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ABSTRACT

The Centers for Disease Control (CDC) is currently attempting to eliminate syphilis in

the United States (US); to ensure that their control strategies will be effective it is

important to understand the transmission dynamics of syphilis. Epidemics of certain

infectious diseases (e.g., influenza) can rise and fall with a well-defined periodicity; this

cycling behavior is important because it can have significant implications for the design

and effectiveness of control strategies. Here we discuss the methodology that has been

used to identify epidemic cycles in longitudinal data sets, and the endogenous and

exogenous mechanisms that generate cycling. We then examine the recently proposed

hypothesis that syphilis epidemics cycle. This hypothesis was proposed based upon the

results of a spectral analysis of a longitudinal data set that had been collected by the

(CDC), and the analysis of a syphilis transmission model. We use spectral analysis to

reanalyze the CDC's data set, as well as to analyze a longitudinal mortality data set

provided by the CDC. We also use published transmission models to predict the expected

dynamics of syphilis epidemics. In contrast to the previous findings we find that: (i) that

neither of the CDC's data sets provide compelling evidence that syphilis epidemics cycle

and (ii) published transmission models predict that syphilis epidemics should

monotonically decrease (as a function of the treatment rate) rather than cycle. We explain

the reasons why previous authors had proposed that syphilis epidemics cycle. Finally, we

discuss the implications of our findings regarding the transmission dynamics of syphilis

for the CDC's elimination plan.

Nature Precedings : doi:10.1038/npre.2007.1373.1 : Posted 29 Nov 2007

Introduction

The Centers for Disease Control (CDC) is currently implementing a syphilis

elimination plan in the United States (US); to ensure effectiveness it is important that

their control strategies are based on a quantitative understanding of the transmission

dynamics of syphilis. Several infectious diseases (e.g., influenza) show cyclical behavior.

Therefore multiple epidemics of these diseases rise and fall with a certain periodicity

which can have significant implications for the design, and effectiveness, of control

strategies. Recently it has been proposed that syphilis epidemics intrinsically cycle [1];

this hypothesis is controversial as, if correct, it could potentially reduce the effectiveness

of the CDC's syphilis control program. Epidemic cycling can be detected by both the

observation and statistical analysis (e.g., spectral analysis) of longitudinal data sets, and

mathematical models can be used to determine the conditions that cause cycling. Here we

examine the available evidence to assess the hypothesis that syphilis epidemics cycle.

Firstly we reanalyze, by using spectral analysis, longitudinal syphilis incidence and

mortality data sets from the CDC. Secondly we review the literature on transmission

models of syphilis. We then conclude by discussing the results of our reanalysis and

review in the context of the Centers for Disease Control's (CDC's) current syphilis

elimination plan.

Methodology for identifying epidemic cycles

Multiple cycles are observed in the incidence of many infectious diseases, such as

childhood diseases (e.g., measles [2-4]; see Figure 1A, chickenpox [3, 4]), faecal-oral

infections (e.g., cholera [5]), vector-borne diseases (e.g., malaria [6], dengue [7]),

respiratory infections (e.g., influenza [8], pertussis [9], smallpox [10]) and even some

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sexually transmitted diseases (e.g., gonorrhea [11]); see Figure 1 panel A for a time series

of measles incidence data. Epidemic cycles of various lengths have been observed for

these diseases. The duration of an epidemic cycle (i.e., period) is one year for diseases

such as chickenpox [3, 4], influenza [8] and gonorrhea [11]. Other infectious diseases

such as measles [2-4] (see Figure 1A), pertussis [9] and smallpox [10] show a two-year

cycle, a 2-2.5 year cycle and a 2-3 year cycle, respectively. An even larger cycle of 3-4

years has been documented for dengue [7, 12].

