# Role of Time Scales in the Coupled Epidemic-Opinion Dynamics on Multiplex Networks

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

- (i)
- $S\stackrel{\beta}{\to}I$: a susceptible agent becomes infected with the probability $\beta $.
- (ii)
- $I\stackrel{\gamma}{\to}Q$: an infected agent goes into quarantine with the probability $\gamma $.
- (iii)
- $I\stackrel{\mu}{\to}R$: an infected agent recovers with the probability $\mu $.
- (iv)
- $I\stackrel{\kappa}{\to}D$: an infected agent dies with the probability $\kappa $.
- (v)
- $Q\stackrel{\mu}{\to}R$: an agent in quarantine recovers with the probability $\mu $.
- (vi)
- $Q\stackrel{\kappa}{\to}D$: an agent in quarantine dies with the probability $\kappa $.

## 3. Results

#### 3.1. Role of the Opinion Layer

#### 3.2. Role of Time Scales

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Representation of the opinion–epidemic model. The upper (opinion) layer considers opinion dynamics, and nodes possess two possible states: positive (+1) or negative (−1). This layer also contains additional connections between agents. The lower (epidemic) layer supports the spread of disease. The nodes are the same agents as in the opinion layer, but their states can be (S) susceptible, (I) infected, (Q) quarantined, (D) deceased or (R) recovered.

**Figure 3.**The time evolution of infection rate for different independence probability $p=\{0.01,0.1,0.5\}$ with $N=\{1000,\mathrm{10,000},\mathrm{30,000}\}$. (

**a**) $\beta =0.02$, (

**b**) $\beta =0.2$. We outline the peak of infection ${I}_{max}$ and time when it occurs ${t}_{max}$ in panel (

**a**).

**Figure 4.**(

**a**) The peak of infection ${I}_{max}$ in the function of infection probability $\beta $ with $p=\{0.01,0.1,0.5\}$. The results are averaged over 10 realizations. Error bars are smaller than the symbols’ sizes. (

**b**) Close-ups of smaller $\beta $ values.

**Figure 5.**(

**a**) The time of infection peak ${t}_{max}$ in the function of infection probability $\beta $ with $p=\{0.01,0.1,0.5\}$. Results are averaged over 10 realizations. (

**b**) Close-up to smaller $\beta $ values.

**Figure 6.**The time evolution of infection rate for different q-lobby size $q=\{2,6,9\}$ and independence probabilities $p=\{0.01,0.1,0.5\}$. (

**a**) $\beta =0.02$, (

**b**) $\beta =0.2$.

**Figure 7.**The peak of infection ${I}_{max}$ in the function of initial positive opinion fraction ${o}_{init}$ with $p=\{0.01,0.1,0.5\}$ for (

**a**) $\beta =0.01$, (

**b**) $\beta =0.1$, (

**c**) $\beta =0.5$. Results are averaged over 10 realizations. Error bars are smaller than symbols’ sizes.

**Figure 8.**The peak of infection ${I}_{max}$ in the function of initial positive opinion fraction ${o}_{init}$ for selected timescales ${v}_{step}=\{1,5,20\}$ with $p=0.01$. Each panel corresponds to a different infection probability, (

**a**) $\beta =0.01$, (

**b**) $\beta =0.05$, (

**c**) $\beta =0.1$, (

**d**) $\beta =0.5$. Results are averaged over 10 realizations. Error bars are smaller than symbol size.

**Figure 9.**The time of infection peak ${t}_{max}$ in function of initial positive opinion fraction ${o}_{init}$ for selected timescales ${v}_{step}=\{1,5,20\}$ with $p=0.01$ and $\beta =0.01$. Results are averaged over 10 realizations.

**Figure 10.**The peak of infection ${I}_{max}$ in function of independence probability p for selected timescales ${v}_{step}=\{1,5,10,20\}$. Each panel corresponds to a different infection probability, (

**a**) $\beta =0.02$, (

**b**) $\beta =0.1$, (

**c**) $\beta =0.5$.

**Figure 11.**The peak of infection ${I}_{max}$ in function of independence probability p and group size q. Infection probabilities are shown on the left side of the panels. On top, the labels for ${v}_{step}$ are displayed. First, second and third row show the heatmaps for $\beta =0.02,0.1,0.5$ respectively. Panels (

**a**,

**d**,

**g**) represent models with ${v}_{step}=1$, panels (

**b**,

**e**,

**h**) with ${v}_{step}=5$ and panels (

**c**,

**f**,

**i**) with ${v}_{step}=20$. Each pixel represents the average of 10 model realizations.

**Table 1.**Model parameters with default values. Symbol ${}^{\u2660}$ indicates that a parameter could be changed during experiments.

Parameter | Default Value | Description |
---|---|---|

N | 10,000 ${}^{\u2660}$ | number of nodes |

m | 10 | number of links generated by newly added node in network construction |

${E}_{add}$ | $0.01$ Nm | number of additional links in opinion layer |

p | 0.01 ${}^{\u2660}$ | probability of an agent to act independently in opinion layer |

q | 6 ${}^{\u2660}$ | size of q-lobby in opinion layer |

${o}_{init}$ | 1.0 ${}^{\u2660}$ | initial fraction of agents with positive opinions |

${I}_{init}$ | $0.1$ | initial fraction of infected agents |

${t}_{i}$ | ${x}_{i}\sim \mathcal{N}(10,{5}^{2})$ | duration of infected state for agent i |

$\beta $ | ${}^{\u2660}$ | infection probability |

$\gamma $ | $0.5$ | probability of an agent to enter the quarantine |

$\mu $ | $0.9$ | probability of recovery |

$\kappa $ | $0.1$ | probability of death |

${v}_{step}$ | 1 ${}^{\u2660}$ | number of opinion layer updates per one epidemic layer update |

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**MDPI and ACS Style**

Jankowski, R.; Chmiel, A.
Role of Time Scales in the Coupled Epidemic-Opinion Dynamics on Multiplex Networks. *Entropy* **2022**, *24*, 105.
https://doi.org/10.3390/e24010105

**AMA Style**

Jankowski R, Chmiel A.
Role of Time Scales in the Coupled Epidemic-Opinion Dynamics on Multiplex Networks. *Entropy*. 2022; 24(1):105.
https://doi.org/10.3390/e24010105

**Chicago/Turabian Style**

Jankowski, Robert, and Anna Chmiel.
2022. "Role of Time Scales in the Coupled Epidemic-Opinion Dynamics on Multiplex Networks" *Entropy* 24, no. 1: 105.
https://doi.org/10.3390/e24010105