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.