Epidemic cycling has been detected and the periodicity of epidemic cycles

calculated by analyzing longitudinal data sets using spectral techniques, wavelet

techniques, and time-series models. Periodicity of multiple measles outbreaks (between

1703 and 1917) was investigated as early as 1918 [13]. More recently, spectral

techniques (see Technical Appendix for a brief description of these techniques) have

been used to explore the periodicity of smallpox [10] and cholera [14]. Wavelet analysis

has been used to study epidemiological time-series for measles [15], pertussis [16] and

cholera [17]. Time series models based on the autoregressive integrated moving-average

method (i.e., the Box-Jenkins models [18]) have been used to analyze syphilis and

gonorrhea surveillance data [11, 19] and childhood infectious diseases data [20]. For an

example of how spectral analysis can be used to identify the two-year periodicity of

measles cycles see the power spectrum graph given in Figure 1B (red data). In this graph

one peak is substantially higher than the other peaks (note that the vertical axis is given in

a logarithmic scale); this highest peak represents the principle cyclic component in this

power spectrum (i.e., the two year cycle of measles epidemics) (Figure 1B: red data).

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The mechanisms that explain the cycling of epidemics have been classified into

two major categories: endogenous (i.e., intrinsic) and exogenous (i.e., extrinsic) [21].

Exogenous mechanisms are environmental factors (e.g., temperature) that affect the host

and/or the pathogen separately (e.g., they may drive the abundance of pathogens or the

density of hosts). These factors periodically perturb the epidemic from outside the system

and cause cycling. In contrast, endogenous mechanisms (e.g., immunity [22]) affect

exclusively the host-pathogen interaction. These factors periodically perturb the epidemic

from inside the system, cycling occurs (through a Hopf bifurcation) in the absence of any

external perturbation [23]. The periodicity of an epidemic may be driven by a

combination of both exogenous and endogenous factors. For example, the immune status

of the host population (an endogenous factor) and increased host density during school

terms (an exogenous factor) can explain the biannual cycle of measles in England [2-4,

24, 25]. The annual cycle of cholera in Bangladesh can be explained by the immune

status of the local population (an endogenous factor), and the El Nino southern oscillation

and the Indian Ocean temperature (both exogenous factors) [14, 26]. Koelle et al. have

recently developed a methodology that enables the effects of endogeneous and

exogeneous factors to be independently isolated [26, 27].

The effects of both exogeneous and endogeneous mechanisms on the

epidemiology of an infectious disease have been determined by analyzing mathematical

models of transmission dynamics. Exogenous mechanisms have often been modeled by

changing a parameter, whereas endogenous mechanisms have often been modeled by

changing the structure of the transmission model. For example, seasonality (an

exogenous mechanism) has typically been modeled by a periodic change in the

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infectiousness of the pathogen (e.g. measles [28] and influenza [8]), and, in certain cases,

by a periodic change in the host birth rate as well [29, 30]. Immunity (an endogenous

mechanism) has generally been modeled by adding an additional state variable to the

model.

Using spectral analysis to analyze longitudinal syphilis data sets

Grassly

et al. [1] have recently proposed that syphilis epidemics cycle. Their

hypothesis is based upon the analysis of a short time series (1960 to 1993) selectively

sampled from a long time series (1941 to 2002) of syphilis incidence data collected by

the CDC in the US (Figure 1C). Grassly et al. [1] performed a spectral analysis on the

subset of the data (1960 to 1993) (Figure 1D: blue data) and concluded that syphilis

epidemics cycle with an approximate period of 8 to 11 years in the entire US, and in both

large (e.g., New York and Houston) and small (e.g., Birmingham and Rochester) cities. If

syphilis epidemics cycle it could be extremely difficult for the CDC to achieve syphilis

elimination. Hence before the hypothesis that syphilis epidemics cycle is widely accepted

the methodology used to obtain this result needs to be carefully evaluated. Therefore we

used the same methodology (i.e., spectral analysis) that Grassly et al. [1] had used and we

reanalyzed the same CDC data set.

