# Reconciling the Entomological Hazard and Disease Risk in the Lyme Disease System

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. LD Risk Model

_{i}= the population of population unit i; j = an individual patch within the range of population unit i; J

_{i}= total number of patches within range of population unit i; Exp

_{j}= the human risk of exposure to patch j; Ent

_{j}= the entomological risk associated with patch j.

#### 2.2. Entomological Risk and Human Exposure Functions

- We considered Exp
_{j}as either proportional to the perimeter of patch j that falls within range of i (proportional to the probability of entering patch j, assuming humans in population unit i move by random walk), proportional to the area of patch j that falls within the range of i (proportional to the relative amount of time spent in patch j), or as a constant. - We considered Ent
_{j}as either a negative exponential function of the area of patch j (as hypothesized in Allan et al., 2003 [9]), a linear function of the area of patch j, or as a constant.

#### 2.3. Simulated Landscapes

_{j}) as proportional to intersecting patch perimeter and entomological risk (Ent

_{j}) as a negative exponential function of patch area:

_{i}as the number of non-habitat (i.e., human-occupied) cells it contained. Landscape-level LD risks for each simulation were calculated by summing risk indices for all quadrats and then dividing by total population.

#### 2.4. Landscape Analysis

#### 2.5. Risk Index Calculation

_{i}as U.S. census tracts, delineated by shapefiles obtained from the New York State Civil Boundaries dataset and using population data from the 2010 U.S. census [29]. All forest patches that intersected a 2.4 km buffer surrounding each census tract contributed to that tract’s risk indices (again based on Larsen, et al., 2014 [28]).

- exposure constant, entomological risk as a negative exponential function of patch area:$$\sum}_{i=1}^{I}({P}_{i}{\displaystyle \sum}_{j=1}^{{J}_{i}}1\times {e}^{-{A}_{j}})$$
- exposure directly related to intersecting patch perimeter, entomological risk as a negative exponential function of patch area:$$\sum}_{i=1}^{I}({P}_{i}{\displaystyle \sum}_{j=1}^{{J}_{i}}{B}_{{x}_{j}}{e}^{-{A}_{j}})$$
- exposure directly related to intersecting patch area, entomological risk as a negative exponential function of patch area:$$\sum}_{i=1}^{I}({P}_{i}{\displaystyle \sum}_{j=1}^{{J}_{i}}{A}_{{x}_{j}}{e}^{-{A}_{j}})$$
- exposure directly related to intersecting patch perimeter, entomological risk constant:$$\sum}_{i=1}^{I}({P}_{i}{\displaystyle \sum}_{j=1}^{{J}_{i}}{B}_{{x}_{j}}\times 1)$$
- exposure directly related to intersecting patch area, entomological risk constant:$$\sum}_{i=1}^{I}({P}_{i}{\displaystyle \sum}_{j=1}^{{J}_{i}}{A}_{{x}_{j}}\times 1)$$
- exposure constant, entomological risk directly related to area:$$\sum}_{i=1}^{I}({P}_{i}{\displaystyle \sum}_{j=1}^{{J}_{i}}1\times {A}_{j})$$
- exposure directly related to intersecting patch perimeter, entomological risk directly related to area:$$\sum}_{i=1}^{I}({P}_{i}{\displaystyle \sum}_{j=1}^{{J}_{i}}{B}_{{x}_{j}}{A}_{j})$$
- exposure directly related to intersecting patch area, entomological risk directly related to area:$$\sum}_{i=1}^{I}({P}_{i}{\displaystyle \sum}_{j=1}^{{J}_{i}}{A}_{{x}_{j}}{A}_{j})$$

_{i}= number of forest patches that intersect tract i; ${P}_{i}$ = population of tract i; ${B}_{{x}_{j}}$ = for patch j, length of perimeter intersecting tract i; ${A}_{{x}_{j}}$ = for patch j, area intersecting tract i; ${B}_{j}$ = total perimeter of patch j; and ${A}_{j}$ = total area of patch j.

