Epidemic Model with Isolation in Multilayer Networks

The Susceptible-Infected-Recovered (SIR) model has successfully mimicked the propagation of such airborne diseases as influenza A (H1N1). Although the SIR model has recently been studied in a multilayer networks configuration, in almost all the research the isolation of infected individuals is disregarded. Hence we focus our study in an epidemic model in a two-layer network, and we use an isolation parameter w to measure the effect of quarantining infected individuals from both layers during an isolation period tw. We call this process the Susceptible-Infected-Isolated-Recovered (SIIR) model. Using the framework of link percolation we find that isolation increases the critical epidemic threshold of the disease because the time in which infection can spread is reduced. In this scenario we find that this threshold increases with w and tw. When the isolation period is maximum there is a critical threshold for w above which the disease never becomes an epidemic. We simulate the process and find an excellent agreement with the theoretical results.


I. INTRODUCTION
Most real-world systems can be modeled as complex networks in which nodes represent such entities as individuals, companies, or computers and links represent the interactions between them.In recent decades researchers have focused on the topology of these networks [1].Most recently this focus has been on the processes that spread across networks, e.g., synchronization [2,3], diffusion [4], percolation [5][6][7][8], or the propagation of epidemics [9][10][11][12][13][14]. Epidemic spreading models have been particularly successfully in explaining the propagation of diseases and thereby have allowed the development of mitigation strategies for decreasing the impact of diseases on healthy populations.
A commonly-used model for reproducing disease spreading dynamics in networks is the susceptible-infected-recovered (SIR) model [15,16].It has been used to model such diseases as seasonal influenza and the SARS and AIDS viruses [17].This model groups the population of individuals to be studied into three compartments according to their state: the susceptible (S), the infected (I), and the recovered (R).When a susceptible node comes in contact with an infected node it becomes infected with a probability β and after a period of time t r it recovers and becomes immune.When the parameters β and t r are made constant, the effective probability of infection is given by the transmissibility T = 1 − (1 − β) tr [5,18].At the final state of this process, the fraction of recovered individuals R is the order parameter of a second order phase transition with a control parameter T .When T ≤ T c , where T c is the epidemic threshold, there is an epidemic-free phase with only small outbreaks.However when T > T c an epidemic phase develops.
In isolated networks the epidemic threshold is given by T c = 1/(κ − 1), where κ is the branching factor that is a measure of the heterogeneity of the network.The branching factor is defined as κ ≡ k 2 / k , where k 2 and k are the second and first moment of the degree distribution, respectively.
Because real-world networks are not isolated, in recent years scientific researchers seeking a better representation of interactions between real-world social networks have focused their attention on multilayer networks, i.e., on "networks of networks" [19][20][21][22][23][24][25][26][27][28][29][30][31][32].In multilayer networks, individuals can be actors on different layers with different contacts in each layer.This is not necessarily the case in interacting networks.Dickinson et al. [33] studied numerically the SIR model in two interacting networks.In their system, the net-works interact through inter-layer connections given by a degree distribution and there is a probability of infection between nodes connected through inter-layer connections.They found that, depending on the average degree of the inter-layer connections, one layer can be in an epidemic-free state and the other in an epidemic state.Yagan et al. [34] studied the SIR model in two multilayer networks in which all the individuals act in both layers.
In their model the transmissibility is different in each network because one represents the virtual contact network and the other the real contact network.They found that the multilayer structure and the presence of the actors in both layers make the propagation process more efficient, thus increasing the risk of infection above that found in isolated networks.This can enable catastrophic consequences for the healthy population.Buono et al. [35] studied the SIR model, with β and t r constant, in a system composed of two overlapping layers in which only a fraction q of individuals can act in both layers.In their model, the two layers represent contact networks in which only the overlapping nodes enable the propagation, and thus the transmissibility T is the same in both layers.They found that decreasing the overlap decreases the risk of an epidemic compared to the case of full overlap (q = 1).This case in which q = 1 in a multilayer network we will designate the "regular" SIR model.All of the above research assumes that individuals, independent of their state, will continue acting in many layers.In a real-world scenario, however, an infected individual may be isolated for a period of time and thus may not be able to act in other layers, e.g., for a period of time they may not be able to go to work or visit friends and may have to stay at home or be hospitalized.As a consequence, the propagation of the disease is reduced.This scenario is more realistic than one in which an actor continues to participate in all layers irrespective of their state [34,35].As we will demonstrate, compared to the risk measurement produced by "regular" multilayer network SIR models, our measurement approach more accurately indicates a decreased risk of epidemic propagation.

