# Control of Multiple Viruses Interacting and Propagating in Multilayer Networks

Authors: Xiao Tu; Guo-Ping Jiang; Yurong Song; Xiaoling Wang
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Experimental studies involving control against virus propagation have attracted the interest of scientists. However, most accomplishments have been constrained by the simple assumption of a single virus in various networks, but this assumption apparently conflicts with recent developments in complex network theory, which details that each node might play multiple roles in different topological connections. Multiple viruses propagate through individuals via different routes, and thus, each individual component could be located in various positions of differing importance in each virus propagation process in each network. Therefore, we propose several control strategies for establishing a multiple-virus interaction and propagation model involving multiplex networks, including a novel Multiplex PageRank target control model and a multiplex random control model. Using computer experiments and simulations derived from actual examples, we exploit several actual cases to determine the relationship of the relative infection probability with the immunization probability. The results demonstrate the differences between our multiple-virus interaction and propagation model and the single-virus propagation model and verify the effectiveness of our novel Multiplex PageRank target control strategy. Moreover, we use parallel computing for simulating and identifying the relationships of the immunization thresholds with both interaction coefficients, which is beneficial for further practical applications because it can reduce the multiple interactions between viruses and allows achieving a greater effect through the immunization of fewer nodes in the multilayer networks.


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## Body

## 1. Introduction

The structure and dynamics of multilayer networks have attracted much attention from scientific communities [1–24]. Comprised of a set of networks combined with interacting layers, these multilayer networks properly describe a variety of realistic complex systems, such as financial networks [18], ecological networks [19], information networks [20], and transportation networks [21].Recently, many studies have attempted to discover the dynamics of viral propagation in multiplex networks. Along this line, various methods aiming at studying the virus propagation process in multilayer networks have been proposed and explored, and some examples include competing epidemics [25], the effect of the interconnected network structure [17], and the interaction of viruses and information [26]. More explicitly, several virus spreading models in partially overlapped networks have been proposed, and these networks are described as distinct networks that contain partially identical individuals [27–29]. By considering one virus spreading via multiple routes [6] or one virus spreading in multiple species [30], corresponding virus spreading models in multilayer networks have been proposed. Interestingly, these measures are not considerably affected by multiple viral interactions.As mentioned above, although some accomplishments have been achieved by focusing on the effect of multilayer topology on virus dynamics and the resulting threshold, the influence of such improved complexity on control strategies remains unexplored [31–36]. In conventional research on virus control, the vaccinated candidate nodes are commonly chosen randomly or spontaneously according to their topological properties, such as degree, betweenness, k-shell, and PageRank [37–42]. Therefore, an interesting problem presents itself, which we aim to address in this work. If we consider the basic control cases in multilayer networks, how do they affect virus propagation?.Here, using the multiple-virus spreading model in multilayer networks, we study the performance of some novel control strategies, including novel Multiplex PageRank targeted control and multiplex random control strategies, in multilayer networks. Based on the generating function theory, extensive computational simulations are performed to assess our measure in various cases, and we find that the effectiveness of the proposed control strategies relies on the interaction relationship between multiple viruses in multilayer networks.


