Joint work with R. Sanches and C.P. Ferreira

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Resonances and bi-modality
in a model for cholera
Roberto A. Kraenkel
Joint work with Claudia Pio Institute for Theoretical Physics - UNESP
Ferreira and Rosângela Sanches São Paulo, Brazil
( Botucatu)
SMB Meeting - 2010

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Cholera

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Cholera

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Cholera
•Water-borne disease

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Cholera
•Water-borne disease
•Etiological agent: Vibrio Cholerae

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Cholera
•Water-borne disease
•Etiological agent: Vibrio Cholerae
•Symptoms vary greatly, from asymptomatic to intense diarrhea

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Cholera
•Water-borne disease
•Etiological agent: Vibrio Cholerae
•Symptoms vary greatly, from asymptomatic to intense diarrhea
•Most important transmission route: environmental-to-human

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Cholera
•Water-borne disease
•Etiological agent: Vibrio Cholerae
•Symptoms vary greatly, from asymptomatic to intense diarrhea
•Most important transmission route: environmental-to-human
•3–5 million cholera cases and 100 000–120 000 deaths due to
cholera every year

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Endemic Cholera

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Endemic Cholera
• Cholera is endemic in many regions, mainly
in the Indian sub-continent, Asia, Africa &
Latin America

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Endemic Cholera
• Cholera is endemic in many regions, mainly
in the Indian sub-continent, Asia, Africa &
Latin America
• Usually displays one or two annual peaks

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Endemic Cholera
• Cholera is endemic in many regions, mainly
in the Indian sub-continent, Asia, Africa &
Latin America
• Usually displays one or two annual peaks
• Spatial correlation suggests environmental
causes for periodicity.

14. Models of Cholera
ift-unesp
epidemics

15. Models of Cholera
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epidemics
• First model proposed by Capasso(1979)

16. Models of Cholera
ift-unesp
epidemics
• First model proposed by Capasso(1979)
• Two equations - infected individuals &
aquatic population of V. Cholerae

17. Models of Cholera
ift-unesp
epidemics
• First model proposed by Capasso(1979)
• Two equations - infected individuals &
aquatic population of V. Cholerae

18. Models of Cholera
ift-unesp
epidemics
• First model proposed by Capasso(1979)
• Two equations - infected individuals &
aquatic population of V. Cholerae
• Model by Codeço : S I R B ( susceptibles,
infected, removed and bacteria)

19. Codeço Model
ift-unesp

20. Codeço Model
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C.T. Codeço
BMC Infectious Diseases
(2001) 1:1

21. Codeço Model
ift-unesp
C.T. Codeço
BMC Infectious Diseases
(2001) 1:1
Epidemic and Endemic solutions are possible

22. Codeço Model
ift-unesp
C.T. Codeço
BMC Infectious Diseases
(2001) 1:1
Epidemic and Endemic solutions are possible
Easily incorporates seasonality through time-dependence in the
parameters: endemic peaks, once a year, no bi-modal solutions.

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Extended model

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Extended model
• incorporates loss of immunity

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Extended model
• incorporates loss of immunity
• seasonality

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Extended model
• incorporates loss of immunity
• seasonality
• control mechanisms

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Results
• If no seasonality and control are present
endemic equilibrium is stable if R_0 > 1

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Effects of seasonality
• Approach to endemic equilibrium is
through damped oscillations
• Seasonal forcing introduces sustained
oscillations.
• The amplitude of oscillations depends on
and .

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Resonance approach

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Resonance approach
• plot the local maximum in terms of p

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Resonance approach
• plot the local maximum in terms of p

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Resonance approach
• plot the local maximum in terms of p
Resonance
peak

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Resonance approach
• plot the local maximum in terms of p
Resonance
peak
bimodality

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Solutions

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Solutions

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Solutions
uni-
modal

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Solutions
uni- bi-
modal modal

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Control mechanisms

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Control mechanisms

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Control mechanisms
hand-
washing,
education..

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Control mechanisms
hand-
washing,
education..
sanitation

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Control mechanisms
hand-
washing,
education.. water
treatment
sanitation

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Control mechanisms
vaccination
hand-
washing,
education.. water
treatment
sanitation

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Control mechanisms
vaccination
hand-
washing,
education.. water
treatment
sanitation
efficiency index

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Control mechanisms
vaccination
hand-
washing,
education.. water
treatment
sanitation
efficiency index
model allows to obtain critical values for eradication

49. eradication thresholds
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hand-washing,
water treatment
education..
continous
continous
no eradication with periodic
no eradication with periodic
sanitation vaccination
continous
no eradication with periodic periodic

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Thanks for your attention
Roberto Kraenkel
kraenkel@ift.unesp.br
http://www.ift.unesp.br/users/kraenkel