key: cord-0941995-nn542drp
authors: Zhu, Peican; Wang, Xinyu; Li, Shudong; Guo, Yangming; Wang, Zhen
title: Investigation of epidemic spreading process on multiplex networks by incorporating fatal properties
date: 2019-10-15
journal: Appl Math Comput
DOI: 10.1016/j.amc.2019.02.049
sha: 900dde3dc5646686fe481d388e8dd5be375aaba8
doc_id: 941995
cord_uid: nn542drp

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.

The study of complex networks has attracted much attention and provides us an opportunity to understand the relationship between system topology and network activities (or dynamics) [1] . The entities and relationship between them are represented by vertices and edges respectively. During the past decades, various networks have been investigated, such us, the World Wide Web, transportation network, the networks of scientific and movie actors collaborations, as well as the epidemic spreading process [2] [3] [4] [5] . Nevertheless, in practice, most networks are not isolated; and they are interconnected with each other due to the existence of common components. Various infrastructures, like power supply system, water supply system, transportation system, are interacted with each other. Thus, numerous researchers recently have devoted efforts to the study of novel networks in order to investigate phenomena being absent in isolated networks [6] . Furthermore, the real-world networks usually exhibit layered properties, i.e. the links in different layer represents the connections in various environments. In this sense, these types of networks are encapsulated into the framework of multiplex networks, which seem to be applicable to modeling lots of real-world systems, like the European air transport system [7] , the global cargo ship network [8] .

Among dynamics on networks, the epidemic spreading process is an evolving realm and has attracted considerable effort s. In [9, 10] , numerous studies of spreading process are conducted. Nevertheless, with fast development of hi-tech products, individuals contact each other via various fashions (either in real-life contact or online communication with the adoption of modern social tools), which indicates that information exchanging process will occur across geographic borders [11] . The epidemic spreading process is likely to be affected by the information transmitting process. Hence, the epidemic spreading process on multiplex networks is attracting much attention.

In order to mimic the propagation of recurrent diseases, the susceptible-infected-susceptible (SIS) model is presented in [11] where certain individual in the investigated community is likely to be in two states: susceptible and infected (i.e., the recovered individual will become susceptible again). The critical point for the epidemic spreading across multiplex networks is much lower than that of isolated networks [12] [13] [14] . Stochastic analyses are performed in order to predict the long run behaviors of SIS model efficiently in [15] with the adoption of the stochastic computational approach.

Nevertheless, there exist some other types of diseases (for instance, H5N5 flu [16] or the SEVERE Acute Respiratory Syndrome (SARS) [17] ) in which the infected individual is either dead or become immune to future infections once recovered. Here, the infected individuals might be recovered due to the adoption of medical treatments or self-rehabilitation. For this class of diseases, the susceptible-infected-recovered (or succinctly, SIR) model is presented in order to mimic the corresponding epidemic spreading process [18, 19] . In [20] , the SIR model is adopted to investigate the epidemic spreading process in two interacting networks numerically. Then, corresponding epidemic spreading thresholds in different communities and internetworks are determined accordingly. An analytical approach is developed in [21] , where two SIR propagations over multiplex networks are thoroughly investigated. According to previous discussions, the awareness of a susceptible individual plays an important role in the epidemic spreading process [14, 22, 23] . Hence, the SIR model is applied to multiplex networks consisting of two different layers, i.e., a virtual layer and a physical one [24] . The information is transmitting on the virtual layer through internet tools while the epidemic spreading process is performed on the physical layer. Furthermore, the dynamics of the epidemic spreading process is further investigated either simulation-based or theoretically [25] [26] [27] [28] [29] based on the traditional SIR or SIS model.

