1
SUPPLEMENTARY METHODS
Nicholas C Grassly, Christophe Fraser and Geoff P Garnett (2004) “Increasingly synchronised
epidemics of syphilis across the USA are driven by host immunity”
Detailed description of data.
Reported annual numbers and rates of syphilis and gonorrhoea cases treated in the
major cities of the United States (population size >200,000) have been routinely
collected since 1941. These data are maintained by the Centers for Disease Control and
Prevention (CDC) with surveillance reports summarising levels and trends currently
published annually 1. We compiled reported cases and rates of primary and secondary
syphilis and gonorrhea for the period 1941 to 2002 for the 68 major cities. Cases prior
to 1969 are recorded by fiscal year after which calendar years are used. The reporting of
both case numbers and rates (per 100,000 population) allows the denominator city size
estimate used in the past to be calculated and compared to estimates based on the
decennial census 2. These calculated denominator city sizes show occasional sudden
changes in contrast to the census estimates, which may be a result of changes in the area
from which cases are reported, or simply the use of different sources for the estimates. It
is not possible to distinguish these causes and we therefore choose to analyse the
published rates rather than re-calculate rates based on the city size estimates from the
decennial census, since these published rates contain information about changes in the
reporting area that would otherwise be lost.
Frequency analysis.
Differencing the data prior to spectral analysis removes any long-term changes in the
mean rate (non-stationarity) remaining after the linear transformation, resulting in a
suppressed spectral density for low frequency ‘noise’ 3. The spectrum of differenced
2
data
S
is biased, and for annual data is related to the spectrum of the original
untransformed data S by S (T )  S (T )[ 2  2Cos(2 / T )] , where T is the period. For the
periods of interest, this bias is minimal, and the peak in the spectra for syphilis and
gonorrhea is the same for the differenced and untransformed data, in part reflecting
limits in the resolution of the spectra due to the number of years of observations. In the
paper we present spectra for the differenced data, where the spectral density for low
frequency fluctuations is suppressed (any remaining power at low frequencies indicates
non-stationarity in the differenced data).
Tests for significant periodicity in each city spectrum were based on the spectra of
1000 random permutations of the differenced data for each city. Randomization
removes any periodicity and is equivalent to the hypothesis of random noise in the
differenced data. The resulting spectra are flat and similar to the distribution seen for
white noise, which is approximated by the χ2 distribution 4. However, in this case the
95th percentile is somewhat higher for the bootstrapped distribution. A hypothesis of
white noise in the differenced data is equivalent to a random walk in the untransformed
data. Comparison of the observed spectrum with the distribution of spectra for the
randomized data therefore allows the hypothesis of an aperiodic random walk to be
tested.
Cross-correlation and city size.
Typically average correlation may be plotted against some measure of distance between
locations to derive a correlogram 5. In ecological studies the distance measure is often
simply physical distance. However, in infectious disease epidemiology it is the contact
of people with one another or disease vectors that is important. Throughout the latter
half of the twentieth century the spread of directly transmitted infections including
3
sexually transmitted infections has been strongly hierarchical 6,7. Disease spread is
through the size hierarchy of cities, in the USA spreading rapidly among the largest
cities such as New York and Los Angeles, and then more slowly to the smaller cities. In
some cases this may be followed by more local suburban diffusion. This pattern of
disease spread closely maps to patterns of air travel 8 and is also likely to relate to
cultural diffusion of behaviours that may relate to the transmission of disease.
