Hierarchical Coarse-grained Approach to the Duration-dependent Spreading Dynamics in Complex Networks

Various coarse-grained models have been proposed to study the spreading dynamics in the network. A microscopic theory is needed to connect the spreading dynamics with the individual behaviors. In this letter, we unify the description of different spreading dynamics on complex networks by decomposing the microscopic dynamics into two basic processes, the aging process and the contact process. A microscopic dynamical equation is derived to describe the dynamics of individual nodes on the network. The hierarchy of a duration coarse-grained (DCG) approach is obtained to study duration-dependent processes, where the transition rates depend on the duration of an individual node on a state. Applied to the epidemic spreading, such formalism is feasible to reproduce different epidemic models, e.g., the susceptible-infected-recovered and the susceptible-infected-susceptible models, and to associate with the corresponding macroscopic spreading parameters with the microscopic transition rate. The DCG approach enables us to obtain the steady state of the general SIS model with arbitrary duration-dependent recovery and infection rates. The current hierarchical formalism can also be used to describe the spreading of information and public opinions, or to model a reliability theory in networks.

Various coarse-grained models have been proposed to study the spreading dynamics in the network. A microscopic theory is needed to connect the spreading dynamics with the individual behaviors. In this letter, we unify the description of different spreading dynamics on complex networks by decomposing the microscopic dynamics into two basic processes, the aging process and the contact process. A microscopic dynamical equation is derived to describe the dynamics of individual nodes on the network. The hierarchy of a duration coarse-grained (DCG) approach is obtained to study duration-dependent processes, where the transition rates depend on the duration of an individual node on a state. Applied to the epidemic spreading, such formalism is feasible to reproduce different epidemic models, e.g., the susceptible-infected-recovered and the susceptible-infected-susceptible models, and to associate with the corresponding macroscopic spreading parameters with the microscopic transition rate. The DCG approach enables us to obtain the steady state of the general SIS model with arbitrary duration-dependent recovery and infection rates. The current hierarchical formalism can also be used to describe the spreading of information and public opinions, or to model a reliability theory in networks.
Introduction.-The epidemics [1][2][3][4][5], rumors or information [6][7][8][9][10][11], and public opinions [12][13][14][15], e.t.c, usually spread in the complex network with predefined structures. The spreading dynamics is strongly affected by the characteristic of the structural networks [16,17]. The utilization of susceptible-infected-susceptible (SIS) and the susceptible-infected-recovered (SIR) model initiated the study of the spreading dynamics of epidemics on the network with simple microscopic mechanism [2,3]. The network structure, known as the degree distribution, affects an epidemic threshold, which is an index to determine whether the disease is capable to spread over the society [17][18][19][20][21][22][23][24][25]. The spreading dynamics is also affected by the microscopic mechanism, namely, the rules of the state change and the transition rates of the basic processes. For the complicated mechanism, the transition rates may not be a constant during the evolution, which might relate to the duration of the node in a certain state. For example, the infection rate of the disease is typically related the duration of an infected individual [26][27][28][29][30][31][32][33][34][35][36]. Currently a unified spreading model with combining both the network structure and microscopic mechanism remains missing. In this Letter, we propose a unified formalism to describe the spreading dynamics on the network with different microscopic mechanism.
In our formalism, we decompose the spreading dynamics into two basic processes, the aging process describing the self-evolution of an individual node (single-body process), and the contact process describing the state change of two connected individual nodes (two-body process). The two processes are modeled here as a continuous-time stochastic process among a set of discrete states with the adoption of the reliability theory [37][38][39]. From the microscopic model, we introduce the duration density func-tion (DDF) in the coarse-grained models to study the duration-dependent effect of the transition rates, and derive a duration coarse-grained (DCG) equation of the DDF for the spreading dynamics. The DCG equation allows us to derive the steady state of the general SIS model with duration-dependent recovery and infection rates. Through a further coarse-grain procedure without distinguishing the degrees of the nodes, we recover the compartmental epidemic models [40,41] at the macroscopic level.
