compartmental model for chlamydia, a sexually transmitted disease
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Compartmental Model for Chlamydia, a Sexually
Transmitted Disease
Figueroa-Monsanto, Héctor L.1
Pedrogo-Flores, Leonardo2
Cruz-Aponte, Mayteé3
Department of Chemistry1, Department of Natural Sciences2, and
Department of Mathematics-Physics3
University of Puerto Rico at Cayey
+Biological Disease: Chlamydia
Bacterium: Chlamydia trachomatis
Chlamydia is the most common sexually transmitted
disease (STD) in the world (Siam, 2012)
Transmission:
Direct contact with any infected area of the
Child Birth
Incubation period: 5 - 21 days
Treatment: Antibiotics
www.allposters.com
+Disease Symptoms
Females
Blood flow between periods
Pain: periods, sex, abdomen, and urination
Vagina itch
Males
Pain: testicles and urination
Penis: swelling, pain, burn, itch, and transparent discharge
Urethritis: 55%
Recovered people could be susceptible again.
+First Research Question
What is the behavior of the model on a
bigger scale?
Bigger scale – population dynamic of disease spreading.
How important is Chlamydia in our community?
+Future Research Question
What is the behavior of the disease in a
lower scale?
Lower scale – on a cellular level within the host using
compartments for the progression of the incubation.
Can our model be more precise using equations that
better model the incubation period?
+Purpose
By this we intend to develop a much more
precise model that simulates the disease at
its best and contributes in a better scale to
the scientific community.
Better define the behavior of the chlamydia
disease on a lower scale.
+ Schematic of disease dynamics:
Mathematical Model
S
E
I
R
βSI/N
αE
γI
μR
+ Schematic of disease dynamics:
Parameters
Parameter Description Range/Value References
β Probability of obtaining
the bacteria 67% Potterat (1999)
1/α Incubation Period 5 - 21 days http://www.antimicrobe.org/m04.asp
1/γ Days of Recovery if
treated
7 days with
treatment
http://www.cdc.gov/std/treatment/2010
/chlamydial-infections.htm
1/μ
# days after recovery
that the person waits to
be sexually active again
7 days http://www.cdc.gov/std/treatment/2010
/chlamydial-infections.htm
+Results: Chlamydia Disease Trend
+Mathematical Analysis
Equilibrium points:
−𝛽𝑆𝐼
𝑁+ 𝜇𝑅 = 0 → −
𝛽𝑆
𝑁+ 𝛾 𝐼 = 0
𝛽𝑆𝐼
𝑁− 𝛼𝐸 = 0 →
𝛽𝑆
𝑁− 𝛾 𝐼 = 0
𝛼𝐸 − 𝛾𝐼 = 0 → 𝐸 =𝛾𝐼
𝛼
𝛾𝐼 − 𝜇𝑅 = 0 → 𝑅 =𝛾𝐼
𝜇
Disease Free Equilibrium 𝑁, 0,0,0 "Trivial Solution”
Unstable
Endemic Equilibrium𝛾𝑁
𝛽,𝛾𝐼∗
𝛼, 𝐼∗,
𝛾𝐼∗
𝜇, 𝐼∗ ≠ 0
Stable
+Results
Using the endemic equilibrium point
and the fact that our population is constant:
Infected Individuals at the endemic stage:
Range:
Between 16% – 29%
of the population.
+Endemic Equilibrium Point
Solving for I we established the general equation:
𝐼∗ = 𝑁𝛽 − 𝛾
𝛽
𝛼𝜇
𝜇𝛾 + 𝜇𝛼 + 𝛼𝛾
The specific case of our scenario:
𝐼∗ = 1 −1
.67(7)
17𝑥
149
+ 217𝑥
Varying our incubation period (5,15, 21) days:
𝐼∗ 5 = 1 −1
.67(7)
1
7(5)1
49+2
1
7(5)
= 0.29
𝐼∗ 15 = 1 −1
.67(7)
1
7(15)1
49+2
1
7(15)
= 0.19
𝐼∗ 21 = 1 −1
.67(7)
1
7(21)1
49+2
1
7(21)
= 0.16
+Endemic Equilibrium Point Graph
+Graphical perturbation and
stability of the equilibrium point.
+R0 Basic Reproductive Number
(Using the Next Genertation Operator)
ℱ =𝛽𝑆𝐼
𝑁0
⇒ ℱ =0 𝛽0 0
𝒱 =𝛼𝐸
−𝛼𝐸 + 𝛾𝐼⇒
𝛼 0−𝛼 𝛾
⇒ 𝒱−1 =
1
𝛼0
1
𝛾
1
𝛾
R0=𝜌 ℱ𝒱−1 = 𝜌0 𝛽0 0
1
𝛼0
1
𝛾
1
𝛾
R0= 𝛽
𝛾= 7(.67)= 4.69
+Basic Reproductive Number For
Known Diseases
+Future Work
Develop and agent-base model for a given
population.
Modify schematic model by inserting new
equations that specify on the behavior of
the disease during its period of incubation.
+Bibliography
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.
+Bibliography
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.
+Acknowledgments
Mentors:
Dr. Mayteé Cruz-Aponte
Danilo Pérez-Rivera
+
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