# Generalizing the Wells–Riley Infection Probability: A Superstatistical Scheme for Indoor Infection Risk Estimation

## Abstract

**:**

## 1. Introduction

## 2. Model Construction

#### 2.1. Preliminaries

#### 2.2. Airborne Exposure Risk Statistics

#### 2.2.1. Homogeneous Indoor Air Environment

#### 2.2.2. Heterogeneous Indoor Air Environment

#### 2.3. Infection-Activation Considerations

#### 2.4. Indoor Infection Dynamics

## 3. Insights Gained from Computational Analysis

#### 3.1. Scrutinising the Generalised WRIP

#### 3.2. Refining the IIRE

#### 3.3. Evaluation of the Six-Foot Rule

## 4. Discussion

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Probing the Spatial Configuration of VCAPs

#### Appendix A.2. Schematic Illustration of the Contaminated-Air-Sharing Scenario

**Figure A1.**Investigating contaminated-air-sharing scenarios. We consider the same setup as that in Figure 1, but for V partitioned in two different ways. For simplicity, we assume that $v={\xi}^{3}$. The susceptible occupants’ breathing zones are highlighted and outlined in light grey and blue, respectively. On the

**left**subfigure, we partitioned V into $i=1,2,\dots ,\mathsf{\Omega}=n4$ subvolumes of size $v=V/\mathsf{\Omega}$, so that ${v}_{br}=v/4$, implying that $\xi >\sqrt[3]{{v}_{br}}$. The susceptible occupants’ breathing zones were embedded within the subvolume indexed with $i=3$. Because $\xi >\sqrt[3]{{v}_{br}}$, the epidemiological status of A and B may be influenced by any of the surrounding subvolumes $i=1,2,3,4$. In fact, during ${\tau}_{rel}$, a VCAP suspended within any of $i=1,2,3,4$ could reach A or B. Hence, it is likely that A and B are sharing contaminated air; imagine a VCAP originating from any of $i=1,2,3,4$ being displaced towards and inhaled by either A or B during ${\tau}_{rel}$. On the

**right**subfigure, we partitioned V into $i=1,2,\dots ,\mathsf{\Omega}=n64$ subvolumes of size $v=V/\mathsf{\Omega}$, so that ${v}_{br}=4v$, implying that $\xi <\sqrt[3]{{v}_{br}}$. Those indexed with $i=33,34,41,42$ and $i=35,36,43,44$ were embedded within the breathing zone of A and B, respectively. Because $\xi <\sqrt[3]{{v}_{br}}$, the epidemiological status of A and B is unlikely to be influenced by any other surrounding subvolumes except from $i=33,34,41,42$ and $i=35,36,43,44$, respectively. Hence, A and B are also unlikely to share contaminated air, since VCAPs suspended in any of $i=33,34,41,42$ and $i=35,36,43,44$ are unlikely to reach B and A, respectively, during ${\tau}_{rel}$.

#### Appendix A.3. Discretisation of the Tsallis Entropic Functional

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**Figure 1.**Schematic illustration of the proposed spatial-epidemiology model. We consider an indoor space of volume V occupied by one infector (i.e., $F=1$) and two susceptible (i.e., ${S}_{0}=2$) individuals, shown in red and blue, respectively. Round dots represent steady-state VCAP densities in V. The classical WRIP scheme assumes that the indoor air environment is homogeneous, i.e., that VCAPs are roughly uniformly spaced in V (see subfigure on the

**left**). On the other hand, the generalized WRIP scheme does not rely on the homogeneity assumption (e.g., as we can see in the subfigure on the

**right**, VCAP density can be higher near an infectious source). To systematically capture deviations from homogeneity, we partition V into $i=1,2,\dots ,\mathsf{\Omega}=n16$ subvolumes of size $v=V/\mathsf{\Omega}$, where n denotes the number of subvolume layers used to fill V. For clarity, we illustrate only the front layer containing the first 16 subvolumes (subvolume boundaries are highlighted in black). Supposing that the epidemiological status of susceptible occupants is determined by a single surrounding subvolume (i.e., if $\kappa =1$), then the subvolumes indexed with $i=9$ and $i=10$ correspond to the breathing zones of susceptible occupants A and B with ${\lambda}_{9}={x}_{9}$ and ${\lambda}_{10}={x}_{10}$, respectively, representing the values of the corresponding realizations of $\lambda $ with $\lambda \sim \mathrm{Gamma}(1,\mu )=\mathrm{Exp}\left(\mu \right)$ (see Equations (11) and (14)).