_{j}and Exp

_{j}as separate, our model is incapable of distinguishing between entomological risk and exposure when applied to real landscapes. In other words, although we may conceptualize Equation (3) as representing entomological risk in the form of a negative exponential function of patch area and exposure as a constant, it equally well represents the inverse.

#### 2.6. County LDI

#### 2.7. Model Evaluation

## 3. Results

#### 3.1. Simulated Landscapes

#### 3.2. Landscape Characterization

#### 3.3. Spatial Structure of LDI

#### 3.4. Model Evaluation

^{2}of 0.67–0.68 (Table 2).

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Simulated landscapes. (

**a**) Modified Random Clusters landscapes with varying habitat (forest) occupancy and percolation probability. Forest cells are yellow, non-forest are white; (

**b**) Simulated landscape with overlaid quadrats.

**Figure 2.**Model schematic and data. (

**a**) Cartoon of model. The square represents the peridomestic region being considered. The gray circle represents an intersecting forest patch. When calculating entomological risk, the entire gray region is considered, as the patch’s entomological risk is a product of total area rather than area falling within the peridomestic region. When calculating exposure risk, only the portion of perimeter falling within the peridomestic region is considered (illustrated by bolded portion of perimeter); (

**b**) Study landscape, including perimeter of 12-county region, 1.6 km buffer (black outer border in image), and intersecting forest patches (gray raster cells) derived from the National Land Cover Database 2011.

**Figure 3.**Lyme disease (LD) risk of simulated landscapes. Predicted LD risk as a function of habitat occupancy (forest cover). Each curve is evaluated for a landscape with a different percolation probability p.

**Figure 4.**Predicted Lyme disease risk as a function of mean minimum distance between patch edges, (

**a**) across all simulated landscapes and (

**b**) stratified by habitat occupancy A. Each curve connects landscape scores evaluated at sequential values of percolation probability p (0.05–0.5).

**Figure 5.**Predicted Lyme disease risk as a function of mean patch area. (

**a**) LDI as a function of patch area across all simulated landscapes; (

**b**) LDI as a function of area stratified by percolation probability p, with axes rescaled to better show maxima. Each curve connects landscape scores evaluated at sequential values of habitat occupancy A (0.1–0.9).

**Table 1.**County-level forest and LD statistics. Forest patch statistics only include deciduous and mixed forest patches with areas greater than 1 ha.

County | Population 2010 | Mean LDI 2000–2015 | Forest Area (ha) 2011 | Forest Perimeter (km) 2011 | No. Patches 2011 |
---|---|---|---|---|---|

Albany | 304,204 | 7.40 × 10^{−4} | 65.1426 | 378.75 | 1374 |

Columbia | 63,096 | 6.76 × 10^{−3} | 83.0822 | 407.078 | 1637 |

Dutchess | 297,488 | 2.49 × 10^{−3} | 111.7104 | 516.086 | 1835 |

Greene | 49,221 | 3.12 × 10^{−3} | 125.3981 | 410.046 | 747 |

Orange | 372,813 | 1.06 × 10^{−3} | 118.2992 | 446.79 | 1853 |

Putnam | 99,710 | 1.66 × 10^{−3} | 43.4138 | 157.504 | 349 |

Rensselaer | 159,429 | 1.44 × 10^{−3} | 91.6401 | 455.258 | 1560 |

Rockland | 311,687 | 5.30 × 10^{−4} | 21.4342 | 71.91 | 548 |

Schenectady | 154,727 | 1.70 × 10^{−4} | 26.1887 | 142.528 | 548 |

Sullivan | 77,547 | 4.62 × 10^{−4} | 204.5404 | 604.91 | 539 |

Ulster | 182,493 | 1.73 × 10^{−3} | 228.011 | 598.846 | 1241 |

Westchester | 949,113 | 3.06 × 10^{−4} | 47.8913 | 267.724 | 1150 |

Formula | Exp_{j} | Ent_{j} ~ Area | Coefficient | SE | p | λ | p_{λ} | L | R^{2} | |
---|---|---|---|---|---|---|---|---|---|---|

