Forecast and control of epidemics in a globalized world

Abstract

The rapid worldwide spread of severe acute respiratory syndrome demonstrated the potential threat an infectious disease poses in a closely interconnected and interdependent world. Here we introduce a probabilistic model that describes the worldwide spread of infectious diseases and demonstrate that a forecast of the geographical spread of epidemics is indeed possible. This model combines a stochastic local infection dynamics among individuals with stochastic transport in a worldwide network, taking into account national and international civil aviation traffic. Our simulations of the severe acute respiratory syndrome outbreak are in surprisingly good agreement with published case reports. We show that the high degree of predictability is caused by the strong heterogeneity of the network. Our model can be used to predict the worldwide spread of future infectious diseases and to identify endangered regions in advance. The performance of different control strategies is analyzed, and our simulations show that a quick and focused reaction is essential to inhibiting the global spread of epidemics.

Fig. 1.

Global aviation network. A geographical representation of the civil aviation traffic among the 500 largest international airports in >100 different countries is shown. Each line represents a direct connection between airports. The color encodes the number of passengers per day (see color code at the bottom) traveling between two airports. The network accounts for >95% of the international civil aviation traffic. For each pair (i,j) of airports, we checked all flights departing from airport j and arriving at airport i. The amount of passengers carried by a specific flight within 1 week can be estimated by the size of the aircraft (We used manufacturer capacity information on >150 different aircraft types) times the number of days the flight operates in 1 week. The sum of all flights yields the passengers per week, i.e., M_ij in Eq. 7. We computed the total passenger capacity ∑ M_ij of each airport j per week and found very good agreement with independently obtained airport capacities.

Fig. 2.

Global spread of SARS. (A) Geographical representation of the global spreading of probable SARS cases on May 30, 2003, as reported by the WHO and Centers for Disease Control and Prevention. The first cases of SARS emerged in mid-November 2002 in Guangdong Province, China (17). The disease was then carried to Hong Kong on the February 21, 2003, and began spreading around the world along international air travel routes, because tourists and the medical doctors who treated the early cases traveled internationally. As the disease moved out of southern China, the first hot zones of SARS were Hong Kong, Singapore, Hanoi (Vietnam), and Toronto (Canada), but soon cases in Taiwan, Thailand, the U.S., Europe, and elsewhere were reported. (B) Geographical representation of the results of our simulations 90 days after an initial infection in Hong Kong, The simulation corresponds to the real SARS infection at the end of May 2003. Because our simulations cannot describe the infection in China, where the disease started in November 2002, we used the WHO data for China.

Fig. 3.

Demonstration of the impact of fluctuations on inhomogeneous networks. Two confined populations with exchange of individuals are shown. In each population, the dynamics is governed by the SIR reaction scheme (6). Individuals travel from one population to the other at a rate γ. Parameters are N_A = N_B = 10,000, R₀ = 4, and an initial number of infecteds I₀ = 20 in population A. (Left) The probability p(γ) of an outbreak occurring in population B as a function of transition rate γ. (Insets) Histograms of the time lag T between the outbreaks in A and B for those realizations for which an outbreak occurs in B. The circles are results of the simulations of 100,000 realizations; the solid curve is the analytic result of Eq. 8. (Right) A star-shaped network with a central population A connected to M – 1 populations B₁,..., B_M-1 with rates γ₁,..., γ_M-1. The cumulated variance (see text) for a star network with 32 populations is depicted as a function of the average transmission rate . Two cases are exemplified: equal rates (circles) and distributed rates according to Eq. 10 with γ_max/γ_min ≈ 1,000 (squares). The solid lines show the analytical results given by Eqs. 8 and 9. Parameters are N_A = N_B = 10,000, R₀ = 4, and an initial number of infecteds I₀ = 20 in population A. The numerical values are obtained by calculating the variance of the fluctuations of 100 different realizations of the epidemic outbreak for each .

Fig. 4.

Inhomogeneity of the aviation network and control strategies. (A and B) Geographical representation of the results of two simulations of hypothetical SARS outbreaks 90 days after an initial infection in New York (A) and London (B) for the same parameters and color code as in Fig. 2. (C) Impact and control of epidemics. The probability p_n(v) of having to vaccinate a fraction v of the population to prevent the epidemic from spreading, if an initial infected individual is permitted to travel n = 1 (red), 2 (blue), and 3 (green) times. The probability p_n(v) is estimated by placing the infected individual on a node i (black dot) of the network. The fraction v_i associated with node i is given by the number of susceptibles in a subnetwork that can be reached by the infected individual after n = 1, 2 and 3 steps. Histogramming v_i for all nodes i yields an estimate for p_n(v). The light-blue curve depicts the strong impact of isolating only 2% of the largest cities after an initial outbreak (n = 2) and is to be compared to the blue curve.