## 2. Multiple-Virus Interaction and Propagation

Unlike the existing research on single-virus propagation in overlapping networks and double virus propagation in conventional networks, we focus on multiple viruses propagating through individuals, who are integrated into all layers of a multilayer network formed by different propagation routes. In our model, we briefly exploit a double-layer network containing networkA and network Bof the same size N in which the nodes in both layers adopt one-to-one links. A schematic of the multilayer network is shown in Figure 1. The dashed lines represent the inner links of integrated individuals and their correspondence in the other layer. For the simulation, we construct each layer using a BA scale-free network model consisting of N nodes; this network was primarily constructed based on a random network of m0 nodes, and a new node with m links to the existing nodes was then introduced until the network contained a total of N nodes. In our model, we assume that the two viruses propagate according to their respective transmissibility in each layer, and the simulation is initiated by infecting several randomly chosen individuals in the multilayer network.Figure 1
Multilayer network; the dashed lines show the inner links of the integrated nodes. Prior to the initiation of propagation, all individuals are in the susceptible state, which is represented by a green circle.The propagation process follows a multiple-virus propagation modelSAIASA−SBIBSB in a multilayer network, where each node can be divided into four compartments relevant to the two viruses: the individuals who are susceptible to both the virus in network A and the virus in network BSASB, the individuals who are susceptible to the virus propagated in A while infected with the virus propagated in BSAIB, the individuals who are infected with the virus propagated in A while susceptible to the virus propagated in BIASB, and the individuals who are infected with both the virus propagated in A and the virus propagated in BIAIB. An individual at state SA in A can be converted to state IA if it comes in contact with any neighbor at state IA, which occurs with the probability αASB if the individual is at state SB in B. An individual at state SA in A can be converted to state IA with the probability αAIB if the individual is at state IB in B. Similar to A, an individual at state SB in B can be converted to state IB with the probabilityαBSA if the individual is at state SA in A. An individual at state SB in B can be converted to IB state with the probability αBIA if the individual is at state IA in A. An individual at state IA might be cured and returned to state SA with the probability μA, whereas an individual at state IB might be cured and return to state SB with the probability μB.Moreover,αASB and αAIB are related by the interaction coefficient cAIB such that αAIB=cAIBαASB, where cAIB indicates the influence of the virus in B on the spreading probability of the virus in A, whereas αBSA and αBIA are related by the interaction coefficient cBIA such that αBIA=cBIAαBSA, where cBIA refers to the influence of the virus in Aon the propagation probability of the virus in B. Notably, cAIB≥0, αASB∈0,1, and αAIB∈0,1. In particular, cAIB=0 indicates that the virus in B is immune to the virus in A; 0<cAIB<1 indicates