Investigation of epidemic spreading process on multiplex networks by incorporating fatal properties

Abstract

Numerous efforts have been devoted to investigating the network activities and dynamics of isolated networks. Nevertheless, in practice, most complex networks might be interconnected with each other (due to the existence of common components) and exhibit layered properties while the connections on different layers represent various relationships. These types of networks are characterized as multiplex networks. A two-layered multiplex network model (usually composed of a virtual layer sustaining unaware-aware-unaware (UAU) dynamics and a physical one supporting susceptible-infected-recovered-dead (SIRD) process) is presented to investigate the spreading property of fatal epidemics in this manuscript. Due to the incorporation of the virtual layer, the recovered and dead individuals seem to play different roles in affecting the epidemic spreading process. In details, the corresponding nodes on the virtual layer for the recovered individuals are capable of transmitting information to other individuals, while the corresponding nodes for the dead individuals (which are to be eliminated) on the virtual layer should be removed as well. With the coupled UAU-SIRD model, the relationships between the focused variables and parameters of the epidemic are studied thoroughly. As indicated by the results, the range of affected individuals will be reduced by a large amount with the incorporation of virtual layers. Furthermore, the effects of recovery time on the epidemic spreading process are also investigated aiming to consider various physical conditions. Theoretical analyses are also derived for scenarios with and without required time periods for recovery which validates the reducing effects of incorporating virtual layers on the epidemic spreading process.

Keywords: Epidemic information spreading; Multiplex networks; Theoretical validation; UAU-SIRD model.

Fig. 1

Illustration of a community modeled by two-layered multiplex models. Dashed lines indicate the correspondences of individuals on the physical layers and nodes on the virtual ones, while solid lines show connections at each layer.

Fig. 2

Possible transitions between the states of certain individual on the physical layer.

Fig. 3

(a) Transition probability tree for a susceptible and unaware individual. (a) Transition probability tree for a susceptible and aware individual. (c) Transition probability tree for an infected individual. I: infected; S: susceptible; D: dead; R: recovered.

Fig. 4

Possible transitions between the state combinations for certain individual from a given state combination at time step t to the possible successor state at time step t + 1.

Fig. 5

(a) An illustration of a community represented by multiplex networks, here a susceptible individual (represented by a white node) is randomly selected to be infected (indicated by a node with I); (b) An illustration of the infection and information transition process via inter-connections; (c) An illustration of the scenario that an individual is dead (indicated by a node with D); (d) An illustration of the community with recovered individuals (indicated by a node with R).

Fig. 6

An illustration of the revised recovery model.

Fig. 7

(a) Illustration of the results obtained by different approaches. (b) Illustration of the range of individuals being affected through simulation-based approach for scenarios with and without virtual layer. Here, t_m is not considered.

Fig. 8

An illustration of the relationship between ${Pro}_R+{Pro}_D$ and the parameter combination of μ and λ through (a) simulation-based approach. (b) theoretical analysis (the other parameters are the same as those of Fig. 7).

Fig. 9

(a) Evolution of proportions of individuals with different states at different time steps for the scenario without t_m; (b) Evolution of proportions of individuals with different states for the scenario with t_m = 10.

Fig. 10

(a) Fraction of recovered individuals for different t_m; (b) Fraction of dead individuals for different t_m; (c) The relationship between recovered/dead individuals and t_m.

Fig. 11

The relationship between the fraction of dead individuals and the parameter combination of μ_d and λ by (a) theoretical analysis and (b) simulation-based approach for the revised recovery model (as indicated by Fig. 6). The relationship between the fraction of dead individuals and the parameter combination of γ and λ by (c) theoretical analysis and (d) simulation-based approach for the revised recovery model. (Here, t_m = 4 while the other parameters are the same as those in Fig. 7).