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Cholera in South Africa
2000/01
Tomoko Ueno
Rui Zhang
Supervisor: Professor Gerda de Vries
Outline
Background
SI Model
SIB Model
Results
Conclusion
Further Discussion
What is Cholera?
Cholera is a severe diarrheal disease
caused by the bacterium Viblio Cholerae.
It infects people’s intestines, causing
diarrhea, vomiting and leg cramps.
What is Cholera?
Vibrio cholerae
- consists of Gramnegative straight or
curved rods
- produces cholera
toxin
- can be found in the
aquatic environment
How do people get this disease?
Transmission
Infected People
water
Susceptible People
Why is it so serious?
Can be mild or even without symptoms,
but a severe case can lead to death
without immediate treatment.
The diarrhea and vomiting brought on by
the infection quickly leaves the body
without enough fluid.
The dehydration and shock can kill a
person within hours.
Can it be treated?
Yes, very simply and effectively.
A mixture of sugar and certain salts that
the body needs must be mixed with clean
water and drunk large amounts
immediately after the first symptoms show
up, then the disease can be completely
cured.
(Susceptible) – (Infected)
S-I model
Infected People
α
Susceptible People
dS/dt = - α * S
dI/dt = α *S - d*I
dD/dt = d*I
d
die
Phase Portrait
S-I model (result)
Initial conditions
i0 = 1900/0.15
S0 = 8417083 - i0
(population in
Kwazulu-Natal (South
Africa) 2000/01)
Parameters
α = 0.0003435
d = 0.0004
Error Analysis
residuals = data from real world – modeling value
S-I-B model
I
α
d
die
S
e
r
0.2
recover
B
S-I-B model
dS/dt = - α *B*S +0.2*r*I
dI/dt = α *B*S-d*I-r*I
dB/dt = β *B(1-B/b)+e*I
S - susceptible people
I – infected people
B – bacteria population
b – carrying capacity of bacteria
d – death rate
r - recovery rate
e – ratio of I transfer bacteria to water
α - ratio of S infected
β– growth rate of bacteria
Find Parameters
α is one of the most important parameters in this
model
α is related to good food hygiene, cooking
washing hands after defection and before meals
β is also an important parameter in this model
β is related to hygienic disposal of human faces
Varying the Parameters
Alpha = 5e-5, 5e-6
Beta = 0.02, 0.005
Least - Square Method
Sum of squared of the error:
SSE = sum( (observed value – estimated value)2)
Find the point where SSE has the least value.
S-I-B model
d = 0.02,
r = (1 - d) = 0.08
e = 0.000002
alpha = 0.00001
beta = 0.01
B = 100000
s0 = 8417083
i0 = 1900/0.15
b0 = 10
S-I-B model
Conclusions
SSE (SIB model) / SSE (S-I model)=28.4%
SIB model is better than SI model in this
particular case
However, both models do not fit to the given
data
Further Discussion
Seasonal Effects
Continuing improvement of the treatments
α
α exp(-t/100) good food hygiene,
washing hands
B
B exp(-t/50) adequate supply of
water
Exploration for the parameters of previous
model
Seasonal Effects
Extensions of the Days
The Parameters of the Previous Model
DS/DT = -αBS + &λI
DI/DT = αBS – λI + θI
DB/DT = βB(1-B) + µI