II. MODEL AND SIMULATION RESULTS
We consider the case of a multilayer network represented by two layers, A and B, of equal size N.The degree distribution in each layer is given by P i (k), with i = A, B and where k min and k max are the minimum and the maximum degree that a node can have.
In our epidemic model, an infected individual-depending on the severity of their illness-is isolated from both layers with a probability w during a period of time t w .At the initial stage, all individuals in both layers are susceptible nodes.We randomly infect an individual in layer A. At the beginning of the propagation process, each infected individual is isolated from both layers with a probability w for a period of time t w .The probability that an infected individual is not isolated from both layers is thus 1 − w.At each time step, a non-isolated infected individual spreads the disease with a probability β during a time interval t r after which they recover.When an individual j after t w time steps is no longer isolated they revert to two possibles states.When t w < t r , j will be infected in both layers for only t r − t w time steps and the infection transmissibility of j is reduced from 1 − (1 − β) tr to 1 − (1 − β) tr−tw , but when t w ≥ t r , j recovers and no longer spreads the disease.At the final stage of the propagation all of the individuals are either susceptible or recovered.The overall transmissibility T * ≡ T * β,tr,tw,w is the probability that an infected individual will transmit the disease to their neighbors.This probability takes into account that the infected is either isolated or non-isolated in both layers for a period of time and is given by Here the second and third term takes into account non-isolated and isolated individuals and represents the probabilities that this infected individual does not transmit the disease during t r and t r − t w time steps respectively.
Mapping this process onto link percolation, we can write two self-consistent coupled equations, f i , i = A, B, for the probability that in a randomly chosen edge traversed by the disease there will be a node that facilitates an infinite branch of infection throughout the multilayer network, i.e., where G i 0 (x) = kmax k=k min P i (k)x k is the generating function of the degree distribution and G i 1 = kmax k=k min P i (k) kx k−1 is the generating function of the excess degree distribution in layer i.Using the nontrivial roots of Eq. ( 2) we compute the order parameter of the phase transition, which is the fraction of recovered nodes R, where R is given by Note that in the final state of the process the fraction of recovered nodes in layers A and B are equal because all nodes are present in both layers.From Eqs. ( 1) and (2) we see that if we use the overall transmissibility T * as the control parameter we lose information about w, the isolation parameter, and t w , the characteristic time of the isolation.In our model we thus use β ≡ β T * as the control parameter, where β is obtained by inverting Eq. ( 1) with fixed t r [36].
Figure 2 shows a plot of the order parameter R as a function of β for different values of w, with t r = 6 and t w = 4 obtained from Eq. ( 3) and from the simulations.For (a) we consider two Erdős-Rényi (ER) networks [37], which have a Poisson degree distribution and an average degree k A = k B = 2, and for (b) we consider two scale free networks with an exponential cutoff c = 20 [7], where , with λ A = 2.5 and λ B = 3.5.We use this kind of SF network because it accurately represents structures seen in real-world systems.
In the simulations we construct two networks of equal size using the Molloy-Reed algorithm, and we randomly overlap one-to-one the nodes in network A with the nodes of networks B. We assume that an epidemic occurs at each realization if the number of recovered individuals is greater than 200 for a system size of N = 10 5 [38].Realizations with fewer than 200 recovered individuals are considered outbreaks and are disregarded.Figure 2 shows an excellent agreement between the theoretical equations (See Eq. 3) and the simulation results.The plot shows that the critical threshold β c increases with w, which indicates that the risk for an epidemic decreases with the isolation parameter w.Note that above the threshold but near it R decreases as the isolation w increases, indicating that isolation for even a brief period of time reduces the propagation of the disease.The critical threshold β c is at the intersection of the two Eqs.( 2) where all branches of infection stop spreading, i.e., f A = f B = 0.This is equivalent to finding the solution of the system det(J − I) = 0, where J is the Jacobian of the coupled equation and I is the identity, and where κ A and κ B are the branching factor of layers A and B, and k A and k B are their average degree.By numerical evaluations of the roots of Eq. ( 4) we found the physical and stable solution for the critical threshold β c , which corresponds to the smaller root of Eq. ( 4) [39].Figure 3 shows a plot of the phase diagram in the plane β − w for (a) two ER multilayer networks [37] with average degree k A = k B = 2 and (b) for two power law networks with an exponential cutoff c = 20 [7], with λ A = 2.5 and λ B = 3.5.In both w c = 0.88, above which there is only an Epidemic-free phase.
The regions below the curves shown in Fig. 3 correspond to the epidemic-free phase.
Note that for different values of t w those regions widen as w increases.Note also that when t r = t w there is a threshold w c above which, irrespective of the risk (β c ), the disease never becomes an epidemic.For t w = 0 and w = 0 we recover the regular SIR process in a multilayer network that corresponds to β c ≈ 0.043 with k min = 1 and k max = 40 [40] in Fig. 3 As expected and confirmed by our model, the best way to stop the propagation of a disease before it becomes an epidemic is to isolate the infected individuals until they recover, which corresponds to t w = t r and w > 0. Because this is strongly dependent upon the resources of the location from which the disease begins to spread and on each infected patient's knowledge of the consequences of being in contact with healthy individuals, the isolation procedure can be difficult to implement.The phase diagram indicates that the regular SIR model applied to multilayer networks, which corresponds to the case t w = 0, overestimates the risk β c of an epidemic.This overestimation could have significant consequences if a health service declared an epidemic when it was not in fact occurring.In the limit t w = 0 and w → 0 we revert to the regular SIR model in multilayer networks [35].As w increases and when t w = 0 there is always an overestimation of the risk.Note that the plot shows that when the percentage of infected individuals who are hospitalized or isolated in their homes is between 40 and 60 percent, the most-used regular SIR model indicates double the actual risk of infection.The declaration of an epidemic by a government health service is a non-trivial decision, and can cause panic and chaos and negatively effect the economy of the region.Thus any epidemic model of airborne diseases that spread in multilayer networks, if the projected scenario is to be realistic and in agreement with the available real data, must take into account that some infected individuals will be isolated for a period of time.In particular, in such diseases as the recent outbreak of Ebola in Western Africa, in which the hospitalization of patients is a significant factor strongly affecting the propagation of the outbreak, research take this hospitalization into account [42][43][44].Note also that this isolation can also delay the onset of the peak of the epidemic and thus allow health authorities more time to make interventions.This is an important topic for future investigation.