**Figure 2.**Classical WRIP as an upper asymptotic bound of its generalisation. (

**a**), we plot (17) for $\mu =1/\psi $ and $\kappa =1/\psi $ with ${\tau}^{\prime}=\tau =2$ and $v\in (0,2]$ (i.e., $\psi \in (0,1]$). The black arrow indicates that the generalised WRIP measure approaches the classical one as v decreases. The shaded area between the two curves visualises the image of the generalised WRIP function for $\psi \in (0,1]$. An estimation of the exposure risk difference (ERD), i.e., the difference between the generalised and the classical WRIP measures, calculated by ${(1+\frac{t}{\mu})}^{-\kappa}-\mathrm{exp}(-t)$ for $v\in (0,2]$ and $t=1$ (=60 min) is shown in the inset graph. Colorful markers in the inset graph indicate the v key values considered in (

**a**). In the same inset graph, the ratio $\mathcal{S}/{\mathcal{S}}_{q\to 1}$ is plotted, where $\mathcal{S}$ is calculated by using the discretised version of (27) and with ${\mathcal{S}}_{q\to 1}$ denoting the Boltzmann–Gibbs entropy (see Appendix A.3, Equations (A3) and (A4), respectively). (

**b**), for a hypothetical volume of size $V=60$, we plot $i=1,2,\dots ,30$ randomly-chosen realisations of $\lambda $ for $v\to 0$, $v=0.2$, and $v=2$ implying that $\mathsf{\Omega}\to \infty $, $\mathsf{\Omega}=300$, and $\mathsf{\Omega}=30$, respectively, since $v=\frac{V}{\mathsf{\Omega}}$ (see (5)). (

**c**), The corresponding gamma distributions are shown. Different colors are used to visualise the distributions obtained for the key values appearing in (

**a**,

**b**).

**Figure 3.**Instances of a VCAP distribution. We plot $g\left(k\right)$ for $\mu =1/\psi $ and $\kappa =1/\psi $ with ${\tau}^{\prime}=\tau =2$ and $v\in (0,2]$ (i.e., $\psi \in (0,1]$).

**Figure 4.**Refined biosafety scoring. (

**a**), s, e, i traces are plotted for $\mu =1/\psi $ and $\kappa =1/\psi $ with $v={\tau}^{\prime}=\tau =12$ (i.e., $\psi =1$), ${\beta}^{\ast}=0.1$, ${R}_{0}=0.01$, and $\zeta \in [0.1,10]$. The shaded areas highlight the images of the E and I for $\zeta \in [0.1,10]$. Green circular markers highlight the maxima of E (their values and locations both decreased with increasing $\beta $, indicating that the transfer of occupants from the E subgroup to the I subgroup was accelerated). The black arrow indicates that the sum $E+I$ approached the classical WRIP for $v\to 0$. $\mathsf{\Phi}$-dependent dynamics of R are illustrated in the inset graph. Magenta points in the inset graph highlight R values corresponding to the tunable inflection point ${\mathsf{\Phi}}_{infl}=1/\zeta $. The shaded area in the inset graph visualises the image of R for $\zeta \in [0.1,10]$. The main and inset graphs use the same line styles to account for $\zeta =0.1,1,10$. (

**b**), same as (

**a**), but for ${\beta}^{\ast}=10$.

**Figure 5.**Evaluation of the six-foot rule. (

**a**), we plot ${i}_{\tau}$ as a function of $\rho $ for $r=0.54$, $\zeta \in [{\zeta}_{min}=0.1,{\zeta}_{max}=10]$, ${\beta}^{\ast}=1$, $\xi ={d}_{safe}$, ${R}_{0}=0.01$, and $\tau =0.25,0.5,0.75,1$. Continuous and dashed lines highlight the upper and lower boundaries of ${i}_{\tau}$ values range obtained for ${\zeta}_{min}=0.1$ and ${\zeta}_{max}=10$, respectively. (

**c**), same as (

**a**), but for $r=1.38$. (

**e**), same as (

**a**), but for $r=3.3$. Inset graphs in (

**a**,

**c**,

**e**) illustrate how the first-order differences trace ${\Delta}_{\rho}{i}_{\tau}={i}_{\tau ;\rho +\Delta \rho}-{i}_{\tau ;\rho}$ “flattens” for increasing $\rho $. The following computational criterion was used to decide whether a ${\Delta}_{\rho}{i}_{\tau}$-trace has “flattened”: find ${\rho}^{\ast}$ so that ${\Delta}_{\rho}{i}_{\tau}<\u03f5$, $\u03f5=1\times {10}^{-4}$, is satisfied for any $\rho \ge {\rho}^{\ast}$. Once all ${\rho}^{\ast}$ values had been gathered for a specific r-value, we found their maximum, ${\rho}_{max}^{\ast}=\underset{\zeta ,\tau ,{\beta}^{\ast}=1}{\mathrm{max}}\left\{{\rho}^{\ast}\right\}$. ${\rho}_{max}^{\ast}$ is provided here for convenience: it serves as a baseline value for probing the behaviour of ${i}_{\tau}$-values with respect to $1/{\beta}^{\ast}$. Round and square markers in (