Two basic processes.-We consider a structural network with N T nodes, which are connected with links to represent the network. The connection between nodes l and m is described by the adjacent matrix A lm , i. e., A lm = 1 for connection and A lm = 0 for no connection. The node state is picked from the state set i ∈ {0, 1, 2, ...}. The spreading dynamics describes the state evolution with two basic processes, the aging process and the contact process.
The aging process describes the state change i α i ,i −→ i of one single node independent of other nodes, as illustrated in Fig. 1(a). The transition rate α i ,i (τ i ) generally relates to the duration τ i on the state i [38]. The maximum entropy principle can be used to estimate the most probable transition rate [41,42], when limited information, e.g., the mean infection time, is provided about the process.
The contact process describes the joint state change i + j β i j ,ij −→ i + j of two linked nodes, as illustrated in Fig. 1(b). The transition rate β i j ,ij (τ i , τ j ) relates to the duration τ i and τ j of the nodes on two states i and j. Different patterns exist for the contact process, e.g., the exchange process i + j Most of the widely used models can be constructed with the two basic processes above. In the SIS model, two states 0 and 1 are the susceptible and the infected states. The basic processes include an aging process 1 α(τ1) −→ 0 with the recovery rate α(τ 1 ), and a contact process 0 + 1 β(τ0,τ1) −→ 1 + 1 with the infection rate β(τ 0 , τ 1 ). The duration-dependent infection rate reflects the change of both the vulnerability of the susceptible state and the transmissibility of the infected state. An example evolution of one individual node is shown in Fig. 1(c). At the initial time t = t 0 , the node stays in the state 0 with zero duration τ = 0. The node state changes with resetting of the duration τ at time t 1 and t 2 due to the contact and the aging processes. In the application to the epidemic spreading, the infection rate is typically independent of the susceptible duration τ 0 , which implies the infection process is dominated by the infected state. In the typical model of rumor spreading [7,8], three states 0, 1 and 2 are the ignorant, spreading, and stifling states. Three basic processes are 0+1 The transition rates β 2 (τ 1 , τ 1 ) and β 3 (τ 1 , τ 2 ) generally depend on the duration of the both states. Currently, such duration-dependent effects have seldom been considered in previous studies.
The coarse-grain of the microscopic model.-The conventional model [2] with only recording the node states is not enough to describe the spreading dynamics with the duration-dependent transition rates. In Fig. 2(a), we introduce the probability density function (PDF) ρ l,i (τ i , t) of the duration for the node l in the microscopic model. (Color online) Hierarchy of the microscopic, the mesoscopic, and the macroscopic models of the spreading dynamics. The information of the duration distribution is recorded by the probability density function ρ l,i (τi, t) of the duration, the duration density function f k,i (τi, t) and the gross duration density function fi(τi, t), respectively.
The probability for the node l in the state i follows as P l,i (t) = ∞ 0 ρ l,i (τ i , t)dτ i . By neglecting the correlation between nodes, the state of the network is described by the PDF ρ l,i (τ i , t). For the node l, we introduce the total transformation rate Γ l,i (τ i , t) of leaving the state i. The equation of the PDF reads (see the derivation in supplementary materials [43]) The total transformation rate for the node l is Γ l,i (τ i , t) = i γ l,i i (τ i , t). The transformation rate γ l,i i (τ i , t) from the state i to the state i is explicitly determined by the transition rates α i ,i (τ i ) and β i j ,ij (τ i , τ j ) as which contains the contribution from all the basic processes involved the transformation from the state i to the state i . The connecting condition for the PDF at τ i = 0 is determined by the flux to the state i as ρ l,i (0, t) = Φ l,i (t) = i φ l,ii (t), where φ l,ii (t) = ∞ 0 γ l,ii (τ i , t)ρ l,i (τ i , t)dτ i is the probability of the node l transforming from the state i to the state i in unit time.