The authors of the original analysis aggregated syphilis incidence data over the

entire US from many subpopulations that differed considerably in geography, sexual

orientation, race, gender etc [31]. It is possible that their conclusion that syphilis

epidemics cycle could be an artifact of data aggregation. To investigate this possibility

we stratified the CDC data (for New York City) by gender and then used spectral

analysis. If syphilis epidemics appear to cycle in an aggregated data set, then epidemic

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cycles (of the same periodicity) should also be visible when these data are stratified.

However we find that the period in the female incidence data (Figure 2A: red data) is

twice as long as the period in the male incidence data (Figure 2A; blue data), indicating

that it is unlikely that syphilis epidemics cycle. Recent analyses of the trends in syphilis

epidemics in subpopulations (defined on the basis of geography, gender, race, sexual

orientation etc) by Peterman et al. have also shown that there is little evidence that

syphilis epidemics cycle [31].

To ensure the reliability of spectral analysis in identifying cycles the time series

analyzed needs to be significantly longer than the period of the cycle. If only a short time

series is analyzed a bias, known as aliasing, can occur and any apparent periodicity in the

incidence data is spurious. For example, a spectral analysis of 33 years of a constant

incidence rate would show a spurious periodicity of 7.3 years due to aliasing (see

Technical Appendix for further explanation). To determine whether a spectral analysis

has reliably identified cycles the spectrum of the selected time series should be compared

with the spectrum of the entire data set. If these two spectra closely match, it can be

concluded that spectral analysis of the selected time series reliably identifies cycles. For

example, the spectrum of the selected time series of measles incidence delimited by the

dotted lines in Figure 1A and the spectrum of the entire measles data set shown in Figure

1A match very well (Figure 1B; compare red data with blue data); the normalized root

mean square difference is 0.11. Hence it can be concluded that the spectral analysis of the

selected time series can reliably identify cycles of measles epidemics and determine that

these cycles have a period of two years (Figure 1B) (see Technical Appendix for further

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explanation). A short time series can be used to reliably identify cycles of measles

epidemics as they cycle every two years.

Grassly

et al. [1] concluded that syphilis epidemics cycle and have a periodicity

of 8 to 11 years by analyzing a selected 33 year time series (1960 to 1993) from the CDC

syphilis incidence data set. To determine whether their results reliably identify syphilis

cycles and determine their periodicity we compared the spectrum (blue data) of the entire

CDC syphilis data set (1947 to 2002) with the spectrum (red data) of the subset of the

data set (1960 to 1993) selected by Grassly et al. [1] (Figure 1D). These two spectra

match poorly; the normalized root mean square difference is ~0.21 (i.e., twice as much as

for measles). Hence these results indicate that the selected 33 year time series is too short

to reliably identify cycles with a periodicity of 8 to 11 years. The apparent periodicity

that is observed in the spectrum of the selected subset of the syphilis incidence data could

simply be the result of aliasing. We also performed a spectral analysis of a longitudinal

syphilis mortality data set that was collected by the CDC between 1900 and 1971; the

data set is shown in Figure 2A. We conducted this analysis to determine if mortality rates

show oscillations, since any oscillations in incidence rates should be reflected in the

mortality rates. However, the calculated spectrum (Figure 2B) for this data set shows no

evidence that syphilis epidemics cycle as the maximum peak in the power spectrum is at,

or close to, the origin (see Technical Appendix for further explanation).

Is there any theoretical evidence to suggest that syphilis epidemics cycle?

A detailed biologically realistic epidemic model of syphilis was developed and

analyzed by Garnett et al. [32]. They modeled each stage of syphilis separately (primary,

secondary, latent and tertiary) and also explicitly included treatment. Their modeling

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showed that, if syphilis was not treated, incidence rates would show damped oscillations

(but not cycles) over time and stabilize within 25 to 100 years (Figure 3A). Since syphilis

has been around for many hundreds of years the results of Garnett et al. [36] imply that,

even in resource-constrained countries, current incidence rates would not be expected to

show even damped cycles. Garnett et al. [32] also used their model to investigate the

impact of treatment on syphilis epidemics. They showed that treatment would cause

syphilis incidence rates to dramatically and quickly decrease (without cycling) to a lower

endemic level. They determined that the new stable endemic incidence level would be

determined by the treatment rate (i.e., the fraction of cases treated) (Figure 3B).

