## Model

The present model recognizes not only the discreteness of virions and its fluctuations but also that of the inhaled residues/droplets which vector them and hence, introduces fluctuations in the entire size spectrum17–19. Thus, the probability of at least one virus being deposited in a lung is proportional to the likelihood of inhaling at least one droplet containing at least one virion. A double Poisson distribution function is used to represent these probabilities, which is illustrated here:

Table 1

Comparison of infection risk models

| Nicas et al.6 | Buonano et al.7 | Buonano et al.15 | Dhawan et al.11 | Peng et al.9 | Mizukoshi et al.8 | Azuma et al.16 | Sussman et al.12 | Netz et al.10 | Present study |

Emission | Coughing, sneezing | Breathing at different conditions | Speaking, breathing, counting | Sneezing, coughing, speaking | Speaking, breathing, singing, exercise | Coughing, speaking | Coughing, speaking | Breathing, Speaking, Coughing | Speaking | Breathing, Speaking, Coughing, Sneezing |

Size distribution | Discrete | Discrete | Total volume | Continuous | Total volume | Discrete | Discrete | Total volume | Continuous | Continuous |

Virusol | Same viral load as the sputum | Poisson distribution |

Evaporation of droplets | Parameterization, 50% reduction assumed | Not considered | Dehydrated volume is considered | Calculated for each size | Not considered | Not considered | Not considered | Started with evaporated droplet nuclei | Calculated for each size | Calculated for each size |

Ventilation effect | Uniformly mixed, first-order rate | Uniformly mixed, first-order rate | Uniformly mixed, first-order rate | Diffusion, gravitational settling, and ambient air flow | Uniformly mixed, first-order rate | Uniformly mixed, first-order rate | Uniformly mixed | Uniformly mixed | Settling under gravity with evaporation | Falling-to-Mixing-Plate-out model |

Gravitational settling | Stirred settling formula | Constant, 0.24 h− 1 | Constant, 0.24 h− 1 | Neglected | First-order rate | - | **-** |

Airborne inactivation | Yes, first-order rate | Constant, 0.63 h− 1 | Constant, 0.63 h− 1 | Considered | Neglected | Considered | Yes + sterilization | Considered | Neglected | Considered |

Air-cleaning factor | Yes, first-order rate | No | No | No | Yes, first-order rate | No | No | No | - | - |

Respiratory deposition | Yes | No | No | Yes, ICRP model | No | No | No | No | No | Yes, ICRP model |

Risk model | Single-hit & multiple-hit | Single-hit | Single-hit | Single-hit | Single-hit | Single-hit | Single-hit | Single-hit, | No | Single-hit & multiple-hit |

Infectivity factor | No | Yes | Yes, PFU | Yes | Quanta emission | Yes, PFU | Quanta emission | Infective quanta | No | Yes |

Variation of input parameters | Constant | Constant | Prob. density function | Yes, for some parameters | Yes | Yes | Yes | Yes | Yes | Yes |

Protection factor | No | No | No | Mask & distance | Mask | Mask | No | Discussed qualitatively | No | Yes |

Re-suspension | No | No | No | No | No | No | Introduced | No | No | No |

Suppose a person inhales a typical number of droplets (\({N}_{d}\)), each of which is expected to contain an average number of virions, \({n}_{v}\). The risk of breathing at least one virion is then given by \(R=1-exp\left({-N}_{d}{n}_{v}\right)\) according to the single Poisson fluctuation model, which implies that there are fluctuations in the number of virions in the droplet but not in the number of droplets inhaled. The number of droplets inhaled would, however, fluctuate around the mean value at low droplet concentrations. As a result, the risk formula is modified to factor for both fluctuations, and the modified risk for inhalation of at least one droplet carrying a virion can be expressed as,

$$R\text{'}=1-exp\left({-N}_{d}\left[1-exp\left(-{n}_{v}\right)\right]\right)$$

1

The main difference is that in the modified formula (Eq. (1)), the values \({N}_{d}\) and \({n}_{v}\) appear separately, rather than as a product, which is commonly employed in the literature6–11, 15. In the present study, the above formula has been combined with averaging over the polydisperse size distribution function. The present double Poisson model has the advantage of being applicable even in the case of extremely low risk scenarios, such as inhalation for a brief period or at low droplet number concentrations.

To demonstrate the applicability of this model, a standard inhalation infection risk problem6 is used, in which four discrete droplet diameters (4.2 µm (1200 droplets), 9.0 µm (100 droplets), 14.6 µm (6.2 droplets) and 18.8 µm (1.7 droplets)) are released during coughing event (10 h-1) in room volume of 50 m3 with an air-exchange rate of 0.5 h-1. In this scenario, a viral load of (5 x 106 − 5 x 1010) #/mL in the biological fluid is considered, with 0.1 h-1 inactivation rate. The risk estimates from the current model (Eq. (1)) are compared to those of Nicas et al.6 using these input values, as shown in Table 2. In the comparison table, the implications of separating the fluctuations in inhaled droplets from the likelihood that a droplet is infected are shown.