To effectively describe the spreading dynamics without considering the state of each node, we propose a duration coarse-grained (DCG) approach to study the duration-dependent effect with the duration density functions (DDFs). For example, the infection mechanism of the general SIS model is reflected by the equation of the DDF of the infected individuals [32,34]. In the mesoscopic model, we sort the nodes into different ensembles with the degree k, namely the number of neighbors for a node, as shown in Fig 2(b). The states of the network are coarse-grainedly described by the DDF of the k-degree nodes as f k,i (τ i , t) = l δ k,k l ρ l,i (τ i , t). The population of the k-degree nodes in the state i follows as The population of all k-degree nodes is n k = i n k,i (t), which is a constant for the static network structure. We assume the PDFs of the nodes in one ensemble are identical, namely ρ l,i (τ i , t) = ρ l ,i (τ i , t) for k l = k l . The PDFs is rewritten with the DDF as ρ l,i (τ i , t) = f k l ,i (τ i , t)/n k l . For the k-degree nodes, the transformation rate approximates as the average of all the k-degree nodes γ k,i i (τ i , t) = l δ k,k l γ l,i i (τ i , t)/n k . The equation of the DDF is obtained from Eq. (1) as The total transformation rate is where the degree correlation P (k |k) describes the degree distribution of one neighbor of a k-degree node, and relates to the adjacent matrix A lm as P (k |k) = l,m δ k,k l δ k ,km A lm /(kn k ) [43]. The connecting condition for the DDF is is the flux of the k-degree nodes transforming from the state i to the state i. An example with explicit equations of DDFs in the general SIS model can be found in the supplementary materials [43] or in Ref. [32].
At the macroscopic level, a further coarse-grained procedure introduces the gross DDF f i (τ i , t) = ∞ k=1 f k,i (τ i , t) of all the nodes to simplify the spreading dynamics, as shown in 2(c). The population of the nodes in the state i follows as N i (t) = ∞ 0 f i (τ i , t)dτ i , and the total node number is N T = i N i (t). This approximation is suitable for the homogeneous network with similar degrees for different nodes. The dynamics is then regarded to be independent of the degree with the DDF as f k,i (τ i , t) = P (k)f i (τ i , t) and the transformation rate as where P (k) = n k /N T is the degree distribution. The equation of the gross DDF is obtained from Eq. (3) as The total transformation rate is Γ i (τ i , t) = i γ i i (τ i , t), and the transformation rate γ i i (τ i , t) is obtained from Eq. (4) as The effect of the network structure on the spreading dynamics is reflected by the average degree k = ∞ k=1 kP (k). The connecting condition for the gross dτ i is the gross flux transforming from the state i to the state i. The details of the coarsegrained procedures are shown in the supplementary materials [43]. One advantage of our spreading models is its generality for application in different problems. The states and the nodes have different meanings in different models. For example, the node state can be the disease of the individual in an epidemic model [2], or the state of the device in a reliability model [39]. The transformation rates and the connecting conditions are given accordingly from the explicit microscopic models. For the constant transition rates, our models recover the conventional models describing the spreading dynamics with the probabilities P l,i (t) or the populations n k,i (t) and N i (t). The derivation to such recovery is given in the supplementary materials [43].
As follows, we apply our spreading models to the epidemics. In Tab. I, we list the dictionary for constructing the general SIS and SIR models with the transformation rates, the fluxes and the connecting conditions in the mesoscopic model. The two models are uniformly described by the same partial differential equations with different coupling forms of the connecting conditions.
The macroscopic model of spreading dynamics recovers the normal compartmental SIS model [40,44,45] with the constant recovery α and infection rate β, where the populations of the susceptible and the infected individ- In Ref. [42], the effect of the duration-dependent recovery rate α(τ 1 ) has been studied in an extended compartmental model with the integrodifferential equations. In the supplementary materials [43], we derive both the normal and the extended compartmental model through the duration coarse-grained approach.
SIS model in a network.-We apply the current DCG approach to solve the spreading dynamics of the general SIS model with duration-dependent infection mechanism  on an uncorrelated network, whose the degree correlation satisfies P (k |k) = k P (k )/ k [17]. The DCG approach enables us to obtain the steady state with arbitrary duration-dependent recovery and infection rates by solving a self-consistent equation.