The previous works either consider the recovery phenomena or the dead effect; nevertheless, the effects of recovery and death are totally different in practice. For an infectious individual, it will be immune once recovered and will not be affected in the future. Nevertheless, the dead individual will be removed from the investigated community forever. Though the epidemic is incapable of being transmitted through either recovered or dead individuals, the recovered individual is capable of informing corresponding neighbors about the fatal epidemic. Hence, preventive measures can be taken to reduce the probability of infection, such as, reducing the frequency of outside activities, wearing face masks, washing hands frequently, taking the vaccination, doing exercises, reducing contacts with other individuals and so forth [30] [31] [32] . In this work, in order to clarify the difference between recovery and death, a susceptible-infected-recovered-dead (SIRD) model is considered to couple with the awareness dynamics, i.e., unaware-aware-unaware (UAU) model in virtual layers, being referred as a UAU-SIRD model. In this model, recovered individuals with awareness can inform susceptible neighbors who will take some preventive measures to reduce the infection probability. Nevertheless, if an infectious individual is dead, corresponding connections should be removed. Furthermore, to validate the accuracy of simulations, theoretical formulas are also derived.

In previous researches, an infectious individual is assumed to be recovered with certain probability; which indicates that recovery occurs immediately after being infected due to selfhealing or medical treatments. However, this seems inconsistent with empirical case: the recovery usually needs several days, or even months even under effective medical treatment. Hence, in this manuscript, the infected individual will transmit the disease to its susceptible neighbors with a prespecified probability and the infected individual is recovered after a fixed time. This recovery model is adopted and a modified probability transition tree by incorporating recovery and death is presented. Furthermore, the required recovery time steps are not necessarily to be a fixed value even under the same medical treatment. This is due to the fact of different physical conditions of the infected individuals. We make some revision to the proposed UAU-SIRD model by considering the recovery property, while theoretical analyses are also performed for validation.

The rest of the paper is organized as follows. Section 2 presents some assumptions in this manuscript. While some fundamentals are presented in Section 3 ; then the spreading process is considered by coupling SIRD model and awareness dynamics via an illustrative example. Later, we also consider the fact of necessary time periods before fully recovered. Furthermore, theoretical analyses are also performed to validate the correctness of proposed models. Simulations of different scenarios are conducted in Section 4 with the results of focused variables being presented. Finally, Section 5 concludes the paper. 

In this section, some necessary assumptions are listed as follows:

(1) For simplicity, in this manuscript, death is solely incurred by the investigated fatal disease, whereas other reasons are not incorporated; (2) Due to information transition process on the virtual layer, an individual becomes aware of the epidemic with a probability of λ; (3) If an individual is infected, then the probability of passing away is μ d ; (4) For the infected individual, the probability of being recovered is represented by μ. Once recovered, the individual is immune to the epidemic forever; (5) A susceptible individual is likely to forget the awareness (i.e., becoming unaware again) and not to care about it, which occurs with probability δ; (6) Because of physical contacts between susceptible and infectious individuals, a susceptible individual is affected with probability p ; (7) Once certain individual is aware of the epidemic, then preventive measures might be taken. Hence, corresponding infection probability is reduced by a factor γ (0 < γ < 1).

A general community is usually modeled with the adoption of the two-layered multiplex network model (i.e., a virtual layer and a physical one); each node on the physical layer has a proxy on the virtual one, referred as completely overlapped networks (if partially overlapped networks [24] are considered, we need some local revisions). Each layer can be indicated by a graph G = ( V , E ) where V and E represent the node and link sets respectively. The link set is usually indicated by an adjacent matrix, i.e., if a link exists between nodes i and j , then a ij = 1. Otherwise, a ij = 0. For the virtual layer, links indicate the virtual connections among individuals through internet tools (e.g., Facebook, twitter); while information about the epidemic is exchanged through these connections. Whereas on the physical one, links denote the physical contacts through which biological elements carrying the virus of epidemic are transmitted. A community consisting of six individuals is presented as in Fig. 1 for an illustration. Here, the community is composed of two layers; while each layer has six nodes and different numbers of connections.

