Epidemic process over the commute network in a metropolitan area

Abstract

An understanding of epidemiological dynamics is important for prevention and control of epidemic outbreaks. However, previous studies tend to focus only on specific areas, indicating that application to another area or intervention strategy requires a similar time-consuming simulation. Here, we study the epidemic dynamics of the disease-spread over a commute network, using the Tokyo metropolitan area as an example, in an attempt to elucidate the general properties of epidemic spread over a commute network that could be used for a prediction in any metropolitan area. The model is formulated on the basis of a metapopulation network in which local populations are interconnected by actual commuter flows in the Tokyo metropolitan area and the spread of infection is simulated by an individual-based model. We find that the probability of a global epidemic as well as the final epidemic sizes in both global and local populations, the timing of the epidemic peak, and the time at which the epidemic reaches a local population are mainly determined by the joint distribution of the local population sizes connected by the commuter flows, but are insensitive to geographical or topological structure of the network. Moreover, there is a strong relation between the population size and the time that the epidemic reaches this local population and we are able to determine the reason for this relation as well as its dependence on the commute network structure and epidemic parameters. This study shows that the model based on the connection between the population size classes is sufficient to predict both global and local epidemic dynamics in metropolitan area. Moreover, the clear relation of the time taken by the epidemic to reach each local population can be used as a novel measure for intervention; this enables efficient intervention strategies in each local population prior to the actual arrival.

Figure 1. Commuter flow data for the Tokyo metropolitan area.

(A) Geographical location of the Tokyo metropolitan area within Kanto region, Japan. The framed rectangle shows the central part of the Tokyo metropolitan area. (B) Distribution of the station sizes on a double-logarithmic plot. Blue line, distribution of home-node stations; red line, distribution of work-node stations. (C) and (D) Geographical distributions of the sizes of home- and work-node stations, respectively, within the central part of the Tokyo metropolitan area. The color indicates the size of the station: black, commuters; blue, commuters; green, commuters; red, commuters. All numbers are from the 139,841 collected questionnaires of UTC. The red-colored stations in the middle of (D) correspond to Tokyo's inner urban area (along the loop of the Yamanote line); the 2 red stations in the lower left of (D) are the Kawasaki and Yokohama stations. The longitude and latitude of each station were acquired from the Station Database [http://www.ekidata.jp].

Figure 2. Schematic representation of the population size class model (PSCM).

(A) Geographical distribution of the railway network; each node corresponds to a station and each line corresponds to a commuter railway of commuter trains. Every station has a home population (those who reside in the area) and a work population (those who travel to as a workplace/school in the area). There are multiple commuters using the commuter train between each pair of populations. (B) The commuter network utilized in the individual-based model (IBM) calculations. The nodes correspond to the home and work populations of each station, forming a bipartite network in which each line denotes a connection via commuter flow between home and work populations. Local populations with different population sizes are represented by different colors and sizes. (C) The commuter network utilized in the PSCM calculation. Local populations with similar population sizes are grouped into population size classes, which form the nodes, while the total commuter flows between pairs of population size classes form the lines. (D) Joint distribution of the home and work population sizes of commuters in the Tokyo metropolitan area. The number of commuters that live in a home population of size class and commute to a work population of size class is plotted as a density plot. The data were obtained from the Urban Transportation Census (UTC) commute data (Ministry of Land, Infrastructure, Transport and Tourism, The 10th Urban Transportation Census Report, 2007; in Japanese).

Figure 3. The probability that a single infected host causes global epidemic.

The probability of a global epidemic, , as a function of the home population size (horizontal axis; initial home population) and work population size (vertical axis; initial work population) of the initially infected host for various infection rate . (A1–4) The results of the individual-based model (IBM) simulations of the spread of infectious disease over the commute network of the Tokyo metropolitan area which starts with a single infectious individual commuting from a randomly chosen home population to a randomly chosen work population. Each panel corresponds to a different infection rate, and the population sizes are plotted on logarithmic scales. (B1–4) The corresponding results obtained using a branching process formula in the population size class model (PSCM). (C1–4) The corresponding results of the IBM simulations using the random reconnection model (RRM). In each panel, the contour plot represents interpolation of the results calculated using the data of 184 combinations of initial home and work population sizes.

Figure 4. The probability of global epidemic as a function of the basic reproductive ratios in the initially infected home and work populations.

The probability of a global epidemic observed in the individual-based model (IBM) simulations based on the commute network data for the Tokyo metropolitan area plotted as a function of , where the independent variable is the sum of the single population basic reproduction ratios of the initial home and work populations ( and : home and work population sizes of the initially infected individual, respectively). Each point corresponds to a different set of epidemic parameters, and the color represents the infection rate . Black line, the probability of a global epidemic in the single population model with population size , i.e., that from (main text for details).

Figure 5. The final size and the peak time of global epidemic.

The final size of the global epidemic (A) and the time until an epidemic initiated by a single host reaches its peak (B) plotted against the infection rate . (A1) and (B1): results observed in the individual-based model (IBM) simulations; each point gives the Monte Carlo ensemble average value corresponding to different epidemic parameters, and the color indicates the sum of the sizes of the initially infected home and work populations. Here, the cases for initial extinction of disease are excluded from the ensemble. (A2) and (B2): corresponding results from the population size class model (PSCM) calculations.

Figure 6. The final size of epidemic and the arrival time of epidemic at local populations.

The final size of the local epidemic (A) and the time until the infected individuals first appear in the local population (B) (i.e., the arrival time of epidemic) plotted against the local population size. The population size is on a logarithmic scale. (A1–2) and (B1–2): results of the individual-based model (IBM) simulations; each point (dots) gives the mean value of the Monte Carlo ensemble averaged over 100 Monte Carlo runs for each local population, and the blue and red dots correspond to the results for the home and work populations, respectively. The black lines in (A1–2) give the mean value of the final size of the local epidemic for each population size class. The black lines in (B1–2) represent the regression line of the arrival time of the epidemic in the local population versus the logarithm of the population size. The regression line for the arrival time in the -th home population with population size , , was highly significant, with a P-value of in the test ( with the degrees of freedom (1, 1084)), . The estimated intercept and slope and their confidence intervals (CIs) are ( CI) and ( CI). The same was true for the arrival times in the work population; the regression was highly significant (, with ), with estimated intercept and slope ( CI) and ( CI), respectively. (A3) and (B3): corresponding results obtained from the population size class model (PSCM); the blue line shows the result for the home population and the red line the result for the work population (refer main text for details). The infection rate was . A person commuting from “Gyotoku” station to “Aoyama-itchome” station was designated the initially infectious individual.

Figure 7. The intercept and slope of the linear relationship between the arriving time of epidemic and logarithmic local population size.

The intercept (A1–2) and slope (A3–4) of linear regression line for the arrival time of epidemic, and the corresponding results calculated from the exponential growth approximation of the linearized population size class model (B1–2) and (B3–4), respectively (see main text for details of approximation) are shown. The results are plotted as the functions of infection rate and the color of each line indicates the sum of the population sizes of the initially infected home and work populations. (A1–4): regression coefficient statistically estimated from the results of the individual-based model (IBM) simulations. (A1) and (A2): estimated intercepts and , respectively. (A3) and (A4): estimated (sign reversed) slopes and , respectively. All regressions were statistically significant according to the P-value of the regression coefficient ().