# An Advanced Risk Modeling Method to Estimate Legionellosis Risks Within a Diverse Population

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## Abstract

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## 1. Introduction

#### 1.1. Fundamental Background of L. pneumophila and Disease Endpoints

#### 1.2. The Evolution of L. pneumophila Causes Water Systems Control Challenges and Drives Pathogenesis

#### 1.3. Health Effects Within Diverse Populations Are Not Equal

#### 1.4. QMRA Can Optimize L. pneumophila Interventions

#### 1.5. Current QMRA Modeling Methods Are Inadequate for L. pneumophila

- The host is exposed to an average dose in the environment (d), modeled as a random discrete Poisson dose (red in Equation (1)).
- From this d, there are j pathogens that are delivered to a susceptible location within the host, for example, alveoli for L. pneumophila.
- From those j organisms, k survive to initiate an infection, which is a Boolean outcome and, thus, the binomial distribution (blue in Equation (1)). These likelihoods are coupled to develop the joint probability that d will result in an infection, which is then simplified to the exponential dose-response function at the far right (green in Equation (1)).

#### 1.6. Scope and Purpose of Research

## 2. Modeling Methods

#### 2.1. Two-Dimensional Simulation for Risk Modeling

#### 2.2. Exposure Model and Probability of Infection

- Concentration of L. pneumophila in the premise plumbing water;
- Concentration of L. pneumophila in the air during showering, within a shower duration;
- Generation of aerosols sized ≤5 µm;
- Delivered dose to alveoli of human lungs, accounting for intermediate losses in the previous two regions of the respiratory system.

_{Qx}, which is the fraction of aerosols that are ≤5 µm, generated after mixing hot and cold water at flow rate x. This is performed using fractional proportions of 43 °C water for hot water and 24 °C for cold water. This models the aerosol fractions for each flow rate and each broad temperature classification, where AF

_{QxC}and AF

_{QxH}are the cold and hot water aerosol fractions for each flow rate x (5.1, 6.6, and 9.0 L min

^{−1}), respectively. This level of granularity of the aerosolization has not been included in a QMRA previously; therefore, these data were chosen to provide for this option.

_{mix}and H

_{mix}, which are fractional portions of cold and hot water used to develop a safe and comfortable showering temperature. The mixed water temperature modeled is monitored to ensure that it does not exceed 43 °C to prevent scalding [54]. Distributions for C

_{mix}and H

_{mix}were chosen by iteratively solving for each mixture value (C

_{mix}and H

_{mix}) at each temperature setting for an upper value of 43 °C and a low value of 30 °C, using 0.5 °C increments. Then probability distributions are optimized to the iteratively solved mixture values (AICw in Table 1). Distributions and parameter values for AF

_{QxC}, AF

_{QxH}, C

_{mix}, and H

_{mix}can be seen in Table 1.

_{QX}estimates to model the concentration of L. pneumophila in the air within aerosols ≤5 µm aerosols. For Equation (3), the variability of concentration in the water (C

_{w}) is modeled as a uniform distribution parameterized from the open literature [47,48]. The concentration of L. pneumophila in these studies is serogroup-1 and based on a culture method [47,48]. L. pneumophila concentrations vary greatly in municipal water, and a QMRA model developed for the real world uses site-specific sampling results to inform the concentration. This model is intended as a generalized risk model rather than a model for a specific municipality or region. Thus, as in all QMRA models, the values used in the distributions should not be used for a policy or engineering application but rather for the method used. Then, a partitioning coefficient (PC) allows the estimation of L. pneumophila concentration in the air during a shower (C

_{aQx}; Table 1).

_{s}) and the inhalation rate dependent on age group (R

_{I,age}; normalized to t

_{s}) models ${D}_{{R}_{0}Qx}$.