For each individual on the physical layer, it is anticipated with one of the following four states, i.e., susceptible ( S ) in which the individual is free of the epidemic but likely to be infected via direct contacts with infected neighbors, infected ( I ) in which susceptible individuals are infected and affect other susceptible individuals, recovered ( R ) in which the infected individuals are recovered and can not transmit the disease to other susceptible individuals (i.e., immune), death ( D ) in which the infected individuals pass away due to the fatal effect of the epidemic. Based on the above discussion, the state transition process for an individual on the physical layer is presented in Fig. 2 , where the outgoing arrow from certain state at time t points to the possible successor state at t + 1. For example, a susceptible individual is likely to remain susceptible or be infected because of direct contacts with infected neighbors. Once infected, the individual might be in I , R or D at the next According to transitions in Fig. 2 , the transition probability tree for the epidemic spreading process on the physical layer can be indicated by Fig. 3 . Here, through practical contact, a susceptible individual might be infected; nevertheless, the probability of being infected is also affected by the state of corresponding node on the virtual layer. For instance, if a susceptible individual is unaware of the epidemic, then it is affected by the infected neighbor with probability p. Otherwise, it remains to be susceptible. This is indicated by Fig. 3 (a) . Nevertheless, if a susceptible individual is aware, then preventive measures, such as, avoiding direct contacts, might be adopted. This indicates the corresponding infection probability will be reduced, i.e., implemented by multiplying by a factor γ (0 < γ < 1) illustrated as Fig. 3 (b) . Based on the possible transitions in Fig. 3 (c) , the infectious individual might pass away with probability μ d ; nevertheless, s/he is also likely to be recovered with a probability of μ. Otherwise, the individual remains infected (this occurs with probability 1-μ d − μ).

For certain node on the virtual layer, it might be one of the two states, i.e., aware ( A ) in which corresponding individual is aware of the epidemic and unware ( U ) in which no information about the epidemic is known by the individual. The unaware individual will be aware due to the information spreading process, i.e., U → A . Nevertheless, the epidemic is usually seasonal, and an aware individual is likely to forget the awareness, i.e., become unaware again in the future indicated by A → U . In this manuscript, the aware and susceptible individual might forget the epidemic while the infected and recovered individual is unable to be unaware.

Overall, in this manuscript, each individual falls into the following combinations, i.e., US , UI , AS , AI and AR . Possible state combination transitions are illustrated in Fig. 4 . Here, X VA Y PA indicates that the state of certain individual (for community A ) on the virtual layer is X VA (here, X VA ∈ { A , U }) and the state of corresponding node on the physical layer is Y PA (here, Y PA ∈ { S , I , R , D }). In particular, U VA R PA is neglected because of the assumption that a recovered individual will not forget the awareness; while the state combination of X VA D PA is deleted as it is unmeaningful to consider corresponding node on the virtual layer for a dead individual.

Because of the fatal property (i.e., occurrence of death incurred by the epidemic), the number of individuals in the investigated community varies. For a dead individual, corresponding node on the virtual layer becomes unmeaningful and should be removed. Hence, inter-layer connection related with these nodes should also be removed. This indicates the investigated system topology varies from time to time.

An example of the epidemic spreading process on two-layered multiplex networks is presented in Fig. 5 . For simplicity, the investigated community consists of 10 susceptible individuals; then there are 10 corresponding nodes on the virtual layer who are assumed to be unaware of the epidemic initially. In Fig. 5 (a) , a random individual on the physical layer is infected, indicated by a node with I; thus, corresponding node on the virtual layer is aware of the epidemic immediately (indicated by a node with A). In Fig. 5 (b) , a susceptible individual is infected via physical contacts and the information transition process on the virtual layer is also presented. Then, at the next time step, certain infectious individual might be dead due to the fatal property of the epidemic; this is illustrated as Fig. 5 (c) (represented by a node with D ). If an infectious individual is dead, all the links with this dead individual will be naturally removed (denoted by lines with × ). Furthermore, corresponding node on the virtual layer should be removed and the connections should also be deleted. Moreover, the infected individual might be recovered, as indicated by a node with R in Fig. 5 (d) . Once recovered, corresponding node on the virtual layer keep aware of the epidemic forever and can exchange the information with other nodes. Due to the fatal effect, the community varies from time to time.