that the virus in B inhibits the virus in A; cAIB=1 indicates that the virus in B has no effect on the virus in A; and cAIB>1 indicates that the virus in B enhances the virus in A. In addition, the spreading probability αAIB must satisfy αAIB≤1; therefore, if cAIBαASB>1, then αAIB=1. The conditions in B are similar to those inA.Because we useA and B to denote the two virus propagation routes, Aji and Bji are denoted as the edges from individual j to individual i by the routes of each propagated virus, which are equal to 1 if edges exist and 0 otherwise. We adopt the process of a continuous time Markov chain to describe the interaction of multiple propagating viruses by defining Xit and Yit as measures of the random variables X and Y of node i at time t, respectively. For each i∈1,2,…,N, the random variables Xi and Yi satisfy Xi: SA,IA⟶0,1 and Yi: SB,IB⟶0,1. Therefore, the continuous time Markov chain equation of the node state evolution over time is represented as follows:(1)PrXit+1=1,Yit+1=0Xit=0,Yit=0=αASBΔt∑j∈Ni1Xjt=1+οΔt,PrXit+1=1,Yit+1=1Xit=0,Yit=1=αAIBΔt∑j∈Ni1Xjt=1+οΔt,PrXit+1=0,Yit+1=0Xit=1,Yit=0=μAΔt+οΔt,PrXit+1=0,Yit+1=1Xit=0,Yit=0=αBSAΔt∑j∈Ni1Yjt=0+οΔt,PrXit+1=1,Yit+1=1Xit=1,Yit=0=αBIAΔt∑j∈Ni1Yjt=1+οΔt,PrXit+1=0,Yit+1=0Xit=0,Yit=1=μBΔt+οΔt.where Pr is the probability and Δt>0 is a time step, 1χ, which is equal to 1 if χ is true or 0 otherwise. If limΔt⟶0fΔt/Δt=0, fΔt can be written as οΔt. In particular, 1Xjt=1 can be represented by PrXj=1. Thus, the continuous Markov chain equation (1) can be transformed to a new random stochastic equation:(2)PrXit+1=1,Yit+1=0Xit=0,Yit=0=αASBΔt∑j∈NiPrXjt=1+οΔt,PrXit+1=1,Yit+1=1Xit=0,Yit=1=αAIBΔt∑j∈NiPrXjt=1+οΔt,PrXit+1=0,Yit+1=0Xit=1,Yit=0=μAΔt+οΔt,PrXit+1=0,Yit+1=1Xit=0,Yit=0=αBSAΔt∑j∈NiPrYjt=1+οΔt,PrXit+1=1,Yit+1=1Xit=1,Yit=0=αBIAΔt∑j∈NiPrYjt=1+οΔt,PrXit+1=0,Yit+1=0Xit=0,Yit=1=μBΔt+οΔt.The value of the nodes reflects the average probability of the four states at timet, which are, respectively, denoted by piSASBt, piSAIBt, piIASBt, and piIAIBt and satisfy(3)piSASBt+piSAIBt+piIASBt+piIAIBt=1.Thus, the average probability of the virus inA at time t is represented as piIAt=piIASBt+piIAIBt and that of the virus in B at time t is represented as piIBt=piSAIBt+piIAIBt. We defined the new state as Zi∈piSASB,piSAIB,piIASB,piIAIB. According to the correlation theorem and the properties of the continuous time Markov chain, the transfer matrix Q of the stochastic process can be obtained as follows:(4)Q=−αASB∑jAjipjIA−αBSA∑jBjipjIB−αAIB∑jAjipjIA+μBμA−αBSA∑jBjipjIBμA+μB−αASB∑jAjipjIA+αBSA∑jBjipjIB−αAIB∑jAjipjIA−μBμA+αBSA∑jBjipjIBμA−μBαASB∑jAjipjIA−αBSA∑jBjipjIBαAIB∑jAjipjIA+μB−μA−αBSA∑jBjipjIB−μA+μBαASB∑jAjipjIA+αBSA∑jBjipjIBαAIB∑jAjipjIA−μB−μA+αBSA∑jBjipjIB−μA−μBT.Therefore, the Kolmogorov forward differential equations of the stochastic process (2) can be represented as follows:(5)Zi•=ZiQ,i∈1,2,…,N.We can expand the differential equations of nodei to obtain(6)p•iSASB=−αASB∑jAjipjIA−αBSA∑jBjipjIBpiSASB+−αAIB∑jAjipjIA+μBpiSAIB+μA−αBSA∑jBjipjIBpiIASB+μA+μBpiIAIB,p•iSAIB=−αASB∑jAjipjIA+αBSA∑jBjipjIBpiSASB+−αAIB∑jAjipjIA−μBpiSAIB+μA+αBSA∑jBjipjIBpiIASB+μA−μBpiIAIB,p•iIASB=αASB∑jAjipjIA−αBSA∑jBjipjIBpiSASB+αAIB∑jAjipjIA+μBpiSAIB+−μA−αBSA∑jBjipjIBpiIASB+−μA+μBpiIAIB,p•iIAIB=αASB∑jAjipjIA+αBSA∑jBjipjIBpiSASB+αAIB∑jAjipjIA−μBpiSAIB+−μA+αBSA∑jBjipjIBpiIASB+−μA−μBpiIAIB.


