Abstract
The ease of travelling between cities has contributed much to globalization. Yet, it poses a threat on epidemic outbreaks. It is of great importance for network science and health control to understand the impact of frequent journeys on epidemics. We stress that a new framework of modelling that takes a traveller’s viewpoint is needed. Such integrated travel network (ITN) model should incorporate the diversity among links as dictated by the distances between cities and different speeds of different modes of transportation, diversity among nodes as dictated by the population and the ease of travelling due to infrastructures and economic development of a city and roundtrip journeys to targeted destinations via the paths of shortest travel times typical of human journeys. An example is constructed for 116 cities in China with populations over one million that are connected by highspeed train services and highways. Epidemic spread on the constructed network is studied. It is revealed both numerically and theoretically that the traveling speed and frequency are important factors of epidemic spreading. Depending on the infection rate, increasing the traveling speed would result in either an enhanced or suppressed epidemic, while increasing the traveling frequency enhances the epidemic spreading.
Introduction
Controlling an epidemic, e.g. severe acute respiratory syndrome (SARS), H1N1 swine influenza and Ebola, in the midst of frequent movements of infected persons via cars, trains and aeroplanes poses a challenging problem. In network science, much effort and progress has been made on understanding epidemics in singlelayered networks^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20} and multilayered networks^{21,22,23,24,25,26,27,28}. In singlelayered static networks with an immobile agent at each node, for example, no finite epidemic threshold exists for scalefree (SF) networks and a tiny initial infection eventually spreads^{11}. A delicate balance between the number of high degree nodes and the topological distance between them^{29} is shown to be crucial. The same result holds for reactiondiffusion models with random diffusion of agents among nodes with infections only among the agents momentarily on the same node^{6}. Recently, how human dynamics affects an epidemic has become the focus of research^{14,18,30,31,32,33}, but the diversity of links and the time spending on journeys are largely ignored. Reallife networks, e.g. power grids and the internet, are often multilayered networks^{34,35}, with their mutual influence and cascades being hot research topics^{36,37}. Epidemics in twolayered networks also received much attention^{21,22,23,24,25,26} and the layer for infection processes actually shares the same set of nodes with the layer for information exchanges.
For diseases spreading through human contacts, it is most important to understand the impact of frequent journeys. There exist many single and multilayered transportation network models^{38,39,40,41,42,43}, with the layers representing networks of airports, railways, highways, etc. coupled together. To incorporate epidemics, however, random diffusion of people on such networks will be an oversimplification, as a journey involves a planned route to a destination using mixed modes of transportation. These directed movements should be incorporated in studying epidemics.
The ease and speed of intercity travels offered by the growth in the airline and highspeed train^{44} industries and better highways has contributed to making our Earth a global village. These intercity travels readily spread a disease to different places. However, the big populations in major cities and densely packed travellers on multiple means of transportation of various speeds add further complications. A reliable framework for studying the effects of travelling on epidemics has yet to be constructed. Earlier works on epidemics in airport and railway networks often modelled journeys as random diffusion of agents^{4,5,45}. The obvious shortcomings are: (i) real journeys typically involve multiple means of transportation instead of agents all travelling the same way; (ii) neighboring stations have different distances that affect the chance of infection instead of identical distance between adjacent nodes; (iii) real journeys are roundtrip with an destination instead of random diffusion. It should be noted that intracity travel is also inhomogeneous. It is, therefore, of fundamental importance to construct a framework incorporating the differences in travelling means and distances between cities. We propose here such a framework to incorporate inhomogeneity among the links and roundtrip journeys with intended destination. It is found that infections at the links greatly affect the epidemic threshold and the traveling speed and frequency are key factors in determining the extent of an epidemic.
Results
An integrated travel network (ITN) model
Our integrated travel network (ITN) model accounts for different means of transportation by different kinds of links. Figure 1(a) shows schematically an intercity transportation network emphasizing its link inhomogeneity: Links of faster transportation (dashed lines), e.g. airlines and highspeed trains, connecting major cities and links of slower transportation (solid lines), e.g. highways, connecting to surrounding cities (blue nodes) via part of a highway network.
