This document describes a compartmental model for Chlamydia, a sexually transmitted disease. The model divides a population into susceptible, incubation, infected, and recovered compartments. Parameters like infection probability and incubation period are used to model disease flow between compartments. Simulation results show that a longer 13-day incubation period leads to more susceptibility and less infection/recovery over time compared to a 5-day period. The model demonstrates how Chlamydia spreads in a population over time.

1. University of Puerto Rico
Cayey Campus
2015 RISE Program
Compartmental Model for Chlamydia, a Sexually Transmitted Disease
Pérez-Ayala, Michelle C.1, Figueroa-Monsanto, Héctor L.2 and Cruz-Aponte, Maytee3
Department of Biology, University of Puerto Rico at Cayey1, Department of Chemistry,
University of Puerto Rico at Cayey2 and Department of Mathematics-Physics, University
of Puerto Rico at Cayey3
1. Introduction
Chlamydia is the most common sexually transmitted disease (STD) in the world
(Siam, 2012). Actually, people obtain the disease by having direct contact with body
fluids, sperm and/or any area of the body with the bacterium Chlamydia trachomatis,
which is known to produce another sexual transmitted disease called Lymphogranuloma
venereum (LGV) (Borges, 2015). This microorganism is transmitted through sex,
including oral, thus it could be easily spread in a population, mainly those that do not use
protection. Both men and females are susceptible, but young individuals less than 25
years of age are the most infected (Occhionero et al. 2015). Another way to get infected
with Chlamydia is when a pregnant woman infects her child while he is being foaled
through the infectious vagina. Once the person obtains C. trachomatis, the incubation
period inside the human takes between five and twenty one days. It is uncertain if the
person with the ompA gene that codifies for this bacterium (Occhionero et al. 2015) could
infect his or her partner, while the bacterium is still incubating.
Another factor that makes this disease a high risk is that the symptoms are not
commonly shown in the individual, even after a complete incubation. Thus the diseased
person does not receive treatment (antibiotics) and if the individual continues to have
sexual intercourse more people could receive this bacterial disease. At the time that the

2. symptoms appear, the effects are reflected on both females and males, which can have
harmful and painful infections, such as trachoma if combined with C. psittaci (Nittler et al.
2014). For instance, ladies can have blood flow between periods, itching vagina, painful
menstrual periods, discomfort during sex, abdomen pain and ache during urinations, as
well as many other burning and uncomfortable consequences. For instance, women can
contain an ectopic pregnancy and infertility. Actually, De Borborema-Alfaia (2013) study
reflected a 3.4 % of premature membrane rupture in Brazilian women around the age of
30 or less. On the other hand, men can have pain in the testicles and in urination, a
swelling, painful, itching and transparent discharge through the penis, 55% get urethritis
and many other effects. Al-Mously (2015) studied the effect of Chlamydia trachomatis on
sperms to detect if the bacterium caused sub-fertility in men. The results showed that
among the sperm properties that were affected were viability and motility, which cannot
be prevented by In vitro fertilization.
In 2012, the Department of Health of Puerto Rico reported 5,007 cases of women
infected and 1,124 diseased men (Departamento, 2012). These statistics show that
women are more susceptible to this disease. But the fact is that this female statistic could
be the result of the more frequent visits of women to gynecologists. Thus, it can be
presumed that there can be a high number of unreported cases of Chlamydia by men to
urologists.
The population of Puerto Rico between the ages of 20 and 24 are the more
susceptible to be infected, which may be due to lack of orientations. So, the young group
who are sexually active without knowing that their infected they continue transmitting the
disease. Since this STD does not usually present symptoms, people do not get treated
with antibiotics and re-infections can occur. In addition, it is uncertain if the bacteria
isolates in the gastrointestinal tract, even though treatment is performed or if the
incubation period occurs again in the person that was infected.
This investigation aims to construct and use a compartmental model, in order to
demonstrate the susceptible, incubation period, infected or re-infected and recovered
Puerto Ricans from Chlamydia trachomatis. As a result, more precautions should be
taken, due to this method of informing the public about an STD that causes painful

3. consequences and even infertility. Thus, the number of infected Puerto Ricans could
lessen.
2. Materials and methods
2.1 Chlamydia Parameters and Model
The SIR model, which consists of Susceptible, Infected and Recovered individuals
in a population, was the base of the compartmental model of Chlamydia, but an Incubation
compartment was added. This is due to the fact that this biological disease is obtained by
having contact with Chlamydia trachomatis. The compartmental model shows the flow of
the disease from susceptible to recovered and from recovered to susceptible in any given
population.
According to Figure 1, the different parameters were used to create formulas that
aid to mathematically demonstrate the stages of the disease. These take a particular
place in the model. The model includes the susceptible (S) individuals of a given
population that have 0.67 changes of obtaining the bacteria (α), the incubated (E) period
(1/ β) of the prokaryote inside the person, Infected (I) people and the individuals that
recuperate (R). The mathematical model is shown in Figure 2.
Parameter Description Range/Value References
α Probability of obtaining
the bacteria
0.67 Potterat (1999) [1]
1/ β Incubation Period 5-21 days http://www.antimicrobe.org/
m04.asp [2]
1/γ Days of Recovery if
treated
7 days with
treatment
http://www.cdc.gov/std/treat
ment/2010/chlamydial-
infections.htm [3]
1/μ # days after recovery
that the person waits to
be sexually active again
7 days http://www.cdc.gov/std/treat
ment/2010/chlamydial-
infections.htm [3]

