# On Spatiotemporal Overdispersion and Macroparasite Accumulation in Hosts Leading to Aggregation: A Quantitative Framework

## Abstract

**:**

## 1. Introduction

## 2. Framework

- Variance-to-mean ratio, where if this ratio is approximately equal to 1, then a Poisson (random) distribution could characterize the distribution of parasites in the host population. If it is greater than 1, then parasite aggregation may occur. Smaller values (<1) may represent a distribution following the binomial distribution; if the value is near zero, the parasite distribution could be uniformly or evenly distributed. The variance-to-mean ratio is related to the index of dispersion ($D$), which can be described by the following [6,23]:$$D=\frac{{\sigma}^{2}}{\mu}\left(n-1\right)$$

- Negative binomial parameter $k$, which can be described as$$k=\frac{{\mu}^{2}}{{\sigma}^{2}-\mu}.$$

- Taylor’s Power Law $b$, in which $b$ is the regression slope described by the following [6,17]:$${\sigma}^{2}=a+{\mu}^{b}\mathrm{or}\phantom{\rule{0ex}{0ex}}\mathrm{log}\left({\sigma}^{2}\right)=\mathrm{log}\left(a\right)+b\mathrm{log}\left(\mu \right)$$Here, $a$ and $b$ are fitted against the collected data. The distribution of the parasites in the host population could be uniform if $b$ is zero, random if $b$ is approximately equal to 1, and aggregated if $b$ is significantly greater than 1.

## 3. Method

- What is the degree distribution of the contact or interaction network (e.g., from contact tracing)?
- What are characteristics of the hosts (e.g., age, sex, body size, and foraging behavior)?
- What are the characteristics of the parasites (e.g., age, body size, and complex life cycle)?
- What are the characteristics of the environment (e.g., microlocality, mixing dynamics, temperature, and food distribution)?
- What are the characteristics of the host–parasite interaction (e.g., preferential attachment and host immunity)?
- What is the spatiotemporal parasite load of sampled hosts?

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Concepts related to spatiotemporal overdispersion (see Table 1 for the glossary). Each factor can be sufficient to drive parasite load overdispersion in hosts. Social network with hubs, heterogeneity in the characteristics of actors (e.g., hosts), and complexity in host–parasite dynamics may lead to a distribution with variance greater than the mean. This distribution can characterize parasite aggregation in the host population. This aggregated pattern may lead to differential risk, such as superspreading events. These concepts, when investigated, can provide insights about the level of spatiotemporal overdispersion in the system and when tracked, can aid in identifying strategies for infectious disease prevention or control. Overdispersion is the link among concepts presented here. For example, Heterogeneity and Aggregation are both linked to Overdispersion.

**Figure 2.**Example distribution where there is parasite aggregation in host population. Variance > Mean. In this distribution, many have zero or low parasite burden, and only a few have high parasite burden.

**Figure 3.**Multiscale investigation of overdispersion. At the top part of the figure, overdispersion can happen at the species population network and also at the interspecies (or intercommunity) network. Food chain or web, predation, parasitism, and other interactions can characterize interspecies network. At the middle part of the figure, overdispersion can be investigated at a certain locality and at the macroscale (global) spatial level. Spatial investigation can be analyzed using a grid system, depending on the availability of data and on the spatial homo/heterogeneity of the interaction networks. For simplicity, a 2D spatial representation is presented here, but a 3D representation can also be performed. Moreover, spatial overdispersion can be dynamic through time. The distribution of parasites in host population can vary and evolve spatially and temporally.

**Figure 4.**Effect of heterogeneity and overdispersion in epidemics. In both figures, $k$ near zero represents more heterogeneity and high overdispersion. (

**a**) 90% probability, $k=1$ is assumed to be equivalent to a homogeneous transmission; (

**b**) $k\to \infty $ represents random transmission (e.g., Poisson-distributed spread of infectious disease).

**Figure 5.**Interaction network among hosts. Blue line denotes transmission from Host species A to Host species B (e.g., parasite eggs from main host to intermediary host). Red line denotes transmission from Host species B to Host species A (e.g., parasite larva from intermediary host to main host). The lines can also be weighted based on volume of parasite loading. (

**a**) sample network; (

**b**) network is converted to a bipartite graph; (

**c**) graph projection where black lines denote transmission interaction via an intermediate host. Note that a multipartite (e.g., tripartite) interaction network can also be studied to account for the role of the parasites.

**Figure 6.**Force of infection as risk is represented as a triple (f1, f2, and f3) meaning value of disease hazard, value of exposure, and value of transmissibility. (

**a**) example of a risk point, which characterizes status of one individual host for a specific time frame; (

**b**) collection of risk points, which characterizes host population.

**Figure 7.**Simplex of risk points that could characterize host population with parasite aggregation. High-risk individual hosts could have high parasite burden. Each red dot could represent an individual host.

**Table 1.**Glossary of terms related to parasite aggregation. These terms, while related to each other, should not be used interchangeably.

Term | Definition |
---|---|

Aggregation | Clustering of parasites in few hosts, while many other hosts have few or none. |

Complexity | A characteristic of systems with many dynamic and interacting components. The interaction among the components usually results in an emergent behavior. The interaction among components can be modeled using networks. |

Heterogeneity | Presence of variability in the system. Variance is not zero. |

Overdispersion | Variance is greater than the mean. Usually, an overdispersed distribution is often characterized by the negative binomial or by the power law. |

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**MDPI and ACS Style**

Rabajante, J.F. On Spatiotemporal Overdispersion and Macroparasite Accumulation in Hosts Leading to Aggregation: A Quantitative Framework. *Diseases* **2023**, *11*, 4.
https://doi.org/10.3390/diseases11010004

**AMA Style**

Rabajante JF. On Spatiotemporal Overdispersion and Macroparasite Accumulation in Hosts Leading to Aggregation: A Quantitative Framework. *Diseases*. 2023; 11(1):4.
https://doi.org/10.3390/diseases11010004

**Chicago/Turabian Style**

Rabajante, Jomar Fajardo. 2023. "On Spatiotemporal Overdispersion and Macroparasite Accumulation in Hosts Leading to Aggregation: A Quantitative Framework" *Diseases* 11, no. 1: 4.
https://doi.org/10.3390/diseases11010004