Generalized Epidemic MeanField Model for Spreading Processes over MultiLayer Complex Networks


 Winfred Garrison
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1 1 Generalized Epidemic MeanField Model for preading Processes over MultiLayer Complex Networks Faryad Darabi ahneh, Caterina coglio, Piet Van Mieghem Abstract Meanfield deterministic epidemic models have been successful to elucidate several important dynamical properties of stochastic epidemic spreading processes over complex networks. n particular, individualbased epidemic models isolate the impact of the network topology on the spreading dynamics. n this paper, the existing models are generalized to develop a class of spreading models which includes the spreading process in multilayer complex networks. We provide a detailed description of the stochastic process at the agentlevel where the agents interact through different layers, each represented with a graph. The set of differential equations that describes the time evolution of the state occupancy probabilities has an exponentially growing state space size in terms of the number of the agents. Using a meanfield type approximation, a set of nonlinear differential equations is developed which has linearly growing state space size. We find that the latter system, referred to as the generalized epidemic meanfield (GEMF) model, has a simple structure. t is characterized by the elements of the adjacency matrices of the network layers and the Laplacian matrices of the transition rate graphs. everal examples of epidemic models are presented, including spreading of virus and information in computer networks, and spreading of multiple pathogens in a host population. ndex Terms Epidemic spreading, Markov process, mean field theory, complex networks. NTRODUCTON EPDEMC are critical phenomena in the current world, not only from a biological viewpoint, as referred to infectious diseases, but also from a technological viewpoint, as referred to malware propagation. Epidemics can produce huge damage, and the development of accurate and effective models for epidemics is imperative. Epidemic modeling has a long history in biological systems. Recently, these biologically inspired epidemic models have attracted a substantial attention in modeling propagation phenomena in communication networks [1] [3]. Epidemic spreading, like many other processes (see, e.g., [4] [6]) on complex networks, can be modeled as a network of coupled stochastic agents. The populationbased or networkbased epidemic models (cf., [7] [9]) lead to individualbased epidemic models [10] [14]. A common approach is to consider Markovian interacting agents (i.e., dynamics of the agents satisfy the Markov property [15], [16]) This work was supported by National Agricultural Biosecurity Center at Kansas tate University. F. D. ahneh and C. coglio are with the Electrical & Computer Engineering Department, Kansas tate University, K, 66506, UA ( P. Van Mieghem is with the Faculty of Electrical Engineering, Mathematics and Computer cience, Delft University of Technology, 2600 GA Delft, The Netherlands ( while the interaction is represented by a generic graph. This approach avoids the use of random network models (e.g., ErdösRéyni [17], BarábasiAlbert [18], etc.) which may fail to properly represent engineered networks [19]. Being stochastic in nature, the study of the dynamical behavior of epidemic spreading processes on graphs is very challenging, even for simple scenarios. For example, the system governing the state occupancy probabilities has an exponentially growing space size in terms of the number of the agents. Therefore, the problem becomes soon intractable as the number of the agents increases. Through a meanfield approach, the size of the governing equations reduces dramatically at the expense of exactness. Meanfield epidemic models have been very successful in finding several interesting results for individualbased epidemic spreading processes. For example, it has been shown that the epidemic threshold in model is actually the inverse of the spectral radius of the adjacency matrix of the contact graph [10], [12]. n the existing individualbased epidemic models, the interaction is driven by a single graph. However, in order to study epidemics in communication networks and cyberphysical systems, a more elaborate description of the interaction is required. everal researchers from computer science, communication, networking, and control communities are working on describing this complex interaction by using multiple interconnected networks [20], [21]. The study of the spreading of epidemics in interconnected networks is a major challenge of complex networks [22] [25]. n this paper, we provide a novel and generalized formulation of the epidemic spreading problem and a modeling solution. We consider a spreading process among a group of agents which can be in different compartments, and where the agents interact through a multilayer network, explained in detail in ection V. We follow a rigorous methodology towards the development of a general epidemic spreading model. The modeling starts with a simple agentlevel description of the underlying stochastic process. The exact Markov equations, which describe the time evolution of the state occupancy probabilities, are linear differential equations however with exponentially growing state space size in terms of the number of the agents. Through a meanfield type approximation, the state space dramatically reduces. The approximate system is a set of nonlinear ordinary differential equations which we call the generalized epidemic meanfield (GEMF) model. We will apply GEMF to interesting problems, such as: (a) the spread of infection among a population where
