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
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
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/
treatment/2010/chlamydial-
infections.htm [3]
1/μ # days after recovery
that the person waits to
7 days http://www.cdc.gov/std/
treatment/2010/chlamydial-
be sexually active again infections.htm [3]
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
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.
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 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&tabid=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.
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.