In the general SIS model, the evolution of the DDFs f k,0 (τ 0 , t) and f k,1 (τ 1 , t) is governed by Eq. (3) with the transformation rates and the connecting conditions listed in Tab. I. The epidemic spreading is typically assessed by the fraction r 1 (t) = ( ∞ k=1 n k,1 (t)) / ( ∞ k=1 n k ) of the infected nodes. In a typical spreading process, the vulnerability of the susceptible node can be regarded unchanged with the duration τ 0 . And it is reasonable to assume the infection rate only depends on the duration τ 1 of the infected node as β(τ 0 , τ 1 ) = β(τ 1 ). This assumption leads to the transformation rate Γ k,0 (t) independent of the susceptible duration τ 0 .
On the uncorrelated network, the transformation rate of the contact process is simplified as Γ k,0 (t) = kΘ(t) with For the steady state ∂f k,i (τ i , t)/∂t = 0 of Eq. (3), the DDFs of the steady state are solved as and where Φ k = n k kΘ/(1 + kΘτ 1 ) is the steadystate flux with the average infection which is the self-consistent equation for the quantity Θ of the steady state. Here, Υ is the refined spreading rate for the general SIS model as The steady-state fraction of the infected nodes is which is determined by the refined spreading rate Υ via the quantity Θ and the average infection durationτ 1 [46]. The effect of network structure is explicitly reflected via the degree distribution P (k). At the case with the constant recovery and infection rates, the refined spreading rate Υ returns to the effective spreading rate Υ = β/α used in the duration-independent SIS model [2]. The non-zero solution to Eq. (10) satisfies the condi- The existence of the non-zero solution requires the refined spreading rate Υ to exceed a critical value, defined as the epidemic threshold Υ c = k / k 2 , which is solely determined by the network structure. When the spreading rate exceeds the epidemic threshold Υ > Υ c , the system reaches the epidemic steady state with non-zero infection nodes. At the situation Υ < Υ c , the system reaches the disease-free steady state with zero infected nodes. A necessary condition to ensure a disease-free steady state is k ≤ 1/Υ, which implies the contacts of people need to be controlled according to the spreading ability of the disease.
To validate the current coarse-grained model, we simulate the general SIS model in an uncorrelated scale-free network with the continuous-time Monte Carlo method [47]. The details of the simulation are illustrated in the supplementary materials [43]. The uncorrelated scalefree network is generated by the configuration model [48]. The degree sequence {k l } is generated according to the degree distribution P (k) = c/k 3 , where k ranges from the minimal degree k min = 11 to the maximal degree k max = 22 with the normalized constant c = 1/( kmax k =kmin 1/k 3 ) of the degree distribution. The total node number is set as N T = 500. The maximal degree k max fulfills the condition k max ≤ √ N T to ensure an uncorrelated network [48]. All nodes are randomly linked respecting the assigned degrees without multiple and self-connection.
We carry out the simulation with the Weibull distribution of the recovery and the infection time, where the recovery and the infection rates are set as α( In each simulation, we run the evolution for sufficient events to ensure the steady state at the end of the simulation. The steady-state fraction r 1 of the infected nodes is then calculated for each run and averaged with 100,000 events. In Fig. 3, the steady-state fraction r 1 of the infected nodes is plotted as the function of the refined spreading rate Υ for the DCG approach (solid curve) and the continuous-time Monte Carlo simulation results (dots). In the simulation, the effects of duration-dependent recovery and the infection rates are considered with the parameters a α = 1.0, 1.5, b α = 0.5, 1.0, a β = 0.5, 1.0, 1.5, and b β ranging from 1.0 to 10.0 with interval 1.0. The agreement between the analytical and the simulation results validates that the steady-state fraction of the infected nodes is effectively described with the refined spreading rate Υ by Eq. (11). The curve shows clearly the existence of the epidemic threshold Υ c = 0.067 (gray grid-line), which matches the theoretical prediction of the epidemic threshold Υ c = k / k 2 . The current model shows the availability of the refined spreading rate Υ for justifying the spreading ability of a disease.