Rather than develop a detailed biologically realistic model Grassly et al. [1]

modeled primary, secondary, latent and tertiary syphilis as one state and used a

deterministic and stochastic version of a simple transmission model: the Susceptible-

Infected-Recovered-Susceptible (SIRS) model (see Technical Appendix for model

description). By simulating the deterministic version of the model they found that,

depending on parameterization, syphilis incidence approached a stable level through a

series of damped oscillations (Figures 4A and 4B). However, when they simulated the

stochastic version of the model they found that incidence rates followed the damped

trajectory generated by the deterministic version for only a brief period, after which

incidence rates cycled (Figure 4A). This cycling phenomenon is called coherence

resonance or autonomous stochastic resonance [33, 34]. We repeated the analysis of the

stochastic version of the SIRS model using a kinetic Monte Carlo model [35, 36] and the

same parameterization as Grassly et al. [1] (see Technical Appendix for modeling

details). We determined that epidemic dynamics are strongly affected by population size.

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If the population size is 10

as used by Grassly et al. [1], cycles are clearly visible (i.e.,

coherence resonance occurs) (Figure 4A; green data). Sustained cycles are also present if

the population size is 10

, but their amplitude is substantially reduced (Figure 4B; red

data). However, for a population size of 10

, that is typical for a large US city such as

New York, the effects of coherence resonance are hardly visible (these results are not

shown as the path taken by the stochastic dynamics lies on top of that found by the

deterministic dynamics). Our results imply that if Grassly et al. [1] had used a realistic

population size for their transmission dynamic modeling, they would not have concluded

that syphilis epidemics cycle.

Conclusions and implications for syphilis control

As we have discussed, and illustrated with the presentation of a longitudinal data

set of measles and the spectral analyses of this data set, infectious diseases can show

cyclical behavior with a clearly defined periodicity. This cyclical behavior is caused by a

variety of endogenous and/or endogenous mechanisms that affect the transmission

dynamics. For the infectious diseases that have been observed to cycle, biologically

realistic transmission models of the specific-disease dynamics have been developed and

analyzed in order to determine the effect of these mechanisms in generating cycling

behavior. Here we have investigated the recent controversial claim that syphilis

epidemics intrinsically cycle [1]. We have used spectral techniques to reanalyze

longitudinal syphilis incidence and mortality data sets from the CDC, and found that

these analyses do not support the claim that syphilis epidemics cycle. We have also

examined the results of the analysis of transmission dynamic models of syphilis

epidemics and found that the predictions from these models (if they are correctly

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parameterized) also do not support the claim that syphilis epidemics cycle. Thus we

conclude that spectral analyses of CDC data sets and predictions from published

transmission modeling studies do not provide evidence that syphilis epidemics cycle.

The CDC is currently attempting to eliminate syphilis in the US. Their Syphilis

Elimination Effort for the US (The 3-By-3 approach to syphilis elimination) was

redesigned in 1995. Interim elimination targets are to reduce rates of primary and

secondary syphilis cases, by 2010, to less than 2.2 per 100,000 population, congenital

syphilis to less than 3.9 per 100,000 live births; and Black:White racial disparities to a

ratio of less than 3:1. The CDC has specified three goals for reaching their elimination

targets: enhancement of public health services, evidence-based interventions that are

culturally appropriate; and accountability. Their plan is intended to guide and assist local,

state, and national health agencies to focus on achieving syphilis elimination in the most

cost-effective, ethical, and acceptable way manner. The success of the CDC's elimination

plan will be greatly affected by the transmission dynamics of syphilis. Clearly outbreaks

of syphilis have occurred over recent decades, but occasional outbreaks are very different

from cycling behavior. Syphilis outbreaks can be explained by a variety of factors:

changes in sexual behavior, the gay liberation movement in the 1970's, the HIV

epidemic, the sexual revolution, and changes in the intensity of syphilis control programs

[37]. If syphilis epidemics cycle, and cycle intrinsically as Grassly et al. [1] have

proposed then it could be very difficult for the CDC to achieve their goal of elimination.

