key: cord-0434497-ij9ut8g5
authors: Fujiwara, Naoya; Sonawane, Abhijeet R.; Iwayama, Koji; Aihara, Kazuyuki
title: Contribution of hidden modes to nonlinear epidemic dynamics in urban human proximity networks
date: 2015-12-07
journal: nan
DOI: nan
sha: ea00a3307b67794b0189e371d5df4ce06711e7ba
doc_id: 434497
cord_uid: ij9ut8g5

Recently developed techniques to acquire high-quality human mobility data allow large-scale simulations of the spread of infectious diseases with high spatial and temporal resolution.Analysis of such data has revealed the oversimplification of existing theoretical frameworks to infer the final epidemic size or influential nodes from the network topology. Here we propose a spectral decomposition-based framework for the quantitative analysis of epidemic processes on realistic networks of human proximity derived from urban mobility data. Common wisdom suggests that modes with larger eigenvalues contribute more to the epidemic dynamics. However, we show that hidden dominant structures, namely modes with smaller eigenvalues but a greater contribution to the epidemic dynamics, exist in the proximity network. This framework provides a basic understanding of the relationship between urban human motion and epidemic dynamics, and will contribute to strategic mitigation policy decisions.

Epidemics of infectious diseases in the human population, e.g. the SARS outbreak in 2002-2003 [1] , 2009 H1N1 influenza pandemic [2] , and Ebola outbreak [3] , can be a serious factor in human mortality, and have a significant socio-economic impact in terms of reducing a population's healthy years because of morbidity. In recent years, the mathematical modelling of the outbreak and spread of infectious diseases has mainly been performed using two approaches: agent-based models [4] [5] [6] [7] [8] [9] [10] and structured metapopulation models [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] . Both techniques incorporate real or synthetic data on the long-range mobility and migration of populations, but vary in their granularity of the population, from individuals to sub-groups of society. Various spatial scales have been studied, ranging from specific geographic locales such as cities [4, 5] to nations [6-8, 23, 24] and intra-/inter-continental regions [15, 16] .

In particular, it is imperative to study the mechanisms of an epidemic's spread in urban scenarios, considering restrictions on available space and transportation methods (e.g. traffic fluxes, travel routes, timescales of modes of transport), which result in high density and heterogeneous contact patterns among residents. These factors make the populace vulnerable to epidemics of diseases that spread through exhaled aerosols, e.g. [25] . To understand the details of epidemic dynamics within cities, it is crucial to exploit the heterogeneous contact patterns derived from human mobility data. Fortunately, recent technical advances in data acquisition methods enable us to incorporate human mobility data obtained from various sources [26] [27] [28] into mathematical models [16] .

Network epidemiology [20, [29] [30] [31] [32] [33] is a useful means of uncovering the dependence of the contagion dynamics on the heterogeneous contact pattern. The approach of tracking connections between people provides a comprehensive framework with which to study the effects of the underlying network of people in the region of interest (e.g. cities). The present understanding of epidemic or social contagion processes has been bolstered by many network-based studies [16, 29, 34] , which treat cities as social petri-dishes [35] .

Analytical approaches such as heterogeneous mean field (HMF) theory [36] [37] [38] [39] and moment closure approximation [40] [41] [42] provide an important bridge between the network topology and the spread of epidemics in complex networks. For example, the epidemic threshold is absent in scale-free networks with no degree correlation [36] . However, certain simplifications made to obtain analytical results, e.g. neglecting the degree correlation in the HMF [37] , could be superfluous, and these approximations may not necessarily hold in realistic networks. Therefore, a novel analytical approach is required to better understand the spread of epidemics in heterogeneous populations.

In this paper, we outline a framework for modelling a proximity network from transportation data, and propose an analytical method to estimate the infection probability of each individual from stochastic simulations. The prevalence, i.e. the average number of recovered agents who were once infected before the spread of the epidemic subsides, is of particular interest.

