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