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Study and Analysis of Cholera
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MALLESSA YEBOAH
OLIVIA HYLTON
MENTOR: DR. URMI GHOSH -DASTIDAR
Outbreak in Haiti
Following the earthquake of January 2010 in Haiti the first cholera
outbreak in almost a century was announced in October of that year.
To date, over 470,000 cases of cholera have been reported in Haiti
with over 7,000 attributable deaths.
This marks the worst cholera outbreak in recent history, as well as the
best documented cholera outbreak in modern public health.
Since the beginning of the outbreak the CDC (Center of Disease
Control) has worked closely with the Haitian MSPP (Ministry of Public
Health and Population) to combat the cholera epidemic and reduce
the impact of the disease.
Modeling
◦ Epidemic models attempts to understand then prevalence and distribution
of a disease, together with the factors that determine incidence, spread, and
persistence.
◦ They allow us to conceptualize transmission dynamics in a quantitative way,
while also giving a platform where different hypotheses can be tested.
◦ The 1st differential equation models of infectious disease were the work of
Daniel Bernoulli. (1766)
◦ During the 1900s the dynamical systems approaches were applied.
SIR Model
Stability Analysis
What is stability analysis and why is it important?
The Jacobian stability approach is used to prove the stability of the Equilibriums.
Linearization a model's equations give the Jacobian matrix and tells us about the
behavior near equilibrium.
An equilibrium is determined to locally asymptotically stable (LAS) if all the
eigenvalues have negative real parts.
In order to show that an infection/disease is independent of the sample population
size used for the model one must show that the equilibrium is globally-asymptotically
stable (GAS).
Disease Free Equilibrium is when no presence of the infection. The infected class
will be zero (I=0) and the entire population will consist of only susceptible people.
Endemic Equilibrium is a situation where there is presence of the infection and
the disease invades. I≠0
Basic Reproductive Number
The expected number of new infections created during its by an infected
individual under the best conditions for transmission.
Calculated by taking infectivity rate over the infectivity period.
◦ The below case holds try for true for the our sample SIR model:
The value of the reproductive number will determine the presence of the
disease.
If
< 1 there exist some conditions under which the disease cannot grow,
therefore the disease dies out.
If
> 1 there exist some conditions under which the disease can grow,
therefore the disease invades.
Calculating the Basic
Reproductive Number
◦ Next Generation Method
◦ Next Generation Matrix K=
Note that F is associated with new infections and V is associated with infected
population going out.
◦
represents the expected number of secondary cases in compartment i put
in by an individual in compartment j
◦
the next generation operator:
◦ Gives rate at which individuals in compartment j generate new
infections in compartment i multiplied by the average length of time
that individuals spend in a single visit to compartment j
◦
is given by the dominant(largest) eigenvalue of
Disease Free Equilibrium
D.F.E Point: (1, 0, 0)
 Since infection is not present there is no recovery
Stability Analysis of Disease Free
Equilibrium
Evaluation at D.F.E: (1,0,0)
• The D.F.E is stable for all negative real parts of the eigenvalues.
 Here the system is stable when μ > 0 and β < γ+μ
 β / (γ+μ ) < 1 is equivalent to
Endemic Equilibrium
E.E Point:
Stability Analysis of Endemic
Equilibrium
Evaluate at E.E point:
SIVWR Model
𝑑𝑆
= n(S+IA+IS+V+R) + θV(t) -bWHWHS - bWLWLS- bISSIs - bIASIA 𝑑𝑡
𝑑𝐼𝐴
= p(bWHWHS - bWLWLS- bISSIs - bIASIA) - γAIA-μIA- dIA= 0
𝑑𝑡
𝑑𝐼𝑆
= (1-p)(bWHWHS - bWLWLS- bISSIs - bIASIA)- γSIS-μIS- dIS= 0
𝑑𝑡
𝑑𝑉
= ØS - θV(t)- μV= 0
𝑑𝑡
𝑑𝑊𝐻
= αAIA + αsIs - χWH= 0
𝑑𝑡
𝑑𝑊𝐿
= χWH – εWL= 0
𝑑𝑡
𝑑𝑅
= γAIA+ γSIS – μR- ωR = 0
𝑑𝑡
N = S+IA+IS+V+WH+WL+R
ØS – μS +ωR = 0
[1A]
[2A]
[3A]
[4A]
[5A]
[6A]
[7A]
Disease Free Equilibrium
IA IS =0
dR
dt
= γAIA+ γSIS – μR- ωR = 0
R(μ+ ω) =0
R=0
WH WL= 0
n(S+0+0+V+0) + θV(t) -0 - 0- 0 - 0 - ØS – μS + 0 = 0
n(S+ V) + θV -S(𝜃 + Ø) = 0
(n- μ- Ø)S + (n+θ)V = 0
[plug in 4B]
𝑆Ø
(n- μ- Ø)S + (n+θ)
=0
(𝜃+𝜇)
S(n- μ- Ø + (n+θ)
Ø
(𝜃+𝜇)
)=0
[Assuming S ≠0]
(n- μ- Ø)(n+θ)+ (n+θ)Ø = 0
dV
=
dt
ØS - θV(t)- μV= 0
S=
(𝜃−𝜇)𝑉
𝑆Ø
V= (𝜃+𝜇)
nθ - μθ- Øθ +nμ-μ2-μØ+nØ+θØ = 0
-μ2-μ (Ø+θ)+ n(Ø+μ+θ) = 0
μ (μ+Ø+θ) = n(μ+Ø+θ)
Ø
[4B]
[1A]
[μ+Ø+θ ≠0]
disease free non-equilibrium exists if μ = n
(natural death rate = natural birth rate)
Disease Free Equilibrium
Assume μ = n
V=
𝑆Ø
(𝜃+𝜇)
N-S=
[V+S=N]
𝑆Ø
S(1+(𝜃+𝜇)
)=N
SD =
S(𝜃+𝜇+Ø
)=N
(𝜃+𝜇)
𝑁(𝜃+𝜇)
(𝜃+𝜇+Ø)
𝑁(𝜃+𝜇)
V= (𝜃+𝜇+Ø)(𝜃+𝜇) Ø
VD =
Disease free equilibrium:
𝑆Ø
(𝜃+𝜇)
𝑁Ø
(𝜃+𝜇+Ø)
[from 4B]
(SD, VD, IAD, ISD, WH , WL , R)=
(
𝑁(𝜃+𝜇)
𝑁Ø
,
,
(𝜃+𝜇+Ø) (𝜃+𝜇+Ø)
0, 0, 0, 0, 0)
Endemic Equilibrium
•substitutions were used to get solutions for
each system of equations.