Table 2

Comparison of risk with Nicas et al.6

Initial droplet diameter, µm | Single-hit risk |

Cv **= 5 x 10****6** **mL****− 1** | Cv **= 5 x 10****8** **mL****− 1** | Cv **= 5 x 10****10** **mL****− 1** |

**Present model** | **Nicas et al.**6 | **Present model** | **Nicas et al.**6 | **Present model** | **Nicas et al.**6 |

4.2 | 4.89E-04 | 1.93E-03 | 4.73E-02 | 1.76E-01 | 8.97E-01 | 1.00E + 00 |

9.0 | 5.21E-03 | 5.89E-03 | 3.81E-01 | 4.46E-01 | 9.47E-01 | 1.00E + 00 |

14.6 | 1.03E-03 | 9.07E-05 | 6.99E-02 | 9.03E-03 | 1.27E-01 | 5.96E-01 |

18.8 | 4.00E-04 | 4.62E-06 | 1.96E-02 | 4.62E-04 | 2.42E-02 | 4.52E-02 |

When compared to Nicas et al.6, the projected risk from the current model for the 4.2 µm droplet is four times lower for \({C}_{v}\le\) 5 x 108 mL-1 (Table 2), owing to the difference in final droplet size, which determines its lifetime and lung deposition characteristics. In the case of larger size droplets (14.6 µm and 18.8 µm), the risk estimates from these two models differ by a ratio of ~(8 to 90). According to the current simulation results, the final droplet diameter is decreased to ~ 1/5th of the original droplet (Fig. 1) as opposed to 50% reduction in Nicas et al.6 due to evaporation. The difference in the equilibrium droplet size is mainly due to the solid content in the saliva/droplet apart from other ambient conditions. In the present work, a solid content of 8 g/L is assumed as against 88 g/L in Nicas et al.6, and the evaporation of droplet is modelled precisely and coupled with the other processes seamlessly. Also, Nicas et al.6 assumes that the droplets released are instantaneously mixed in the room environment and hence the concentration is uniform whereas, the present studies include the effect of ventilation induced turbulence to simulate the dynamics of the droplets in the room.

A recent study by Lieber et al.20 also shown that a mass concentration of salts and proteins of 0.8% in the saliva droplets will result in a ratio between equilibrium and initial diameter of 20%. This difference in the final droplet size leads to different sedimentation velocity that modifies the residence time of droplets in the indoor environment, and lung deposition fraction to estimate the inhalation risk. Also, the fluctuation in the low droplet number concentration (for 14.6 µm and 18.8 µm sizes) contribute to the variation in the risk estimation. In the case of higher viral load, the risk estimates from these two models are closer since \(1-exp\left(-\mu \right)\) tends to 1 due to high viral load, and other effects compensate each other. Although final risk estimates from these two models are nearly same in some cases, large difference is observed in handling individual processes. Hence, it is recommended to couple the physical processes as much as possible and run the dynamic model to arrive at realistic estimates. In the present study, falling-to-mixing-plate-out model21 is implemented, which allows a droplet's residence time (τ) to smoothly transition from a gravity-dominated (larger particles, diameter > 50 µm, τ < 100 s) to a turbulence-dominated (small particle, diameter < 5 µm, τ > 3000 s) regime as shown in Fig. 1. It's worth mentioning that turbulent mixing extends the particle residence time for droplets of intermediate size. The variation of droplet lifetime with RH is significant only for large particles of diameter in the range of 20–80 µm, mainly due to the effect of evaporation and gravitational settling in this size regime. The lifetime of virusols in the indoor environment is mostly determined by deposition; however, viral deposition in the lungs is completely determined by viral load and aerosol physics. The current study provides a common factor for risk estimation, bringing universality to the modified risk formula.

## Single-hit risk and reproduction number

Present study attempts to calculate the exposure time required to achieve a tangible single-risk for a given expiratory event as well as the reproduction number (\({R}_{0}\)) for the given input parameters. Coughing and sneezing will be specific to the sick and symptomatic patients, although breathing and speaking are normal expiratory processes relevant to all subjects. Table 3 lists the parameters of expiratory emission, such as droplet size distribution, frequency of emission, virion concentration in emitted droplets, etc.

