Cholera in South Africa 2000/01 Tomoko Ueno Rui Zhang Supervisor: Professor Gerda de Vries

Outline Background SI Model SIB Model ResultsConclusion 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 Gram- negative straight or curved rods - consists of Gram- negative straight or curved rods - produces cholera toxin - produces cholera toxin - can be found in the aquatic environment - 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 Susceptible People Infected People die α d dS/dt = - α * S dI/dt = α *S - d*I dD/dt = d*I

Phase Portrait Phase Portrait

S-I model (result) Initial conditions i0 = 1900/0.15 S0 = i0 (population in Kwazulu-Natal (South Africa) 2000/01) Parameters α = d =

Error Analysis residuals = data from real world – modeling value

S-I-B model S I die recover B α d r e 0.2

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 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-6Beta = 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 = alpha = beta = 0.01 B = s0 = 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 Exploration for the parameters of previous model α α exp(-t/100) good food hygiene, washing hands B B exp(-t/50) adequate supply of water

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 DS/DT = -αBS + &λI DI/DT = αBS – λI + θI DB/DT = βB(1-B) + µI