2 2 the infection spreads through a contact network while agents respond to the spreading by learn about the existence of the infection through information dissemination networks, and (b) the bispreading of two types of interacting viruses in a host population demanding different transmission routes for the infection propagation. The contribution of this paper is twofold. First, a general epidemiclike spreading Markov model is proposed with multicompartment agents dynamics and multilayer interaction network. econd, we propose GEMF as a generalized epidemic meanfield model suitable for a large class of individualbased spreading scenarios. GEMF is rigorously derived from an agentlevel description of the spreading process and is elegantly expressed (see equation (27)). t is expressed in terms of the adjacency matrix of each network layer and of the Laplacian of the transition rate diagrams. n GEMF, there is no approximation on the network topology; the only approximation is a meanfield type approximation on the dynamics of the agents. The impact of the this approximation is a function of the network topologies and epidemic parameters. For completeness of our developments, we have also explicitly derived the exact Markov equations in the Appendix. The rest of the paper is organized as follows. Motivations for developing GEMF are provided in ection through examples. The basic definitions for the spreading problem are the topic of ection. n ection V, the agentlevel Markov description of the spreading process is provided. GEMF is developed in ection V. The paper is concluded in ection V by applying GEMF to some existing and novel spreading scenarios.. MOTVATNG EXAMPLE n this section, first, we review some of the existing individualbased epidemic models. At the end, we discuss what generalizations are important to develop a general class of epidemic models. A. ndividualbased Model n the individualbased model (cf., [11] [13]) each agent can be either susceptible or infected. Hence, the number of compartments, denoted by, in the model is =2. A susceptible agent can become infected if it is surrounded by infected agents. The infection transition rate for an agent with one infected neighbor is. The infection processes are independent of each other. Therefore, for an agent with more than one infected agent in its neighborhood, the infection rate is times the number of the infected agents. The neighborhood of each agent is determined by a graph G, which represents the contact network. n addition to the infection process, there exists also a curing process. An infected agent becomes susceptible with a curing rate. A schematics for the Nntertwined model is shown in Fig. 1. B. A preading Model The A model was developed in [26] to incorporate for agents reaction to the spread of the virus. n the A spreading model, each agent can be either susceptible, infected, or Neighbors of Node i Node i Node i Y it Fig. 1. chematics of a contact network along with the agentlevel stochastic transition diagram for agent according to the epidemic spreading model (explained in ection A). The parameters and denote the infection rate and curing rate, respectively. () is the number of the neighbors of agent which are infected at time. Neighbors of Node i Node i Node i Y it A Y it ay it Fig. 2. imilarly as in Fig. 1, the A epidemic is sketched (see B) on a contact network. n addition to the infection rate and the curing rate, parameters and denote the alerted infection rate and the alerting rate, respectively. () is the number of the neighbors of agent which are infected at time. alert. Hence, the number of compartments in A model is =3. The curing process is the same as the model, and is characterized by curing rate. The infection process of a susceptible agent is also similar to that of the model which is determined by infection rate and the contact graph G. However, in the A model, a susceptible agent can become alert if it senses infected agents in its neighborhood. n the A model, the alerting transition rate is times the number of infected agents. An alert agent can also become infected. This infection process is similar to the infection process of a susceptible agent. However, the infection rate for alert agents is lower due to the adoption of increased security for computer networks or better hygiene for human population. The alert infection rate is denoted by.nfig.2,aschematics for the A spreading model is represented. C. Generalization of Epidemic Models The and A models are good examples of how a simple compartmental model at the node level along with a network topology can lead to very rich and complex dynamics. While following the structure and underlying assumptions of existing epidemic models, we propose to develop a generalized
3 3 individualbased spreading model where: (a) the node model has multiple compartments, and (b) the network topology has multiple layers. Both generalizations are important. For example, many epidemic models can be created by adding new compartments to the basic or R epidemic models. Also, for applications in cybersocial and cyberphysical systems, more network layers needs to be taken into account (see Fig 3). For example, in the A model, the agents have the chance to observe the infection status of their neighbors in the contact network. However, a more realistic scenario is that agents learn about the infection status of other agents through an infection information dissemination network, represented by G, which can be very different from the contact network. We can also take into account an alert information dissemination network among the agents, represented by G. Through this network, agents can become alert if some of their neighbors (determined by G ) are alert. n this case, the network topology has three layers. n ection VC, we develop an A model with information dissemination. Multilayer epidemic modeling can also have applications in biological networks. Consider the scenario where two pathogens are spreading among the host population. nfection by one pathogen can effectively influence the infection process by the other pathogen. ince the infection transmission routes may be different, the contact networks for each virus can potentially be separate from each other. n ection VD, we develop an individualbased bispreading model with separate contact network for each pathogen. The GEMF class of models developed in this paper allows not only to incorporate an arbitrary number of compartments, but also to account for multiple network layers.. DEFNTON The network consists of interacting agents. Each agent can be in one