Table 3

Emission characteristics of expiratory events

Expiratory events | Size distribution parameters | Number release rate* | References |

**Breathing** | CMD = 1.6 µm; GSD = 1.3 | 14 s− 1 (continuous) | Johnson et al.32 |

**Coughing** | CMD = 14 µm; GSD = 2.6 | 28 s− 1 (10 cough/h) | Duguid17 |

**Speaking** | CMD = 4 µm; GSD = 1.6 | 270 s− 1 (5 min/h) | Johnson et al.32; Alsved et al.33 |

**Sneezing** | GM − 8.1 µm; GSD = 2.3 | 2778 s− 1 (10 sneezes/h) | Duguid17 |

## * - long-time averaged droplet release rate

For each expiratory event, numerical computations are used to determine the exposure time for different risk levels (0.1%, 1%, 10%, and 50%) and AERs (0.5 h-1 – 10 h-1). In the exposure time calculations, it is assumed that the emissions are continuous with the given rate and the value is estimated for a given risk. The model findings (Fig. 2) reveal that, up to a critical viral load, the exposure duration decreases linearly with the viral load in the log-log graph. Although the findings are not shown here, the slope of the linear component increases with emission rate (*S*0). The critical viral load in this case is, 1013 #/mL for breathing, 1011 #/mL for coughing, 1010 #/mL for sneezing, 1012 #/mL for speaking for a risk of 0.1%. Beyond the critical viral load, the risk becomes a constant or invariant w.r.to viral load.

Alternative to the exposure time estimates, single-hit risk is estimated under the influence of all the four expiratory events occurring simultaneously at given emission rates. The joint risk probability is then given by,

$${R}^{\text{'}}=1-{P}_{0,B}\times {P}_{0,Sp}\times {P}_{0,C}\times {P}_{0,Sn}$$

2

where \({P}_{0,B}=exp\left({-N}_{d}\left[1-exp\left(-{n}_{v}\right)\right]\right)\) is the probability of zero-hit for breathing expiratory process, the suffices *Sp*, *C* and *Sn* denotes speaking, coughing and sneezing events respectively. It is to be noted that transmissibility of a virus is measured via single-hit risk probability, dominated by the aerosol route of exposure. Also, it has been argued often that the transmissibility of the virus is linked with the viral load22,23, and hence, the risk of transmission to a susceptible individual is estimated as a function of viral load for specified exposure times (Fig. 3).

Numerical results (Fig. 3a) shows that the risk is less than 1% for viral loads < 108 RNA copies/mL for 1-hour exposure period. But the risk rapidly approaches higher value (ex. 50% for 1010 RNA copies/mL and 10-min exposure), which demonstrates the high transmissibility of Delta and possibly Omicron variants which are reported to give rise to higher viral loads24–28 (Table 4). The disease's actual severity, on the other hand, is linked to its biological infectivity. Thus, the present study clearly demonstrates the dependence of risk on the viral load irrespective of variants. The model also explores the effect of ventilation rate on indoor infection risks (Fig. 3b). When the air-exchange rate is increased from 0.5 h-1 to 10 h-1 for a 10-minute exposure time, the single-hit risk decreases approximately by an order. This is primarily due to the elimination of airborne viruses from the indoor environment via ventilation. However, when viral load increases, the effect of enhancing ventilation reduces because smaller particles contribute to the risk as well. The ambient RH has only a minor impact on the risk; higher RH leads to larger final droplet sizes, which reduces their lifetime and therefore infection risk, as seen in Fig. 3b.

Figure 3a: **Variation of single hit risk for susceptible persons as a function of viral load for different time of exposure**

Table 4

Typical viral load of SARS-CoV-2 variants24–28

SARS-CoV-2 variant | Viral load (RNA copies/mL) |

Wild | ~ 105 − 108 |

Delta | ~ 106-109 |

Omicron | ~ 106-109.5 |

Another essential metric to describe infection risk is the reproduction number (*R*0), which is computed by multiplying the infection risk during the exposure time of each susceptible person by the number of susceptible people exposed for a specific exposure scenario. The following two scenarios are studied in this work to demonstrate how the model can be used: a) 25 students in a classroom with an infected subject exposed for four hours; b) 4 employees in an office environment with an infected subject exposed for eight hours. The *R*0 value approaches 2 when the viral load of infected person in the class room exceeds 5 x 107 #/mL, as shown in the results; also, the *R*0 value shall remain \(\le\) 1 if the viral load is less than 2.5 x 107 #/mL for the given input and environmental parameters, as shown in Fig. 4. Similarly, if the viral load is \(\le\) 7 x 107 #/mL for the given exposure conditions in an office setting, the *R*0 value will be \(\le\) 1. These findings imply that if the viral load is less than a certain value or if the contact period is limited for the specified emission and indoor settings, the reproduction number will remain less than one. Alternatively, the limit on number of people can also be estimated using the present approach for a given virus variant and the exposure duration. Hence, these studies can be used as a tool to aid decision/policy making as the spread of the disease can be directly predicted based on the viral load and other physically measurable input parameters.