# Phase Transitions in Spatial Connectivity during Influenza Pandemics

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## Abstract

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## 1. Introduction

## 2. The Simulation Framework

#### 2.1. Population Generation

#### 2.2. Disease Description

#### 2.3. Epidemic Synchrony and Bimodality

## 3. Methods

#### 3.1. Percolation Phase Transitions

#### 3.2. Fisher Information

## 4. Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The natural history of the disease used in the AceMod simulator [5]. The infectivity of an agent is modelled with an initial linear increase, followed by a linear decrease as the host recovers. Time 0 indicates the time at which an individual is infected. For the first day of infection, all infected individuals are in the Latent state and are not infectious. 33% of cases are Asymptomatic, following the Asymptomatic infectivity curve (in blue). Of the Symptomatic individuals, 30% of individuals become fully infectious on day 1 (solid line), 50% on day 2 (orange dashed line) and 20% on day 3 (blue dotted line).

**Figure 2.**A typical epidemic curve tracing the average local prevalence fraction for $\kappa =2.5$. Each of the dotted lines indicate the times of the two local maxima, ${t}_{1}$ and ${t}_{2}$ (urban and rural peaks of the epidemic).

**Figure 3.**Comparison of global and averaged local prevalences for $\kappa =3.33$. Local measures accentuate the presence of the second (rural) peak due to relative differences in population between urban and rural locations.

**Figure 4.**The relationship between the attack rate and $\kappa $ (blue curve) and the relationship between the critical threshold ${R}_{0}$ and the scaling parameter $\kappa $ (circles) estimated from 65,000 initial seeds. The standard error of the mean is smaller than the marker size. The black dotted line indicates the line of best fit ${R}_{0}=0.5345\kappa +0.0247$ with an ${R}^{2}$ value of 0.9998. This gives a critical value of ${R}_{0}^{*}\approx 0.67$ for ${\kappa}^{*}$ between $1.15$ and $1.25$. Shown inset are the distributions of secondary cases from random primary cases which are used in calculating ${R}_{0}$.

**Figure 5.**Mean peak times $\mathbb{E}\left[{t}_{1}\right]$ and $\mathbb{E}\left[{t}_{2}\right]$ respectively calculated from 100 runs. The AceMod simulator caps simulations at 195 days and so low values of $\kappa $ indicate that the epidemic has not developed sufficiently within the 195 day period. The vertical shaded region ${\kappa}^{*}\in [1.15,1.25]$ marks the critical interval of $\kappa $ for which we observe non-vanishing attack rates (cf. Figure 4). The shaded region around the points indicate the standard deviation in the distribution of peak times ${t}_{i}$ while the error bars indicate the standard error of the mean $\mathbb{E}\left[{t}_{i}\right]$.

**Figure 6.**Mean difference in peak times $\mathbb{E}[{t}_{2}-{t}_{1}]$. Error bars indicate the standard deviation in the difference ${t}_{2}-{t}_{1}$, whereas the blue shaded region represents the standard deviation of the samples. The vertical shaded region ${\kappa}^{*}\in [1.15,1.25]$ marks the critical interval of $\kappa $ for which we observe non-vanishing attack rates (cf. Figure 4). Directly after the critical region, we observe a decoupling in the timing of the epidemic peaks, illustrated by the rapid increase in this difference in the post-critical phase.

**Figure 7.**Thresholded infection map of the Australian epidemic at the primary peak of infection of Figure 2. The regions in black represent SA2s with greater than average prevalence fraction $\langle I\left({t}_{1}\right)\rangle $, at time ${t}_{1}$, whereas the regions in white show the areas with infection below this threshold.

**Figure 8.**Thresholded infection map of the Australian epidemic at the secondary peak of infection Figure 2. The regions in black represent SA2s with greater than average prevalence fraction at time ${t}_{2}$, $\langle I\left({t}_{2}\right)\rangle $, whereas the regions in white show the areas with infection below this threshold.

**Figure 9.**Temporal comparison of local measures of infection the mean local prevalence and $\mathbb{E}\left[\widehat{r}\right]$ for $\kappa =2$ (

**a**) and $\kappa =3.33$ (

**b**). We note that the standard deviation of $\langle \widehat{r}\rangle $ (dotted line) lags behind $\mathbb{E}$[$\langle \widehat{r}\rangle $] and the epidemic curve. We observe a large amount of variability in the connectivity of the secondary wave of the epidemic, whereas the first wave is relatively stable. Underneath each averaged time-series are two extremal examples of the time series $\langle \widehat{r}\rangle $$\left(t\right)$ where the second wave is very poorly connected (

**c**,

**e**) or extremely highly connected (

**d**,

**f**).

**Figure 10.**The average $\mathbb{E}$[$\langle \widehat{r}\rangle $] over 100 runs, traced across a range of $1\le \kappa \le 4$, where $\langle \widehat{r}\rangle $ is the normalised mean cluster size to which a randomly selected location belongs at each of the two epidemic peaks. The first peak corresponds to the urban wave of the epidemic whereas the second peak reflects the rural epidemic wave. The error bars indicate the standard error of the mean $\langle \widehat{r}\rangle $. The vertical shaded area for ${\kappa}^{*}\in [1.15,1.25]$ indicates the critical interval of $\kappa $.

**Figure 11.**The Fisher information of $\widehat{r}$, the cluster size to which a randomly selected site belongs, based on a bin width of 5. The Fisher information identifies the peak observed in $\mathbb{E}$[$\langle \widehat{r}\rangle $] concurring with critical interval of $\kappa $.

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**MDPI and ACS Style**

Harding, N.; Spinney, R.; Prokopenko, M.
Phase Transitions in Spatial Connectivity during Influenza Pandemics. *Entropy* **2020**, *22*, 133.
https://doi.org/10.3390/e22020133

**AMA Style**

Harding N, Spinney R, Prokopenko M.
Phase Transitions in Spatial Connectivity during Influenza Pandemics. *Entropy*. 2020; 22(2):133.
https://doi.org/10.3390/e22020133

**Chicago/Turabian Style**

Harding, Nathan, Richard Spinney, and Mikhail Prokopenko.
2020. "Phase Transitions in Spatial Connectivity during Influenza Pandemics" *Entropy* 22, no. 2: 133.
https://doi.org/10.3390/e22020133