_{1}; Table 1). The ${D}_{{R}_{1}Qx}$ estimates and deposition fraction for region-2 (DF

_{2}; Table 1) by modeling the dose lost to region-2 at flow rate x (${D}_{{R}_{2}Qx}$; Equation (6)). Equation (7) uses these previous results from Equations (4–6) to model the dose to the alveolated region of the lungs at flow rate x (${D}_{{R}_{3}Qx}$) using the deposition fraction for region-3 (DF

_{3}; Table 1). ${D}_{{R}_{3}Qx}$ is the dose that deposits within the alveoli, which is the dose that is used to estimate the risk of infection for flow rate x (P

_{inf, x}). The deposition fractions used are from a standard text used throughout inhalation toxicology as well as QMRA modeling [30,55]. P

_{inf, x}is modeled using Equation (8), the exponential dose-response model (Equation (8)). The exponential has one parameter k (Table 1), where k is the probability of a pathogen surviving to initiate an infection in the host [26].

#### 2.3. Risk Characterization: Probability of Illness and DALYs

#### 2.4. Method to Estimate Morbidity Ratio Proxies and Dynamic Health Effect QMRA Models

_{ill}) given exposure and infection [56,57]. Using data from the morbidity and mortality weekly report (MMWR) [13], the incidence rate for each demographic group is used to calculate a morbidity ratio proxy for that group ($\widehat{M{R}_{G}}$). This will allow for the P

_{ill}to be modeled for each group and overcome the assumed homogeneity of health effects from the dose-response model derivation.

_{G}) from the MMWR [59], and the legionellosis incidence rate for the total US population (IR

_{P}) during the years of the MMWR study. All $\widehat{M{R}_{G}}$ values can be seen in Table 1. A brief logic explanation is provided in the supplementary material.

_{ill,G}for the following groups: children (P

_{ill,C}), adults (P

_{ill,A}), Elderly (P

_{ill,E}), American Indians (P

_{ill,AI}), Asian (P

_{ill,Asian}), Black (P

_{ill,B}), White (P

_{ill,W}), Other race (P

_{ill,O}), females (P

_{ill,F}), and males (P

_{ill,M}).

_{S}) and sequela from LD ($DAL{Y}_{{P}_{x}}$) at flow rate x utilizing the associated disability weights published by the World Health Organization. Equation (13) multiplies the annualized risks from Equation (12) (P

_{ill,G,A}) by the disability weights for moderate illness, DW

_{M}, or severe DW

_{S}(termed DW

_{a}in Equation (12)). The value of $DAL{Y}_{{P}_{x}}$ is estimated using Equation (14) to account for post-acute illness (DW

_{P}) and DW

_{S}disability weights. DWs is used in Equation (14) since post-acute is estimating sequela for survivors of LD.

## 3. Results

^{−1}flow rate will be used to demonstrate the results. The plots depict the same fundamental trends for the other two flow rates, 6.6 and 9.0 L min

^{−1}. Results of the statistical comparisons across flow rates using the ANOVA are presented below. All plots for the other two flow rates can be seen in the supplementary information. Full model plotting and analysis source code are available in the supplementary information.

#### 3.1. Effects of Flow Rate

^{−5}), there is a significant difference between the first two flow rates (5.1 and 6.6 L min

^{−1}) when comparing the probability of illness. There is also a significant difference in probability of illness or DALY between the lowest and highest, 5.1 and 9 L min

^{−1}p-value of 1 (10

^{−4}), and medium and highest 6.6 and 9.0 L min

^{−1}, p-value of 1 (10

^{−5}). With higher flow rates, there is an increased concentration of aerosolized L. pneumophila in the shower air resulting in higher inhalation doses, thereby higher risks.

#### 3.2. Age Demographic Risks

_{ill}and DALY, there is a significant difference between ages for probability of illness—p = 4 (10

^{−4}), PDALY—p = 3 (10

^{−4}), SDALY—p = 3 (10

^{−4}), and MDALY—p = 4 (10

^{−4}).

#### 3.3. Race Demographic Risks

^{−4}), Black and Other risks—p = 5 (10

^{−5}), Black and White risks—p = 1 (10

^{−3}), and White and Other risks—p = 1 (10

^{−5}).

#### 3.4. Sex Demographic Risks

^{−4}) and PDALY p = 1 (10

^{−3}).

## 4. Discussion

#### 4.1. Interpretation of Results

_{ill}and DALY as compared to P

_{inf}is highlighted best for elderly populations. The elderly population is the group best known clinically to demonstrate a significant increase in susceptibility to L. pneumophila, including the likelihood of LD and mortality. The method developed in this research derives morbidity ratio proxies to model P

_{ill}and health outcome specific DALYs for each demographic group. This is the first instance of this level of resolution for specific demographics in QMRA modeling. This level of resolution is important to target future research towards sampling location for exposure intervention or research to explain sex and racial differences in LD risks.

