Super-spreaders and the rate of transmission of the SARS virus

Abstract

We describe a stochastic small-world network model of transmission of the SARS virus. Unlike the standard Susceptible-Infected-Removed models of disease transmission, our model exhibits both geographically localised outbreaks and "super-spreaders". Moreover, the combination of localised and long range links allows for more accurate modelling of partial isolation and various public health policies. From this model, we derive an expression for the probability of a widespread outbreak and a condition to ensure that the epidemic is controlled. Moreover, multiple simulations are used to make predictions of the likelihood of various eventual scenarios for fixed initial conditions. The main conclusions of this study are: (i) "super-spreaders" may occur even if the infectiousness of all infected individuals is constant; (ii) consistent with previous reports, extended exposure time beyond 3-5 days (i.e. significant nosocomial transmission) was the key factor in the severity of the SARS outbreak in Hong Kong; and, (iii) the spread of SARS can be effectively controlled by either limiting long range links (imposing a partial quarantine) or enforcing rapid hospitalisation and isolation of symptomatic individuals.

Keywords: Severe Acute Respiratory Syndrome (SARS); Small-world network; Super-spreader event; Transmission dynamics.

(a) Reported.

(b) Revised.

(a) State transition flow graph.

(b) Small-world network structure.

Fig. 3

Probability of complete infection. The probability of an endemic epidemic for various values of $p_{1,2}$ and $r_1$ with $r_0=0.1$, $n_1=4$ and $\mu =7$. The probability is estimated from 1000 simulations of the model and according to Eq. (5) with $k=1$ (upper dashed line), $k=\frac{3}{4}$ (middle dashed) and $k=\frac{1}{2}$. The convex shape of each of these plots is extremely interesting. This indicates that neither purely local nor purely non-local infection results in the greatest probability of overall infection, rather an endemic outbreak is most likely when both local and non-local infection is possible.

Fig. 4

Level curves for probability of complete infection. The probability of an endemic epidemic for various values of $p_{1,2}$ and $r_1$ with $r_0=0.1$, $n_1=4$ and $\mu =7$. The probability is estimated according to Eq. (5) with $k=\frac{3}{4}$. The four panels correspond to level curves for the probability of the disease being self-contained of 0.01, 0.1, 0.5 and 0.9, respectively. The curves and the contours for $r=0.1$, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, and 0.5. The smooth dependence of $P_{\mathrm{safe}}$ on each of the parameters is evident.

Fig. 5

Probability of control of infection. The probability of the epidemic terminating without intervention for various values of $p_{1,2}$ and $\mu$ with $r_0=0.1$, $r_1=0.25$ and $n_1=4$. The probability is estimated according to Eq. (5) with $k=\frac{3}{4}$. The four panels correspond to $\mu =1$, 2, 4 and 7, respectively. Reducing $\mu$ greatly increases the probability of being able to control the epidemic, but increasing $\mu$ beyond moderate values (i.e. $\mu \ge 4$) does not change the behaviour substantially.

Fig. 6

Unconstrained growth of the infectious population. The upper panels show the number of individuals infected after 50 days; the lower plots show the number of distinct clusters detected after the same time. The parameter $p_1=0.135-\frac{7}{4}{p}_2$, $r_0=0.1$, $n_1=4$ and $\mu =7$. The left hand plots are for $r_1=0.25$ (i.e. no nosocomial transmission) and the right panels are for $r_1=0.165$ (a mean infection period of 6 days). The results are median, 70% and 90% confidence intervals from 1000 simulations.

Fig. 7

Outbreak of clusters in a single simulation. The top four panels show the location of infected individuals after 50, 100, 150 and 250 days. Removed, infected and prone individuals are illustrated as solid dots (R are black, I are red, and P are blue). Parameter values used are $r_1=0.25$, $p_1=0.093$ and $p_2=0.06$, and the results for these parameter values are typical. The bottom two panels show the number of prone and infected individuals over the same period as time series. One can clearly see the irregular bursts of activities in the time series corresponding to the explosion in clusters. The total number of infections (R+I+P) is 22, 40, 117, and 309 after 50, 100, 150, and 200 days, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 8

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 is plotted as a heavy solid line.

(a) $x_t$.

(b) $x_t$.

Fig. 10

Probability distribution on the total number of infections. The probability distribution on the total number of infections for 1000 simulations of the model in Fig. 8 is shown on a linear (top plot) and a logarithmic scale.

Fig. 11

Probability distribution infection dynamics. The probability distribution of the daily number of infections for 1000 simulations of the model in Fig. 8 are shown on a logarithmic scale. Blue represents low probability, while red represents high probability of a particular infection tally for any number of days after onset.(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)