ift-unesp              Resonances and bi-modality                      in a model for cholera                             ...
ift-unespCholera
ift-unespCholera
ift-unesp                       Cholera•Water-borne disease
ift-unesp                         Cholera•Water-borne disease•Etiological agent: Vibrio Cholerae
ift-unesp                      Cholera•Water-borne disease•Etiological agent: Vibrio Cholerae•Symptoms vary greatly, from ...
ift-unesp                      Cholera•Water-borne disease•Etiological agent: Vibrio Cholerae•Symptoms vary greatly, from ...
ift-unesp                      Cholera•Water-borne disease•Etiological agent: Vibrio Cholerae•Symptoms vary greatly, from ...
ift-unesp
ift-unespEndemic Cholera
ift-unesp     Endemic Cholera• Cholera is endemic in many regions, mainly  in the Indian sub-continent, Asia, Africa &  La...
ift-unesp     Endemic Cholera• Cholera is endemic in many regions, mainly  in the Indian sub-continent, Asia, Africa &  La...
ift-unesp     Endemic Cholera• Cholera is endemic in many regions, mainly  in the Indian sub-continent, Asia, Africa &  La...
Models of Cholera                    ift-unesp   epidemics
Models of Cholera                                          ift-unesp      epidemics• First model proposed by Capasso(1979)
Models of Cholera                                           ift-unesp      epidemics• First model proposed by Capasso(1979...
Models of Cholera                                           ift-unesp      epidemics• First model proposed by Capasso(1979...
Models of Cholera                                              ift-unesp      epidemics• First model proposed by Capasso(1...
Codeço Model               ift-unesp
Codeço Model                               ift-unesp                    C.T. Codeço               BMC Infectious Diseases ...
Codeço Model                                                        ift-unesp                                             ...
Codeço Model                                                                 ift-unesp                                    ...
ift-unespExtended model
ift-unesp     Extended model• incorporates loss of immunity
ift-unesp     Extended model• incorporates loss of immunity• seasonality
ift-unesp     Extended model• incorporates loss of immunity• seasonality• control mechanisms
ift-unesp
ift-unesp
ift-unesp
ift-unesp               Results• If no seasonality and control   are present endemic equilibrium is stable if R_0 > 1
ift-unesp Effects of seasonality• Approach to endemic equilibrium is  through damped oscillations• Seasonal forcing introd...
ift-unespResonance approach
ift-unespResonance approach• plot the local maximum in terms of p
ift-unespResonance approach• plot the local maximum in terms of p
ift-unesp     Resonance approach      • plot the local maximum in terms of pResonance  peak
ift-unesp     Resonance approach      • plot the local maximum in terms of pResonance  peak                               ...
ift-unespSolutions
ift-unespSolutions
ift-unesp        Solutions uni-modal
ift-unesp        Solutions uni-                bi-modal               modal
ift-unespControl mechanisms
ift-unespControl mechanisms
ift-unesp        Control mechanisms  hand- washing,education..
ift-unesp        Control mechanisms  hand- washing,education..              sanitation
ift-unesp        Control mechanisms  hand- washing,education..                  water                           treatment ...
ift-unesp        Control mechanisms                           vaccination  hand- washing,education..                  wate...
ift-unesp        Control mechanisms                                      vaccination  hand- washing,education..           ...
ift-unesp        Control mechanisms                                              vaccination  hand- washing,education..   ...
eradication thresholds                                                               ift-unesp     hand-washing,          ...
ift-unespThanks for your attention                         Roberto Kraenkel                     kraenkel@ift.unesp.br     ...
