3.1 SIR model

Cholera in South Africa 2000/2001

Catalina Anghel

Andrea Borowski

3.1 SIR model

Overview

1. What is cholera?

2. Cholera in South Africa1. Data

3. Models1. SIR model

2. Modified SIR model according to Torres Codeco

4. Conclusions

3.1 SIR model

1. What is cholera?

Acute intestinal infection caused by the bacterium Vibrio cholerae

Infection results in a diarrhoea that can lead to dehydration and death

Infected people: excrete bacteria between 7 and 14 days <20% show typical symptoms of cholera

Rarely person to person transmission

3.1 SIR model

2. Cholera in South Africa in 2000/2001

Total number of infected: 106 159

Start in mid august 2000 Affected region: KwaZulu-

Natal population: 9.3 mi. most people use rivers

and wells as their water supply

Heavy rainfall starting in November 2000

3.1 SIR model

2.1. Data

total infected cases and deaths from October 13, 2000 to April 16, 2001

2.1 Data

ratio of total deaths to total number of infected people decreased, then stabilized

3.1 SIR model

3. Models

3.1 SIR model

3.1 SIR model

Fitted α and β numerically using a integration method embedded within a minimization routine

α = 0.0086 [1/day] β = 2.2 x 10-7 [1/day]

IStC

ItR

IIStI

IStS

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

7

8

9x 10

4 accumulative infectives c(t), fitted using SIR, first trial

days

num

ber

of in

fect

ed

raw datafitted data

3.1 SIR model

Whole susceptible population becomes infected Number of infected increases and decreases

over time

3.1 SIR model

Fitted α, β and S0 using the same method

α = - 0.025 [1/day] β = 1.5 x 10-7 [1/day] S0= 82,500

Un-physical results: α < 0 S0< total infected.0 20 40 60 80 100 120 140 160 180 200

0

1

2

3

4

5

6

7

8

9x 10

4 accumulative infectives c(t), fitted using SIR, 2nd trial

days

num

ber

of in

fect

ed

raw datafitted data

3.2 Modified SIR model (Torres Codeco)

3.2 Modified SIR model (Torres Codeco)

Susceptible (S)

Infected

(I)

Aquatic toxigenic V. cholerae (B)

a

eλ(B)

Environmental factors

m

r

3.2 Modified SIR model (Torres Codeco)

BK

BB

SBadt

dC

eImBdt

dB

rISBadt

dI

SBadt

dS

Symbol Description

λ probability to catch cholera (cells/ml)

B concentration of toxigenic V. cholerae in water (cells/ml)

a rate of exposure to contaminated water (day-1)

r rate at which people recover from cholera (day-1)

m growth rate of V. cholerae in aquatic environment (day-1)

e contribution of each infected person to the population of V. cholerae in the aquatic environment (cells/ml day-1 person-1)

K concentration of V. cholerae in water that yields 50% chance of catching cholera (cells/ml)

3.2 Modified SIR model (Torres Codeco)

Fitted a, r and e: a = 1.332 (Codeco: a = 1) r = 0.012 (Codeco: r = 0.2)

Different definition for cholera cases

e = 0.0654 (Codeco: e = 10) Hospitalizing

3.2 Modified SIR model (Torres Codeco)

S and I plots similar to SIR model

3.2 Modified SIR model (Torres Codeco)

‘Heavy rains and flooding have increased the risk of contaminating rivers and drinking wells.’

‘The risks are particularly high because many villages have no latrines, and human waste mixes with floodwaters.’

‘10 million households were cut off from water in 2001.’

3.2 Modified SIR model (Torres Codeco)

modified e over time to take rainy season into consideration:

365

2sin1 tete

3.2 Modified SIR model (Torres Codeco)

a = 0.110, r = 0.319 e =1.566* (1+sin(2*π/365 -0.029))

3.2 Modified SIR model (Torres Codeco)

Number of susceptibles decreases to 320,000 and stays constant

Infected population decreases at the beginning (start of the epidemic in august!)

3.2 Modified SIR model (Torres Codeco)

Control mechanisms

Hygienic disposal of human feces Adequate supply of safe drinking water Hygienic measures: washing hands, cooking

food

3.2 Modified SIR model (Torres Codeco)

How do control mechanisms influence the parameters?

Good sanitation reduces parameter e and water treatment reduces parameter a.

The smaller these parameters are, the larger must be the susceptible pool in order to a cholera outbreak to

develop.

3.1 SIR model

4. Conclusions

Simple SIR model does not consider transmission through water supply.

SIR model taking into account the aquatic reservoir corresponds to our data

Through changing parameter e over time, we included the onset of the rainy season in the model.

FUTURE: Taking into account the onset of the epidemic (August-October) Seasonal changes also in parameter a Independent estimates for parameters to compare with fitted

parameters Using the model to fit data from other cholera outbreaks

3.1 SIR model

References

www.who.int www.textbookofbacteriology.net Endemic and epidemic dynamics of cholera: the role of

the aquatic reservoir, Claudia Torres Codeco, BMC Infectious Diseases, 2001