## 3. Multiplex Control Strategy

In contrast to the control of each noninteracted virus in each conventional network individually, the multiplex control strategy aims to immunize fewer nodes but achieve a greater effect on both of the interacting viruses in the multilayer network simultaneously. Thus, we compare our novel Multiplex PageRank target control with multiplex random control in a multilayer network.Recently, researchers have focused on measuring the centrality of multiplex networks. The eigenvector multiplex centrality hypothesizes that the centrality of a node in one layer is impacted by its centrality in other layers by an overlapped influence matrix [43]. The versatility of nodes highlights the relevance of related nodes in different layers and applies to multilayer networks in which the corresponding nodes in different layers are connected by interlinks [44]. The Multiplex PageRank centrality utilizes the correlations among the degrees of nodes in different layers by means of a random walk subject to teleportation [45–49].Among these centrality measures, the versatility of nodes is the only measure that considers the interlinks, whereas both the Multiplex PageRank centrality and the eigenvector multiplex centrality stipulate one-to-one links among the nodes in different layers, which are denoted inner links throughout the rest of the manuscript paper. An inner link induces a coupling relationship that will impact the distribution of centrality inside the same node. The major challenge when identifying the centrality of a node in a multiplex network with inner links is that the centrality depends on the relationship among the distinct types of links, which are also known as traditional links, between different nodes in the same layer. This study attempts to address the issue of the centrality distribution through a generalization of PageRank by considering the coupling relationship.Our novel Multiplex PageRank measure is derived from a random population migration in an urban multilayer transport network, including a flight networkA and a railway network B, to obtain the rankings of the hub cities in terms of transportation. Because we aim to obtain the rankings of the hub cities in terms of transportation, two or more train stations or airports in a same city are considered single stations or airports. The connections between train stations or airports in different cities are identified by the existence of railways or airlines. The population is migrating according to the distribution coefficients βA and βB, which satisfy 0≤βA+βB≤1. This relationship indicates that some of the population βA prefer travelling by aircraft, some of the population βB prefer travelling by train, and some of the population 1−βA−βB refuse to travel because they are choosing to remain in cities due to personal interests. The total population of city i at time t is defined as Sit. Therefore, we can obtain Sit+1 as(7)Sit+1=∑jAjiβASjtHjA+∑jBjiβBSjtHjB+1−βA−βBvi,where HjA=∑rAjr+δ0,∑rAjr, HjB=∑rBjr+δ0,∑rBjr, and δx,y is the Kronecker delta, which equals one if x=y and zero if x≠y and can be exploited to control city j without any transportation to other cities to keep the population of j in itself, and vi is the personalized interests of the population staying in city i. By initializing each Si0 by the value N−1, as determined by the number of cities and repeating the migration process, we can obtain the rankings of the hub cities in terms of transportation and derive the rankings in terms of node importance in a multilayer network.Based on our urban multiplex transport network, which has two layers, the generated Multiplex PageRank centrality measure is applied to a multiplex network ofM layers with N nodes in each layer as follows:(8)Sit+1=∑L∑jgjiLαLSitHjL+1−∑LαLvi,where HjL=∑rgjrL+δ0,∑rgjrL,(9)p•iSASB=−αASB∑jAjipjIA−αBSA∑jBjipjIBpiSASB1−wi+−αAIB∑jAjipjIA+μBpiSAIB1−wi+μA−αBSA∑jBjipjIBpiIASB1−wi+μA+μBpiIAIB1−wi,p•iSAIB=−αASB∑jAjipjIA+αBSA∑jBjipjIBpiSASB1−wi+−αAIB∑jAjipjIA−μBpiSAIB1−wi+μA+αBSA∑jBjipjIBpiIASB1−wi+μA−μBpiIAIB1−wi,p•iIASB=αASB∑jAjipjIA−αBSA∑jBjipjIBpiSASB1−wi+αAIB∑jAjipjIA+μBpiSAIB1−wi+−μA−αBSA∑jBjipjIBpiIASB1−wi+−μA+μBpiIAIB1−wi,p•iIAIB=αASB∑jAjipjIA+αBSA∑jBjipjIBpiSASB1−wi+αAIB∑jAjipjIA−μBpiSAIB1−wi+−μA+αBSA∑jBjipjIBpiIASB1−wi+−μA−μBpiIAIB1−wi.We start the multiplex target control strategy by immunizing the topϕ∈0,1 ranked individuals in the multilayer network or the multiplex random control strategy by immunizing an arbitrary number of ϕ individuals in the multilayer network before virus propagation begins. An immunized individual is aware of, takes preventive action against, and is immunized to both viruses in both layers and thus cannot be infected or is infectious throughout the propagation process. We define the immunization state of an individual i as wi, which is equal to 1 if i is immunized and to 0 otherwise.After immunization, we propagate both viruses in the multilayer network, and this process is initiated by infecting several random susceptible nonimmunized individuals in each layer. We then obtain expansion of the Kolmogorov forward differential equations of the stochastic process of the continuous time Markov Chains for the multiple-virus propagation model, which is constrained by the multiplex control strategy for each nodei as represented in (9).


