# Quantifying Invasive Pest Dynamics through Inference of a Two-Node Epidemic Network Model

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Data

#### 2.2. The Two-Node Model

#### 2.3. Linear Noise Approximation

#### 2.4. Bayesian Inference

## 3. Results

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

LNA | linear noise approximation |

MJP | Markov jump process |

ODE | ordinary differential equation |

OPM | Oak processionary moth |

SDE | stochastic differential equation |

SIR | susceptible, infected, removed |

## References

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**Figure 1.**Satellite images with eastings and northings (obtained from EDINA Digimap Aerial © Getmapping Plc [20]) of (

**a**) Bushy Park and (

**b**) Richmond Park with locations of nest removals between 2013 and 2021. (

**c**) The ‘Removed’ prevalence time series, $R\left(t\right)\equiv {R}_{t}$ (the cumulative numbers of tree locations where nest removal has taken place) is used as our observational data set. Bushy Park is shown in blue with open diamonds and Richmond park is shown in orange with filled circles. (

**d**) A subsection of the Ordnance Survey map OS Open Data [21]) with eastings and northings showing the location of the two parks.

**Figure 2.**Illustration of the two-node epidemic network model. In each node, the tree population transitions through the available states: Susceptible (S), infested (I), and removed (R). The transition between I and R is governed by the removal rate parameter, $\gamma $, which (in this case) is chosen to be identical within each node. For the transition between S and I, there is the standard infestation rate parameter, $\beta $, plus an additional infestation pressure resulting from the other node, with ${\alpha}_{12}$ corresponding to the infestation pressure from node 1 on node 2, and ${\alpha}_{21}$ the infestation pressure from node 2 on node 1. The resulting stochastic differential equation model for this scenario is given in Equations (1)–(5).

**Figure 3.**Inference results for the two-node network model applied to Bushy and Richmond Park. The within-sample posteriors for (

**a**) the susceptible tree time series for (

**i**) Bushy Park, ${S}_{1,t}$, and (

**ii**) Richmond Park, ${S}_{2,t}$, (

**b**) the infested tree time series, ${I}_{1,t}$ and ${I}_{2,t}$, and (

**c**) the removed tree time series, ${R}_{1,t}$ and ${R}_{2,t}$. The median of the within-sample posteriors is shown for Bushy Park with blue diamonds and Richmond Park with orange circles. In all cases, the shaded error bars represent the 95% credible interval. In (

**c**), the observed time series for ${R}_{1,t}$ and ${R}_{2,t}$ are shown as black dashed lines. In (

**d**) (

**i**–

**v**) the posterior parameter distributions are shown for $\beta ,\gamma ,\sigma ,{\alpha}_{12}$ and ${\alpha}_{21}$, respectively.

**Figure 4.**The estimated intra-park ($\beta {I}_{i}{S}_{i}$) and inter-park ($\beta {\alpha}_{ij}{I}_{i}{S}_{j}$) infestation components (see (2)) using the median posterior estimations for I and S in each year for (

**a**,

**b**) Bushy Park, and (

**c**,

**d**) Richmond Park.

**Figure 5.**An example scenario for possible new infestations occurring in (

**a**) Bushy and (

**b**) Richmond Park, averaged over 50 simulations of the stochastic two-node epidemic model with the median parameters estimated through the inference scheme (shown in Table 2). In both panels, the solid lines show the new infestations resulting from within the park (intra-park), with dashed lines showing new infestations resulting from the neighbouring park (inter-park). Error bars show the 95% credible intervals over the 50 simulations.

**Figure 6.**Forward simulations for the removal prevalence in (

**a**) Bushy and (

**b**) Richmond Park from the stochastic two-node epidemic model with the median parameters estimated through the inference scheme (shown in Table 2). In both panels, the solid lines show the mean removal prevalence with error bars showing the 95% credible intervals over 50 simulations. Black dashed lines show the observed data.