Conclusion.-In this Letter, we presented the microscopic description of the spreading dynamics on the network, and show hierarchical emergence of the widely-used coarse-grained models. The spreading dynamics is derived as a unified equation for both the aging and the contact process with the duration-dependent microscopic mechanism. The unified formalism enables us to recover different spreading models, e. g. the SIS and the SIR model. With the current formalism, we prove that the steady state of the infection is solely determined by the refined spreading rate Υ, which is a coarse-grained parameter of the microscopic mechanism details. We show the existence of the epidemic threshold Υ c = k / k 2 to determine the fate of an epidemic is solely given by the network structure for a general SIS model.  The document is devoted to providing detailed discussions and derivations to support the discussions in the main content. In Sec. I, we build the microscopic spreading model by introducing the probability density function (PDF) of the duration for each node in the network. In Sec. II, we show the emergence of the duration coarse-grained (DCG) approach to obtain the mesoscopic and macroscopic models. In Sec. III, we apply the DCG approach to the susceptible-infected-susceptible (SIS) model. In Sec. IV, we show the macroscopic SIS model recovers the normal [1] and the extended compartmental models [2]. In Sec. V, we solve the steady state of the mesoscopic SIS model in an uncorrelated network with duration-dependent recovery and infection rates. In Sec VI, we provide the details of the continuous-time Monte Carlo simulation of the SIS model in an uncorrelated scale-free network.

I. SPREADING DYNAMICS IN MICROSCOPIC MODEL
In the basic processes, the node transforms from one state to another state. In the microscopic model, we use the probability distribution to describe the state for each node. The probability of the node l staying in the state i is P l,i (t), with the normalization condition i P l,i (t) = 1. We assume the states of different nodes are uncorrelated: the probability for the node l in the state i and the node m in the state j can be written in the product form P l,i (t) × P m,j (t).
A. Probability density function ρ l,i (τi, t) of the duration and its equation of the evolution Under the uncorrelated assumption, we introduce the probability density function (PDF) ρ l,i (τ i , t) for each node with the duration time τ i to describe the state of the network. For the node l, the probability in the state i with the duration between τ i and τ i + δτ i is ρ l,i (τ i , t)δτ i . The probability P l,i (t) is equal to the integral of the PDF ρ l,i (τ i , t) as The total transformation rate Γ l,i (τ i , t) from the state i to the other states is where γ l,i i (τ i , t) is the transformation rate from the state i to the state i . In the small time step dt, the node l transforms to other states with the conditional probability Γ l,i (τ i , t)dt if it stays in the state i. At the time t + dt, the probability in the state i with the duration between τ i + dt and τ i + δτ i + dt is ρ l,i (τ i + dt, t + dt)δτ i . The change of the probability is caused by the transformation, namely, With the above equation, we obtain the differential equation for the PDF as The transformation rate γ l,i i (τ i , t) relates to the basic processes with the transformation from the state i to the state i . In the aging process i α i ,i −→ i , the contribution to the transformation rate is given directly by the transition rate In the contact process i+j β i j ,ij −→ i +j , the transformation depends on the states and the duration of all the neighbors m as where A lm is the adjacent matrix of the network: A lm = 1 if the nodes l and m are linked, otherwise A lm = 0. Including the contribution from both the aging and the contact processes, the overall transformation rate follows as which is Eq. (3) in the main content.

