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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