h, the infection risk was below 5%. The nature of the Wells-Riley equation determines that the infection risk decreased with the respiratory protection efficiency.

Fig. 10. The variations of estimated $\mathit{R}$ and ${\mathit{R}}_0$ under some typical masks in the classroom.

Discussion

The comparison of the quanta values

The estimation of quanta generation is vitally important to accurately predict the infection risk of SARS-CoV-2 in different confined spaces. Although the world was still in a pandemic, scholars studied the quanta value of SRAS-CoV-2. Figure 11 demonstrates the comparison between the predicted quanta numbers in previous studies and the results of our study. (Miller et al. 2020) calculated quanta values as high as 970 quanta/h for an outbreak case in the United States using the conventional backward calculation method. However, (Hota et al. 2020) employed the same method to calculate the quanta value for an outbreak among nurses in a hospital, resulting in only 0.025 quanta/h. Further, (Buonanno, Stabile, et al. 2020; Buonanno, Morawska, et al. 2020) proposed a forward calculation method to estimate the quanta value of confined spaces such as pharmacies, hospitals, restaurants, and gyms. The results ranged from 1 to 150 quanta/h. On this basis, the actual viral load distribution and the fitted droplets number concentration were both considered in our paper, with results ranging from 20.49 to 454.87 quanta/h. As mentioned in Section “Estimation method and steps”, (Dai and Zhao 2020) obtained quanta values in a range of 16 ∼ 48 quanta/h. However, more data on virus characteristics are still needed to research the quanta values further.

Fig. 11. Comparison of quanta generation rate in different scenarios.

The role of the controlling measures

We analyzed the transmission risk of different confined spaces on the university campus from the individual infection risk and public disease prevention perspectives. The infection risk and $R_0$ should be evaluated comprehensively instead of separately. The calculated results showed that for scenarios with small volumes such as dormitory, $R_0$ was low, but the infection risk was high. As for scenarios with large volumes, the results were the opposite. Usually, as an essential parameter to assessing the epidemic, $R_0$ was meaningful when the population density was as high as possible (Kermack and McKendrick 1927). If $R_0$ was controlled below a specific threshold value, it meant the epidemic was effectively controlled. We adopted one as the threshold value as most of the studies did. However, if the susceptible number was low, this value may be inappropriate. Only a few studies focused on the $R_0$ applications for confined spaces from a perspective viewpoint. There are still many problems that need to be solved. Therefore, although the infection risk of both the dormitory and the classroom was highest, all scenarios should be critical places for epidemic prevention and control.

In classrooms and libraries, wearing a mask could effectively prevent SARS-CoV-2 but reduce the learning efficiency and sustained attention. As personal protection equipment, masks are also an essential means of public health intervention, recommended by WHO self-protection guidelines (WHO 2020a). The US CDC also suggested wearing masks with low efficiency but N95 masks (CDC 2021b). (Chen et al. 2021) studied the efficacy of masks with different efficiencies on public transportation. The results illustrated that wearing homemade masks could reduce the infection risk by 67%. (Wei et al. 2021) discovered that conventional masks with low efficiency have high efficacy. Even if wearing a low-efficiency mask, the susceptible individual also needs to wear it correctly during the exposure time. However, (Feng, Liu, et al. 2021) found that only 29.0% of medical students wore masks in enclosed spaces, crowded spaces and, medical establishments. The reasons for not wearing the mask were accessibility (45.6%) and comfort (11.9%). In addition, compliance, violent damage, and improper wearing were also the qualifications. (Liu et al. 2020) discovered that wearing a mask for a long time may increase the discomfort significantly in a warm environment. Therefore, the effectiveness of wearing masks on university campuses needs more research.