## 4. Simulation

Considering the interaction relationships between the two viruses, several actual cases are investigated through computer experiments and simulations. By adoptingρϕ as the infection probability at an immunization probability ϕ at the steady state and ρ0 as the infection probability without control at the steady state, we can obtain the immunized probability versus the steady infection probability for the following. The common parameters are set to N=10000, αASB=0.2, αBSA=0.2, μA=0.8, and μB=0.8, and ϕ ranges from 0 to 1 at 1% intervals. Moreover, we adopt m0=20 and m=8 to construct network A, and m0=30 and m=12 are used to construct network B. Moreover, IA means the nodes infected with the virus in network A, IB means the nodes infected with the virus in network B, and IA∪IB means the union of IA and IB, i.e., all infected nodes without considering the types of viruses. The control results are averaged by simulating 100 time steps 1000 times in 50 multilayer networks.Because an individual who is affected with whooping cough or measles will infect another individual [50], we adopt cAIB=0 for whopping cough or cBIA=0 for measles. Because there are more propagation routes for measles than whooping cough, we adopt A to denote the propagation routes of whooping cough and B to denote the propagation routes of measles. By deciding to mainly immunize against measles and to immunize against whooping cough indirectly, we adopt the distribution coefficients βA=0.35 and βB=0.5 because measles is more dangerous than whooping cough. The simulation results are shown in Figure 2. As shown in Figure 2(a), the immunization threshold of the multiplex random control of whooping cough is in a high value about 0.8, and the decreasing rate of the curve is smaller and smaller, whereas the immunization threshold of the multiplex target control of whooping cough is constrained to a very low value, which is affected by the control of measles with our novel Multiplex PageRank method and the limitations caused by the existence of fewer propagation routes for whooping cough compared with those of multiplex random control. In addition, as shown in Figure 2(b), the immunization threshold of the multiplex random control of whooping cough is about equal to that of measles, but the decreasing rate of the curve is constant, whereas a turning point can be found in the curve of the multiplex target control of measles before the immunization threshold is reached. This finding indicates that measles benefits from a greater number of propagation routes, and thus, the infection probability of our Multiplex PageRank target control decreases at a lower rate compared with that observed with multiplex random control. The turning point occurs at the value of approximately 0.12 when the population has been completely immunized against whooping cough, and thus, the control of the population to whooping cough results in a decrease in the immunization rate of measles. Figure 2(c) presents the union of infections of whooping cough and measles in the multilayer network. The decreasing infection probability rate remains at a steady rate. As shown by the results, the Multiplex PageRank target control strategy immunizes fewer nodes and has a greater effect on both viruses simultaneously in the multilayer network.Figure 2
Relative sizeρϕ/ρ0 of infected clusters versus the immunization probability ϕ for multiple targets or random control in a double layer network that involves whooping cough and measles: (a) whooping cough-infected clusters, (b) measles-infected clusters, and (c) clusters infected with both whooping cough and measles. The arrow denotes the immunization threshold ϕc of our novel Multiplex PageRank target control.