However, because there is little evidence to support this claim, and since mass treatment

has been shown to be extremely effective in reducing syphilis levels it is quite possible

that the CDC could be successful in eliminating syphilis in the next few decades.

Nature Precedings : doi:10.1038/npre.2007.1373.1 : Posted 29 Nov 2007

Figure Legends

Figure 1: Spectral analysis of measles and syphilis incidence time series. A. Time series

of the monthly measles incidence cases in Glasgow between 1901 and 1916 [25]. B. The

spectral density of the complete measles data set in panel A is shown in red while the

spectral density of the subset of measles data within the dotted lines is shown in blue.

Note that the two spectra match well; the normalized root mean square difference

between the graphs is 0.11. C. Time series of the annual syphilis incidence rate per

100,000 in New York between 1941 and 2002 [1] provided by the CDC. Data collected

before 1947 (when penicillin became widely available for treating syphilis [38]) are

shown in magenta. D. The spectral density of the black data in panel C is shown in red

while the spectral density of the subset of data delimited by the dotted lines (which is the

interval Grassly et al. [1] selected for their analysis) is shown in blue. Note that the two

spectra do not match as well as those in Figure 1B; the normalized root mean square

difference between the graphs is 0.21 (i.e., double).

Figure 2: Syphilis mortality and incidence data sets provided by the CDC A. Time series

of the annual syphilis incidence rate per 100,000 in New York between 1967 and 2003

stratified by gender. The blue data are for the male cases while the red data are for the

female cases. Note that the number of red peaks is less than the number of blue peaks. B.

Time series of the syphilis mortality rate in the US. The data collected before 1943 (when

penicillin was introduced for the treatment of syphilis [38]) are shown in magenta. C. The

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spectral density of the magenta data in panel B. Note that no significant peaks exist (See

Technical Appendix for explanation).

Figure 3: Incidence of primary (blue) and secondary (red) syphilis generated by the

model developed by Garnett et al. [32]. A. Simulations from this model, without

including the effects of treatment, show damped oscillations before reaching a steady

endemic state. B. When treatment is introduced at the steady state, incidence is

dramatically and quickly reduced. In this simulation, treatment would eventually

eliminate syphilis; under lower treatment levels, incidence could stabilize at a low

endemic level.

Figure 4: Syphilis incidence per capita dynamics obtained from the SIRS model used by

Grassly et al. [1]. A. The dynamics of the deterministic version of the SIRS model is

shown by the black curve. A simulation of the corresponding stochastic version of the

SIRS model using a population size of 10

is shown in green. B. SIRS dynamics of the

deterministic version of the model is shown by the black curve. A simulation of the

corresponding stochastic version of the model using a population size of 10

is shown in

red.

Nature Precedings : doi:10.1038/npre.2007.1373.1 : Posted 29 Nov 2007

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Technical Appendix:

The CDC's syphilis elimination plan & the transmission dynamics of syphilis

Romulus Breban, Virginie Supervie, Justin Okano, Raffaele Vardavas, and Sally Blower

Semel Institute for Neuroscience and Human Behavior, David Geffen School of Medicine,

University of California, Los Angeles, California 90095-1555

Here we layout some of the technical methodologies that we have used in the main text. In

particular we explain the SIRS model as formulated deterministically via Ordinary Differential
Equations (ODEs). We also explain how to construct a Monte Carlo algorithm for the SIRS model
that takes into account all of its inherent probabilistic processes.