C. Connecting condition
For the node l, we define the flux from the state i to the state i as which is the probability for the transformation from the state i to the state i in unit time. The total flux to the state i from all other states is In the small time step dt, the probability ρ l,i (0, t)dt of the transformation to the state i is equal to Φ l,i (t)dt due to the conservation of the probability as The change of the probability relates to the fluxes as In the above derivation, we have used the equation of the evolution by Eq. (4) and the condition ρ l,i (∞, t) = 0. For constant transition rates with α i ,i (τ i ) = α i ,i and β i j ,ij (τ i , τ j ) = β i j ,ij , the flux φ l,i i (t) is directly given by the probability as

II. HIERARCHICAL DURATION COARSE-GRAINED APPROACH
Typically, the spread is usually evaluated through the populations of different states. The duration coarse-grained (DCG) approach enables us to derive the coarse-grained models with the populations from the microscopic model with probability. Here, we supplement the derivation of the duration coarse-grained approach in the main content and show the hierarchy among the microscopic, mesoscopic and macroscopic models.

A. Mesoscopic model
In the mesoscopic model, the state of the network is coarse-grainedly described by the duration density function (DDF) f k,i (τ i , t) for the k-degree nodes, which relates to the PDF of each node as where k l is the degree of the node l. The differential equation of the PDF by Eq. (4) leads to that of the DDF as We assume the PDFs of the nodes with the same degree is identical, namely ρ l,i (τ i , t) = ρ l ,i (τ i , t) for k l = k l . With this identical assumption, we obtain where n k is the number of the nodes with the degree k. The right hand side of Eq. (14) is simplified as t). The corresponding transformation rate for the kdegree nodes is and Then, the differential equation of the DDF is rewritten as Plugging the transformation rate γ l,i i (τ i , t) of the node l into Eq. (17), we obtain the transformation rate γ k,i i (τ i , t) for the k-degree nodes in the main content as where M kk = l,m (δ k,k l δ k ,km A lm ) /(1 + δ k,k ) is the number of the edges linked two nodes with the degrees k and k . We have used the identical assumption ρ l,i (τ i , t) = f k l ,i (τ i , t)/n k l in Eq. (19). For a k-degree node, the conditional probability of having a k -degree neighbor is described by the degree correlation P (k |k), which is explicitly determined by the edge number M kk as The number of the edges linked to a k-degree node relates to the number of k-degree nodes as We obtain the transformation rate for the k-degree nodes as which is Eq. (4) in the main content. According to Eq. (13), the connecting condition of the DDF is f k,i (0, t) = l δ k,k l ρ l (0, t), which leads to the mesoscopic flux as Under the identical assumption, the mesoscopic flux is determined by the DDF as The total flux follows The connecting condition of the DDF is rewritten as The change of the population of k-degree nodes in the state i is obtained as For the constant transition rates α i ,i (τ i ) = α i ,i and β i j ,ij (τ i , τ j ) = β i j ,ij , the mesoscopic flux is directly given by the populations as where n k,i = ∞ 0 f k,i (τ i , t)dτ i is the population of the k-degree nodes in the state i.

B. Macroscopic model
At the macroscopic level, we introduce the gross DDF to describe the nodes in the state i without distinguishing the degrees as The differential equation of the gross DDF follows from Eq. (18) as To obtain the equation of the gross DDF f i (τ i , t), we need to estimate the DDF f k,i (τ i , t) of the k-degree nodes with the gross DDF f i (τ i , t). For the homogeneous network with similar degrees of all the nodes, the PDF of each node approximates the same ρ l,i (τ i , t) ρ l ,i (τ i , t), which leads the DDF to satisfy f k,i (τ i , t)/n k f k ,i (τ i , t)/n k with different degrees k and k . The DDF is estimated with the gross DDF as where the degree distribution P (k) = n k /N T gives the fraction of the k-degree nodes. The right hand side of Eq.
The corresponding transformation rate follows as and The differential equation of the DDF is rewritten as Plugging Eq. (23) into γ i i (τ i , t) = ∞ k=1 P (k)γ k,i i (τ i , t), we obtain the transformation rate γ i i (τ i , t) from the state i to the state i as where N T is the total node number. Here, we have used the normalization condition k P (k |k) = 1.