We can construct an individual-based network using a dataset of human mobility. The People flow data [28] record the one-day movement of individuals living in the Kanto region of Japan, which includes the Tokyo metropolitan area. Details of the dataset are given in the Methods section. We randomly chose the mobility track of N individuals from this dataset.

Time-dependent proximity networks were then constructed by connecting individuals when they came within a certain geographical proximity, defined by a distance threshold D (Fig. 1 ).

Let us denote the adjacency matrix of the proximity network as A(t). All results in the main text were obtained with D = 1, 000 m and N = 10, 000. Note that we observed the percolation-like transition and the scaling relation of the giant cluster component size ratio as S(N, D)/N ∼ f (N 0.605 D). Using this scaling relation, D = 1, 000 m with N = 10, 000 corresponds to D ≈ 8.3 m with the real population in the corresponding region, where N ≈ 27, 500, 000. A detailed discussion can be found in Supplementary Information SI1.

We simulated the disease propagation using the agent-based SIR model [43] on the abovementioned network, which assigns susceptible (S), infected (I), or recovered (R) states to each agent. We assumed that each agent repeats the same trip pattern every day in the same way as recorded in the People flow data. When a susceptible agent is connected with an infected one, the former is infected with probability β∆t for a time interval ∆t. If a susceptible agent is connected with d infected agents, they are infected with probability [29] . Further details of the stochastic simulation are given in the Methods section. In this study, β∆t is sufficiently small which enables us to perform the simulation with the discretised time step. An infected agent recovers with probability µ∆t over this time interval. Once A(t), β, and µ are given, stochastic simulations can be conducted to determine the epidemic dynamics. We denote the probabilities that agent j is in state S, I, or R at time t as s j (t), i j (t), and r j (t), respectively; the equality s j (t) + i j (t) + r j (t) = 1 holds. We define the infection probability of agent j as r j (∞), because a recovered agent has experienced the infected state before recovering.

The epidemic dynamics can be characterized by three different stages [44] : the initial stage, in which stochastic fluctuations are dominant, the exponential growth stage, and the final stage, where nonlinearity suppresses the further spread of the disease. The latter two stages representing the dynamics of the outbreak are of particular interest. In these stages, the time evolution of the epidemic can be approximated by the following deterministic differential equations:

Here, we approximate the time-dependent adjacency matrix A(t) with its time averaged form A [45] . This averaging gives a sufficient approximation of the time evolution of s j , i j , and r j if the parameters β and µ are sufficiently small, as discussed in Supplementary Information SI2. We also assume that node j being susceptible and node k being infected are statistically independent events [46] . Although this assumption does not hold for small networks [43] , we have verified that the numerical solution of equation (25) gives a good approximation of the dynamics if D is much greater than the percolation transition point and the giant cluster component is sufficiently large.

One of the most common methods used to analyse the contagion processes within networks is HMF [36] [37] [38] [39] . This approach assumes that, in a statistical sense, nodes grouped by the degrees behave in the same way. Under this assumption, one can derive the equation for the infection probability of each agent, and, in the absence of degree correlation, the epidemic threshold, i.e. the critical value of the epidemic's spread, can be represented as a function of the first and second moments of the degree distribution [37] . One question that persists is whether such an approximation is valid in realistic networks of human contact. Actually, the accuracy of HMF depends on factors such as the mean degree and first-neighbour degree [47] . For the simplest case of the absence of degree correlation, where HMF is often applied, the component of the first eigenvector is proportional to the degree [44] . However, this assumption does not hold for the proximity network (Fig. 2a) . This highlights the need to develop an analytical framework to identify people with a high infection probability along with the overall infection propagation pattern. In this study, we have developed a method based on spectral decomposition and mode truncation which demonstrates that a small number of dominant modes, which may not necessarily have the largest eigenvalues, give a higher contribution in the prevalence of epidemics.