of states (compartments). The stochastic transitions of an agent not only depend on its own state but also on the states of the other agents. The group of agents is assumed to be jointly Markovian: the collective system is a Markov process. The state of the collective system, which will be referred to as network state, is actually the joint state of all the agents states. Assuming that all the agents can take values among compartments, the size of the network state space is. n the following, the agent state and network state are precisely defined. A. Agent tate and Network Markov tate One of the generalizations of GEMF concerns the compartment set. Each agent can be in one compartment in the set = { 1 2 }. For example in the model for epidemic spread, =2and = { usceptible nfected } From now on, without loss of generality, each compartment is labeled with a number from 1 to. Thestate () of agent at time is () = if the agent is in compartment at time. Here, is the th standard unit vector in the R Euclidean space, i.e., all entries of are zero except for the th entry which is equal to one., [00 {z} 1 00] R (1) entry The definition () = illustrates that each entry of () is a Bernoulli random variable. Therefore, the expected value of () is in fact the compartment occupancy probability vector, i.e., [ ]=[Pr[ = 1 ] Pr[ = ]] (2) The above property is very important in the future developments, particularly in (14), (18), and (2527). There are other possibilities for defining the node state (). For example, one might define = 1 if node is in compartment. Bythisdefinition, takesvaluesfrom0 to 1. Thisdefinition is particularly useful if =2.nthis case, is a binary random variable. Van Mieghem et. al. [12] used this definition for the Nntertwined model. The dynamics of an individual agent depend on the states of the other agents. Therefore, the state of a single agent is not enough to describe the evolution of the agent state. nstead, the joint state of all the agents follows a Markov process. The network state at time, denoted by (), is the joint state of all the agents defined as [27] () = O () = 1 () 2 () () (3) =1 where is the Kronecker product. By (3), () is an 1 random vector with exactly one element equal to one and the rest equal to zero. The expected value of () is the joint probability distribution function of the network state. For example, for the model, the first element of the expectation of () is the probability that all the agents are simultaneously susceptible. One could define the network state as a 1 vector, [ 1 2 ]. However, in this case, the expectation of () will only provide the marginal probability distribution of the node states. As shown in ection VA, the information about marginal probabilities at a given time is not enough to describe the evolution of the marginal probabilities and the joint probability distribution is required. Hence, we adopt the definition (3) for the network state. B. MultiLayer Network Topology The other generalization in GEMF concerns the topology. n most epidemic models, the interaction among the agents is represented by the contact network. However, as discussed in ection C, the types of the interaction can be different in a complex network. For our modeling purpose, we represent the topology by layers of graphs G (N ),wheren is the set of nodes, denoting the agents, and is the set of edges which represents the interaction between each pair of individuals in the th layer. These graphs have the same nodes but the edges can be different. The adjacency matrix corresponding to the graph G is denoted by A =[ ]. f agent can influence agent in the layer, =1otherwise =0.n ection VA2, we define precisely what influence implies in
4 4 Network Layers Fig. 3. Network layers describe the different types of interactions among agents in GEMF. The vertical dotted lines emphasize that all graphs have same nodes but the edges are different. our model. A representation of the network layering structure is depicted in Fig. 3. V. AGENTLEVEL DECRPTON OF THE MARKOV PREADNG PROCE The network state () follows a continuoustime Markov process. Knowing that the network is in state () at time, what is the network state ( + ) at time +? n a network of interacting agents, this question can be very complicated. nstead, a more natural approach is to describe the agent state (+ ) given the network state () at time. The spreading process is fully described if the probability to record a transition from compartment to compartment for agent, conditioned on the network state (), is known for all possible values of,, and. n this section, we focus on deducing an expression for Pr[ ( + ) = () = ()], which will be used later on to develop the GEMF model. The challenge in deducing an expression for Pr[ (+ ) = () = ()] is that there are too many possibilities for the dependence of the transition of the individual agent on the network state. Here are a few examples: the transition happens completely independently from the states of other agents, the transition happens if the number of other agents in compartment are more than the number of agents in compartment, the transition happens if agents 1 and 2 are both in compartment and the rate of the transition is the logarithm of the number of agents in compartment. All of these examples are legitimate so far. However, we need to specify the transition possibilities properly to develop a coherent and consistent epidemic spreading model. A. Epidemic preading Process Modeling The Nntertwined model [12] gives very good insights in how to properly define the transition possibilities to describe an epidemic spreading process. n the model, there are two transitions. The curing process, which is basically the transition from infected state to susceptible, occurs independently from the states of other agents. nstead, the infection process, which refers to transition from susceptible state to infected state, happens through a different mechanism. A susceptible agent is in contact with some other agents. During the time interval ( + ], the susceptible agent receives the infection from its infected neighbor with some probability. The process of receiving the