Contents

I. Notes on Spectral Analysis

A. What is a Power Spectrum graph?

B. Decibel or Logarithmic scale of Amplitude

C. Aliasing

II. The Deterministic

SIRS model

III. The Stochastic

SIRS model and kinetic Monte Carlo Integration

A. Parameters and Initial Conditions

B. Results

References

NOTES ON SPECTRAL ANALYSIS

What is a Power Spectrum graph?

A power spectrum graph represents a mathematical transformation of the yearly incidence data into weighted

components of sinusoidal waves of different frequencies. Slow repeating patterns appear to the left of the graph while
fast repeating patterns appear to the right. These fast repeating patters in the incidence data become less significant
as one approaches the frequency at which the original data is sampled at. For example if data is sampled four times
per year then peaks that appear at frequencies larger than 4 years

-1

are physically or biologically non significant.

The amplitude on the vertical axis categorizes the weight given to the frequencies or cycles of the repeating patterns.

Decibel or Logarithmic scale of Amplitude

Often, the vertical axis of a power spectrum graph is given in units of Decibel which is a logarithmic scale of

amplitude. This scale helps to reveal the other frequency components that would not be apparent if a linear scale
was used. The highest peak on a power spectrum graph with a vertical axis given in logarithmic scale, represents
the principle cyclic component. If the maximum peak is at or close to the origin then the data is best described with
sinusoidal waves of very large wavelengths. If these wavelengths are greater than the length of the time series then it
implies that there is no periodicity in the data.

Aliasing

When analyzing the available data set that is given within a certain time window of duration we can assume

either that (i) points outside the time window are set to zero or (ii) the data set outside the time window repeats
indefinitely with period . This however introduces discontinuities at the beginning and end points of the available
data set. This is not a big problem if there are many data points within the time window. However, often the number

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of available data points within the time window that one needs to analyze is quite limited. The frequency spectrum
obtain by the spectral analysis of the data set is affected by this boundary effect. For example if we have a limited
number of data points and we assume that the data set outside the time window repeats indefinitely with period
then besides having introduced an artificial frequency component of frequency 1/ in our spectrum we have added
higher frequency components that describe the discontinuity in the data at the boundaries. This problem is referred
to aliasing-error.

In the Main Text, we mention that the syphilis data (that are yearly sampled) could be vulnerable to aliasing-

error. This is because the number of data points is very limited and the boundary points is not a very small fraction
of the available points. Whereas, for the case of measles incidence data (that are sampled numerous times per year)
the boundary points represent a very small fraction of the available points.

II.

THE DETERMINISTIC SIRS MODEL

We begin by describing the deterministic version of the

SIRS model used by Grassly et al.[1] We denote S(t), I(t)

and R(t) to represent the susceptible, infected and recovered populations at time t and N (t) = S(t) + I(t) + R(t)
represents the total population. We can then write the density of these populations s(t) = S(t)/N (t), i(t) = I(t)/N (t)
and r(t) = R(t)/N (t). According to the SIRS model the deterministic dynamics of these densities are represented by
the following equations:

ds(t)/dt = µ - s(t)(µ + i(t)) + r(t),

(1)

di(t)/dt = s(t)i(t) - (µ + )i(t),

(2)

dr(t)/dt = i(t) - ( + µ)r(t),

(3)

where is the transmission parameter, the rate of recovery from infection, the rate of loss of immunity and
µ the rate of birth/death. Notice that in the above formulation the sum the equations is equal to zero and thus
dN (t)/dt = 0, this represents the situation where we are at a demographic equilibrium. The basic reproductive
number of this model is given by R

= /( + µ). The deterministic dynamics of the SIRS model approaches an

equilibrium state

, i

, r

} in two possible ways depending on the parameters: i) monotonically and ii) via damped

oscillations. In the later case it can be shown that the period of these damped oscillations is given by

T =

( + µ) -

( + + µ)

(4)

where

= (R

- 1)(µ + ), and = ( + µ)/( + µ + ).

Nature Precedings : doi:10.1038/npre.2007.1373.1 : Posted 29 Nov 2007

III.