${\displaystyle \sum}_{i=1}^{I}}({P}_{i}{\displaystyle {\displaystyle \sum}_{j=1}^{{J}_{i}}}{e}^{-{A}_{j}})$ | Const. | Neg. Exp. | 5.47 × 10^{−2} | 1.01 × 10^{−2} | 5.70 × 10^{−8} *** | −2.43 × 10^{−2} | 9.39 × 10^{−1} | −10.0 | 0.68 | |

${\displaystyle \sum}_{i=1}^{I}}({P}_{i}{\displaystyle {\displaystyle \sum}_{j=1}^{{J}_{i}}}{B}_{{x}_{j}}{e}^{-{A}_{j}})$ | Perim. | Neg. Exp. | 2.55 × 10^{−2} | 5.01 × 10^{−3} | 3.66 × 10^{−7} *** | −7.85 × 10^{−3} | 9.80 × 10^{−1} | −10.5 | 0.67 | |

${\displaystyle \sum}_{i=1}^{I}}({P}_{i}{\displaystyle {\displaystyle \sum}_{j=1}^{{J}_{i}}}{A}_{{x}_{j}}{e}^{-{A}_{j}})$ | Area | Neg. Exp. | 8.43 × 10^{−1} | 1.67 × 10^{−1} | 4.76 × 10^{−7} *** | −9.00 × 10^{−4} | 9.98 × 10^{−1} | −10.5 | 0.67 | |

${\displaystyle \sum}_{i=1}^{I}}({P}_{i}{\displaystyle {\displaystyle \sum}_{j=1}^{{J}_{i}}}{B}_{{x}_{j}})$ | Perim. | Const. | 1.95 × 10^{−2} | 1.31 × 10^{−2} | 1.38 × 10^{−1} | 3.76 × 10^{−1} | 5.56 × 10^{−1} | −15.1 | 0.16 | |

${\displaystyle \sum}_{i=1}^{I}}({P}_{i}{\displaystyle {\displaystyle \sum}_{j=1}^{{J}_{i}}}{A}_{{x}_{j}})$ | Area | Const. | 5.83 × 10^{−4} | 3.94 × 10^{−4} | 1.39 × 10^{−1} | 3.77 × 10^{−1} | 5.75 × 10^{−1} | −15.1 | 0.16 | |

${\displaystyle \sum}_{i=1}^{I}}({P}_{i}{\displaystyle {\displaystyle \sum}_{j=1}^{{J}_{i}}}{A}_{j})$ | Const. | Lin. | 1.53 × 10^{−5} | 2.96 × 10^{−4} | 9.59 × 10^{−1} | 6.45 × 10^{−1} | 1.32 × 10^{−1} | −15.7 | 0.016 | |

${\displaystyle \sum}_{i=1}^{I}}({P}_{i}{\displaystyle {\displaystyle \sum}_{j=1}^{{J}_{i}}}{B}_{{x}_{j}}{A}_{j})$ | Perim. | Lin. | −1.56 × 10^{−8} | 1.82 × 10^{−7} | 9.32 × 10^{−1} | 6.59 × 10^{−1} | 1.24 × 10^{−1} | −15.7 | 0.025 | |

${\displaystyle \sum}_{i=1}^{I}}({P}_{i}{\displaystyle {\displaystyle \sum}_{j=1}^{{J}_{i}}}{A}_{{x}_{j}}{A}_{j})$ | Area | Lin. | −5.21 × 10^{−7} | 6.07 × 10^{−6} | 9.32 × 10^{−1} | 6.59 × 10^{−1} | 1.24 × 10^{−1} | −15.7 | 0.025 | |

No. patches < 2 ha | 1.35 × 10^{−1} | 2.62 × 10^{−2} | 2.46 × 10^{−7} *** | −1.36 × 10^{−2} | 9.66 × 10^{−1} | −10.3 | 0.67 |

_{i}= number of forest patches that intersect tract i; ${P}_{i}$ = population of tract i; ${B}_{{X}_{i}}$ = for patch j, length of perimeter intersecting tract i; ${A}_{{X}_{i}}$ = for patch j, area intersecting tract i; ${B}_{j}$ = total perimeter of patch j; = total area of patch j; λ = spatial autocorrelation parameter; p

_{λ}= p value associated with λ; L = log-likelihood. “***” indicates significance to p < 0.001.