As in Fig. 5 (c) , an infected individual is recovered with certain probability; it is also likely for the infected individual to be dead immediately. Nevertheless, this is usually not in accordance with empirical observations. In practice, once medical treatments are conducted, the infected individuals can not recover or die immediately. Thus, different from Fig. 5 (c) , the recovery model is revised in Fig. 6 , where certain time steps (i.e., t m ) are required for the recovery process. Once a susceptible individual is infected, a variable of t u is set, which indicates time period of suffering from the epidemic for the infected individual. In the following each step, an infected individual may be dead with probability μ d ; otherwise, the infectious state continues. If the infected individual does not pass away; then, t u increases by 1 after. Repeating the above process, when t u reaches t m which is a pre-specified time period for the recovery process, the infectious individual is regarded as being eventually recovered.

Because of various body conditions, t m might vary for different individuals even if same medical treatments are adopted. Furthermore, t m seems to be related with various factors, such as age, absorbing capability, physical quality. In this sense, t m is likely to follow certain distributions. For simplicity, t m is a randomly selected value from a provided range in this manuscript. Then, corresponding analyses can be performed to investigate the effects of varying t m .

Given parameters and topology of the multiplex networks, analyses can be performed through Monte Carlo (MC) simulation. For certain experiment trial, the number of dead individuals (i.e., N dead ) can be determined easily. Then, the fraction of dead individuals is obtained as

where N represents the number of individuals in the investigated community. Similarly, the fractions of currently infected and recovered individuals are represented as Pro I and Pro R , respectively. It is obvious when initial infected individuals vary, corresponding epidemic spreading process also changes accordingly. Hence, Pro D , Pro I and Pro R are affected by various factors, such as infectivity probability, death-incurring probability, information transition probability, system topology (i.e., degree distribution). Here, in order to investigate the relationship between focused variables such as Pro D and the epidemic transition probability p, other parameters are fixed. Furthermore, the relationship still varies if the initial infected individuals vary. For instance, if the initial infected individual is an isolated one, then no individuals will be infected. At variance, once a hub node is infected, the disease will spread rapidly. Hence, simulations are performed for a number of trials to obtain sufficient accuracy.

In this section, theoretical analysis is also performed for the presented model. The introduced SIRD model is adopted to describe the dynamic process of epidemic spreading process. Given a population containing a group of individuals, they are divided into four categories: ( S ) susceptible, ( I ) infectious, ( R ) recovered and ( D ) dead individuals. Here, s ( t ), i ( t ), r ( t ) and d ( t ) denote the fractions of individuals in different states at time t for a population with N individuals (i.e., s (t) + i (t) + r(t ) + d(t ) = 1 for any time t ). Moreover, a ( t ), u ( t ) represent the fractions of individuals with and without awareness at time t .

Firstly, we perform theoretical analysis for SIRD model illustrated by Fig. 3 . Here, M A is defined as the fraction of individuals with awareness and M A (t) = (a (t )) / (a (t ) + u (t)) . Given the infection probability p , the number of susceptible and aware individuals being infected and switching to I state is γ p < k > s ( t ) i ( t ) M A ; whereas the number of susceptible and unaware individuals being infected is p < k > s (t ) i (t )(1 − M A ) . Furthermore, we assume a fraction μ of infectious individuals are recovered during a unit time, the number of individuals swithing from I state to R state is μ i ( t ). However, during a unit time, if a fraction (i.e., μ d ) of infectious individuals pass away, then the number of individuals switching from I state to D state is μ d i ( t ). Hence, the amount of increased infectious individuals are obtained as −d Next, we also perform theoretical analysis of the scenario with recovery time t m (i.e., revised recovery model). Due to the incorporation of t m and the recovery model shown in Fig. 6 , an infectious individual can only be recovered after t m ; then the theoretical analysis varies after the appearance of recovery. If t < t m + 1, the dynamics on the physical layer is depicted by Eq. (4) ; otherwise, corresponding dynamics on the physical layer are described by Eq. (5) . As to the information spreading process on the virtual layer, corresponding dynamics can be described by Eq. (6) . Through performing iterative analysis of the formulas 4 -6 , the epidemic spreading system with the incorporation of t m can be conducted. theoretical analysis (the other parameters are the same as those of Fig. 7 ) .

where, < k > represents the average degree; the meanings of the other parameters can be referred to the assumption section.