III. DISCUSSION
In summary, we study a SIR epidemic model in two multilayer networks in which infected individuals are isolated with probability w during a period of time t w .Using a generating function framework, we compute the total fraction of recovered nodes in the steady state as a fraction of the risk of infection β and find a perfect agreement between the theoretical and the simulation results.We derive an expression for the epidemic threshold and we find that β c increases as w and t w increase.For t w = t r we find a critical threshold w c above which any disease can be stopped before it becomes an epidemic.From our results we also note that as the isolation parameter and the period of isolation increases the overestimation increases.Our model enables us to conclude that the regular SIR model of multilayer networks overestimates the risk of infection.This finding is important and highly relevant to the work of researchers developing epidemic models.Our results can also be used by health authorities when implementing policies for stopping a disease before it becomes an epidemic.
Figure 1 shows a schematic of the contributions to Eqs. (2).

FIG. 1 :
FIG. 1: Schematic of a multilayer network consisting of two layers, each of size N = 12.The black nodes represent the susceptible individuals and the red nodes the infected individuals.In this case, we consider t w < t r .(a) The red arrows indicate the direction of the branches of infection.All the branches spreads through A and B because the infected nodes are not isolated and thus interact in both layers.(b) The gray node, represents an individual who is isolated from both layers for a period of time t w .(c) After t w time steps the gray node in (b) is no longer isolated, and can infect its neighbors in A and B, if they were not reach by another branch of infection, during t r − t w time steps (Color on line).

FIG. 2 :
FIG. 2: Simulations and theoretical results of the total fraction of recovered nodes R, in the final state of the process, as a function of β, with t r = 6 and t w = 4, for different values of w.The full lines corresponds to the theoretical evaluation of Eq. 3 and the symbols corresponds to the simulations results, for w = 0.1 ( ) in green, w = 0.5 (✷) in blue and w = 1 (✸) in violet.The multilayer network is consisted by two layers, each of size N = 10 5 .For (a) two ER layers with k A = k B = 2, k min = 1 and k max = 40 and (b) two scale free networks with λ A = 2.5, λ B = 3.5 and exponential cutoff c = 20 with k min = 2 and k max = 250 (Color online).
(a) and β c ≈ 0.019 with k min = 2 and k max = 250 in Fig. 3(b).Although in the limit c → ∞, β c → 0, most real-world networks are not that heterogeneous and exhibit low values of c [9, 41].

Figure 3 (FIG. 4 :
FIG. 4: Ratio of β c (t w ) to β c (0) as a function of w.For t w = 1, 2, 3, 4, 5, 6 from bottom to top for (a) two ER networks with k A = k A = 2 with k min = 1 and k max = 40 and (b) two power law networks with λ A = 2.5 and λ B = 3.5 with k min = 2 and k max = 250, with exponential cutoff c = 20.In both Figures, the limitw → 0 correspond to a regular SIR process, and as w increases the overestimation increases.