#### 4.2. Model Limitations

#### 4.3. Other Potential Methods for Future Research

#### 4.4. Broader Impacts of This Research

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**log-10 Illness and DALY results delineated by racial demographics for 5.1 L min

^{−1}flow rate.

**Figure 5.**log-10 Illness and DALY results delineated by sex demographics for 5.1 L min

^{−1}flow rate.

**Figure 6.**Sensitivity analysis of the example quantitative microbial risk assessment (QMRA) model for adults.

Model Parameter | Distribution Variability or Uncertainty ^{a}; AICw | Value(s) | Units | Citation |
---|---|---|---|---|

H_{mix} | Truncated Normal ^{b}—VAICw = 0.350 | µ ^{c} = 0.5778; σ ^{d} = 0.0399UB ^{e} = 0.5094;LB ^{f} = 0.6472 | Unitless | Estimated in this research |

C_{mix} | Cauchy—V AICw = 0.65 | Location = 0.363; Scale = 0.0217 | ||

AF_{Q1H} | Triangular—V AICw = NA ^{h} | Min = 0.0097; Median = 0.0013 Max = 0.0069 | Distribution optimized in this research, data used for optimization from [46] | |

AF_{Q2H} | Triangular—V AICw = NA ^{h} | Min = 0.0014; Median = 0.0027 Max = 0.0128 | ||

AF_{Q3H} | Triangular—V AICw = NA ^{h} | Min = 0.0010; Median = 0.0015 Max = 0.0061 | ||

AF_{Q1C} | Truncated Weibull—V AICw = 0.349 | Scale = 2.754; shape = 0.0308 UB = 0.0478; LB = 1(10 ^{−15}) | ||

AF_{Q2C} | Truncated Weibull—V AICw = 0.460 | Scale = 4.125; shape = 0.0373 UB = 0.0473; LB = 1(10 ^{−15}) | ||

AF_{Q3C} | Triangular—V AICw = NA ^{h} | Min = 0.0097; Median = 0.0128 Max = 0.0173 | ||

C_{w} | Uniform—U AICw = NA ^{h} | Min = 1; Max = 1(10^{6}) | CFU L^{−1} | Min [47], Max [48] |

PC | Uniform—V AICw = NA ^{h} | Min = 0.25; Max = 0.65 | Unitless | Value from [29] |

t_{s} | Uniform—V AICw = NA ^{h} | Min = 0.0333; Max = 0.25 | Hours | Estimated in this research |

R_{I,Child} | Triangular—V AICw = NA ^{h} | Min = 0.0076; Median = 0.0111 Max = 0.013 | m^{3} h^{−1} | Distributions chosen in this research, data from US EPA exposure factors handbook [35] |

R_{I,Adult} | Triangular—V AICw = NA ^{h} | Min = 0.012; Median = 0.0124 Max = 0.013 | ||

R_{I,Elderly} | Point value—V AICw = NA ^{h} | 0.012 | ||

DF_{1} | Truncated Logistic ^{b}—UAICw = 0.525 | Location = 0.06777; Scale = 0.04305 UB = 0.01; LB = 0.195 | Unitless | Distributions from this research, data from [49] |

DF_{2} | Truncated Weibull ^{b}—UAICw = 0.658 | Scale = 0.4134; shape = 0.2102 UB = 0.00; LB = 0.41 | ||

DF_{3} | Truncated lognormal ^{b}—UAICw = 0.429 | µ = –1.052; σ = 0.4350 UB = 0.159; LB = 0.62 | ||

k | Triangular—U AICw = NA ^{h} | Median = 0.00599; Min = 0.00326; Max = 0.131 | Distribution from this research. Values from [43] | |

$\widehat{M{R}_{C}}$;$\widehat{M{R}_{A}}$;$\widehat{M{R}_{E}\text{}}$^{g} | Point values AICw = NA ^{h} | 0.0077; 0.097; 0.75 for Child Adult and Elderly respectively | Calculated in this research Data from [50] | |