(a)(b)(c)A computer that is affected by countermeasures, which are also classified as infectious malware in China, can avoid infection by some computer viruses [51]. In addition, a computer affected by these computer viruses is not affected by the propagation of countermeasures; thus, we adopt cAIB=1 for countermeasures and cBIA=0 for computer viruses. Because the propagation routes of computer viruses are increased compared with those of countermeasures, we adopt A to denote the propagation routes of countermeasures and B to denote the propagation routes of computer viruses. By deciding to mainly immunize against computer viruses and to immunize against countermeasures indirectly, we adopt the distribution coefficients βA=0.35 and βB=0.5 because computer viruses are more dangerous than countermeasures. The simulation results are shown in Figure 3. The results shown in Figure 3(a) demonstrate that the immunization threshold of the multiplex target control of the countermeasures is constrained to a lower value than that of multiplex random control. Even if we mainly concentrate on immunizing against computer viruses, the propagation of countermeasures is constrained by the decreased propagation routes. However, the immunization threshold of the multiplex random control of the countermeasures is at a high value about 0.8, and the decreasing rate of the curve is constant. In addition, Figure 3(b) shows the existence of several turning points in the curve of multiplex target control against computer viruses prior to attaining the immunization threshold. However, the immunization threshold of the multiplex random control of computer viruses is in a high value about 0.8, and the decreasing rate of the curve is constant. This finding indicates that the computer viruses are first immunized by the countermeasures. The first turning point is found at a value of approximately 0.2 when the countermeasures propagate within the top-ranked nodes are mostly immunized, which results in an increase in the probability of infection with computer viruses. However, the infection probability soon begins to decrease at a value of approximately 0.27 after the population is fully immunized against countermeasures, which also immunizes against computer viruses. Figure 3(c) presents the union of the infections by countermeasures and computer viruses in the multilayer network. The decreasing rate of the infection probability remains unchanged, and the results show that our Multiplex PageRank target control strategy involves the control of fewer nodes to achieve greater effects on both viruses simultaneously in the multilayer network.2Figure 3
Relative sizeρϕ/ρ0 of infected clusters versus the immunization probability ϕ for multiplex target or random control in a double-layer network that includes countermeasures and computer viruses: (a) countermeasures-infected clusters, (b) computer virus-infected clusters, and (c) clusters infected with both countermeasures and computer viruses. The arrow denotes the immunization threshold ϕc of our novel Multiplex PageRank target control.
(a)(b)(c)Assuming that an individual who is affected with hepatitis C or B virus will show inhibition against the other virus and that the inhibition of hepatitis B virus by hepatitis C virus is stronger than that of hepatitis C virus by hepatitis B virus [52], we adopt cAIB=0.5 for hepatitis C virus and cBIA=0.3 for hepatitis B virus. Because there are more propagation routes for hepatitis B virus than for hepatitis C virus, we adopt A to denote the propagation routes of hepatitis C virus and B to denote the propagation routes of hepatitis C virus. Based on the decision to mainly immunize against hepatitis B virus and sometimes immunize against hepatitis C virus, we adopt the distribution coefficients βA=0.35 and βB=0.5 because hepatitis B virus is more dangerous than hepatitis C virus. The simulation results are shown in Figure 4. As shown in Figure 4(a), the immunization threshold of the multiplex random control of hepatitis C virus is at a high value about 0.8, and the decreasing rate of the curve is constant, whereas the immunization threshold of the multiplex target control of the hepatitis C virus is constrained to a lower value based on control with our novel Multiplex PageRank, the suppression of hepatitis B virus, and the limitation of the propagation routes of hepatitis C virus compared with multiplex random control. Moreover, the results in Figure 4(b) demonstrate a change in the decreasing rate of the infection probability obtained with multiplex target control of hepatitis B virus before attaining the immunization threshold. However, the immunization threshold of the multiplex random control of hepatitis B virus is in a high value about 0.8, and the decreasing rate of the curve is constant. These results indicate that hepatitis B virus is first suppressed by hepatitis C virus. The change in the decreasing rate of the infection is probability at a value of approximately 0.24 occurred when the population is completely immunized against hepatitis C virus, which occasioning results in a decline in the decreasing rate of the infection probability of hepatitis B virus. However, the decreasing rate of the infection probability soon increases with an increase in the immunization probability. Figure 4(c) presents the union of infection with hepatitis C and B viruses in the multilayer network. The decreasing rate of the infection probability is maintained at a constant rate. Our Multiplex PageRank target control strategy indicates that the control of fewer nodes can achieve a greater effect on both viruses simultaneously in the multilayer network.Figure 4
Relative sizeρϕ/ρ0 of infected clusters versus the immunization probability ϕ for multiplex target or random control in a double-layer network that includes hepatitis B virus and hepatitis C virus: (a) hepatitis C virus-infected clusters, (b) hepatitis B virus-infected clusters, and (c) clusters infected with both hepatitis C and B viruses. The arrow denotes the immunization threshold ϕc of our novel Multiplex PageRank target control.