infection from one infected neighbor is independent from the process of receiving the infection from another neighbor. All the infected neighbors compete to infect the susceptible agent. The susceptible agent becomes infected when one of the neighbors succeeds to transmit the infection. The transitions in the epidemic model are very similar to the transitions in most existing epidemic models. As a result, we impose a similar structure of independent competing processes to the generalized spreading model. Assumption 1: A transition for agent is the result of several statistically independent competing processes: the process for agent that happens independently of the states of other agents, and the process for agent because of interaction with agent 6=, for each {1 }\{}. According to Assumption (1), the interaction of agent with agent 6= is statistically independent of its interaction with agent { }. Define the auxiliary counting process () () corresponding to the interaction of agent with agent. For convenience of notations, let () () correspond to the transition for agent occurring independently from the states of other agents. According to Assumption 1, conditioned on the network state, these counting processes are statistically independent. The transition occurs in the time interval ( + ] if any of these counting processes record an event. Therefore, Pr[ ( + ) = () = ()] can be written as Pr[ ( + ) = () = ()] = Pr[ {1} s.t. () (+ ) () () 6= 0 ()] (4) Each of the counting processes () () is a Poisson process with the rate () (), to be determined. Therefore, Pr[ () (+ ) () () 6= 0 ()] = () () +( ) (5) The sum of independent Poisson processes is also a Poisson process with aggregate rate equal to the sum of the individual rates (see Th in [16]). Therefore, Pr[ ( + ) = () = ()] = X () ()+( ) (6) =1 The remaining part of this section is dedicated to properly find () (). 1) Nodal Transition: As discussed earlier, a nodal transition occurs independently from the states of other agents. For example, in the model, the curing process is the result of a nodal transition. The curing process happens with rate
5 5 regardless of the infection status of other agents. n general, for the nodal transition, we can consider a rate 1 0, which is actually the rate for the counting process () (), i.e., () () = (7) 2) EdgeBased Transition: The edgebased transitions are different from nodal transitions because they depend on the states of other agents. For example, in the model, the infection process is an edgebased transition. The susceptible agent becomes infected with rate if it is in contact with infected agent. n the model, the existence of contact among agents is determined by the contact network. However, we extend the concept of contact network to multilayer networks. n our formulation, the interactions among agents are not limited to contacts. For example, agents might also exchange information. Corresponding to each layer, there is one influencer compartment. t is possible that the influencer compartment of two distinct layers is the same. For example, recall the extended A model with three network layers proposed in ection C. For the contact network and the infection information dissemination network, infected is the influencer compartment. While for the alert information dissemination network, alert istheinfluencer compartment. The transition from compartment to is characterized by the transition rate 0 for layer. n general, the edgebased transition from to 6= through edge ( ) is described by () () = X 1 {()= } (8) =1 where 1 { } is the indicator function. A transition occurs if a neighbor, i.e., =1, is in. Assigning only one influencer compartment to a graph layer allows the elegant development of the subsequent analysis. However, a more general possibility is that a transition occurs if a neighbor, i.e., =1, is in a subset of the compartments, say 1 or 2. This case can be treated within the same structure of GEMF. n this case, we can count the network layer twice. Therefore, we assume that the first time the graph has the influencer compartment 1 and the second time the graph has the influencer compartment 2. An example of this case is shown in ection VD. B. Transition Rate Graphs n order to make the subsequent developments systematic, we propose to use transition rate graphs defined as follows. A nodal transition rate graph is graph with nodes where each node represents a compartment. A directed link ( ) from to represents the nodal transition weighted by the positive transition rate 0. Corresponding to the nodal transition rate graph, the adjacency matrices of the nodal transition rates is defined as, [ ] (9) 1 Here is a nonnegative scalar which represents nodal transitions. t should not be confused with the Kronecker delta symbol. a) b) Fig. 4. Transition rate graphs in the model: a) nodal transition rate graph; nodes represent the two compartments susceptible and infected, directed link from to represents curing process (a nodal transition) weighted by the curing rate 0, and b) edgebased transition graph of the contact network layer G ; directed link from to represents the infection process (edgebased transition) weighted by the infection rate 0. For the contact network, the influencer compartment is 1 =2, i.e., infected. a) A b) Fig. 5. Transition rate graphs in the A model: a) nodal transition rate graph; nodes represent the three compartments susceptible, infected, and alert, directed link from to represents curing process (a nodal transition) weighted by the curing rate 0, and b) edgebased transition graph of the contact network layer G ; directed link from to represents the infection process (edgebased transition) weighted by the infection rate 0, directed link from to represents the alerting process (edgebased transition) weighted by the alerting rate 0, directed link from to represents the alerted infection process (edgebased transition) weighted by the alerted infection rate 0. For the contact network, the influencer compartment is 1 =2,i.e., infected. An edgebased transition rate graph, corresponding to the network layer G is a graph with nodes where each node represents a compartment. A