THE STOCHASTIC SIRS MODEL AND KINETIC MONTE CARLO INTEGRATION

The stochastic dynamics of the

SIRS model can be represented via a Master Equation (not shown). The Master

equation describs the dynamics in the probability distribution of the state variable

{S, I, R}. This equation is formu-

lated by listing all possible transition events from the state

{S, I, R}. In table I we list all seven possible transition

events.

TABLE I: Supplementary Table 1 Rate of events in the SIRS model with demography.

Event

Transition

Rate

Cumulative Rate

Infection

{S, I, R} {S - 1, I + 1, R} SI/N =

Recovery from infection

{S, I, R} {S, I - 1, R + 1}

I =

Loss of immunity

{S, I, R} {S + 1, I, R - 1}

R =

j=1

Death of susceptible

{S, I, R} {S - 1, I, R}

µS =

j=1

Death of infected

{S, I, R} {S, I - 1, R}

µI =

j=1

Death of immune

{S, I, R} {S, I, R - 1}

µR =

j=1

Entry of susceptible

{S, I, R} {S + 1, I, R}

µN =

j=1

Each of the possible seven events, labeled by the subscript j, occurs with a rate

given in table I. We define the

cumulative rate of event j to be C

k=1

with C

= 0 [2]. The Master Equation can be integrated numerically

via a kinetic Monte Carlo (kMC) algorithm. Starting from a certain number of susceptibles, infected and recovered
that define the state

{S, I, R} at t = 0, we can integrate the dynamics over a number of steps. At each step one event

is chosen randomly. This is done by first generating a random variable X that is uniformly distributed in the range
0 to 1. We then select event j if C

j-1

< C

. Once we carry out event j we increment the step by a unit and

increment the time t by t that is given by

t = -C

-1

log[Y ],

where Y is also a uniformly distributed random variable in the range 0 to 1.

Parameters and Initial Conditions

The

SIRS model we considered used parameter values taken directly from Grassly et al.[1] These are the transmission

parameter = 9.045, the rate of recovery from infection = 6, the rate of loss of immunity = 0.1 and the rate
of birth/death µ = 0.03 all quoted in units (years)

-1

As mentioned in Grassly

et al.[1] the value for the rate of

recovery from infection is large as it takes into account the introduction of treatment. This set of parameters yields
a basic reproductive number R

of 1.5 and a period T of 10.16 years. With these parameters the equilibrium state is

, i

, r

} = {0.66667, 0.00707, 0.32626}.

We simulated the stochastic

SIRS model using these parameters and population sizes of N = 10

and N = 10

We integrated the stochastic

SIRS model starting from the following initial conditions {S(0), I(0), R(0)} = N ×

{0.66667, 0.02, 0.32626}.

Nature Precedings : doi:10.1038/npre.2007.1373.1 : Posted 29 Nov 2007

Results

The deterministic and stochastic

SIRS dynamics obtain using these parameter sets and this initial condition is

shown in Fig. 1. We coded the deterministic dynamics and the stochastic dynamics with Mathematics v5.1.0.0. This
code is available upon request.

0.01

0.02

Incidence per capita

0.01

0.02

Year

Incidence per capita

FIG. 1: Syphilis incidence per capita dynamics obtained from the SIRS model for used by Grassly et al. [1] A. The dynamics
of the deterministic SIRS model is shown by the black curve. A simulation of the corresponding stochastic version of the model
using a population size of 10

is shown in green. B. SIRS dynamics of the deterministic model is shown by the black curve. A

simulation of the corresponding stochastic version of the model using a population size of 10

is shown in red.

[1] N. C. Grassly, C. Fraser, and G. P. Garnett. Host immunity and synchronized epidemics of syphilis across the united states.

Nature, 433(7024):41721, Jan 27 2005. Journal Article Research Support, Non-U.S. Gov't.

[2] Notice that the cumulative rates

change with time as the state

{S, I, R}

Nature Precedings : doi:10.1038/npre.2007.1373.1 : Posted 29 Nov 2007