(a)(b)(c)Because tuberculosis does not affect HIV but an individual who is infected with HIV will be more susceptible to infection by tuberculosis [53], we adopt cAIB=1.5 for tuberculosis and cBIA=1 for HIV. Due to the higher number of propagation routes for HIV compared with tuberculosis, we adopt A to denote the propagation routes of tuberculosis and B to denote the propagation routes of HIV. By deciding to mainly immunize against HIV and only slightly immunize against tuberculosis, we adopt the distribution coefficients βA=0.35 and βB=0.5 because HIV is more dangerous than tuberculosis. The simulation results are shown in Figure 5. The results in Figure 5(a) reveal a small change in the decreasing rate of the infection probability with multiplex target control of tuberculosis before attaining the immunization threshold. However, the immunization threshold of the multiplex random control of tuberculosis is at a high value about 0.8, and the decreasing rate of the curve is constant. This finding indicates that tuberculosis is first enhanced by HIV. The change in the decreasing rate of the infection probability occurs at the value of approximately 0.22 when HIV is propagated in the top ranking nodes, which are mostly immunized, and this finding results in a small increase in the decreasing rate of the infection probability of tuberculosis. However, even if tuberculosis infection has been enhanced, it is constrained by its fewer propagation routes; thus, the population is completely immunized against tuberculosis before it is immunized against HIV. Moreover, as shown in Figure 5(b), the immunization threshold of the multiplex random control of HIV is at a high value about 0.8, and the decreasing rate of the curve is constant. We also find a small change in the rate of the infection probability obtained through multiplex target control for HIV before the immunization threshold is attained. The change in the decreasing rate of the infection is probability obtained at a value of approximately 0.29 when the population is completely immunized against tuberculosis, which results in a greater decreasing rate for control against HIV. Figure 5(c) shows the union of the infection probability of tuberculosis and HIV in the multilayer network. A decreasing rate of the infection probability increases at a value of approximately 0.22 due to the increase in the decreasing rate of tuberculosis and decreases at a value of approximately 0.29 because only the HIV rate decreases. We find that our Multiplex PageRank target control strategy is capable of immunizing fewer nodes to achieve a greater effect on both viruses simultaneously in the multilayer network.Figure 5
Relative sizeρϕ/ρ0 of infected clusters versus the immunization probability ϕ for multiplex target or random control in a double-layer network that includes tuberculosis and HIV: (a) clusters infected with tuberculosis, (b) clusters infected with HIV, and (c) clusters infected with both tuberculosis ϕc of our novel Multiplex PageRank target control.