directed link ( ) from to represents the edgebased transition weighted by the positive transition rate rate 0 in network layer with influencer compartment. Corresponding to the edgebased transition rate graph, the adjacency matrices of the edgebased transition rates is defined as, [ ] (10) For example, in both and A model described in ection A and ection B, only the curing process is a nodal transition. The nodal transition rate graphs for the and A models are shown in Fig. 4 and Fig. 5, respectively. The schematics of the nodal transition rate graph in general is drawninlefthandsideoffig.6.nbothandamodels described in ection A and ection B, respectively, the contact network is the only network layer. Therefore, these two models have one edgebased transition rate graph. The edgebased transition rate graphs for the and A models are shown in Fig. 4 and Fig. 5, respectively. The schematics of the transition rate graphs in general is drawn in Fig. 6. n ection V (see (27), below), the Laplacian (see, [28]) matrices associated to the transition rate graphs appears in the expression of GEM. A a
6 6 a) 1 b) 26 1 According to (11), the probability of remaining in the previous state is Pr[ ( + ) = () = ()] = l,51 l,36 l,32 1 X 6= Pr[ ( + ) = () = ()] (13) Combining (2), (11), and (13) leads to 4 4 l,43 Fig. 6. Transition rate graphs in GEMF: a) nodal transition rate graph; nodes represent compartments, directed link ( ) represent nodal transition weighted by the transition rate 0, and b) edgebased transition graph of network layer G ; directed link ( ) represent the edgebased transition weighted by the transition rate 0 in network layer. The inducer compartment of layer is. C. AgentLevel Markov Description of the preading Process n ection VA, we developed the expressions for the nodal transition and edgebased transitions. ubstituting (7) and (8) into (6) yields Pr[ ( + ) = () = ()] = + X ()+( ) (11) =1 for = {1 } and 6=, where (), X 1 { ()= } (12) =1 is the number of neighbors of agent in G that are in the corresponding influencer compartment. Equation (11) provides an agentlevel description of the Markov process. t can directly be used for Monte Carlo numerical simulation of the spreading process. V. GEMF: GENERALZED EPDEMC MEANFELD MODEL The objective of this section is to derive the time evolution of the state occupancy probabilities. A. Exact Markov Differential Equation n the previous section, the spreading model was described and the corresponding Markov process was derived in (11). The evolution of the state occupancy probabilities, associated to a Markov process, follows a set of differential equations known as the Kolmogorov differential equations. The derivation of the Kolmogorov differential equation of a Markov process is fairly standard (see, [16], [29]) when the transition rates between the states of the Markov process are known. However, the challenge here is that the network states are the actual Markov states and instead of the transition rates between the network states, we have the agentlevel description of the transitions in (11). n this section, we derive the differential equations directly from (11). [ ( + ) () = ()] = 1 + P =1 1 (). ( 1) + P =1 ( 1) () P =1 () (+1) + P + =1 + ( ) (+1) (). + P =1 () (14) where, X 6=, X 6= (15) and ( ) is a function of higher order terms of satisfying the condition ( ) =0 (16) where is the all ones vector with appropriate dimensions. Define the generalized transition matrices R and R with the elements ( ),, (17) Accordingtodefinitions (15) and (17), the matrices and are actually the Laplacian matrices of transition rate graphs, definedinectionvb. Using (14) and the definition (17), [ ( + ) ()] is obtained as [ ( + ) ()] = () X () () =1 + ()+( ) (18) where () is defined in (12). Computing the expected value of each side of (18), we get [ ( + )] = [ ()] X [ () ()] =1 +[ ()] + ( ) (19) where ( ) =[( )] and the formula for iterative expectation (see [30]) rule [[ ]] = [] hasbeenusedto
7 7 find [ ( + )]. Movingthe[ ()] term in (19) to the left side and dividing both sides by yields [ ( + )] [ ()] = X [ ()] [ () ()] + =1 Letting 0 in (20), we obtain [ ()] = [ ()] 1 ( ) (20) X [ () ()] (21) Using (12), the term [ () ()] in (21) can be written as X [ () ()] = [( ) ()] (22) =1 The term [( ) ()] is actually embedded in [ () ()]. Therefore, the evolution of [ ()] depends on [ () ()] term, which is the joint state of pairs of nodes. This means that the marginal information about the compartmental occupancy probabilities is not enough to fully describe the time evolutions of the probabilities. f we continue to derive the evolution law for [ () ()], it turns out that the time derivative of [ () ()] depends on terms of the form [ () () ()], which are the joint states of triplets. This dependency of the evolution of expectation of node groups on expectation of ( +1)node groups continues until reaches to =. As a result, any system describing the evolution of the expected value of the joint state of any group of nodes is not a closed system. When =, the expectation of the joint state of all nodes [ 1 () ()], which according to definition (3) is actually the expectation of the network state, satisfies a differential equation of the form [] = Q [] (23) where Q R is the infinitesimal generator (see [16], [29]) of the underlying Markov process. The Kolmogorov differential equation (23) is, which we refer to as the exact Markov model, is derived explicitly in the Appendix. The exact Markov equation (23) fully describes the system. However, the above differential equation has states. Therefore for large values of, it is neither analytically nor computationally tractable. n the following section, it is shown that through a meanfield type approximation, a differential equation with states can be derived. B. GEMF: Generalized Epidemic MeanField Model One way to reduce the state space size is to use closure approximation techniques. As explained in the previous section, expectations of order depend on expectation of order +1. The goal of closure techniques is to approximate the expectations of order +1, and express them in terms of expectations of order less or equal