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Resonances and bi-modality in a model for cholera

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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

  1. 1. ift-unesp Resonances and bi-modality in a model for cholera Roberto A. Kraenkel Joint work with Claudia Pio Institute for Theoretical Physics - UNESPFerreira and Rosângela Sanches São Paulo, Brazil ( Botucatu) SMB Meeting - 2010
  2. 2. ift-unespCholera
  3. 3. ift-unespCholera
  4. 4. ift-unesp Cholera•Water-borne disease
  5. 5. ift-unesp Cholera•Water-borne disease•Etiological agent: Vibrio Cholerae
  6. 6. ift-unesp Cholera•Water-borne disease•Etiological agent: Vibrio Cholerae•Symptoms vary greatly, from asymptomatic to intense diarrhea
  7. 7. ift-unesp Cholera•Water-borne disease•Etiological agent: Vibrio Cholerae•Symptoms vary greatly, from asymptomatic to intense diarrhea•Most important transmission route: environmental-to-human
  8. 8. ift-unesp 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 tocholera every year
  9. 9. ift-unesp
  10. 10. ift-unespEndemic Cholera
  11. 11. ift-unesp Endemic Cholera• Cholera is endemic in many regions, mainly in the Indian sub-continent, Asia, Africa & Latin America
  12. 12. ift-unesp 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
  13. 13. ift-unesp 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. 14. Models of Cholera ift-unesp epidemics
  15. 15. Models of Cholera ift-unesp epidemics• First model proposed by Capasso(1979)
  16. 16. Models of Cholera ift-unesp epidemics• First model proposed by Capasso(1979)• Two equations - infected individuals & aquatic population of V. Cholerae
  17. 17. Models of Cholera ift-unesp epidemics• First model proposed by Capasso(1979)• Two equations - infected individuals & aquatic population of V. Cholerae
  18. 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. 19. Codeço Model ift-unesp
  20. 20. Codeço Model ift-unesp C.T. Codeço BMC Infectious Diseases (2001) 1:1
  21. 21. Codeço Model ift-unesp C.T. Codeço BMC Infectious Diseases (2001) 1:1Epidemic and Endemic solutions are possible
  22. 22. Codeço Model ift-unesp C.T. Codeço BMC Infectious Diseases (2001) 1:1 Epidemic and Endemic solutions are possibleEasily incorporates seasonality through time-dependence in theparameters: endemic peaks, once a year, no bi-modal solutions.
  23. 23. ift-unespExtended model
  24. 24. ift-unesp Extended model• incorporates loss of immunity
  25. 25. ift-unesp Extended model• incorporates loss of immunity• seasonality
  26. 26. ift-unesp Extended model• incorporates loss of immunity• seasonality• control mechanisms
  27. 27. ift-unesp
  28. 28. ift-unesp
  29. 29. ift-unesp
  30. 30. ift-unesp Results• If no seasonality and control are present endemic equilibrium is stable if R_0 > 1
  31. 31. ift-unesp Effects of seasonality• Approach to endemic equilibrium is through damped oscillations• Seasonal forcing introduces sustained oscillations.• The amplitude of oscillations depends on and .
  32. 32. ift-unespResonance approach
  33. 33. ift-unespResonance approach• plot the local maximum in terms of p
  34. 34. ift-unespResonance approach• plot the local maximum in terms of p
  35. 35. ift-unesp Resonance approach • plot the local maximum in terms of pResonance peak
  36. 36. ift-unesp Resonance approach • plot the local maximum in terms of pResonance peak bimodality
  37. 37. ift-unespSolutions
  38. 38. ift-unespSolutions
  39. 39. ift-unesp Solutions uni-modal
  40. 40. ift-unesp Solutions uni- bi-modal modal
  41. 41. ift-unespControl mechanisms
  42. 42. ift-unespControl mechanisms
  43. 43. ift-unesp Control mechanisms hand- washing,education..
  44. 44. ift-unesp Control mechanisms hand- washing,education.. sanitation
  45. 45. ift-unesp Control mechanisms hand- washing,education.. water treatment sanitation
  46. 46. ift-unesp Control mechanisms vaccination hand- washing,education.. water treatment sanitation
  47. 47. ift-unesp Control mechanisms vaccination hand- washing,education.. water treatment sanitation efficiency index
  48. 48. ift-unesp Control mechanisms vaccination hand- washing,education.. water treatment sanitation efficiency index model allows to obtain critical values for eradication
  49. 49. eradication thresholds ift-unesp hand-washing, water treatment education.. continous continous no eradication with periodic no eradication with periodic sanitation vaccination continousno eradication with periodic periodic
  50. 50. ift-unespThanks for your attention Roberto Kraenkel kraenkel@ift.unesp.br http://www.ift.unesp.br/users/kraenkel
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