Formula | Coefficient | SE | p | λ | p_{λ} | L |
---|---|---|---|---|---|---|

${e}^{-A}$ | −2.92 × 10^{0} | 8.61 × 10^{−1} | 6.95 × 10^{−4} *** | −1.20 × 10^{−1} | 7.62 × 10^{−1} | −13.8 |

$B{e}^{-A}$ | −2.57 × 10^{0} | 3.54 × 10^{0} | 4.67 × 10^{−1} | 6.85 × 10^{−1} | 5.34 × 10^{−2} | −15.5 |

$A{e}^{-A}$ | 1.25 × 10^{1} | 2.98 × 10^{0} | 2.60 × 10^{−5} *** | −6.98 × 10^{−1} | 3.45 × 10^{−1} | −15.3 |

${e}^{-E\left[A\right]}$ | −1.72 × 10^{0} | 1.26 × 10^{0} | 1.73 × 10^{−1} | 5.41 × 10^{−1} | 1.94 × 10^{−1} | −14.9 |

$E\left[B\right]{e}^{-E\left[A\right]}$ | −4.25 × 10^{0} | 5.96 × 10^{0} | 4.76 × 10^{−1} | 6.57 × 10^{−1} | 6.43 × 10^{−2} | −15.5 |

$E\left[A\right]{e}^{-E\left[A\right]}$ | 7.26 × 10^{0} | 3.53 × 10^{0} | 3.98 × 10^{−2} * | 6.50 × 10^{−1} | 6.26 × 10^{−2} | −13.9 |

$E\left[A\right]$ | 1.03 × 10^{−1} | 3.4 × 10^{−1} | 7.64 × 10^{−1} | 6.27 × 10^{−1} | 1.14 × 10^{−1} | −15.7 |

A | 8.74 × 10^{−4} | 4.07 × 10^{−4} | 3.17 × 10^{−2} * | 1.47 × 10^{−1} | 7.39 × 10^{−1} | −14.8 |

% Forest cover | 3.62 × 10^{3} | 1.57 × 10^{3} | 2.14 × 10^{−2} * | 4.79 × 10^{−1} | 2.60 × 10^{−1} | −13.7 |

(% Forest cover) ^{2} | 2.79 × 10^{6} | 1.47 × 10^{6} | 5.79 × 10^{−1} | 5.05 × 10^{−1} | 2.33 × 10^{−1} | −14.3 |

B | 3.89 × 10^{−3} | 1.04 × 10^{−3} | 1.96 × 10^{−4} *** | −2.94 × 10^{−1} | 4.40 × 10^{−1} | −14.0 |

No. patches | 1.75 × 10^{−3} | 3.40 × 10^{−4} | 2.93 × 10^{−7} *** | −7.70 × 10^{−1} | 9.47 × 10^{−2} | −14.2 |

_{λ}= p value associated with λ; L = log-likelihood. “***” indicates significance to p < 0.001, “*” indicates significance to p < 0.05;

^{2}: indicates that the percentage is raised to the second power (a quadratic function).

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**MDPI and ACS Style**

McClure, M.; Diuk-Wasser, M.
Reconciling the Entomological Hazard and Disease Risk in the Lyme Disease System. *Int. J. Environ. Res. Public Health* **2018**, *15*, 1048.
https://doi.org/10.3390/ijerph15051048

**AMA Style**

McClure M, Diuk-Wasser M.
Reconciling the Entomological Hazard and Disease Risk in the Lyme Disease System. *International Journal of Environmental Research and Public Health*. 2018; 15(5):1048.
https://doi.org/10.3390/ijerph15051048

**Chicago/Turabian Style**

McClure, Max, and Maria Diuk-Wasser.
2018. "Reconciling the Entomological Hazard and Disease Risk in the Lyme Disease System" *International Journal of Environmental Research and Public Health* 15, no. 5: 1048.
https://doi.org/10.3390/ijerph15051048