𝐺
𝐹
𝐺
𝐹
=( ,
GØIS∗ pDIS∗IS∗ 𝐺
𝐸
𝐸
,
, IS* , IS* , IS*
𝐹(𝜃+𝜇)
(1−𝑝)
𝐹
χ
ε
GØ
pD
𝐺
,
,
𝐹(𝜃+𝜇) (1−𝑝) 𝐹
𝐸
𝐸
, χ , , F ) IS*
ε
D=
E=
(S*, V*, IA*, IS*, WL* , WH*, R*)
= ( IS* ,
Where:
, F IS* )
γS+μ+dS
γA+μ+dA
αApD+αs 1−p
F=
G=
(1−p)
θ+μ+Ø n−μ S
(θ+μ)
IS
(1−p)
(
θ+μ+ n−μ S
(θ+μ)
− npD − (1−p)(n+nF+ωF))
New Generation Method
• The matrices F and V are calculated using the next generation
method.
• We will need to find
in order to calculate
which will aid
in finding the basic reproductive number.
Future Research
◦ We will calculate
.
◦ After finding
we will find the largest eigenvalues. This will give the
spectral radius which is also the basic reproductive number.
◦ We will work on an extension of this problem by taking u(t) not equal to
zero.
◦ We believe u(t) is a combination of antibiotic treatment and hydration.
Reference
◦ Albert MJ, Neira M, Motarjemi Y., The Role of Food in the Epidemiology of Cholera, World Health Stat Q., 1997; vol. 50: 111-8.
[PMID:9282393]
◦ Cholera Outbreak – Haiti, October 2010, MMWR, CDC
◦ "Cholera Outbreak in Haiti." - International Medical Corps. N.p., n.d. Web. 04 June 2015.
◦ Cholera Prevention and Control, CDC
◦ Cui, Jing'an, Zhanmin Wu, and Xueyong Zhou. "Mathematical Analysis of a Cholera Model with Vaccination." N.p., 13 Feb.
2014. Web.
◦ Dimitrov, Nedialko B. B., Meyers, and Laure Ancel. "Mathematical Approaches to Infectious Disease Prediction and Control."
Mathematical Approaches to Infectious Disease Prediction and Control. N.p., n.d. Web. 04 June 2015.
◦ "Flooding Intensifies Cholera Outbreak in Haiti - New America Media."Flooding Intensifies Cholera Outbreak in Haiti - New
America Media. N.p., n.d. Web. 04 June 2015.
◦ Fung, Isaac Chun-Hai. "Cholera Transmission Dynamic Models for Public Health Practitioners." Emerging Themes in
Epidemiology. BioMed Central, n.d. Web. 04 June 2015.
◦ GHOSH-DASTIDAR, URMI, and SUZANNE LENHART. "MODELING THE EFFECT OF VACCINES ON CHOLERA TRANSMISSION."
World Scientific Publishing Company. N.p., 29 May 2015. Web. 04 June 2015.
◦ Joseph H. Tien and David J.D. Earn, Multiple Transmission Pathways and Disease Dynamics in a Waterborne Pathogen Model,
Bulletin of Mathematical Biology, vol. 72, 1506-1533, 2010
◦ Rachael L. Miller Neilan, Elsa Schaefer, Holly Gaff, K. Renee Fister, and Suzanne Lenhart, Modeling Optimal Intervention
Strategies for Cholera, Bulletin of Mathematical Biology, vol. 72, 2004 – 2018, 2010.
◦ S. M. Moghadas, Modeling the Effect of Imperfect Vaccines on Disease Epidemiology, Discrete and Continuous Dynamical
Systems – Series B, vol. 4, no. 4, pp. 999-1012, Nov. 2004.
◦ Tien et al., Herald Waves of Cholera in Nineteenth Century London, Journal of the Royal Society, 2010.
◦ Tuite et al., Cholera Epidemic in Haiti, 2010: Using a Transmission Model to Explain Spatial Spread of Disease and Identify
Optimal Control Interventions, Annals of Internal Medicine, vol. 154, pp. 593-601, May 2011.
Acknowledgements
We would like to thank Dr. Urmi Ghosh-Dastidar for her support and
guidance throughout our project.
Also we would like to thank Matt, Becky, and especially to the MAA for
funding and DIMACS REU Program for providing us with research
facilities at Rutgers University.