In this section, various simulations are conducted for the purpose of mimicking the epidemic spreading process on a community. The investigated community consists of N individuals (here, N = 10 0 0 for an illustration). The size of the community varies due to the fatal property of the epidemic. Furthermore, the studied community is assumed to be composed of two-layers (i.e., a physical layer and a virtual one). Hence, the effects of incorporating virtual layer on the epidemic spreading process can also be analyzed. The analyses here are conducted through MC simulation, which is furthermore validated through theoretical analyses. Simulation stops if stationary states or anticipated time steps are achieved. For each layer, the network topology is assumed to be a WS network, where each node relates to 4 nodes with the rewiring probability 1.0. Firstly, simulations are conducted for the UAU-SIRD model in Fig. 7 (a) (the model without t m ) . For comparison, theoretical analyses are also performed through the application of the formulas 2 and 3 . We mainly focus on the evolution of fractions of susceptible/dead/infected/recovered individuals. In Fig. 7 (a) , the obtained theoretical results are denoted by solid lines while the simulation results are indicated by solid points. As we see, the accuracy of the UAU-SIRD model is validated by the theoretical analysis. In particular, Pro I increases first which is incurred by the spreading of the epidemic; nevertheless, this value reduces after reaching the peak. This is due to the reason that the infected individuals are either recovered or dead and finally vanish after reaching the stationary state.

Furthermore, in order to reflect the importance of considering virtual layer, the scenario of SIRD model without virtual layer is also shown (the fractions of dead and recovered individuals are illustrated) in Fig. 7 (b) . Compared with the scenario without virtual layer, the proportion of dead individuals with awareness decreases apparently; which is incurred by the information diffusion making susceptible individuals be aware of the epidemic. In such a case, preventive measures can be taken, and the infection probability is multiplied by a factor γ (0 < γ < 1). This indicates that the awareness of epidemic can reduce the infection probability of epidemic, which is also consistent with common sense. Furthermore, we can also conclude that the proportion of recovered individual will increase apparently if the virtual layer is considered. Nevertheless, before the stationary state, the range of individuals being affected/dead with virtual layer seems to be larger than that without virtual layer.

Then, the effects of varying parameters on the focused value (the fractions of individuals being affected (i.e., P ro R + P ro D )) are shown in Fig. 8 . In Fig. 8 (a) , the effect of varying parameters combination of μ and λ on P ro R + P ro D is obtained through simulation-based approach. For validation, theoretical analyses are also presented in Fig. 8 (b) . For fixed λ, the range of individuals being affected reduces with the increase of μ since more individuals will be recovered if the recovering rate increases. For fixed μ, the proportion of individuals being affected reduces with the increase of λ. This is due to the reason that it is more easily for the epidemic information spreading process among nodes on the virtual layer. It indicates that the susceptible individuals are more likely to become aware of epidemic. Hence, the infection probability declines as a susceptible individual with awareness takes preventive measures. This will further reduce the infection probability for susceptible individuals. It thus becomes natural that the range of individuals being affected reduces. Next, if the recovery model depicted by Fig. 6 is investigated, corresponding analyses results performed through simulation-based approach are presented in Fig. 9 (a) . For comparison, the simulation results with t m = 10 are also presented in Fig. 9 (b) . It is clear that the number of dead individuals increases first and reaches a steady value after certain time steps (similar trends exist for the recovered individuals). However, the proportion of dead individuals increases, and the fraction of recovered individuals decreases apparently if t m is considered. Furthermore, the peak of proportion of infected individuals comes later for the scenario with t m yet the peak possesses a larger value.

Moreover, as indicated by Fig. 9 (b) , the landmark of the curve for the proportion of recovered individuals is supposed to be at time step 10 marked by a circle; this is affected by the pre-provided recovery time for the investigated disease. For instance, the initially infected individuals can only be recovered at time step 10 if corresponding death does not occur; thus, the proportion of recovered individuals equals to 0 before 10th time step.