The engineering control measures play an essential role in fighting SARS-CoV-2. In hospitals, (Feng, Wei, et al. 2021) recommended that UV + Filter and combine ionized electric field and fibrous filter applicated. In classrooms, (Zhang 2020) proposed indoor air quality (IAQ) strategies, including source control, ventilation, and air cleaning to reduce the infection risk. (Ding, Yu, and Cao et al. 2020) pointed out that the HEPA filter may be effective according to the experimental results conducted by (Park et al. 2020). Our calculation results also showed that in all scenarios, the combined engineering control measures were more effective. Moreover, the transmission routes consist of contact transmission, droplet transmission, and airborne transmission (Xu et al. 2020). The WHO suggested that droplet transmission is the main route, while some scholars believed that the airborne transmission could not be ignored. We estimated the quanta value based on all the exhaled droplets, which covered part of droplet transmission and airborne transmission (Morawska et al. 2020). With regard to transmissions via large droplets ($\ge 100\mu m)$ and contact, the common measures were disinfection and physical distance (Prather et al. 2020). The UVGI and the HEPA could also disinfect the surface and move out the virus in the air, respectively. Undoubtfully, adopt a single measure may not be effective. We recommended the control measures mainly based on engineering measures such as ventilation, HEPA UVGI, partially based on surgical masks. However, if we know the equivalent infectious removal rate in our mathematical model, the infection risk could be calculated. Therefore, we did not focus on the type or installation location of the UVGI or HEPA filter. They should be studied furthermore.

The limitations of this study

1. Due to the lack of SARS-CoV-2 infectivity data, only the worst case was considered. It may cause the result to be worse than the actual value.

2. Only the infection risk of adopting different measures was analyzed. Issues such as energy consumption, IAQ, noise, initial investment, cost were not involved.

Conclusions

The scope of this paper was to explore the transmission of SARS-CoV-2 in different confined spaces (dormitory, classroom, gym, library, dining hall) on a university campus. Based on previous work, the forward quanta calculation method based on viral load conservation was further refined, and two contributions were considered: one is the viral load distribution based on the cases reported in the literature, the other is to fit the droplets number concentration in consideration of reducing the contingency. Respiratory protection and various engineering control measures were considered and integrated into the improved Wells-Riley equation. The infection risk and the efficacy of the engineering control measure against five exposure scenarios on a university campus were assessed. The findings and recommendations are as follows

1. The quanta generation rates in the dormitory, classroom, library, gym, and dining hall were 83.10 quanta/h, 194.4 quanta/h, 20.49 quanta/h, 454.87 quanta/h, 20.49 quanta/h, respectively. The quanta generation rate is significantly influenced by the activity level and the respiratory intensity. The forward estimation method based on the conservation of viral load still needs more virus characteristics and epidemic data support.

2. The infection risk in the dormitory and classroom was obviously higher than in other confined spaces. The infection risk of students staying in the dormitory for more than 8 hours was close to 100%, while it was 5% after finishing a class. The infection risk in other confined spaces after the design exposure time was below 0.5%. The threshold value of $R_0$ from a prospective perspective should be further studied.

3. Students should reduce mobility in the dormitory building. Places with high mobility, such as the dining hall, should control the susceptible flow. The susceptible flow should also be controlled in a class, and classes should not be arranged continuously. The transmission risk caused by the random movement of people deserves further study.

4. On the university campus, engineering measures such as enhanced ventilation, HEPA filter, UVGI should be adopted. Surgical masks could provide additional protection.

Nomenclature
$\mathit{AER}$	=	air exchange rate [h−1]
$c$	=	quanta concentration [quanta/m3]
$c_i$	=	conversion factor
$c_v$	=	viral load [RNA copies/mL]
$D$	=	Droplet diameter [$\mathrm{\mu }\mathrm{m}$ ]
$D_{\mathit{max}}$	=	maximum droplet diameter [$\mathrm{\mu }\mathrm{m}$ ]
$h_r$	=	equivalent virus removal rate of the filter [h-1]
$N_d\left(D\right)$	=	Droplet number concentration [part./m3]
$p$	=	pulmonary ventilation rate [m3/h]
$q_{\mathit{estimated}}$	=	estimated quanta generation rate [quanta/h]
$r$	=	virus removal rate [h-1]
$R$	=	infection risk [%]
$R_0$	=	basic reproduction number
$S$	=	susceptible
$t$	=	exposure time [h]
$v$	=	total volume of the droplet [m3]
$V$	=	volume of confined space [m3]
$V_d(D)$	=	particle size [m3]
${\eta }_s$	=	respiratory protection efficiency [%]
${\lambda }_{\mathit{Dep}}$	=	rate deposition loss of the infectious particles [h−1]
${\lambda }_{UV}$	=	rate coefficient of inactivation by ultraviolet irradiation [h-1]
${\lambda }_V$	=	viral inactivation [h-1]
${\eta }_r$	=	filtration efficiency [%]

Third-party material statement

I am not using third-party material for which formal permission is required.

Declarations of interest

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (52078314).