$\widehat{M{R}_{AI}}$;$\widehat{M{R}_{A}}$;$\widehat{M{R}_{B}}$;$\widehat{M{R}_{W}}$;$\widehat{M{R}_{O}}$^{g} | Point values AICw = NA ^{h} | 0.0025; 0.010; 0.15; 0.61; 1.4 (10 ^{−5}) for American Indian; Asian; Black; White and Other races respectively | ||

$\widehat{M{R}_{M}}$;$\widehat{M{R}_{F}}$^{g} | Point Values AICw = NA ^{h} | 0.64; 0.36 for male and female respectively | ||

DW_{M} | Triangular—U AICw = NA ^{h} | Min = 0.039; median = 0.051; max = 0.06 | Distributions from this research data from [51] | |

DW_{P} | Triangular—U AICw = NA ^{h} | Min = 0.104; median = 0.125; max = 0.152 | ||

DW_{S} | Triangular—U AICw = NA ^{h} | Min = 0.179; median = 0.217; max = 0.251 |

^{a}—V = Variability, U = Uncertainty,

^{b}—necessary to ensure no negative values, while using the distribution that best fit the data;

^{c}—µ = mean;

^{d}—σ = standard deviation;

^{e}—UB = Upper Bound;

^{f}—LB = Lower Bound; min = minimum; max = maximum;

^{g}—Corrected for population;

^{h}—AICw is not applicable for assumed distributions or point values.

Demographic Group | Statistic | P_{inf} ^{a} | P_{ill} ^{b} | MDALY ^{c} | SDALY ^{d} | PDALY ^{e} |
---|---|---|---|---|---|---|

Child | U_{95} ^{f} | 7.29 (10^{−6}) | 2.39 (10^{−7}) | 1.20 (10^{−8}) | 3.03 (10^{−8}) | 6.36 (10^{−8}) |

M ^{g} | 2.61 (10^{−6}) | 4.98 (10^{−8}) | 2.48 (10^{−9}) | 6.28 (10^{−9}) | 1.32 (10^{−8}) | |

L_{95} ^{h} | 2.69 (10^{−7}) | 4.19 (10^{−9}) | 2.08 (10^{−10}) | 5.28 (10^{−10}) | 1.11 (10^{−9}) | |

Adult | U_{95} | 7.93 (10^{−6}) | 1.84 (10^{−6}) | 9.30 (10^{−8}) | 2.36 (10^{−6}) | 4.88 (10^{−7}) |

M | 2.84 (10^{−6}) | 3.85 (10^{−7}) | 1.91 (10^{−8}) | 4.87 (10^{−8}) | 1.02 (10^{−7}) | |

L_{95} | 2.93 (10^{−7}) | 3.24 (10^{−8}) | 1.60 (10^{−9}) | 4.09 (10^{−9}) | 8.53 (10^{−9}) | |

Elderly | U_{95} | 7.57 (10^{−6}) | 2.64 (10^{−5}) | 1.33 (10^{−6}) | 3.37 (10^{−6}) | 7.00 (10^{−6}) |

M | 2.71 (10^{−6}) | 5.54 (10^{−6}) | 2.75 (10^{−7}) | 7.03 (10^{−7}) | 1.47 (10^{−6}) | |

L_{95} | 2.79 (10^{−7}) | 4.65 (10^{−7}) | 2.30 (10^{−8}) | 5.88 (10^{−8}) | 1.23 (10^{−7}) | |

Combined Ages | U_{95} | 5.95 (10^{−5}) | Not Modeled ^{j} | |||

M | 1.25 (10^{−5}) | |||||

L_{95} | 1.05 (10^{−6}) | |||||

American Indian | U_{95} | NA ^{i} | 8.34 (10^{−6}) | 4.20 (10^{−7}) | 1.07 (10^{−6}) | 2.21 (10^{−6}) |

M | 1.75 (10^{−6}) | 8.70 (10^{−8}) | 2.21 (10^{−7}) | 4.65 (10^{−7}) | ||

L_{95} | 1.47 (10^{−7}) | 7.27 (10^{−9}) | 1.86 (10^{−8}) | 3.91 (10^{−8}) | ||

Asian | U_{95} | 5.56 (10^{−6}) | 2.80 (10^{−7}) | 7.05 (10^{−7}) | 1.48 (10^{−6}) | |