A journey starts from a city i to an intended destination j through intermediate places along the path that takes the shortest time, which necessarily invoke the actual distance between two cities and the mode of transportation. The return journey could follow the same path or an alternative path, as depicted in Fig. 2(a,b). The ITN aims to incorporate the key features of how human travel, namely roundtrip journeys of shortest time through multiple means of transportation. Here, we invoke the travel time, which depends on the distance and the means of transportation, as the key factor, instead of the effective distance^{43}. Instead of emphasizing the multilayered network structure as in previous works, ITN takes a traveller’s viewpoint that journeys take place in a singlelayered undetachable network with a diversity of links connecting cities representing an inhomogeneous transportation network, see Methods for details. It aims to provide a step closer to a realistic description of human journeys and an alternative platform for studying epidemics on which finer and further details on local area transportation could be added.
Epidemic spreading on ITN
Contacts during journeys are important for epidemics. An example is the 2009 H1N1 cases in a Singapore’s hospital that 116 of 152 patients in two months were classified as air travelassociated imported cases^{46}. The time that travellers meet becomes a crucial factor. It is related to the length of a link and how fast agents travel on it. As a minimum model, we consider two speeds v_{s} and v_{f} with v_{s} < v_{f} (see solid and dashed lines in Fig. 1) representing slower and faster transportation. An agent starts a roundtrip journey from a node (home) to a destination chosen randomly (upper Fig. 3) through intermediate (middle) nodes along the path of shortest travel time^{18}. Let r_{ij} be the distance between neighbouring nodes i and j. The time travelling on the link is
with v = v_{s} or v_{f} depending on the type of transportation. To account for travel time, a link from node i to node j is divided into τ_{ij} segments, with τ_{ij} = t_{ij} if mod(r_{ij},v) = 0 and τ_{ij} = int(t_{ij}) + 1 if mod(r_{ij},v) ≠ 0 (lower Fig. 3), where mod(x,y) represents the modulo operation and int(x) taking the integral part of x.
For epidemic on ITN, we invoke the susceptibleinfectedsusceptible (SIS) model^{6,9,10,11,12,13,14,15,16,17}. A susceptible agent will be infected if it contacts an infected agent, with an infectious rate β. There are travelling and nontravelling agents in a population. Generally, people travelling are in closer contact and have a higher infectious rate β_{2} than the nontravelling agents with β_{1}^{47}. An infected agent recovers and becomes susceptible with a recovery rate μ. For travelling agents, we assume that infections take place only among agents in the same segment k_{r} (1 ≤ k_{r} ≤ τ_{ij}) of a link. For nontravelling agents, the SIS process is confined to nontravelling agents at the same node. Explicitly, a nontravelling susceptible agent at node i has a probability 1−(1−β_{1})^{n}_{i,I} to be infected at a time step, when there are n_{i,I} infected nontravelling agents at the node. Similarly, a susceptible agent at a segment of a link has a probability to be infected when there are infected agents at that section k_{r}.
An example of ITN: China’s big city network
Buses on highways and highspeed trains in China together provide an example of ITN. To include a large population and to reduce the number of nodes, we consider 116 cities with population over one million (see Table S1 in Supplementary Information (SI)). From highspeed train schedule, 61 cities are served by routes of highspeed trains. For the remaining 55 cities, we construct the highway links as follows. A highway link is added between two cities in the same province or two neighbouring provinces when there is a highway between them. Finally, highway links are added to connect neighboring highway and highspeed railway nodes in the same province. Figure 4 shows the resulting ITN of 116 cities with two types of links. We give the structural properties in SI. It has a mean degree 〈k〉 = 4.25 and a high clustering coefficient of C = 0.35. The degree distribution is shown in Fig. S1(a) in SI. Table S2 in SI gives the lengths of the links.