The connecting condition of the gross DDF relates to that of the DDF as f i (0, t) = ∞ k=1 f k,i (0, t), which leads to the macroscopic flux as The total flux follows as The connecting condition of the gross DDF is rewritten as The change of the population in the state i is obtained as For the constant transition rates α i ,i (τ i ) = α i ,i and β i j ,ij (τ i , τ j ) = β i j ,ij , the macroscopic flux is directly given by the populations as where N i (t) = ∞ 0 f i (τ i , t)dτ i is the number of the nodes in the state i.

III. DURATION COARSE-GRAINED APPROACH TO SIS MODEL
In the SIS model, the nodes stay in the susceptible state 0 and the infected state 1. The basic processes consist of an aging process 1 α(τ1) −→ 0 with the recovery rate α(τ 1 ) and a contact process 0 + 1 In the mesoscopic model, the node states in the network are coarse-grainedly described by the duration density function (DDF) f k,i (τ i , t) with i = 0, 1. Based on the duration coarse-grained (DCG) approach, the DDF f k,0 (τ 0 , t) satisfies where the transformation rate Γ k,0 of the contact process is The DDF f k,1 (τ 1 , t) satisfies where the transformation rate is determined as the transition rate α(τ 1 ) of the aging process. The connecting condition is given by the flux f k,i (0, t) = Φ k,i (t), where the fluxes are determined by the transformation rates as and

IV. RELATION TO THE COMPARTMENTAL MODELS
In the following, we show the macroscopic model recovers the compartmental SIS model. The duration of all the susceptible and the infected individuals is described by the gross DDFs as The network structure is coarse-grainedly described by the average degree k = k kP (k).
The equations of the gross DDFs for the susceptible and the infected states are obtained from Eq. (35) as and The connecting conditions of DDFs are f i (0, t) = Φ i (t), i = 0, 1, with the fluxes determined as and The populations of the susceptible and the infected individuals are The total population is N T = N 0 (t) + N 1 (t).
A. The extended compartmental SIS model with integral-differential equations The extended SIS compartmental model requires the constant infection rate β(τ 0 , τ 1 ) = β, but the recovery rate α(τ 1 ) can be duration-dependent. The flux Φ 1 (t) by Eq. (50) is simplified as The formal solution of f 1 (τ 1 , t) to Eq. (48) is represented by the connecting and the initial condition as Plugging the solution into the flux Φ 0 (t), we obtain where the first and the second terms in the right-hand side relate to the connecting and the initial condition, respectively. The integral-differential equations in the extended compartmental SIS model [3] are obtained by representing the infection rate as the PDF of the infection duration We assume all the infected individuals get infected at the initial time with the initial condition Then, the flux by Eq. (55) is rewritten as which is the integral-differential equation in the extended compartmental model [3].

V. THE STEADY STATE IN THE MESOSCOPIC MODEL
In this section, we solve the steady state of the SIS model in the mesoscopic model. In the steady state ∂f k,i (τ i , t)/∂t = 0, the populations n k,i remains unchanged with the equal fluxes Φ k,0 = Φ k,1 = Φ k . The equations of the steady-state DDFs are obtained from Eqs (42)- (46) as with the connecting condition f k,i (0) = Φ k,i , i = 0, 1. The constraint of the unchanged number of the k-degree nodes is The solutions to the steady-state DDFs follow explicitly as and where the steady-state fluxes Φ k is given by the constraint of the unchanged node number as The steady-state populations follow as and whereτ 1 is the average infection durationτ A. Steady state in uncorrelated network In an uncorrelated network, the degree correlation is independent of the degree k as [4] P (k |k) = k P (k ) k .
The transformation rate by Eq. (62) is simplified as Γ k,0 (τ 0 ) = kΘ(τ 0 ), where the quantity Θ(τ 0 ) is determined as Therefore, the solution by Eq. (67) is simplified as The steady-state flux, in turn, is rewritten as Plugging Eqs. (66), (68) and (75) into Eq. (74), we obtain the self-consistent equation for Θ(τ 0 ) as B. Simple infection rate β(τ0, τ1) = β(τ1) In the following, we consider the case that the infection rate β(τ 0 , τ 1 ) = β(τ 1 ) only depends on the infection duration τ 1 . The independence of the right-hand side of Eq. (77) on the susceptible duration u 0 results in a constant quantity Θ(τ 0 ) = Θ. The integral on the right-hand side is worked out as The non-zero solution Θ exists for Υ > Υ c , where Υ c = k / k 2 is the epidemic threshold determined by the network structure. The proof is given as follows.