**Figure 7.**OPM nest density. (

**a**) Probability density of the number of recorded removed nests per tree in Bushy Park (blue, open diamonds) and Richmond Park (orange, filled circles). For visualisation purposes, this shows nest numbers up to 20 per tree only. (

**b**) Box plots of the number of logged nests per tree on a logarithmic scale, showing all recorded data.

**Table 1.**The input parameters used in the inference scheme for the two-node model detailed in Section 2.2, Section 2.3 and Section 2.4.

Input Parameters | |
---|---|

Total population | ${N}_{1}=5000$, ${N}_{2}$ = 40,000 |

Initial infested | ${I}_{1,0}=240$, ${I}_{2,0}=1400$ |

Initial model parameters | ${\theta}_{0}=({\beta}_{0},{\gamma}_{0},{\alpha}_{{12}_{0}},{\alpha}_{{21}_{0}},{\sigma}_{0})=(0.00001,0.8,0.3,2.5,1)$ |

Prior distributions | $log{\theta}_{i}\sim $N(0, 1${}^{2}$), $i=1,\dots ,4$ |

Tuning parameter | $\Sigma =\left(\begin{array}{ccccc}0.009& 0.003& -0.059& -0.009& 0.002\\ 0.003& 0.006& 0.011& 0.001& 0.001\\ -0.059& 0.011& 0.712& 0.072& -0.006\\ -0.009& 0.001& 0.072& 0.013& -0.002\\ 0.002& 0.001& -0.006& -0.002& 0.052\end{array}\right)$ |

Initial ODE mean | ${x}_{0}={\eta}_{0}=({N}_{1}-{I}_{1,0}-{R}_{1,0},{I}_{1,0},{N}_{2}-{I}_{2,0}-{R}_{2,0},{I}_{2,0})$ |

Initial ODE variance | ${V}_{0}={0}_{4,4}$ |

Observation matrix | $P=\left(\begin{array}{cccc}1& 1& 0& 0\\ 0& 0& 1& 1\end{array}\right)$ |

**Table 2.**The posterior means, standard deviations, medians, and 95% credible intervals for the inferred model parameters.

Parameter | Mean | Standard Deviation | Median | 95% Credible Interval |
---|---|---|---|---|

$\beta $ | $1.74\times {10}^{-5}$ | $1.40\times {10}^{-6}$ | $1.75\times {10}^{-5}$ | $(1.44,2.02)\times {10}^{-5}$ |

$\gamma $ | $0.78$ | $0.056$ | $0.77$ | $(0.67,0.90)$ |

$\sigma $ | $1.32$ | $0.289$ | $1.28$ | $(0.88,1.99)$ |

${\alpha}_{12}$ | $0.38$ | $0.268$ | $0.32$ | $(0.06,1.05)$ |

${\alpha}_{21}$ | $3.12$ | $0.312$ | $3.08$ | $(2.66,3.83)$ |

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**MDPI and ACS Style**

Wadkin, L.E.; Golightly, A.; Branson, J.; Hoppit, A.; Parker, N.G.; Baggaley, A.W.
Quantifying Invasive Pest Dynamics through Inference of a Two-Node Epidemic Network Model. *Diversity* **2023**, *15*, 496.
https://doi.org/10.3390/d15040496

**AMA Style**

Wadkin LE, Golightly A, Branson J, Hoppit A, Parker NG, Baggaley AW.
Quantifying Invasive Pest Dynamics through Inference of a Two-Node Epidemic Network Model. *Diversity*. 2023; 15(4):496.
https://doi.org/10.3390/d15040496

**Chicago/Turabian Style**

Wadkin, Laura E., Andrew Golightly, Julia Branson, Andrew Hoppit, Nick G. Parker, and Andrew W. Baggaley.
2023. "Quantifying Invasive Pest Dynamics through Inference of a Two-Node Epidemic Network Model" *Diversity* 15, no. 4: 496.
https://doi.org/10.3390/d15040496