Network theory and SARS: predicting outbreak diversity


Abstract

Many infectious diseases spread through populations via the networks formed by physical contacts among individuals. The patterns of these contacts tend to be highly heterogeneous. Traditional “compartmental” modeling in epidemiology, however, assumes that population groups are fully mixed, that is, every individual has an equal chance of spreading the disease to every other. Applications of compartmental models to Severe Acute Respiratory Syndrome (SARS) resulted in estimates of the fundamental quantity called the basic reproductive number R0—the number of new cases of SARS resulting from a single initial case—above one, implying that, without public health intervention, most outbreaks should spark large-scale epidemics. Here we compare these predictions to the early epidemiology of SARS. We apply the methods of contact network epidemiology to illustrate that for a single value of R0, any two outbreaks, even in the same setting, may have very different epidemiological outcomes. We offer quantitative insight into the heterogeneity of SARS outbreaks worldwide, and illustrate the utility of this approach for assessing public health strategies.

Keywords

  • SARS;
  • Epidemiology;
  • Intervention;
  • Contact network;
  • Mathematical model

1. Introduction

More than two years since the first case of severe acute respiratory syndrome (SARS), a respiratory illness caused by a novel coronavirus, occurred in Guangdong province of China (November, 2002) and more than 18 months since the syndrome was first recognized outside of Asia (in Canada on March 13, 2003), its pattern of spread remains an enigma to public health officials and epidemiologists (Cyranoski and Abbott, 2003; Drosten et al., 2003; Ksiazek et al., 2003; Marra et al., 2003; Peiris et al., 2003; World Health Organization, 2003). Mathematical epidemiologists originally estimated the average number of secondary cases emanating from one primary case in a susceptible population (R0) to be in the range of 2.2 to 3.6 for this virus—an estimate well above one, approximating that of a new subtype of influenza (Hethcote, 2000; Lipsitch, 2003; Riley et al., 2003).

Despite this estimate and near-universal susceptibility, SARS has not emerged as a global pandemic. Instead, initial seeding was followed by intense but tightly circumscribed activity in some locales with only scant activity in others. In Canada, for instance, Toronto, Ontario and Vancouver, British Columbia were first affected nearly simultaneously in March 2003. By June 3, 2003, Toronto had experienced more than 209 probable cases and Vancouver had experienced only four probable cases. No other province of Canada reported any probable cases (Health Canada, 2003). The United States with a population more than fifty-fold greater than Toronto reported 69 probable cases—67 imported and only two from secondary spread (Centers for Disease Control and Prevention, 2003).

The discrepancy between the estimates of R0 and the observed epidemiology might stem from early and effective intervention since Rt, the reproductive ratio of a disease at time t, will decrease with the implementation of successful infection control measures. Yet, even during the three and a half months of SARS spread in China between its initial appearance and the broad implementation of public health measures, case counts were much less than expected from such values of R0 ( Xu et al., 2004). By definition, the total number of expected cases of a disease goes up by a factor of R0 for every generation of infection, a generation being the mean time between an individual becoming infected and their infecting others. Based on recorded dates of the first symptoms for 124 pairs of infections in Singapore and Toronto ( Leo et al., 2003; Poutanen et al., 2003), we estimate the average generation time (γ) for SARS to be 9.7±0.3 days. (This estimate clearly depends on the accuracy of the reported data.) Roughly, the cumulative number of SARS cases over D days should be View the MathML source (This is capped by total population size and does not consider the reduction in Rt once a substantial proportion of the population is infected). Thus for R0 ranging between 2.2 and 3.6, this equation predicts that in the first 120 days of transmission in China, there should have been between approximately 30,000 and 10 million cases. In fact only 782 cases were reported during the initial three months ( World Health Organization, 2003), which, using this simple calculation, suggests that R0 should be much lower and closer to 1.6. A subsequent estimate based on data from the Hong Kong and Singapore outbreaks brings R0 down to 1.2, which, by the above formula, predicts approximately 50 cases during the first 120 days of transmission ( Chowell et al., 2003). While this number agrees nicely with the case counts observed in Hong Kong and Singapore ( Chowell and Fenimore, 2003a and Chowell and Hyman, 2003b) it is an order of magnitude lower than that reported for China.

Why do epidemiologists derive such varied estimates of R0 and why were the initial estimates so high in comparison to the observed epidemiology in China? The basic premise of fully mixed epidemiological models—that all individuals in a group are equally likely to become infected (or infect others)—often does not hold and therefore may lead to spurious estimates or estimates that cannot justifiably be extrapolated from the specific setting in which they were measured to the broader community context. Early SARS estimates were based largely on transmission data from closed settings like hospitals and crowded apartment buildings, where there are unusually high rates of contact between individuals (Lipsitch, 2003; Riley et al., 2003). In fact, hospital transmission accounts for 50% of the value of R0 described in Riley et al. (2003). If the contact patterns in these settings were highly heterogeneous, then the estimates for R0 may be inaccurate. Even if the estimates for R0 were indeed appropriate for these specific settings, they probably should not be extrapolated to the population at large. Contact rates may be considerably lower outside hospitals and crowded apartment buildings and thus so may be the general value of R0 for SARS (Yu et al., 2004). Such disparity may account for the discrepancy between the estimates and the slower progress of the outbreak in China. In fact, further studies suggest that the unusually large cluster of infected cases in Amoy Gardens complex in Hong Kong was due to exposure to the virus-laden aerosol plume originating from one of the buildings in that area and not from direct person-to-person contact (Yu et al., 2004). A recent analysis of the impact of SARS isolation interventions in Hong Kong, Singapore and Toronto emphasizes the importance of viewing R0 as a distribution of possible values where the mean and median may vary depending on the setting in which the disease is spreading (Chowell et al., 2004).

SARS, like many other infectious diseases exhibits great heterogeneity in transmission efficiency with certain individuals appearing to be responsible for large proportion of transmission events (Booth et al., 2003; Donnelly et al., 2003; Leo et al., 2003). These individuals may be “superspreaders” with unusually large numbers of contacts or “supershedders” who are unusually effective at excreting the virus into the environment they share with others. In contrast to the fully mixed assumption of standard compartmental models, the contact patterns in a community may be quite diverse. There is an enormous difference between a situation in which all individuals share typical contact patterns and one in which most infected individuals pass the disease on to only one or even zero others, but a small number pass it onto dozens or even hundreds—the mean value of R0 can be the same in both cases, while the epidemiological outcomes are vastly different.