**a**,

**c**,

**e**) highlight the values of ${i}_{\tau}$ at the inflection point ${\rho}_{infl}$ for $\zeta ={\zeta}_{min}$ and $\zeta ={\zeta}_{max}$, respectively. For clarity, only round markers are used in the inset graphs, highlighting the maxima of ${\Delta}_{\rho}{i}_{\tau}$ for $\zeta ={\zeta}_{min}$. (

**b**) We plot the value of ${i}_{\tau}$ obtained for $\rho =n{\rho}_{max}^{\ast}$, $n=1,2,3,\dots ,10$, ${\rho}_{max}^{\ast}=384$ versus $1/{\beta}^{\ast}$ for $r=0.54$, $\zeta \in [{\zeta}_{min},{\zeta}_{max}]$, and $\tau =0.25,0.5,0.75,1$. (

**d**), same as (

**b**), but for $r=1.38$ with ${\rho}_{max}^{\ast}=354$. (

**f**), same as (

**b**), but for $r=3.3$ with ${\rho}_{max}^{\ast}=245$. Different colour intensities in (

**b**,

**d**,

**f**) were used to visualize ${i}_{\tau}$-values in ascending n-order. Black traces appearing in (

**b**) are particularly interesting here, as they indicate that for exposure times as short as 15 min, ${i}_{\tau}$ can take values as high as $\approx \phantom{\rule{-0.166667em}{0ex}}0.2$ if the host’s innate immune system falls short in counteracting the viral threat, i.e., if $1/{\beta}^{\ast}$ is very small.

**Table 1.**We consider two parameter sets, namely, the out-of-host set $\{N,F,V,{v}_{br},\tau ,r,w,W,\xi ,{\tau}^{\prime}\}$ and the within-host set $\{{\mathsf{\Phi}}_{crit},\zeta ,{\beta}^{\ast}\}$. Out-of-host parameters r, w, and W denote the volumetric inhalation rate, the VCAP exhalation rate, and the ventilation rate, respectively, and are introduced in Section 2.2.1. Within-host parameters $\zeta $ and ${\beta}^{\ast}$ codetermine a susceptible occupant’s immunological profile and are formally introduced in Section 2.3. $\lambda $, $\rho $, $\kappa $, and $\mu $ may be considered summary parameters, as they are expressed as combinations of out-of- and within-host parameters. $\lambda $ and $\rho $ represent the exposure rate parameter and VCAP density, respectively, and are introduced and reinterpreted in Section 2.2.1 and Section 2.2.2, respectively. $\kappa $ denotes the number of local air environments (modelled as surrounding subvolumes of volumetric size $\propto {\xi}^{3}$) determining a susceptible occupant’s epidemiological status and is introduced in Section 2.2.2. $\mu $ gives the average time separating any pair of subsequent airborne transmission events and is introduced in Section 2.2.2.

Summary Param. | Out-of-Host Param. | Within-Host Param. |
---|---|---|

$\lambda =\frac{rwF}{W{\mathsf{\Phi}}_{crit}}$ (1/h) | N (nr. of occupants) | ${\mathsf{\Phi}}_{crit}$ (VCAPs) |

$\rho =\frac{Fw}{W}$ (VCAPs/m${}^{3}$) | F (nr. of infectors) | $\zeta $ (dimensionless) |

$\kappa \propto \frac{{v}_{br}}{{\xi}^{3}}$ (dimensionless) | V (m${}^{3}$) | ${\beta}^{\ast}$ (1/h) |

$\mu \propto \frac{{\mathsf{\Phi}}_{crit}}{\rho}\frac{{\tau}^{\prime}}{{\xi}^{3}}$ (h) | ${v}_{br}$ (m${}^{3}$) | |

$\tau $ (h) | ||

r (m${}^{3}$/h) | ||

w (VCAPs/h) | ||

W (m${}^{3}$/h) | ||

$\xi $ (m) | ||

${\tau}^{\prime}$ (h) |

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**MDPI and ACS Style**

Xenakis, M.N.
Generalizing the Wells–Riley Infection Probability: A Superstatistical Scheme for Indoor Infection Risk Estimation. *Entropy* **2023**, *25*, 896.
https://doi.org/10.3390/e25060896

**AMA Style**

Xenakis MN.
Generalizing the Wells–Riley Infection Probability: A Superstatistical Scheme for Indoor Infection Risk Estimation. *Entropy*. 2023; 25(6):896.
https://doi.org/10.3390/e25060896

**Chicago/Turabian Style**

Xenakis, Markos N.
2023. "Generalizing the Wells–Riley Infection Probability: A Superstatistical Scheme for Indoor Infection Risk Estimation" *Entropy* 25, no. 6: 896.
https://doi.org/10.3390/e25060896