To investigate the effects of different recovery time on the epidemic spreading process, analyses of several scenarios are presented in Fig. 10 . Here, the fraction of recovered and dead individuals is mainly focused and three scenarios with different t m (here, t m = 3, 6, 9) are considered.

As indicated by the results, the recovery process of infected individual is advanced for smaller t m . The fraction of recovered individuals is larger if shorter recovery time t m is adopted while similar trends exist for the fraction of dead individuals. Furthermore, we also investigated the proportion of recovered/dead individuals if stationary state is achieved with the increment of t m (1 ≤ t m ≤ 10) in Fig. 10 (c) , respectively. As we see, for larger t m , the increasing of dead individuals or decreasing of recovered individuals becomes slower.

Then, we try to analyze the effects of varying parameters on the focused values for the UAU-SIRD model with t m . For simplicity, the fraction of dead individuals is focused through simulation-based approach. Corresponding simulation results for the fraction of dead individuals and the parameter combination of μ d and λ are provided in Fig. 11 (a) for the scenario with t m = 4. Furthermore, theoretical results are also derived through adopting Eqs. (4) - ( 6 ) , while corresponding results are illustrated in Fig. 11 (b) . During the analysis for Fig. 11 , t m is set to be 4.

Here, we mainly focus on the fraction of dead individuals for different parameter combination of μ d and λ (here, 0 ≤ μ d ≤ 0.5, 0 ≤ λ ≤ 0.7). As per the results in Fig. 11 (a) and (b) , for certain μ d (which is relative smaller), the varying of λ seems to play a trivial role in determining the proportion of dead individuals. This is also validated by the theoretical analyses. Nevertheless, for larger μ d , the increase of λ, to some extent, is capable of reducing the fraction of dead individuals.

Furthermore, we also incorporate the relationship between the fraction of dead individuals and the parameters combination of γ and λ (here, 0 ≤ γ ≤ 0.5, 0 ≤ λ ≤ 0.7). The results obtained through simulation-based and theoretical analyses are presented in Fig. 11 (c) and (d) respectively. As indicated in Fig. 11 (c) and (d) , for a small and fixed γ , the fraction of dead individual decreases with the increase of λ. This also confirms the previous conclusion of incorporating virtual layers (similar to reducing the value of λ) on the epidemic spreading process. Nevertheless, for large γ , the increasing of λ seems to have less impact. This is due to the fact that for large γ , the reduced amount of infection probability is limited, which indicates that it is almost unmeaningful for the individual to be aware as there is less impact with taking preventive measures.

For some fatal diseases, death might be unavoidable, such as, H5N5 flu, the SARS. Thus, the susceptible-infectedrecovered/removed (SIR) model is investigated to study the dynamics of the epidemic spreading process. For an infectious individual, it is likely to become immune and will not be infected any more once recovered. Nevertheless, for a dead individual, it will be removed from the investigated community. Because of the difference between recovery and death, a revised model is considered, referred as coupled SIRD model.

Recently, two-layered multiplex networks (consisting of a physical layer and a virtual one) are studied to investigate the epidemic spreading process. Thus, in this manuscript, this widely adopted model is used to investigate the coevolution of epidemic information and fatal epidemics. Furthermore, we also try to investigate the scenario of required time periods for recovery. To validate corresponding models, theoretical analyses are conducted to determine the analytical formulas for different scenarios.

With given parameters and system topology, MC simulations can be conducted accordingly. Through analysis, relationships between focused variables and the parameters of the community are thoroughly investigated. As indicated by the simulation results, the range of dead individuals will be reduced greatly once the information transition process on the virtual layer is incorporated. This can also be validated by theoretical analysis. Furthermore, the effects of fixed recovery time on the fatal epidemic spreading is investigated; if different physical conditions (i.e., varying recovery time) are considered, analyses can be performed with modifications. In the future, the effects of immunization and isolation such as vaccination [33] [34] [35] can also be taken into consideration. Definitely, if certain individual is affected and becomes dead, this will be a warning to his/her neighbors, this alerting effect will be considered later. Furthermore, if the community movement [36] or weighted property is considered, similar analysis can be conducted with minor revisions.

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