Clustering model for transmission of the SARS virus: application to epidemic control and risk assessment

Abstract

We propose a new four state model for disease transmission and illustrate the model with data from the 2003 SARS epidemic in Hong Kong. The critical feature of this model is that the community is modelled as a small-world network of interconnected nodes. Each node is linked to a fixed number of immediate neighbors and a random number of geographically remote nodes. Transmission can only propagate between linked nodes. This model exhibits two features typical of SARS transmission: geographically localized outbreaks and "super-spreaders". Neither of these features are evident in standard susceptible-infected-removed models of disease transmission. Our analysis indicates that "super-spreaders" may occur even if the infectiousness of all infected individuals is constant. Moreover, we find that nosocomial transmission in Hong Kong directly contributed to the severity of the outbreak and that by limiting individual exposure time to 3-5 days the extent of the SARS epidemic would have been minimal.

Keywords: Disease transmission; Epidemiological methods; Nonlinear dynamics; Severe acute respiratory syndrome; Small world networks.

Fig. 1

Daily reported (upper panel) and revised (lower panel) SARS infection data for Hong Kong. At the end of the epidemic, 105 days after the first recorded case, a total of 1755 confirmed infections had been identified in Hong Kong. The reported data are the daily number of hospital admission of SARS cases, as published in the South China Morning Post (and other local media) during the outbreak. Subsequent to the epidemic (several months later) a revised set of data was released by the local government health authority. These are the “revised” data in the lower panel. As we are unsure of the revision process use, and we believe it to include strong assumptions about the etiological agent involved, we analyze only the reported data in this paper.

Fig. 2

Four compartmental small world network model of disease propagation. The top panel depicts the transmission state diagram: S to P based on the SW structure and the infection probabilities $p_{1,2}$; P to I with probability $r_0$; and, I to R with probability $r_1$. The lower panel depicts the distinction between short-range and long-range network links. The lower panel shows the arrangement of nodes in a small network. The black (infected) node may infect its four immediate neighbors with probability $p_1$ and three other nodes (hashed) with probability $p2$.

Fig. 3

Expected parameter distribution for the model. The top panel shows the distribution on the number of days between transition from P to I (first bar) and from I to R (second bar). Note that there is a possibility of P to I transition in zero days. Omitting this possibility does not significantly affect the modelling results. The lower plot illustrates the exponentially decaying distribution $f_X(x)\propto {\mathrm{e}}^{-(x/\mu )}$ on the number of long distances links $n_2^{(i)}$.

Fig. 4

Expected model behavior for $r_1=0.25$. The top panel shows the distribution on the number of casualties during the first 50 days of transmission. The middle plot shows the number of clusters exhibited by the model over 50 days. The bottom plot shows the probability of an uncontrollable outbreak (defined as more than 3000 infected (I and P) individuals, or infection lasting more than 50 days); and the expected total number of casualties in the cases for which the disease was controlled (i.e., self-terminated). The upper two panels show median and 70% and 90% confidence intervals for 1000 simulations for each value of $p_2$. The bottom panel are mean and incidence rate estimated from the same 1000 simulations.

Fig. 5

Expected model behavior for $r_1=0.16$. The top panel shows the distribution of the number of casualties during the first 50 days of transmission. The middle plot shows the number of clusters exhibited by the model over 50 days. The bottom plot shows the probability of an uncontrollable outbreak (defined as more than 3000 infected individuals, or infection lasting more than 50 days) and the expected total number of casualties in the cases where the disease was controlled (i.e., self-terminated). The upper two panels show median and 70% and 90% confidence intervals for 1000 simulations for each value of $p_2$. The bottom panel shows mean incidence rate estimated from the same 1000 simulations.

Fig. 6

Expected model behavior for $p_1=0.065$ and $p_2=0.04$. The figure shows the distribution on the number of casualties during the first 50 days of transmission. Median and 70% and 90% confidence intervals for 1000 simulations for each value of $p_2$ were calculated.

Fig. 7

Model simulations. The top panel shows the change in parameters $r_1$ and $p_2$ with time (all other parameters are constant: $p_1=0.08$, $n_1=4$ and $\mu =7$). The bottom plot shows five model simulations and the true SARS data for Hong Kong. The five model simulations were selected to ensure that a “full” outbreak occurred (a total number of infections greater than 1000). The true data are plotted as a heavy solid line.