Cholera in South Africa 2000/01

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

2. Outline • Background • SI Model • SIB Model • Results • Conclusion • Further Discussion

3. 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.

4. What is Cholera? • Vibrio cholerae - consists of Gram-negative straight or curved rods - produces cholera toxin - can be found in the aquatic environment

5. How do people get this disease?Transmission Susceptible People Infected People water

6. 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.

7. 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.

8. dS/dt = - α * S dI/dt = α *S - d*I dD/dt = d*I (Susceptible) – (Infected) S-I model Infected People α d Susceptible People die

9. Phase Portrait

10. 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

11. Error Analysis • residuals = data from real world – modeling value

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

13. 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

14. Find Parameters • α is one of the most important parameters in this modelα is related to good food hygiene, cookingwashing hands after defection and before meals • β is also an important parameter in this modelβ is related to hygienic disposal of human faces

15. Varying the Parameters Alpha = 5e-5, 5e-6 Beta = 0.02, 0.005

16. 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.

17. 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

18. S-I-B model

19. 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

20. Further Discussion • Seasonal Effects • Continuing improvement of the treatments • Exploration for the parameters of previous model αα exp(-t/100) good food hygiene, washing hands B B exp(-t/50) adequate supply of water

21. Seasonal Effects

22. Extensions of the Days

23. The Parameters of the Previous Model DS/DT = -αBS + &λIDI/DT = αBS – λI + θIDB/DT = βB(1-B) + µI