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Cholera in South Africa Cholera in South Africa 2000/012000/01
Tomoko UenoTomoko UenoRui ZhangRui Zhang
Supervisor: Professor Gerda de VriesSupervisor: Professor Gerda de Vries
OutlineOutline
BackgroundBackground
SI ModelSI Model
SIB ModelSIB Model
ResultsResults
ConclusionConclusion
Further DiscussionFurther Discussion
What is Cholera?What is Cholera?
Cholera is a severe diarrheal disease Cholera is a severe diarrheal disease caused by the bacterium Viblio Cholerae.caused by the bacterium Viblio Cholerae.
It infects people’s intestines, causing It infects people’s intestines, causing diarrhea, vomiting and leg cramps.diarrhea, vomiting and leg cramps.
What is Cholera?What is Cholera?
Vibrio choleraeVibrio cholerae
- consists of Gram-- consists of Gram-negative straight or negative straight or curved rodscurved rods
- produces cholera - produces cholera toxintoxin
- can be found in the - can be found in the aquatic environmentaquatic environment
How do people get this disease?How do people get this disease?TransmissionTransmission
Infected People water Susceptible People
Why is it so serious?Why is it so serious?
Can be mild or even without symptoms, Can be mild or even without symptoms, but a severe case can lead to death but a severe case can lead to death without immediate treatment.without immediate treatment.
The diarrhea and vomiting brought on by The diarrhea and vomiting brought on by the infection quickly leaves the body the infection quickly leaves the body without enough fluid.without enough fluid.
The dehydration and shock can kill a The dehydration and shock can kill a person within person within hourshours. .
Can it be treated?Can it be treated?
Yes, very simply and effectively. Yes, very simply and effectively.
A mixture of sugar and certain salts that A mixture of sugar and certain salts that the body needs must be mixed with clean the body needs must be mixed with clean water and drunk large amounts water and drunk large amounts immediately after the first symptoms show immediately after the first symptoms show up, then the disease can be completely up, then the disease can be completely cured.cured.
(Susceptible) – (Infected) (Susceptible) – (Infected) S-I modelS-I model
Susceptible People
Infected People
die
α d
dS/dt = - dS/dt = - αα * S * S
dI/dt = dI/dt = αα *S - d*I *S - d*I
dD/dt = d*IdD/dt = d*I
Phase PortraitPhase Portrait
S-I model (result)S-I model (result)Initial conditionsInitial conditions
i0 = 1900/0.15i0 = 1900/0.15S0 = 8417083 - i0S0 = 8417083 - i0
(population in (population in Kwazulu-Natal (South Kwazulu-Natal (South Africa) 2000/01)Africa) 2000/01)
ParametersParameters
αα = 0.0003435 = 0.0003435
d = 0.0004 d = 0.0004
Error AnalysisError Analysis
residuals = data from real world – modeling valueresiduals = data from real world – modeling value
S-I-B modelS-I-B model
S
I
die
recoverB
α d
re
0.2
S-I-B modelS-I-B modeldS/dt = - dS/dt = - αα *B*S +0.2*r*I *B*S +0.2*r*I
dI/dt = dI/dt = αα *B*S-d*I-r*I *B*S-d*I-r*I
dB/dt = dB/dt = ββ *B(1-B/b)+e*I *B(1-B/b)+e*I
S - susceptible peopleS - susceptible peopleI – infected peopleI – infected peopleB – bacteria populationB – bacteria populationb – carrying capacity of bacteria b – carrying capacity of bacteria d – death rated – death rater - recovery rater - recovery ratee – ratio of I transfer bacteria to watere – ratio of I transfer bacteria to waterαα - ratio of S infected - ratio of S infectedββ– growth rate of bacteria– growth rate of bacteria
Find ParametersFind Parameters
αα is one of the most important parameters in this is one of the most important parameters in this modelmodel αα is related to good food hygiene, cooking is related to good food hygiene, cookingwashing hands after defection and before mealswashing hands after defection and before meals ββ is also an important parameter in this model is also an important parameter in this model
ββ is related to is related to hygienic disposal of human faceshygienic disposal of human faces
Varying the ParametersVarying the Parameters
Alpha = 5e-5, 5e-6 Beta = 0.02, 0.005
Least - Square MethodLeast - Square Method
Sum of squared of the error:Sum of squared of the error:SSE = sum( (observed value – estimated value)SSE = sum( (observed value – estimated value)22))
Find the point where SSE has the least value. Find the point where SSE has the least value.
S-I-B modelS-I-B model
d = 0.02, d = 0.02,
r = (1 - d) = 0.08r = (1 - d) = 0.08
e = 0.000002 e = 0.000002
alpha = 0.00001 alpha = 0.00001
beta = 0.01beta = 0.01
B = 100000B = 100000s0 = 8417083s0 = 8417083i0 = 1900/0.15i0 = 1900/0.15b0 = 10b0 = 10
S-I-B modelS-I-B model
ConclusionsConclusions
SSE (SIB model) / SSE (S-I model)=28.4%SSE (SIB model) / SSE (S-I model)=28.4%
SIB model is better than SI model in this SIB model is better than SI model in this particular caseparticular case
However, both models do not fit to the given However, both models do not fit to the given data data
Further DiscussionFurther Discussion
Seasonal EffectsSeasonal EffectsContinuing improvement of the treatmentsContinuing improvement of the treatments
Exploration for the parameters of previous Exploration for the parameters of previous modelmodel
α α exp(-t/100) good food hygiene, washing hands B B exp(-t/50) adequate supply of water
Seasonal EffectsSeasonal Effects
Extensions of the Days Extensions of the Days
The Parameters of the Previous ModelThe Parameters of the Previous Model
DS/DT = -DS/DT = -ααBS + &BS + &λλIIDI/DT = DI/DT = ααBS – BS – λλI + I + θθIIDB/DT = DB/DT = ββB(1-B) + µI B(1-B) + µI