# Criticality and Information Dynamics in Epidemiological Models

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Model Description

#### 2.2. Information Dynamics

#### 2.3. Measuring Information Dynamics in the SIS Model

## 3. Results and Discussion

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Epidemic phase transition. Final size of an epidemic as a function of its basic reproductive ratio ${R}_{0}$, for a susceptible-infected-recovered (SIR) model with a homogeneous network structure, with a number of connections (k) of 4 for each individual. Transmission rate $\beta $ varies between 0 and 3 with recovery rate $\gamma =1$, resulting in ${R}_{0}$ ranging between 0 and 3. The line depicts the analytical results whereas the red dots show the results from stochastic simulations with a population size of ${10}^{4}$. The epidemic does not occur for ${R}_{0}<1$, whereas the final size increases as a function of ${R}_{0}$ for values higher than 1. The analytical results and the simulations are in good agreement.

**Figure 2.**Bias-corrected active information storage ${A}_{X}^{\prime}$ in our simulations as a function of embedding length k. ${A}_{X}^{\prime}$ was calculated and then averaged for all three replicates for each ${R}_{0}$. The mean value (shown in y-axis) was then determined for each k (shown in x-axis) across the ${R}_{0}$ values. The difference increases as the k increases, maximising at $k=7$, and decreasing subsequently.

**Figure 3.**Raw average transfer entropy and average active information storage versus ${R}_{0}$. Transfer entropy (

**left**) calculated by averaging local transfer entropy for each individual across the network and active information storage (

**right**) calculated by averaging local active information storage for each individual across the network. For both measures, the embedding time is $k=7$. The average transfer entropy (${T}_{Y\to X}$) is shown in blue, the average active information storage (${A}_{X}$) is shown in gray, and prevalence is shown in red (note the different y-axes). ${R}_{0}$ is shown on the x-axis. After the critical transition both ${T}_{Y\to X}$ and ${A}_{X}$ increase and reach to a peak (at ${R}_{0}=1.8$ and ${R}_{0}=1.3$, respectively), and subsequently lower down.

**Figure 4.**Raw and bias-corrected average transfer entropy and average active information storage versus ${R}_{0}$. Raw average transfer entropy ${T}_{Y\to X}$ and average active information storage ${A}_{X}$ are shown in dark blue and black, respectively (left panel); bias-corrected average transfer entropy ${T}_{Y\to X}^{\prime}$ and average active information storage ${A}_{X}^{\prime}$ are shown in light blue and gray, respectively (right panel). Note the different y-axes for both graphs. ${R}_{0}$ is shown on the x-axis. Both ${A}_{X}$ and ${A}_{X}^{\prime}$ increase and reach a peak right after the critical transition, and subsequently decrease. ${T}_{Y\to X}^{\prime}$ also increases at the same ${R}_{0}$ value (${R}_{0}=1.2$) as ${A}_{X}^{\prime}$ and plummets thereafter, whereas ${T}_{Y\to X}$ reaches its highest value later, at ${R}_{0}=1.8$

**Figure 5.**Bias-corrected average transfer entropy ${T}_{Y\to X}^{\prime}$ versus bias-corrected average active information storage ${A}_{X}^{\prime}$. Bias-corrected transfer entropy ${T}_{Y\to X}^{\prime}$ (shown in the y-axis) and average active information storage ${A}_{X}^{\prime}$ (shown in the x-axis) are calculated separately for three replicates.

Parameter | Value |
---|---|

Time steps (t) | ${10}^{3}$ |

Population size (N) | ${10}^{4}$ |

Number of contacts | 4 |

Transmission rate ($\mu $) | 0.7–2.0 (with step size 0.1) |

Coefficient for per contact transmission rate (c) | 0.33 |

Recovery rate ($\mu $) | 1.0 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Erten, E.Y.; Lizier, J.T.; Piraveenan, M.; Prokopenko, M. Criticality and Information Dynamics in Epidemiological Models. *Entropy* **2017**, *19*, 194.
https://doi.org/10.3390/e19050194

**AMA Style**

Erten EY, Lizier JT, Piraveenan M, Prokopenko M. Criticality and Information Dynamics in Epidemiological Models. *Entropy*. 2017; 19(5):194.
https://doi.org/10.3390/e19050194

**Chicago/Turabian Style**

Erten, E. Yagmur, Joseph T. Lizier, Mahendra Piraveenan, and Mikhail Prokopenko. 2017. "Criticality and Information Dynamics in Epidemiological Models" *Entropy* 19, no. 5: 194.
https://doi.org/10.3390/e19050194