4. Figure 1. Shows the description of the parameters used to create the mathematical
models and their values.
Figure 2. Demonstrates the compartmental model with the formulas.
The formula of the susceptible demonstrates that the α SI/N will be the people that are
susceptible and interacting with the infected divided by the given population, which will
move from being susceptible to being incubating the bacteria. The βE rate represents the
individuals that have completed their incubation period; present symptoms and can
spread the infection (changing from the E compartment to the I compartment). The γI rate
represents the individuals that have moved from being infected to being recovered in
approximately seven days. Finally, the μR rate is the people that pass from being
recovered to susceptible again since there is no permanent immunity for this infection.
3. Results

5. Essentially when compared both figures we can easily appreciate that when the
incubation period varies for instance at 13.00 days of incubation the rate at which the
susceptible set comes closer to equilibrium and diminishes is slower coming apart at a
range of 50.00 days. The fact that the incubation period is 13.00 days cause an overflow
on the exposed compartment and all other quantities flow freely towards the exposed
compartment but not out of it. Meanwhile when the incubation period is 5.00 days it allows
for the equilibrium to be reached earlier without the overflow in any of the other
compartments.
Figure 3. Demonstrates that when the incubation period is of 13.00 days the set of people
that are susceptible lowers after 50.00 days approximately and maintains a constant
value. While the set of people that are infected and in recovery will range about the same
as the susceptible which is between 20-25 individuals after a time of 55.00 days
approximately has been reached.

6. Figure 4. Has a lower incubation period with 5.00 days. Having a quicker lowering of the
susceptible set by 25.00 days becomes lower than the set of infected people. Becoming
more depth by 45.00days were it reaches equilibrium staying around an approximate of
20 individuals. Were as the infected set reaches a constant value in about 30.00 days
from the disease spread. When infected and recovering the values come very close this
is due to the time in each of the compartments which aren’t very much. Therefore the flow
between compartments will come smoothly and won’t have an overflow.
4. Discussion
Chlamydia is the most common sexually transmitted disease in Puerto Rico. Young
people around the age of 33 or less are more susceptible to obtain the bacterium
Chlamydia trachomatis. Using the disease-related parameters (e.g infections), a
compartmental model could be developed. This model was composed of the susceptible,
incubation, infected and recovered compartments. By using the mathematical formulas
and altering the b (incubation period) parameters, the individuals from a given population
of 100 were visualized, in terms of infections, incubation, immunity and recovery. The
results showed by increasing the incubation from 5 days to 13 days, people were more

7. susceptible, less infected and thus less recovered. This model could be used for other
assigned populations, since this disease is common worldwide. Thus, it could be used to
demonstrate the trajectory of the disease in a given time.
5. References
Al-Mously N, Eley A. 2015. Transient exposure to Chlamydia trachomatis can induce
alteration of sperm function which cannot be stopped by sperm washing. Middle East
Fertility Society Journal 20, 48-53. doi:10.1016/j.mefs.2014.04.003.
Borges V, Gomes JP. 2015. Deep comparative genomics among Chlamydia trachomatis
lymphogranuloma venereum isolates highlights genes potentially involved in
pathoadaptation. Infection, genetics and Evolution 32, 74-88.
doi:10.1016/j.meegid.2015.02.026.
De Borborema-Alfaia APB, de Lima Freitas NS, Filho SA, Borborema-Santos CM. 2013.
Chlamydia trachomatis infection in a simple of northern Brazilian pregnant women:
prevalence and prenatal importance. Braz J Infect Dis 17(5):545-550.
doi:10.1016/j.bjid.2013.01.014.
Departamento de Salud. 2012. Datos de Puerto Rico.
http://www.estadisticas.gobierno.pr/iepr/LinkClick.aspx?fileticket=RVOVIk8ka1A%3D&ta
bid=186.
Knittler MR, Berndt A, Böcker S, Dutow P, Hänel F, Heuer D, Kägebein D, Klos A, Koch
S, Liebler-Tenorio E, Ostermann C, Reinhold P, Saluz HP, Schöfl G, Sehnert P, Sachse
K. 2014. Chlamydia psittaci: New insights into genomic diversity, clinical pathology, host-
pathogen interaction and anti-bacterial immunity. International Journal of Medical
Microbiology 304, 877-893. doi:10.1016/j.ijmm.2014.06.010.

8. Occhionero M, Paniccia L, Pedersen D, Rossi G, Mazzucchini H, Entrocassi A, Vaulet
LG, Gualtieri V, Fermepin MR. 2015. Prevalencia de la infección por Chlamydia
trachomatis y factores de riesgo de infecciones transmisibles sexualmente en
estudiantes universitarios. Rev Argent Microbiol 47(1): 9-16.
doi:10.1016/j.ram.2014.11.003.
Siam EM, Hefzy EM. 2012. The relationship between antisperm antibodies prevalence
and genital Chlamydia trachomatis infection in women with unexplained infertility. Middle
East fertility Society Journal (2012) 17, 93-100. doi:10.1016/j.mefs.2011.09.003.