Typically, travels between major cities and/or nearby cities are more frequent. This was modelled by assigning weights to a link, where N_{i} denotes the population at node i and r_{ij} the distance between nodes i and j^{48,49}. To incorporate factors including transportation infrastructure and convenience, we modified the weight in ITN to
where S_{ij} represents the daily services of highspeed trains between nodes i and j and thus an indication of how convenient it is and S_{ij} = 0 for highway links. Values of S_{ij} as obtained by train schedules are listed in Table S2 in SI. Summing W_{ij} for the k_{i} links give the weight W_{i} of node i as
To set up a model for simulations, we measure population in units of 5000 and distance r_{ij} in kilometers. Thus cities of N_{i} ≥ 200 are considered and N_{i} is of the real population. The corresponding weight distribution is shown in Fig. S1(b) in SI. Sensitivity to the choice of measuring populations in lots of 5000 is tested in Fig. S2 in SI. In each time step, agents starts a roundtrip journey from node i, where the parameter p_{T} is chosen so that , i.e., people travelling are fewer than a city’s residents. It is related to the small fraction f of the total population starting a journey every time step by
An agent from node i picks a destination j according to the probability
and follows the path of shortest travel time. An agent typically travels on slower transportation in the local area before transferring to highspeed train followed by local transportation to the destination. ITN captures the inhomogeneous means of travelling better than multilayered networks. An agent spends some time at the destination before the return trip begins, which is taken to be 5 time steps corresponding to 5 hours^{50,51}. Returning to home city, an agent becomes a nontraveller until the next journey. Figure 3 shows a schematic journey. The travelling dynamics leads to a steady state in which residents among are nontravellers at node i. The number of all nontravellers depends on f (see Fig. S2 in SI) linearly for f ≤ 0.01. We thus take f = 0.01. The values of n_{i} and for the 116 cities are shown in Fig. S3 in SI.
Epidemic spreading on China’s ITN network
Let v_{s} = 100 (km/h) be the highway traffic speed and v_{f}> v_{s} be speed of highspeed train. The speeds and r_{ij} determine the time τ_{ij} of each link. After the travelling population reaches the steady state, the SIS process is initialized by assigning agents randomly as infected at t = 0. Practically, uniformly distributed initial infection speeds up the approach to the steady state. The recovery rate is fixed at μ = 0.1. Let ρ_{I} be the fraction of infected agents. Figure 5(a) shows ρ_{I}(t) for β_{1} = 2 × 10^{−5} and β_{2} = 0.004, for two values of v_{f} = 250 and 500. An epidemic steady state is reached quickly. As a higher shortens the time on the links that the infection rate is higher, ρ_{I} is smaller for higher v_{f}. Figure 5(b) shows the steady state ρ_{I} for β_{1} = β_{2}. There exists a threshold β_{1c} ≈ 4 × 10^{−5} above which ρ_{I} ≠ 0.
As β_{2}> β_{1} generally, Fig. 5(c) shows ρ_{I} (β_{2}) after setting β_{1} = 2 × 10^{−5} < β_{1c}, for two values of v_{f}. Figure 5(d) shows ρ_{I}(β_{2}) for three different values of β_{1} < β_{1c}. It is found that β_{2c} remains unchanged for different β_{1} < β_{1c}. It is reasonable in that when the outbreaks come from infections in journeys, the infection rate β_{1} of nontravellers is irrelevant to the threshold β_{2c}. However, for β_{2}> β_{2c}, a higher β_{1} leads to a higher ρ_{I}.
Next, we set β_{1} = 10^{−4}> β_{1c} and Fig. 6(a) shows that ρ_{I}(β_{2}) increases monotonically with β_{2}, for v_{f} = 250 and 500. Here, ρ_{I} ≠ 0 for all β_{2}. There exists a value β_{2c′} (β_{2c′} = 0.0025 for the case in Fig. 6(a) below (above) which ρ_{I} for v_{f} = 250 is lower (higher) than that for v_{f} = 500.
To summarize the findings in a physical picture, for β_{2} < β_{2c′}, infections among nontravellers at the nodes dominate the epidemic process. A higher v_{f} (e.g. v_{f} = 500) reduces the time that agents spent on journeys and thus promotes infection. For β_{2} > β_{2c′}, infections among travellers on journeys dominate the epidemic process. A higher v_{f} shortens the journey and suppresses infection.
For β_{1} = 2 × 10^{−5} < β_{1c} and β_{2} = 0.006> β_{2c}, infections during journeys dominate. Figure 6(b) shows that ρ_{I} increases monotonically with the fraction of travellers f, with ρ_{I} for v_{f} = 500 smaller than that for v_{f} = 250 due to the shorter journey time.