In this section, we present a method for analysing the final epidemic size based on the spectral analysis of the averaged matrix A, and elucidate the relevant dynamics of the epidemic's spread. The node-wise probabilities s j (t), i j (t), and r j (t) can be expanded as

where φ (a) j denotes the ath eigenvector of A associated with the eigenvalue λ a . All eigenvalues of a real symmetric matrix A are real, and can be labeled in descending order as λ 1 ≥ λ 2 ≥ . . . ≥ λ N . The normalization condition N j=1 (φ (a) j ) 2 = 1 is adopted. Since A is symmetric, the left and right eigenvectors coincide, and the expansion coefficients can be obtained

j , respectively. Equation (25) can be rewritten in terms of these expansion coefficients (Supplementary Information SI3). The time evolution of the average number of infected and recovered agents,

, is plotted in Fig. 2b . The time series of i j (t) is converted to that ofî a (t) as in Fig. 2c .

In the exponential stage of the epidemic spreading, 1 − s j , i j , and r j are small. The exponential growth of each coefficientî a (t) can be described by the following linearised equation [48] dî a dt ≈ (βλ a − µ)î a

(see Supplementary information SI4 for the derivation of this expression from the nonlinear mode coupling equation). Hence,î a (t) =î a (0)e (βλa−µ)t holds. It is evident from this equation that a mode with a larger eigenvalue λ a grows faster in this stage. In particular, the exponential growth rate of the number of infected agents is characterised by βλ 1 − µ.

The stochastic simulation of the agent-based model verifies thatî 1 is indeed dominant in the initial stage (Fig. 2c ). In addition, the epidemic threshold is related to λ 1 as β c = µ/λ 1 ( Fig. 2d ) [48] .

However, as shown in Fig. 2c , the coefficientsî a (t) φ (a) (a = 2, 3, . . . , N) are comparable (1) in the later stages of the epidemic, where v = N j=1 v j /N is the node-wise average of a vector v. The importance of each mode needs to be assessed to ascertain its dominance in the spreading mechanism. This can be achieved by introducing a quantifier of the contribution of each mode to the prevalence as

The definition of this contribution enables us to account for the prevalence as the sum of the contributions of all modes, N j=1 r j (∞) = N a=1 C a . It is worth introducing the contribution C all a for the case where all agents are infected, i.e. r j (∞) = 1 for all j. Since the coefficients satisfyr a (∞) = N φ (a) in this case, one can easily verify that C all a = N φ (a) 2 holds.

Interestingly, the contribution C all a does not monotonically increase with the eigenvalue, although a positive correlation can be observed (Fig. 3a ). Since C a and C all a are functions of the average of the eigenvector components, a mode associated with a smaller eigenvalue may contribute more than one with a larger eigenvalue. Although the larger eigenvalue denotes faster growth in the exponential regime, it does not necessarily mean a larger contribution to the prevalence.

It is interesting to note that the "hidden" important structures, corresponding to modes that make a large contribution to the prevalence but have smaller eigenvalues, are unravelled through the spectral analysis. It is still possible to detect such modes when the prevalence is much smaller than the all-infected case (Fig. 3b) . The heterogeneity in the contribution is evident from the cumulative contribution Γ n ≡ n a=1 C σ(a) and Γ all n ≡ n a=1 C all σ(a) (Fig. 3c) . Here, the modes are sorted such that C all σ(1) ≥ C all σ(2) ≥ · · · ≥ C all σ(N ) , where σ(a) denotes the permutation of the index set {1, 2, 3, . . . , N}. Note that Γ all N = 1 holds, because the infection pattern is represented using all modes. Figure 3c suggests Γ all 1000 ≈ 0.9 for the all-infected case, indicating that approximately 90% of the prevalence is described by the top 10% of modes for the present proximity network. This figure indicates that the number of modes needed to describe the prevalence decreases if the prevalence is smaller than 1.

