# A Gravity-Based Food Flow Model to Identify the Source of Foodborne Disease Outbreaks

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## Abstract

**:**

## 1. Introduction

## 2. Gravity Model

#### 2.1. Method

#### 2.2. Model Inputs

#### 2.2.1. Area of Analysis and Zone Delineation

#### 2.2.2. Inter-Zonal Distance Estimation

#### 2.2.3. Intra-Zonal Distance Estimation

_{r}with n

_{r}retail stores depends on the entity density rate ${\mathsf{\lambda}}_{r}$ (stores per square kilometer) and can be calculated as:

#### 2.2.4. Retailer Revenue Estimation

#### 2.2.5. Consumption Potential Estimation

_{j}denotes the grocery consumption in a postal zone j with population $po{p}_{j}$ and REV represents the total revenue of all food retailers over all postal zones. This consumption estimation assumes that the mean food consumption is equal across different zones.

#### 2.2.6. Observed Trip Data

#### 2.3. Model Calibration

_{i}for each row and B

_{j}for each column need to be calculated (Equations (2) and (3)). An additional parameter β for the frictional impact of distance needs to be adjusted. An appropriate beta value is calibrated to ensure that the modeled average flow distance is equal to the target average flow distance (Equation (1)). Consequently, in a matrix with n zones a total of 2n + 1 parameters are required to calibrate a doubly constrained gravity model [30].

_{i}and B

_{j}, while the latter method helps finding a deterrence factor that matches the modeled flow distance with the target flow distance.

#### 2.4. Gravity Model Results

#### 2.4.1. Food Flow Distribution

- (i)
- How many postal zones are supplied by a retailer zone?
- (ii)
- What proportion of goods are expected to be sold intra-zonally to consumers?

#### 2.4.2. Revenue Estimation of Food Retailers in Affected Regions

#### 2.4.3. Implication of Gravity Model Results

## 3. Application: Retailer Brand Identification

#### 3.1. Retail Brand Source Identification Model

#### 3.1.1. Network Model

_{Q}denotes transient nodes, where food is produced, distributed, and sold. ${V}_{R}$ denotes the set of absorbing (consumption) nodes where food leaves the network and is consumed. The set of edges is of the form (i, j) $\in {V}_{Q}\text{}\times {V}_{Q}{\displaystyle \cup}\text{}{V}_{Q}\times {V}_{R}$ and indicate food flow interactions between nodes. Each edge (i, j) is weighted by the time-average volume of food traded, ${w}_{ij}$.

#### 3.1.2. Transmission Model

- The contaminated quantity is fixed and is composed of individual contaminated units that neither spread nor recover from contamination as they travel through the supply network.
- Each unit travels independently through the supply network.
- Each transition of a unit from one node to the next entails an independent transmission direction.

#### 3.1.3. Traceback Algorithm: Bayesian Inference

#### 3.2. Model Evaluation

#### 3.2.1. Food Network Models

#### Food Network A (with Gravity Model)

#### Food Network B (without Gravity Model)

#### 3.2.2. Outbreak Simulation

#### 3.2.3. Modeling Results

#### 3.2.4. Interpretation of Results

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**Left**): Given consumer zones where outbreaks were reported. (

**Right**): Buffered gravity models around outbreak zones.

**Figure 3.**Retailer in a lattice arranged grid [43].

**Figure 6.**Decomposed food outflows from Figure 5 for retailer zone Wendlingen.

Parameter | Flow Threshold | ||
---|---|---|---|

>0% | >5% | >10% | |

Number of supplied consumer zones | 49 | 5.3 | 2.6 |

Proportion of intra-zonal flows | 28.5% |

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**MDPI and ACS Style**

Schlaich, T.; Horn, A.L.; Fuhrmann, M.; Friedrich, H.
A Gravity-Based Food Flow Model to Identify the Source of Foodborne Disease Outbreaks. *Int. J. Environ. Res. Public Health* **2020**, *17*, 444.
https://doi.org/10.3390/ijerph17020444

**AMA Style**

Schlaich T, Horn AL, Fuhrmann M, Friedrich H.
A Gravity-Based Food Flow Model to Identify the Source of Foodborne Disease Outbreaks. *International Journal of Environmental Research and Public Health*. 2020; 17(2):444.
https://doi.org/10.3390/ijerph17020444

**Chicago/Turabian Style**

Schlaich, Tim, Abigail L. Horn, Marcel Fuhrmann, and Hanno Friedrich.
2020. "A Gravity-Based Food Flow Model to Identify the Source of Foodborne Disease Outbreaks" *International Journal of Environmental Research and Public Health* 17, no. 2: 444.
https://doi.org/10.3390/ijerph17020444