We define a new function as This function y(x) is continuous and monotonously increasing for x > 0 with lim x→∞ y(x) > 0. The existence of the positive solution to y(x) = 0 requires y(0) < 0, namely The critical value gives the epidemic threshold Υ c .
For the large-size scale-free network with the degree distribution P (k) ∝ k −γ , 2 < γ ≤ 3, the divergence of k 2 = ∞ k=1 k 2 P (k) leads to zero epidemic threshold Υ c = 0 of a large scale-free network [5]. The fraction of the infected nodes is defined as (80) With the steady-state population n k,1 by Eq. (71), the steady-state fraction of the infected nodes is obtained as which is positive with Θ > 0.

VI. CONTINUOUS-TIME MONTE CARLO SIMULATION OF THE SIS MODEL
This section shows the numerical simulation of the duration-dependent SIS model in networks. In the previous studies [6], the simulation of the duration-dependent model is formulated by recording all the possible events in the timeline, referred to as the tickets. The states of the nodes are updated through the tickets. New tickets are generated from infected nodes. In our algorithm, instead of recording the tickets which may or may not occur, we only record the final time when the node will leave the current state, which saves the memory and gives the same results.

A. Simulation algorithm
The current time t cur represents the time of the current step. For each node, we record the state of the node x l , the initial time t (l) ini when the node transformed to the current state, and the final time t (l) fin when the node will transform to the other state, as shown in Fig. 1 (a). The susceptible and the infected states are x l = 0 and x l = 1. At the beginning, the current time t cur is set as 0. The state of the network is prepared by assigning the state x l for each node. The initial time t With the prepared state, the evolution of the spread is realized step by step. In each step, an event occurs with the state change of one node. There are two kinds of events in the SIS model: the recovery (infection) event with a node transforming from the state 1 (0) to the state 0 (1). Since the future events are recorded by the final time of the nodes, the next event is obtained by finding the node l with the smallest final time t (l) fin . We give the explicit procedure of the updating for the recovery and the infection event as follows. The pseudo code is shown in Fig. 1 (b).
For either a recovery or an infection event, the current time is updated with the smallest final time as t cur = t (l) fin , which records the time of the current event and prepares for the next event. The new state of the node l is x l = 1 − x l with the new initial time t fin , as shown in Fig. 1 (a). In a recovery event, the node l recovers to the susceptible state x l = 0, and may get infected again from an infected neighbor in the following evolution. The new final time is first set as t The infection time T

B. Transition Rate of Weibull distribution
In the simulation, we consider the recovery and the infected duration satisfy the Weibull distribution. The cumulative distribution function of Weibull distribution is which gives the transition rate as The Weibull distribution returns to the exponential one with a = 1. In the following simulation of the SIS model, the recovery and the infection rates are chosen as and C. Generating the uncorrelated Scale-free network The uncorrelated scale-free network is generated by the configuration model [7] for N T = 500 nodes. The numbers of the k-degree nodes are set as approximation integers explicitly with the values n k = 106, 82, 64, 52, 42,35,29,24,21,18,14,13 with k ranging from the minimal degree k min = 11 to the maximal degree k max = 22. The maximal degree is set as k max ≤ √ N T = 22.4 to ensure an uncorrelated network [7]. With the assigned degree for each node, all the nodes are randomly linked avoiding multiple and self-connection. For an uncorrelated network, the degree correlation is determined by the degree distribution as P (k |k) = k P (k )/ k .

D. Simulation results of single run
We apply the simulation algorithm to simulate the spreading dynamics of the SIS model in the uncorrelated scalefree network. Figure 2 presents the simulation results (colored curves) of the fraction r 1 (t) of the infected nodes in