# Dynamics of Epidemic Spreading in the Group-Based Multilayer Networks

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model

#### 2.1. Structural Representation of Group-Based Multilayer Networks

#### 2.2. Coupled Spreading Processes

## 3. Microscopic Markov Chain Approach

#### 3.1. Theoretical Analysis

#### 3.2. Numerical Simulations

## 4. The Outbreak Threshold of Epidemic Spreading

#### 4.1. Theoretical Discussion

#### 4.2. Numerical Simulations

## 5. Network Propagation Robustness

#### 5.1. Disease Immunization

#### 5.2. Occluded Information Diffusion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of a coupled group-based bilayer network. The upper (lower) layer stands for the information diffusion (epidemic spreading), where each node in the information (physical) layer only has two possible states: aware (A)and unaware (U) [susceptible (S) and infected (I)]. $\tau $ ($\eta $) represents the feedback intensity from information (physical) layer to physical (information) layer. The nodes in the same color belong to the same group in each layer.

**Figure 2.**Transition probability trees for the states (

**a**) AI, (

**b**) UI, (

**c**) AS, (

**d**) US. Each tree includes four stages: information diffusion (UAU process), feedback from the information layer to the physical layer, epidemic spreading (SIS process), and feedback from the physical layer to the information layer.

**Figure 3.**The infection density $\rho $ as a function of the infectivity parameter $\beta $ for different values of $\gamma $ (

**a**) and different values of $\mu $ (

**b**). The result of (

**a**) is obtained for $\mu =0.9$ and that of (

**b**) is for $\gamma =0.1$. And the other parameter values are set to ${\lambda}_{1}=\eta =0.9$, ${\lambda}_{2}=\tau =\delta =0.5$, $\iota =1.1$, $\overline{x}=\overline{y}=3$.

**Figure 4.**The infection density $\rho $ as a function of the mean group size of information layer $\overline{x}$ (

**a**) and physical layer $\overline{y}$ (

**b**), respectively. The rest of other parameter values are fixed as: $\gamma =0.1$, $\beta =\tau =\delta ={\lambda}_{2}=0.5$, $\iota =1.1$, ${\lambda}_{1}=\eta =\mu =0.9$. When studying the effect of $\overline{x}$ ($\overline{y}$) on the spreading prevalence, we set $\overline{y}=3$ ($\overline{x}=3$).

**Figure 5.**Phase diagrams of the group-based coupled propagation processes on the plane of $\beta -\tau $ (

**a**) and $\beta -\lambda $ (

**b**), where the results in the top panels are obtained by MMCA and those in the bottom panels are obtained by MC. For (

**a**), other parameters are set to $\gamma =0.1$, $\mu ={\lambda}_{1}=\eta =0.9$, $\iota =1.1$, $\beta =\delta ={\lambda}_{2}=0.5$, $\overline{x}=\overline{y}=3$. For (

**b**), the parameter values are $\tau =0.5$, $\lambda ={\lambda}_{1}={\lambda}_{2}$, and others are the same as those of (

**a**).

**Figure 6.**(

**a**) The outbreak threshold ${\beta}_{c}$ as a function of the attenuation factor $\gamma $. (

**b**) ${\beta}_{c}$ as a function of the inter-layer feedback parameter $\tau $. The result of (

**a**) is obtained for $\tau =0.5$ and that of (

**b**) is for $\gamma =0.1$. Other parameters are fixed as ${\lambda}_{1}=0.9$, ${\lambda}_{2}=0.5$, $\iota =1.1$, $\overline{x}=\overline{y}=3$.

**Figure 7.**The epidemic spreading threshold ${\beta}_{c}$ as functions of $\overline{x}$ (

**a**) and $\overline{y}$ (

**b**), respectively. Other parameters are: $\gamma =0.1$, $\mu =\eta ={\lambda}_{1}=0.9$, ${\lambda}_{2}=\tau =0.5$, $\iota =1.1$, and $\overline{y}=3$$(\overline{x}=3)$ when studying how the $\overline{x}$$(\overline{y})$ works on the ${\beta}_{c}$.

**Figure 8.**The I-state density $\rho $ (

**a**) and the epidemic threshold ${\beta}_{c}$ (

**b**) as functions of the attacked node proportion ${p}_{a}$ under four attack strategies. The solid and dashed lines represent the variation of the group-based multilayer network and the multiplex network, respectively, where the lines in the same color indicate the same attack strategy. Other parameters are fixed as $\gamma =0.1$, ${\lambda}_{1}=\eta =\mu =0.9$, $\beta =\tau =\delta ={\lambda}_{2}=0.5$, $\iota =1.1$, and $\overline{x}=\overline{y}=3$.

**Figure 9.**The I-state density $\rho $ (

**a**) and the threshold ${\beta}_{c}$ (

**b**) with respective to the attacked node proportion ${p}_{a}$ under four attack strategies. The solid and dashed lines represent the variation of the group-based multilayer network and the multiplex network, respectively, where the lines in the same color represent the same attack strategy. Other parameters are fixed as $\gamma =0.1$, ${\lambda}_{1}=\eta =\mu =0.9$, $\beta =\tau =\delta ={\lambda}_{2}=0.5$, $\iota =1.1$, and $\overline{x}=\overline{y}=3$.

Parameter | Description |
---|---|

$\tau $ | Feedback intensity from information layer to physical layer |

$\eta $ | Feedback intensity from physical layer to information layer |

${x}_{i}$ | Group size of information layer where node i belongs |

${y}_{i}$ | Group size of physical layer where node i belongs |

X | Group-matrix of information layer |

Y | Group-matrix of physical layer |

${\beta}_{1}^{U}$ | Infection probability for intra-group unsuppressed nodes |

${\beta}_{2}^{U}$ | Infection probability for inter-group unsuppressed nodes |

${\beta}_{1}^{A}$ | Infection probability for intra-group suppressed nodes |

${\beta}_{2}^{A}$ | Infection probability for inter-group suppressed nodes |

${\lambda}_{1}$ | Information diffusion probability of intra-group |

${\lambda}_{2}$ | Information diffusion probability of inter-group |

$\gamma $ | Attenuation factor |

$\rho $ | The density of infected individuals |

$\mu $ | Probability of recovery |

$\delta $ | Probability of forgetting |

$\iota $ | Proportion of intra-group and inter-group transmission |

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**MDPI and ACS Style**

Wang, D.; Zhao, Y.; Leng, H.
Dynamics of Epidemic Spreading in the Group-Based Multilayer Networks. *Mathematics* **2020**, *8*, 1895.
https://doi.org/10.3390/math8111895

**AMA Style**

Wang D, Zhao Y, Leng H.
Dynamics of Epidemic Spreading in the Group-Based Multilayer Networks. *Mathematics*. 2020; 8(11):1895.
https://doi.org/10.3390/math8111895

**Chicago/Turabian Style**

Wang, Dong, Yi Zhao, and Hui Leng.
2020. "Dynamics of Epidemic Spreading in the Group-Based Multilayer Networks" *Mathematics* 8, no. 11: 1895.
https://doi.org/10.3390/math8111895