Discussion
We stressed the necessity of establishing a new framework for modelling journeys in modern times and their effects on epidemics. We illustrated the key ideas by presenting an integrated travel network constructed by considering geographic data, population data and transportation infrastructures in China. An example using only the highspeed trains and highways among the 116 cities of over a million population suffices for stressing the points. An ITN should include: (i) diversity among the links due to different distances and different speeds of transportation; (ii) diversity among the cities due to different population sizes and transportation services often reflecting their economic growth; (iii) roundtrip journeys to targeted destination via paths of shortest time; and (iv) different infection rates for travellers and nontravellers. The ITN can readily be extended to include details on local area transportation, multiple means of transportation and journeys among different countries. For example, Fig. 1(b) shows schematically a local transportation network with stations (nodes) served by a subway network (dashed lines) and a bus network (solid lines). A journey includes generally travelling in both Fig. 1(a,b). Effects such as traffic congestion naturally emerge. As far as epidemics are concerned, faster and more convenient intercity journeys would reduce the travel time during which passengers are crowded and thus suppress the chance of being infected, but they would also induce people to make more journeys and to farther places and thus spread a diseases more readily. Our ITN would serve as a good starting point for exploring the interplay of travelling and infection dynamics for many further work.
Methods
Degree and weight distributions of ITN
Highway buses and highspeed trains are the major means of transportation in China. After constructing ITN (see Fig. 4) based on highspeed trains and highways data, the number of links k_{i} is recorded for each node and the degree distribution P(k) is obtained (Fig. S1(a) in SI). The average degree and the clustering coefficient are calculated, where E_{i} is the number of links connecting the k_{i} neighbors of node i^{52}.
For the weights in Eq. (2), we record the actual populations in each node and reduce them to N_{i} in units of 5000 and the distances r_{ij} between pairs of nodes in km according to the China official website. The frequency of highspeed trains S_{ij} is obtained based on the routes and schedules of all highspeed trains. For each route that originates from a city A and terminates at a city B, we record the cities, say A, C1, C2, C3, B, served along the route and the number of services m_{s} per day. Then, all S_{ij}, i.e. S_{A,C1}, S_{C1,C2}, C_{C2,C3} and S_{C3, B}, are augmented by m_{s}. Data for all routes give the final S_{ij} that go into Eq. (2) for the weights of the links W_{ij} and Eq. (3) for the weights of the nodes W_{i} (see Table S1 in SI).
Journeys on ITN
For a journey that starts from the home city, the path of the shortest travel time to the destination is chosen. For a single type of links, i.e., v_{s} = v_{f}, the path of shortest travel time coincides with the shortest path. In ITN with v_{s} < v_{f}, the shortest paths are generally different from the paths of shortest time. As v_{f} > v_{s}, selected paths will involve railways as much as possible. It is convenient to discretize the journeys. The distance r_{ij} between two neighboring nodes i and j are divided into τ_{ij} time steps. At each time step, agents at node i become travellers. The destinations are chosen according to Eq. (5). The journeys are carried out as follows:

1
For every path between the home city i and destination j, the sum of τ_{ij} along the path is obtained. The path of shortest time is the one with the smallest sum.

2
Paths originated from different cities to different destinations may partially overlap. Therefore, in the intermediate nodes (cities) in a journey, some travellers may come in and other travellers may leave.

3
Upon arrival at the destination, an agent stays 5 time steps before the return journey begins.
Initially, the segments 1 ≤ k_{r} ≤ τ_{ij} on the links are empty and they will be occupied only when agents travel. For a node i, there are new travellers starting their journeys in the steady state, making a total new travellers. Each of them has the chance of choosing node i as the destination, giving a total agents arriving per time step in the steady state.
Epidemic spreading measurement on ITN
In the SIS dynamics, we distinguish infections among nontravellers in the cities and among travellers in the same segment of a link with infectious rates β_{1} and β_{2}, respectively. As travellers on trains/buses are densely packed, β_{2}> β_{1}^{47}. An agent is a traveller and nontraveller at different times. When he is a nontraveller in a city, he is exposed to an infectious rate of β_{1}. Once he is on a journey, he is exposed to an infectious rate of β_{2} during each segment of his journey, regardless of the segment being in the middle of a link or a passingby city. Only travelling agents in the same segment k_{r} (1 ≤ k_{r} ≤ τ_{ij}) towards the same direction can infect each other. Thus, SIS on ITN accounts for the continual exchanges of agents on trains and buses due to partial overlaps of agents’ journeys and the spread of a diseases through journeys. A susceptible nontraveller at node i will be infected by the rate 1−(1−β_{1})^{n}_{i,I} when he is in contact with n_{i,I} infected agents. A susceptible traveller at a segment k_{r} of a link will be infected by the rate when he is in contact with infected agents. Each infected agent recovers with a rate μ. The fraction ρ_{I} of infected agents is obtained by , where is over all the segments in all links in both travelling directions and N_{tot} is the total population.
