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Joint work with R. Sanches and C.P. Ferreira
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Resonances and bi-modality
in a model for cholera
Roberto A. KraenkelInstitute for Theoretical Physics - UNESP
São Paulo, Brazil
SMB Meeting - 2010
Joint work with Claudia Pio Ferreira and Rosângela Sanches
( Botucatu)
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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.
ift-unespModels of Cholera epidemics
ift-unespModels of Cholera epidemics
• First model proposed by Capasso(1979)
ift-unespModels of Cholera epidemics
• First model proposed by Capasso(1979)
• Two equations - infected individuals & aquatic population of V. Cholerae
ift-unespModels of Cholera epidemics
• First model proposed by Capasso(1979)
• Two equations - infected individuals & aquatic population of V. Cholerae
ift-unespModels of Cholera 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)
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Codeço Model
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Codeço Model
C.T. CodeçoBMC Infectious Diseases
(2001) 1:1
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Codeço Model
Epidemic and Endemic solutions are possible
C.T. CodeçoBMC Infectious Diseases
(2001) 1:1
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Codeço Model
Epidemic and Endemic solutions are possible
C.T. CodeçoBMC Infectious Diseases
(2001) 1:1
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-modal
bi-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
water treatment
hand-washing,
education..
sanitation
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Control mechanisms
vaccination
water treatment
hand-washing,
education..
sanitation
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Control mechanisms
vaccination
water treatment
hand-washing,
education..
sanitation
efficiency index
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Control mechanisms
vaccination
water treatment
hand-washing,
education..
sanitation
efficiency index
model allows to obtain critical values for eradication
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eradication thresholds
hand-washing, education..
vaccination
water treatment
sanitation
continous
no eradication with periodic
continous
no eradication with periodic
continous
no eradication with periodic
periodic
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Thanks for your attention
Roberto [email protected]
http://www.ift.unesp.br/users/kraenkel