(a)(b)(c)Because an individual who is infected with malaria or HIV will be more susceptible to the other virus and the degree of enhancement of HIV obtained with malaria infection is stronger than that of malaria with HIV [54], we adopt cAIB=1.8 for malaria and cBIA=1.5 for HIV. As there are more propagation routes for HIV than malaria, we adopt A to denote the propagation routes of malaria and B to denote the propagation routes of HIV. Based on the decision to mainly immunize against HIV and only sometimes immunize against malaria, we adopt the distribution coefficients βA=0.35 and βB=0.5 because HIV is more dangerous than malaria. The simulation results are shown in Figure 6. As shown in Figure 6(a), the immunization threshold of the multiplex random control of malaria is at a high value about 0.8, and the decreasing rate of the curve is constant. We also find a change in the decreasing rate of the infection probability of multiplex target control of malaria before the immunization threshold is reached, which indicates that malaria is first enhanced by HIV. The change in the decreasing rate of the infection probability occurs at a value of approximately 0.25 when HIV has propagated to the top-ranked nodes, which are immunized, and this propagation results in an increase in the decreasing rate of the infection probability of malaria. However, even though the enhancement of HIV obtained with malaria is stronger than that of malaria achieved with HIV, malaria is constrained by its fewer propagation routes, and thus, the population is completely immunized against malaria before it is immunized against HIV. Moreover, as shown in Figure 6(b), the immunization threshold of the multiplex random control of HIV is at a high value about 0.8, and the decreasing rate of the curve is constant. We also find some changes in the decreasing rate of the infection probability of multiplex target control of HIV before the immunization threshold is achieved, which indicates that malaria is first enhanced by HIV. The first change in the decreasing rate of the infection probability occurs at a value of approximately 0.25 when malaria has propagated to the top-ranked nodes, which are mostly immunized, and this results in an increase in the decreasing rate of the HIV infection probability. The second change observed at a value of approximately 0.30 refers to an increase in the decreasing rate due to complete control against malaria, which leads HIV without any enhancement. Figure 6(c) presents the union of the infection probabilities of malaria and HIV in the multilayer network. The decreasing rate of the infection probability increases at a value of approximately 0.25 due to increases in both malaria and HIV and declines at a value of approximately 0.30 because only HIV infection remained. The results show that our Multiplex PageRank target control strategy immunizes fewer nodes to achieve a greater effect on both viruses simultaneously in the multilayer network.Figure 6
Relative sizeρϕ/ρ0 of infected clusters versus the immunization probability ϕ for multiplex target or random control in a double-layer network including malaria and HIV: (a) malaria-infected clusters, (b) HIV-infected clusters, and (c) clusters infected with both malaria and HIV. The arrow denotes the immunization threshold ϕc of our novel Multiplex PageRank target control.
(a)(b)(c)Figures2–6 show the relative sizes of the infected clusters versus the immunization probability in several actual cases. It is noted that the multiplex random control strategy in these double-layer networks with different interaction coefficients has little difference. The results demonstrate the effectiveness of our multiplex target control strategy in multilayer networks. Furthermore, the figures present the immunization threshold ϕc of our novel Multiplex PageRank target control versus the interaction coefficients cAIB and cBIA. By setting αASB=0.2 and αBSA=0.2, both the interaction coefficients are limited from 0 to 5 because αAIB≤1 and αBIA≤1. Using 0.1 intervals for the interaction coefficients, we divide both of them into 51 partitions. The immunization probability ϕ ranges from 0 to 1 at a 0.01 interval, which results in the division of the probability into 101 compartments. Adopting 100 time steps 1000 times for 50 networks with a size of 10,000, the computer simulations take a long time. We exploit parallel computing, including multiple CPUs and multiple GPUs, to solve this problem, and the results are shown in Figure 7. As shown in Figure 7(a), the immunization threshold of the virus in A is majorly impacted by the interaction coefficient cAIB. The results shown in Figure 7(b) demonstrate that the immunization threshold of the virus in B is impacted by the ratio of cBIA to cAIB. Figure 7(c) reveals that the immunization threshold of the total infectious is decided by the virus in B because it has more propagation routes.Figure 7
Novel Multiplex PageRank target immunization thresholdsϕc versus interaction coefficients of multiple-virus propagation. (a) Target immunization threshold of IA. (b) Target immunization threshold of IB. (c) Target immunization threshold of IA∪IB.
(a)(b)(c)


## 5. Conclusion

This paper has presented a novel Multiplex PageRank target control strategy on multiple-virus interaction and propagation in multilayer networks compared with the multiplex random control. Using the proposed strategy, we have simulated several computer experiments that allow us to obtain the relative size of the steady probability versus the immunization probability and the critical immunization threshold for several actual cases. Furthermore, we have shown the critical immunization threshold of our novel Multiplex PageRank target control versus different interaction coefficients of our multiple-virus propagation, which is useful for further practical applications.



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*Source: 9014353-2020-07-10.xml*