The above observations led us to the idea of describing the epidemic dynamics with a small number of dominant modes that give a high contribution. In the case of the agentbased SIR dynamics, we can derive the truncated equation for the contributions at the final Newton's method provides a good description of the contribution obtained by the stochastic simulation ( Fig. 3d) . We further note that considering fewer than N modes suffices to approximate the prevalence as obtained by the stochastic simulation (Figs. 2d, 3c) . Hence, this procedure replaces the need to solve an ordinary differential equation in 2N independent variables (25) with the requirement to use information from the M dominant modes.

As discussed above, the essence of theoretical analyses of epidemic spreading within networks is to infer the dynamical information from the network topology. Conventionally, HMF [36] [37] [38] [39] uses degree distributions and degree correlations as topological information.

In contrast, our approach uses spectral information, which can take more general properties of the network topology into account. As described in the Methods, the final size equation

in HMF [37] is derived from equation (5) under the following simple assumptions: (i) degree correlation is absent, and (ii) the first mode almost perfectly governs the dynamics. As shown in Fig. 2a , the first assumption is not satisfied in the proximity network. Moreover, it is evident from Figs. 2d and 3a that considering only the first mode is not sufficient to account for the whole epidemic dynamics, as C all 1 is less than 6%. The contribution of other dominant modes must be taken into account to quantify the epidemic dynamics ( Fig. 3d) . Thus, HMF with no degree correlation is not a sufficient approximation for the proximity network. It would be possible to improve the approximation by considering the degree correlation in HMF [38] , but it is generally difficult to include information about the second-nearest neighbours. In contrast, the proposed approach provides the flexibility to improve the approximation by using an arbitrary number of dominant modes.

Another feature of epidemic spreading in proximity networks is the strong heterogeneity of the infection risk in space and time. It is important to identify such locations that have a high probability of infection, namely "hot spots", for the mitigation and management of the epidemic. The spatial distribution of the hot spots can be mapped to provide an insight into the nature of epidemic spreading in urban areas (Fig. 4 ). More importantly, to uncover the relevant epidemic dynamics, such identification should be conducted with fewer parameters than the number of agents.

For the present model to be predictive, let us infer the infection probability of a test agent from the mobility pattern of other agents. The probability of infection of this agent depends on the presence of other infectious agents in the vicinity, which in turn varies from region to region and at different times of day, 0 ≤ τ ≤ T = 1, 440 min. Let x = (x, y) T be a position vector and x j (t) be the position of agent j at time t. We introduce a kernel function K(x) signifying the presence or absence of any interaction between two agents based on the distance between them. Since agents are connected within the threshold D, the Heaviside step function K(x) = Θ(|x| − D) is adopted as the kernel function. This allows us to define a spatio-temporal risk factor of infection ρ(τ, x) as

which can be interpreted as the weighted sum of probabilities r j (∞). Integrating ρ(τ, x p (τ )) along the trajectory of the test agent x p (τ ), one can obtain the infection probability of this agent. Details of the derivation are discussed in Supplementary Information SI5. Figure 4a depicts the spatial distribution of the risk factor ρ(τ, x) across the Kanto region of Japan. In the present model, which assumes homogeneous parameter values of β and µ, the high-risk area during the daytime is the central business district, where many workers are located.

In the nighttime, certain suburban areas from where people commute to the centre become high risk. Furthermore, people travelling to the central business district from their suburban residences allow the disease to spread over the whole metropolitan area. Figure 4c shows that the infection probability of the test agents can be estimated from the integration of ρ(τ, x p (τ )), obtained from stochastic simulation, along its trajectory.

Instead of counting the infection probability r j (∞) in the neighbourhood of the region of interest, the spectral properties of the averaged adjacency matrix can be incorporated into the risk factor by choosing the first few high-contribution modes. Namely, one can approximate r j (∞) and ρ(τ, x) asr

Hazard maps based on the mode truncation at different times are similar to those given by the stochastic simulation (Fig. 4a) . Moreover, Fig. 4d shows that the integration ofρ (100) (τ, x p (τ )) along the path of the test agents that asked about basic individual information such as gender, age, occupation, places visited, and trip modes on one day. For the sake of privacy, spatial information was recorded with a resolution such that the complete location could not be determined. The questionnaires allowed us to access advanced information such as transportation modes, travel times, transfer points, and trip purposes. Traces of the trips between two places reported in the raw data were interpolated using geographical information systems methods [28] .