to. nthisway,a new set of differential equations is obtained which is closed =1 and has the state space size, which is polynomially growing by. The simplest approximation is the meanfield type approximation [31]. n first order meanfield models [12], the states of nodes are assumed to be independent random variables. t is also possible to consider higher order meanfield approximations. Cator and Van Mieghem [32] used a second order meanfield approximation and found more accurate performance of the model. Another approach is called the moment closure technique, where the joint states of triplets are assumed to have a specific distribution (usually normal or lognormal) [11], [31]. n this way the joint expectation of triplets are expressed in terms of expectations of pairs. Taylor et al. [31] have conducted a comparison of the performances of different approximations. n this paper, we use a fist order meanfield type approximation. Using a first order meanfield type approximation, the joint expected values are approximated in terms of marginal expected values. pecifically, the term [( ) ()] in (22) is approximated by [( ) ()] ' ( [( ) ])[ ()] (24) This approximation assumes independence between the random variables. Using the approximation (24), we can describe the time evolution of the expected values through a set of ordinary differential equations with states. Denote by (), the expected value of at time, i.e., (), [ ()] (25) ubstituting [ () ()] = P =1 ( ()) () in (21), from (22) and (24), yields () = () X X =1 =1 ( ()) () (26) Arranging the terms in (26) specifies our generalized epidemic spreading model GEMF: X X = ( ) = {1 } =1 =1 (27) Having initially ( 0 )=1, the sum of the probabilities is guaranteed to be 1 at any time. The reason is that from (27) does not change over time because = = ( ) X X ( ) =1 =1 X X ( )( ) =1 =1 = 0 (28) The last conclusion is for the fact that =0and =0, since indeed and are the graph Laplacians for which is the eigenvector corresponding to a zero eigenvalue. GEMF has a systematic procedure to develop different spreading meanfield models. For any specific spreading scenario, the compartment set, the network layers, and their
8 8 corresponding influencer compartments should be identified and the transition rate graphs should be drawn. The individualbased meanfield model of the spreading scenario is found by plugging the matrices and, obtained from the transition rate graphs, into GEMF (27). C. Capabilities and Limitations of GEMF GEMF can be used to describe a wide range of spreading scenarios in a systematic way. n GEMF, there is no approximation of the underlying networks. The only approximation belongs to the meanfieldtype approximation (24). How much this approximation results in deviation from exactness is a problem to be investigated and is out of the scope of this paper. The available studies for the meanfield model (see, [33], [34]) can shed some light on this problem. Concerning the model, extensive numerical simulations have shown that for sparser graphs the meanfield model is less accurate, while for graphs with more mixing, the meanfield model is more close to the exact process. For a homogeneous mixing contact network, it has been proved that the meanfield model is asymptotically exact, i.e., as. Furthermore, the accuracy of the meanfield model very much depends on the range of the epidemic parameters. For example, in the spreading process, the meanfield model is accurate for large values of the infection rate for any graph. While for infection rates close to the epidemic threshold, there is considerable difference between the response of the meanfield model and the exact model. Additionally, it has been shown that meanfield models fail to explain the existence of a stable diseasefree absorbing state [35]. f the initial states are seeded according to an uncorrected distribution, i.e., at the initial time equation (24) is actually exact, then the meanfield model performs fairly accurately during the early stages of the system response. The reason for this is that nodes are poorly correlated at the early stage but become more and more correlated as time goes on. Accuracy of the transient response of meanfield models have been reported in [36] for the spreading process. The steady state solution of the meanfield models are also important. For example, the steady state solution of model belongs to the metastable state in epidemic process [12]. f accuracy is of greater concern, then higher order closure techniques can be used. However, this will result in a much larger state space size. GEMF has the smallest state space size to describe the spreading process of the type considered in this paper. Any further reduction of the state space implies essentially adopting approximation on the network structure. One of the great benefits of GEMF model is its analytical tractability. Using mean field model, the epidemic threshold was found to be the inverse of the spectral radius of the contact network. Finding relationships between spectral properties of underlying network layers and the spreading process is a problem of great interest. n particular, optimal design of some network layers given other network layers is very important from a technological view point. For example, ahneh and coglio [37] used a meanfield A model to find optimal topology of the information dissemination network given a contact network to reduce the impact of an epidemic. V. CAE TUDE n this section, we show that GEMF can reproduce the N ntertwined model [12] and the R model [14]. Furthermore, an A model with information dissemination and a model for a scenario where two pathogens are spreading in a host population are developed. A. Nntertwined Model The Nntertwined model [12] was explained in ection A. The number of compartments in this model is =2. The epidemic parameters are the infection rate and the curing rate. n this model, the interaction is only through the contact graph. For the contact graph, infected is the influencer compartment. Hence, = 1 and 1 = 2. The transition rate graphs for the model are shown in Fig. 4. The adjacency matrices corresponding to the nodal and edgebased transition