An approximate theoretical analysis
We make a qualitative analysis of the key behavior and illustrate that the dependence of ρ_{I} on the model parameters in ITN can be captured by meanfield considerations. Let there be M cities. There are pairs of cities that the journey between which is all on highspeed trains. The mean number of sections 〈τ〉 in a link is τ_{s} = int(s/v_{s}) + 1 for highway links and τ_{f} = int(s/v_{f}) + 1 for railway links, where is the mean distance between neighbouring nodes. There are altogether
sections on the links, with d being the mean shortest path length between two nodes. It follows that N_{mid} decreases with m.
There are two processes in one time step: infection and motion. For the step t → (t + 1), SIS processes take place in the time interval t^{+} → (t + 1)^{−} and the motion occurs at (t + 1). At a node , there are n_{i,s} susceptible and n_{i,I} infected agents and n_{i} = n_{i,S} + n_{i,I}. Similarly, there are n_{α,S} susceptible and n_{α,I} infected agents at a section α of a link, with and . The dynamics of the infected agents can be described by
where X_{I} accounts for infected agents arriving at the destination or at home, Y_{I} represents infected agents starting a journey, k_{station} are nodes where agents switch means of transportation and is over the k_{i} links to node i.
The time evolution of ρ_{I} is given by and thus
where N_{tot} is the total population. The set of equations can be iterated in time for the steady state. Further generalizations of ITN can be treated accordingly.
Based on Eq. (8), we make the following observations:
1. For β_{1} = β_{2}: As n_{i} >> n_{α}, we readily have n_{i,I} >> n_{α,I} and the second term in Eq. (8) dominates. Thus, ρ_{I} in Fig. 5(b) comes mostly from infections at the nodes.
2. For β_{1} ≠ β_{2} and β_{1} > β_{1c}: Infections at the nodes give ρ ≠ 0, but the third term in Eq. (8) becomes important when β_{2} > β_{1} and β_{2} > β_{2c}. This gives the behaviour in Fig. 6(a).
3. For β_{1} ≠ β_{2} with β_{1} < β_{1c}: Infections at the nodes alone cannot sustain ρ_{I}. Infections on journeys dominate and ρ_{I} becomes finite at β_{2} = β_{2c}, independent of β_{1} (see Fig. 5d). It follows from the equation for n_{α,I}((t + 1)^{−}) that
indicating that β_{2c} is inversely proportional to the mean number of agents travelling in a segment of a link n_{α}.
4. For different m: The third term in Eq. (8) indicates that ρ_{I} ∝ N_{mid}. As N_{mid} decreases with m (see Eq. 6), ρ_{I} also drops with increasing m and highspeed railways tend to prevent epidemics by shortening travel times. One should note that this captures one effect of having faster transportation. However, an opposite effect of inducing more travellers poses a risk.
Additional Information
How to cite this article: Ruan, Z. et al. Integrated travel network model for studying epidemics: Interplay between journeys and epidemic. Sci. Rep. 5, 11401; doi: 10.1038/srep11401 (2015).
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Acknowledgements
This work was partially supported by the NNSF of China under Grant Nos. 11135001 and 11375066 and 973 Program under Grant No. 2013CB834100.
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Z.L. and Z.R. conceived the research project. Z.R., Z.L. and C.W. performed research. Z.L., Z.R. and P.M.H. analyzed the results. Z.L. and P.M.H. wrote the paper. All authors reviewed and approved the manuscript.
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Ruan, Z., Wang, C., Ming Hui, P. et al. Integrated travel network model for studying epidemics: Interplay between journeys and epidemic. Sci Rep 5, 11401 (2015). https://doi.org/10.1038/srep11401
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DOI: https://doi.org/10.1038/srep11401
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