The proximity network was constructed by placing a link between two individuals at time t if they were within the threshold distance D of one another (Fig. 1b) . We added another constraint by only placing a link between individuals using the same transportation mode; it is impractical to connect individuals walking along the road with those traveling in an automobile or train because of the improbability of the transmission of infection. The connectivity of the network can be described by the time-dependent adjacency matrix A(t),

where A jk (t) = A kj (t) = 1 if individuals j and k are connected at time t and A jk (t) =

A kj (t) = 0 otherwise. This allowed us to obtain networks for N agents at every ∆t time interval. Proximity networks are spatially embedded temporal networks, which appear in various forms of transportation and infrastructure networks [49] .

The agent-based SIR model simulation was performed with the same time-dependent adjacency matrix A(t). The time evolution of the probabilities s j (t), i j (t), and r j (t) was computed using 500 independent stochastic simulation runs with different random initial conditions. We also assumed that the human mobility patterns had a periodicity of 24 h owing to the circadian nature and economic situation of people going to their place of work and back home. Parameter values were taken as ∆t = 10 min, µ = 2.0 × 10 −4 min −1 , and β = 1.5 × 10 −5 min −1 , except for Fig. 2d , in which β varies.

One of the main results of this article, the equation for the asymptotic solution of equation (5), is derived as follows. Substituting the expansion (26) into equation (25), we obtain

Let us define

so that the solution of this equation is given as

or equivalently,ŝ

For t → ∞, i j (∞) should vanish, and the conservation of probability condition is given as s j (∞) + r j (∞) = 1. This is equivalent tô

Substituting equation (11) into this expression, we obtain

This is a transcendental equation forr a (∞), and one can determine the infection probability by solving this equation instead of solving the differential equation. We choose M (≤ N) modes and neglect the other modes to obtain

where σ(a) and σ(b) denote arbitrary permutations of modes. Finally, we multiply both sides by φ (σ(a)) , and obtain equation (5) by choosing the top M modes with respect to their contribution.

The relationship between the equation for the expansion coefficients (13) and that obtained with HMF can be described as follows. We concentrate on the simplest case, where the probability P dd ′ that a node with degree d and one with degree d ′ are connected is proportional to the product of their degrees, i.e.

the degree distribution of the network satisfying N −1 d=1 P (d) = 1. In this case, the largest eigenvalue of the adjacency matrix is λ 1 = d 2 / d , and the component of the corresponding

Here, we denote the degree of node j as d j .

We assume that the probability of nodes with the same degree is identical, i.e. r j (t) = r d j (t) holds.

Substituting the expression for the first eigenvector into equation (13) and neglecting other modes with the initial condition s j (0) ≈ 1, we get

where the sum in the second equation is taken for degree d. Note that

holds, where we have defined

Substituting this equation into equation (16) and taking the limit t → ∞, we obtain

This is the equation for Ψ(∞) in HMF derived in [37] . * Electronic address: fujiwara@csis.u-tokyo.ac.jp 

We discuss an approximation to replace the time-dependent adjacency matrix with the time-averaged one, and determine the condition under which this approximation is valid.

Let us consider a general example where a dynamical variable X(t) evolves via contact with other nodes asẊ

where the parameter β represents the coupling strength and A(t) is the instantaneous adjacency matrix of a temporal network at time t. Equation (21) can be solved as

where exp T denotes the time-ordered exponential. If A(t) changes with the time interval τ , equation (22) can be rewritten as

where A = L−1 l=0 A(t + lτ )/L is the time-averaged adjacency matrix. The above equation can be interpreted as follows. The terms of β m in the power series expansion in equation (22) consist of time-ordered products of m adjacency matrices of the temporal network. These terms collect paths of length m, namely all possible indirect interactions occurring m times between two nodes at different times taking into account the time order of the interactions.