rate graphs follow from Fig. 4, = 0 = (29) 0 0 Based on the GEMF (27), the compartment occupancy probability vectors satisfy the following differential equation = X 2 =1 (30) for {1 },wherethe and matrices, corresponding to and respectively, are 0 0 = = (31) 0 0 Denote the probabilities of being susceptible by and being infected by, i.e., = [ ]. Therefore, the evolution of these probabilities can be described as 0 0 X = ( ) 0 0 =1 ( P =1 = ) + ( P =1 (32) ) ince, + =1, the differential equation = + (1 )( X ) (33) =1 is obtained for and {1 }, which are the governing equations for the Nntertwined model in [12]. B. R Nntertwined Model Youssef and coglio [14] developed the R Nntertwined model where each agent can be either susceptible, infected or recovered. Therefore, the number of compartments in this model is = 3. A susceptible agent can become infected if it is surrounded by infected agents. The infection process is characterized by the infection rate. Furthermore, an infected agent becomes recovered with rate. Unlike the model, a recovered agent does not become infected again in R model. imilar to, there is only =1contact
9 9 a) b) Fig. 7. Transition rate graphs in the R model: a) nodal transition rate graph; nodes represent the two compartments susceptible, infected, and recovered, directed link from to represents curing process (a nodal transition) weighted by the curing rate 0, and b) edgebased transition graph of the contact network layer G ; directed link from to represents the infection process (edgebased transition) weighted by the infection rate 0. For the contact network, the influencer compartment is 1 =2, i.e., infected. graph and 1 =2. The transition rate graphs, shown in Fig. 7, illustrate that = = (34) According to GEMF (27), the compartment occupancy probability vector satisfies the following differential equation = X 2 (35) =1 for {1 }, wherethe matrices are = = (36) based on (34). Denote the probabilities of being susceptible, infected, and recovered by and, respectively, i.e., = [ ]. The evolution of these probabilities can be described as = X ( ) = ( P =1 ) = ( P =1 ) (37) R R ince, + + =1, the differential equation = + (1 )( X ) =1 = (38) is obtained for and which is the R Nntertwined model in [14]. C. A Model with nformation Dissemination Consider the A model in ection C. Assume that a susceptible individual becomes alert not only if there are infected individual in its neighborhood, but also if there are alert individuals in the neighborhood. Assume that the latter happens with rate. The interaction is through the contact network, infection information dissemination network, and the alert information dissemination network. For both the contact network and the sensor network, alert is the influencer compartment. For the information dissemination network, alert istheinfluencer compartment. Hence, =3 and 1 =2, 2 =2, 3 =3. From Fig. 8, = = = = (39) From GEMF (27), the compartment occupancy probability vector satisfies the following differential equation = =1 X (40) =1 X X =1 for {1}, wherethe matrices are = = = = (41) accordingto(39). A model very similar to (40), where there are only two layers of graphs, namely, the contact network and the infection information dissemination network, is used in [37] to assess the effectiveness of the information networks in reducing the impact of an epidemic. A novel information dissemination metric is introduced which measures the impact of information network on improving the resilience of the system against epidemic spreading. The developed information dissemination metric leads to a rigorous and elegant solution that determines the optimal topology of the information network to minimize the impact of an epidemic. D. Multiple nteracting Pathogen preading The problem of multiple pathogen spreading has recently attracted substantial attention (see e.g. [38] [41]). Most models consider a full cross immunity between pathogens, i.e., a node infected by one type of pathogen cannot be infected with any other type of pathogens at the same time. Beutel et al. [40] considered the case where the pathogens also have interacting
10 10 a) A b) A a B0 a) b) c) A0 A0 B0 A1 B1 A1 B1 c) d) A Fig. 8. Transition rate graphs in the A model: a) nodal transition rate graph; nodes represent the three compartments susceptible, infected, and alert, directed link from to represents curing process weighted by the curing rate 0, directed link from to represents the unalerting process weighted by the unalerting rate 0, and b) edgebased transition graph of the contact network layer G ; directed link from to represents the infection process (edgebased transition) weighted by the infection rate 0, directed link from to represents the alerted infection process (edgebased transition) weighted by the alerted infection rate 0. For the contact network, the influencer compartment is 1 =2, i.e., infected. c) edgebased transition graph of the infection information dissemination network layer G ; directed link from to represents the alerting process weighted by the alerting rate 0, For the infection information dissemination network, the influencer compartment is 1 =2,i.e., infected. d) edgebased transition graph of the alert information dissemination network layer G ; directed link from to represents the alerting process weighted by the alerting rate 0, For the alert information dissemination network, the influencer compartment is 1 =3,i.e., alert. effect on each other. n the model introduced by Marceau et al. [41], each pathogen has a separate contact network. n the following, we apply GEMF to develop an individualbased bispreading model for epidemic spreading of multiple interacting pathogens, very similar to [40], where each pathogen, as in [41], has a different contact network. Consider a spreading scenario where two pathogens and are spreading among a host population. The contact network for virus is G,while spreads through G. The transition rates for the pathogens depend on each other. For example, from pathogen spreading point of view, a susceptible individual has different infection rate if it is infected by versus being susceptible to. n general, we assume the