Different measures of distance between cities that reflect their position in the size
hierarchy may be derived. However, a correlogram for syphilis based on such measures
would be problematic since smaller cities typically have fewer case reports and greater
sampling error. This makes it extremely difficult to test the significance of any observed
correlation or synchronisation distance threshold. Instead we compare the average
correlation of the 10 largest to the 10 smallest cities (based on their average rank in the
size hierarchy over the period 1960-96). Significant correlation of the log-transformed
rates of primary and secondary syphilis for each city pair within the two groups was
tested using Pearson’s correlation coefficient. This was then repeated for the 10 largest
cities but where the reported rates were resampled 10,000 times from a binomial
distribution with mean equal to the reported rate and sample size equal to that for the
smallest cities (the population size for each city for each bootstrap replicate was
randomly sampled without replacement from the distribution of sizes of the smallest
cities averaged over 1960-96). This gives the distribution of the number of significant
pairwise correlations expected for the largest cities but in the presence of additional
sampling error corresponding to that expected for the smallest cities. If this distribution
excludes the number seen for the 10 smallest cities, then the largest and smallest cities
can be said to have significantly different levels of cross-correlation in rates of reported
syphilis over the period of interest.
4
Detailed analysis of SIRS dynamics.
The SIRS model describes the process of becoming infected (I), recovery to an immune
class (R) and subsequent loss of immunity to return to the susceptible class (S). It
extends what has been termed the ‘classic endemic model’9 - the SIR model with
demography (births and deaths) - by allowing for loss of immunity. We define X, Y and
Z as the number of susceptible, infected and recovered/immune individuals in a
population of total size N = X + Y + Z. We use lower case x, y and z to denote the
fraction of the population in each of these categories (i.e. x = X/N). The deterministic
SIRS model is described by three ordinary differential equations
dx / dt    ( y   ) x  z
(1)
dy / dt  yx  (   ) y
(2)
dz / dt  y  (   ) z
(3)
where β is the transmission parameter, ν the rate of recovery from infection, γ the rate of
loss of immunity and μ the rate of birth/death. This model can be re-parameterised in
terms of the basic reproductive number R0   /(   ) which gives the number of
infections a single infectious individual would generate in an entirely susceptible
population. At equilibrium x*  1 / R0 and y*  q( R0  1) / R0 where
q  (   ) /(     ) .
Stability analysis using Taylor series expansion for small perturbations from x*
and y* leads to a quadratic for the eigenvalues  of the linearized system
2  [qs     ]  (   )s  0
(4)
where s  (   )( R0  1) is the exponential rate of growth in incidence for an outbreak
within a fully susceptible subpopulation . If the solution for the eigenvalues contains an
5
imaginary part ( 4s  (   )[1  s /(     )]2 ) then the system shows damped
oscillations towards the endemic equilibrium with characteristic damping time
TD  2 /( qs     )
(5)
and period T given by the imaginary part
T  2 / s (   )  0.25(    qs ) 2
(6)
The stochastic version of the SIRS model given in equations (1)-(3) can be solved
numerically by taking small time steps t such that the rates of each event (infection,
death etc…) can be considered independent of one another. The number of events of
type k that have rate rk in a population of size N is then binomially distributed with
mean Nrk t . The seven possible events for the SIRS model and associated rates are
given in Supplementary Table 1.
The marginal distribution of the number of infected individuals in the quasistationary state is approximately normal in the stochastic SIRS model for reasonably
large N 10. The moments of this distribution can be found using a diffusion
approximation. Assuming recovery from infection is rapid compared to the loss of
immunity, the variance in the number of infected individuals is
 Y2  N ( R0  1) / R02
(7)
A measure of the magnitude of this variability is the ratio of variance to mean
number of infections, which is 1 / qR0 : thus low values of the basic reproduction number,
such that R0  1 / q , result in an over-dispersed distribution of infections compared to
the Poisson.
In sharp contrast to the deterministic case, the stochastic SIRS model results in
sustained oscillations due to the continued perturbation of the system by random
6
events11 (Supplementary Figure 1a). The period of these oscillations is given by T
(equation 6), the amplitude by the variance derived above (equation 7), while the phase
of these oscillations is subject to drift. Thus for a single population stochastic SIRS
model, in the absence of random extinction, sustained oscillations can be observed with
a period T that is dependent on the parameters of the infection and population
birth/death rates. As the rate of loss of immunity   0 the SIR model is recovered, in
which case oscillations can still occur with the period given by equation (6). In contrast
allowing    such that there is no longer any protective immunity results in the SIS
model and oscillations in prevalence do not occur. Failure to develop protective
immunity in all those infected can be described by allowing a fraction φ of individuals
recovering from infection to directly re-enter the susceptible population. In the
deterministic case the ordinary differential equations are modified such that the νy term
in equation (3) is replaced by (1-φ)νy and the term + φνy added to equation (1)
describing the dynamics of susceptibles. In this case, as   1 periodicity in
oscillations gradually disappears and with   1 only random noise is apparent.