If we consider the case of a small value of β, the time evolution can be approximated up to the first order. This corresponds to assuming that the interaction only takes place by direct contact between agents, and all multi-step interactions during the time interval T = Lτ are negligible. In such a case, the time-averaged network is sufficient to describe the dynamical processes in the temporal network. Note that the time-order information from one day does not appear in the first-order terms.

One can derive the mode coupling equation from the following differential equations (1) in the main text:

Let us substitute equation (2) in the main text,

into this equation. Taking into account the eigenvalue equation for the eigenvector φ

it is easy to verify that

Multiplying φ (a) j on both sides of the above equations and taking the sum over j to use the orthonormal condition

where δ ac is Kronecker's delta, we obtain

j . Note that each element of G abc consists of the product of the eigenvector components and is time-independent.

It is easy to extend this equation to the case where the static adjacency matrix A is asymmetric. Let us define the left and right eigenvectors of A for the eigenvalue λ a as ψ (30) holds

By considering M (≤ N) modes and neglecting the other modes in equation (30), we obtain the transient dynamics of the epidemic from a small number of modes.

Equations (30) contain mode coupling terms, but the modes are decoupled in the linearised equation. Let us take s j (t) = 1 − ∆s j (t) and focus on the initial stage of the epidemic spreading. Assume that ∆s j (t) and i j (t) are small, and approximate s j i j ≈ i j by neglecting the product ∆s j i j . Since s j appears only as the product s j i j , the linearised approximation allows us to put s j ≈ 1. Under this assumption, we can take

Substituting this into the evolution equation forî a in equation (30), the linearised equation

By taking the summation over b using equation (31), we obtain

= β N c=1 λ cîc δ ac − µî a

= (βλ a − µ)î a .

In this way, we derive equation (3) in the main text.

We now discuss how to estimate the infection probability of a test agent. Let x j (t) = (x j (t), y j (t)) T be the position of agent j at time t. Since the infection probability depends on the distance between agents, we introduce the kernel function K(x). In this study, we used the Heaviside step function K(x) = Θ(|x| − D), where Θ(x) = 1 for x ≥ 0 and Θ(x) = 0 otherwise. Then the element of the adjacency matrix A jk (t) is given as

We define the spatio-temporal risk factor of infection as

where we take 0 ≤ τ ≤ T = 1440 min, because we assume the mobility pattern is periodic over 24 h and ρ(τ + T, x) = ρ(τ, x) holds.

We want to estimate the risk of infection of a test agent p who travels inside the urban area with an arbitrary trajectory and returns to the original location after time T . Equation (1) in the main text can be solved as s j (t) = s j (0)e −βΨ j (t) ,

where

For t → ∞, i j (∞) should vanish, and the conservation of probability condition is given as r j (∞) = 1 − s j (∞). Therefore, the infection probability of this agent is

This equation is transcendental and cannot be solved analytically. However, if N is sufficiently large and the contribution of the test agent to ρ(τ, x) is negligible, we can approximate the infection probability of the test agent p using ρ(τ, x). Since the timeaveraged adjacency matrix is given in terms of the risk factor as 

we obtain

by taking s j (0) = 1 and r k (0) = 0. Therefore, the final infection probability of a susceptible agent traveling along an arbitrary trajectory is given by the time integral of the risk factor along its trajectory.

If the mode truncation with M (≤ N) modes r j (t) ≈r 

The infection probability is approximated with the mode truncation

by replacing ρ(τ, x) withρ (M ) (τ, x) in equation (44) .

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All authors designed the research. N.F. and A.R.S. carried out the numerical simulations.N.F., A.R.S, and K.I. analysed the results. N.F., K.I., and K.A. contributed to the analytical calculations. N.F. and A.R.S. wrote the bulk of the manuscript. K.I. and K.A. edited the manuscript.

The authors declare no competing financial interests.