transition rates are For example, if an individual is infected by but is not infected by, thenitrecoversbyrate 0.Whereas,ifitis also infected by, disease gets cured by rate 1. imilar arguments apply for other rate terms. For this spreading scenario, =4compartments can be defined to model the problem. Agent is in compartment 1 if it is susceptible to both and. tis2 if it is susceptible to but infected by. tis3 if infected by and susceptible to. Andfinally, it is 4 if it is infected by both and. The nodal and edgebased transitions are shown in Fig. 9. A Fig. 9. Transition rate graphs in the bispreading model: a) nodal transition rate graph; nodes represent the four compartments,,,, and, directed links from to and from to represents curing process for virus weighed with curing rates 0 and 1, respectively and the directed links from to and from to represents curing process for virus weighted by the curing rates 0 and 1, respectively, and b) edgebased transition graph of the contact network layer G for virus ; directed link from to and from to represents infection process for virus weighed with infection rates 0 and 1, respectively. For the contact network G,the influencer compartment is =3 4, i.e., and. c) edgebased transition graph of the contact network layer G for virus ; directed link from to and from to represents infection process for virus weighed with infection rates 0 and 1, respectively. For the contact network G,theinfluencer compartment is =2 4, i.e., and. t follows from Fig. 9, = = = (42) Therefore, according to the GEMF model (27), the compartment occupancy probability vector satisfies the following differential equation = X ( ) =1 X ( ) (43) =1 for {1}, wherethe matrices are = =
11 = (44) V. CONCLUON nspired by existing individualbased epidemic models, the generalized epidemic meanfield (GEMF) model is proposed. While using the same common assumptions of most of the existing individualbased epidemic models, GEMF is capable to model more complex scenarios with multiple compartment and multiple network layers. The set of differential equations that fully describes the time evolution of the compartment occupancy probabilities has equations. Even though the system is linear, it is both computationally and analytically intractable. Through a meanfield type approximation, a set of nonlinear differential equations is developed. The latter system, referred to as GEMF, has a simple structure. t is characterized by the Laplacian of the transition rate graphs and the elements of the adjacency matrices of the network layers. A systematic procedure for developing the model is proposed that culminates in the GEMF governing equations (27). GEMF model is rigorous, allows analytical tractability, and is simple to be adopted to many specific spreading processes, as shown in the several examples presented in this paper. We believe that the GEMF framework has the potential to allow the development of many different and novel individualbased epidemic models considering new compartments and multiple complex interaction structures. REFERENCE [1] J. Kephart and. White, Directedgraph epidemiological models of computer viruses, in ociety ymposium on Research in ecurity and Privacy, may 1991, pp [2] J. Kleinberg, Computing: The wireless epidemic, Nature, vol. 449, no. 7160, pp , [3] M. Meisel, V. Pappas, and L. Zhang, A taxonomy of biologically inspired research in computer networking, Computer Networks, vol.54, no. 6, pp , [4] A. Barabási and Z. Oltvai, Network biology: understanding the cell s functional organization, Nature Reviews Genetics, vol. 5, no. 2, pp , [5] C. LindemannandA. 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Kooij, Virus spread in networks, EEE/ACM Transactions on Networking, vol. 17, no. 1, pp. 1 14, [13] A. Ganesh, L. Massoulie, and D. Towsley, The effect of network topology on the spread of epidemics, in Proceedings EEE NFOCOM, vol. 2, 2005, pp [14] M. Youssef and C. coglio, An individualbased approach to R epidemics in contact networks, Journal of Theoretical Biology, vol. 283, no. 1, pp , [15]. Karlin and H. Taylor, A second course in stochastic processes. Academic Pr, [16] P. Van Mieghem, Performance analysis of communications networks and systems. Cambridge Univ Press, [17] P. Erdős and A. Rényi, On the evolution of random graphs. Akad. Kiadó, [18] A. Barabási and R. Albert, Emergence of scaling in random networks, science, vol. 286, no. 5439, pp , [19] J. Doyle, D. Alderson, L. Li,. Low, M. Roughan,. halunov, R. Tanaka, and W. 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12 12 APPENDX DERVATON OF EXACT MARKOV EQUATON n this section, we explicitly derive the expression for Q in (23). The idea is to derive the expression for [( + )] as a function of [()]. For this end, first we find the expression for the conditional expectation [(+ ) ()]. Then, the expression for [( + )] is found by averaging out the conditional. For small values of, wecanassume that only one transition happens at each time step, i.e., starting at network state at time, the network state can only go to a new state at time + for which only the state of a single node has been changed. Given the network state () =, state () = of each agent are can be determined and we have = 1 (A.1) ince only at most one single node can make a transition, the conditional expected value of the network state at time + is and Q is such that its th column is col(q )= X =1 1 X ( 1 { = } ) =1 (A.6) By letting 0 in (A.5), the time evolution of [] can be fully described by [] = Q [] (A.7) where Q is defined as X Q = Q + Q (A.8) The differential equation (A.7) is referred to as the exact Markov equation. =1 [( + ) () = ]= X 1 =1 [ ( + ) () = ] (A.2) where from (18), the expression for [ ( + ) () = ] is [ ( + ) () = ]= X X 1 {= } =1 =1 + ()+( ) (A.3) By averaging over all of the possible network states, the expected value of the network state at time + is found [( + )] X = [( + ) () = ]Pr[() = ] =1 X = ( X =1 1 =1 [ ( + ) () = ] )Pr[() = ] (A.4) ubstituting for [ (+ ) () = ] from (A.3), [(+ )] is deduced to be [( + )] = Q [()] X Q [()] + [()] + ( ) =1 (A.5) where Q = X =1
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