M | 1.16 (10^{−6}) | 5.78 (10^{−8}) | 1.48 (10^{−7}) | 3.09 (10^{−7}) | ||

L_{95} | 9.80 (10^{−8}) | 4.86 (10^{−9}) | 1.24 (10^{−8}) | 2.60 (10^{−8}) | ||

Black | U_{95} | 3.45 (10^{−5}) | 1.74 (10^{−6}) | 4.9 (10^{−6}) | 9.19 (10^{−6}) | |

M | 7.24 (10^{−6}) | 3.63 (10^{−7}) | 9.15 (10^{−7}) | 1.92 (10^{−6}) | ||

L_{95} | 6.09 (10^{−7}) | 3.04 (10^{−8}) | 7.68 (10^{−8}) | 1.62 (10^{−7}) | ||

White | U_{95} | 2.34 (10^{−5}) | 1.71 (10^{−6}) | 2.97 (10^{−6}) | 6.20 (10^{−6}) | |

M | 4.91 (10^{−6}) | 2.45 (10^{−7}) | 6.21 (10^{−7}) | 1.30 (10^{−6}) | ||

L_{95} | 4.13 (10^{−7}) | 2.04 (10^{−8}) | 5.21 (10^{−8}) | 1.10 (10^{−7}) | ||

Other | U_{95} | 1.48 (10^{−7}) | 7.38 (10^{−9}) | 1.87 (10^{−8}) | 3.93 (10^{−8}) | |

M | 3.10 (10^{−8}) | 1.55 (10^{−9}) | 3.91 (10^{−9}) | 8.22 (10^{−9}) | ||

L_{95} | 2.61 (10^{−9}) | 1.30 (10^{−10}) | 3.29 (10^{−10}) | 6.90 (10^{−10}) | ||

Female | U_{95} | 2.10 (10^{−5}) | 1.06 (10^{−6}) | 2.69 (10^{−6}) | 5.61 (10^{−6}) | |

M | 4.41 (10^{−6}) | 2.19 (10^{−7}) | 5.59 (10^{−7}) | 1.17 (10^{−6}) | ||

L_{95} | 3.71 (10^{−7}) | 1.84 (10^{−8}) | 4.69 (10^{−8}) | 9.80 (10^{−8}) | ||

Male | U_{95} | 3.85 (10^{−5}) | 1.92 (10^{−6}) | 4.93 (10^{−6}) | 1.02 (10^{−5}) | |

M | 8.07 (10^{−6}) | 4.03 (10^{−7}) | 1.02 (10^{−6}) | 2.14 (10^{−6}) | ||

L_{95} | 6.79 (10^{−7}) | 3.37 (10^{−8}) | 8.56 (10^{−8}) | 1.79 (10^{−7}) |

^{a}—Probability of infection;

^{b}—Probability of illness;

^{c}—Moderate disability-adjusted life years (DALY) (Pontiac Fever);

^{d}—Severe DALY (Legionaries’ Disease);

^{e}—Post Acute DALY;

^{f}—Upper 95th Percentile;

^{g}—Median;

^{h}—Lower 95th Percentile;

^{i}—Combined Ages Pinf using equation 8 used;

^{j}—due to data from MMWR not age-adjusted.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Weir, M.H.; Mraz, A.L.; Mitchell, J.
An Advanced Risk Modeling Method to Estimate Legionellosis Risks Within a Diverse Population. *Water* **2020**, *12*, 43.
https://doi.org/10.3390/w12010043

**AMA Style**

Weir MH, Mraz AL, Mitchell J.
An Advanced Risk Modeling Method to Estimate Legionellosis Risks Within a Diverse Population. *Water*. 2020; 12(1):43.
https://doi.org/10.3390/w12010043

**Chicago/Turabian Style**

Weir, Mark H., Alexis L. Mraz, and Jade Mitchell.
2020. "An Advanced Risk Modeling Method to Estimate Legionellosis Risks Within a Diverse Population" *Water* 12, no. 1: 43.
https://doi.org/10.3390/w12010043