However, significant periodicity remains even for reasonably large φ (Supplementary
Figure 1b).
Oscillatory dynamics in the SIRS model are robust to the addition of realistic
model complexity. Inclusion of more realistic distributions for the infectious period still
result in regular oscillations in prevalence but with a larger amplitude than that
predicted by the simple SIRS model 12. Also, while the model population size N will
vary according to the number of individuals sufficiently sexually active to be at risk of
infection, the predicted period of oscillations is independent of N. Perhaps more
important are the heterogeneities in sexual activity and contacts patterns within the
population. However, SIR infection dynamics defined on a contact network show
periodic oscillations for all but the lowest level of network disorder13. Furthermore,
stratification of the population into different risk groups can be considered analogous to
7
the geographic stratifications captured in metapopulation models, where oscillatory
dynamics occur with synchronisation determined by levels of coupling14.
References.
1.
Centers for Disease Control and Prevention. Sexually Transmitted Disease
Surveillance, 2002. (U.S. Department of Health and Human Services, Centers for
Disease Control and Prevention, Atlanta, GA, 2003).
2.
Gibson, C. Population of the 100 largest cities and other urban places in the
United States: 1790-1990. Working Paper No. 27. (Population Division, US Census
Bureau, Washington, D.C., 1998).
3.
Bjørnstad, O. N., Champely, S., Stenseth, N. C. & Saitoh, T. Cyclicity and
stability of grey-sided voles, Clethrionomys rufocanus, of Hokkaido: spectral and
principal components analyses. Phil Trans R Soc Lond B 351, 867-875 (1996).
4.
Chatfield, C. The analysis of time series: an introduction. 6th edition (Chapman
& Hall/CRC, Boca Raton, 2003).
5.
Bjørnstad, O. N., Ims, R. A. & Lambin, X. Spatial population dynamics:
analyzing patterns and processes of population synchrony. Trends Ecol Evol 14, 427432 (1999).
6.
Cliff, A. D., Haggett, P. & Smallman-Raynor, M. Deciphering global
epidemics: analytical approaches to the disease records of world cities, 1888-1912
(eds. Baker, A. R. H., Dennis, R. & Holdsworth, D.) (Cambridge University Press,
Cambridge, 1998).
7.
Wallace, R., Huang, Y. S., Gould, P. & Wallace, D. The hierarchical diffusion of
AIDS and violent crime among US metropolitan regions: Inner-city decay, stochastic
resonance and reversal of the mortality transition. Social Science & Medicine 44, 935947 (1997).
8
8.
US Dept of Transportation. Airline Origin and Destination Survey. (Bureau of
Transportation Statistics, 2003).
9.
Hethcote, H. W. The mathematics of infectious diseases. SIAM Rev 42, 599-653
(2001).
10.
Nåsell, I. Stochastic models of some endemic infections. Math Biosci 179, 1-19
(2002).
11.
Bailey, N. T. J. The mathematical theory of infectious diseases and its
applications. 2nd edition. (Griffin, London, 1975).
12.
Lloyd, A. L. Realistic distributions of infectious periods in epidemic models:
Changing patterns of persistence and dynamics. Theor Popul Biol 60, 59-71 (2001).
13.
Kuperman, M. & Abramson, G. Small world effect in an epidemiological model.
Phys Rev Lett 86, 2909-2912 (2001).
14.
Lloyd, A. L. & Jansen, V. A. A. Spatiotemporal dynamics of epidemics:
synchrony in metapopulation models. Math Biosci 188, 1-16 (2004).