???as?a sta ?a??

1
???as?a sta ?a??µat??? (5? e??µ???)

• ?p?µ??e?a
• ??µß?? ?a????p?
• ?.?.? 4263

2
???t??a d??d?s?? ep?d?µ??? (SIR models)
3

• ???sµ?? ep?d?µ?a?
• ? e??p??s? a???st?a? se µ?a pe????? se s??t?µ?
  ?????
• ??t? p?? ???e? µ?a ep?d?µ?a
• ?? ep?d?µ?e? e??a? ?esp?sµata µ???sµat???? ?
  µetad?t???? as?e?e??? p?? ????? ep?pt?se?? se
  µe???? µ??? e??? p????sµ?? p?? ???? se ?d?a?te?e?
  ??se?? se s???e???µ????? ???????

4
???a? apa?a?t?t? ?a e?et?s??µe t?? ep?d?µ?e? ?p??
eµfa??st??a? st?? ?st???a ??at?

1. ???a? s?µa?t???? ep?????? sta ????? ep?peda
   ???s?µ?t?ta? st? pa?e????
2. ?? ?st?????? ep?d?µ?e? ep?t??p??? st???
   ep?st?µ??e? ?a µe?et?s??? t?? p???? ep?d?µ???
   a???????a
3. ?? µe??te? t?? ?a?? te?µ????µ???? ep?d?µ??? p??
   eµfa??st??a? st?? ????p? st? pa?e???? µp????? ?a
   µa? ß????s??? ?a ?ata??ß??µe t? s?µpe??f???
   e?e???? t?? µ???sµat???? as?e?e??? p?? e??a?
   a??µa e?e???? st?? ????te?? a?apt??µ??e? ???e?
   s?µe?a. (??st???? ??? ??a HIV)

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?? ep?d?µ?e? st?? ?st???a

• ? pa?????a t?? 14? a???a st?? ????p? s??t?se 25
  e?at?µµ???a
• ?? ??t???? ??asa? t? µ?s? t??? p????sµ? (ap? 3,5
  e?at?µµ???a p?? ?ta?) ap? t?? e??????
• 20 e?at?µµ???a ?????p?? p??a?a? ap? t?? ?? t??
  ???p?? t? 1919

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A?t? t? st??µ?

• 1 e?at?µµ???? ???at?? et?s??? ???? t?? e????s?a?
• 1 e?at?µµ???? ???at?? et?s??? ???? t?? ??a???
• 2 e?at?µµ???a ???at?? et?s??? ???? t?? f?µat??s??
  
• 3 e?at?µµ???a ???at?? et?s??? ???? HIV
• ???a? ???p?? d?se?at?µµ???a p?? ????? µ?????e? µe
  ??e? a?t?? t?? as???e?e?

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• ???a e?d? as?e?e??? µp????? ?a p???a??s???
  ep?d?µ?e?
• ?? p?? ???st?? ep?d?µ???? as???e?e? d?ad?d??ta?
  ap? t?? ?µes? epaf? µeta?? µ???sµ???? ?a? µ? p??
  e??a? ta e?pa?? p??s?pa.
• p.?. ??a??, e??????, ???p?, f?µat??s?,
  p????µ?e??t?da, VIH/AIDS, ?a? se???a????
  µetad?d?µe?e? µ????se?? (STIs) ?p?? ? s?f???.
• ??a s?µa?t??? de?te?? ?at?????a ep?d?µ????
  as?e?e??? e??a? a?t?? p?? d?aß?ß????ta? st???
  a????p??? ap? ??t?µa.
• ?e???? pa?ade??µata (?a? ?? µetaf??e?? t???)
  ???ß????? pa?????a (??????), e????s?a (??????p?),
  t?f?? (?e??a s?µ?t??) ?a? ??t????? p??et??
  (??????p?).
• ??a t??t? ?µ?da pe???aµß??e? e?e??e? t??
  ep?d?µ???? as???e?e? p?? ep?????????? ap? t?
  ?e??- ? ap? t??f??? pa?????a (ßa?t???d?a, ???? ?
  ?????? µ???????a??sµ??), p.?. ?????a, t?f?e?d??
  ?a? d?se?te??a.

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• ?p????? ??e? ?? as???e?e? ?a p???a??s??? µ?a
  ep?d?µ?a
• ???, de? e??a? ??e? ?? as???e?e? se ??s? ?a
  p???a??s??? ep?d?µ?e?. ?? d??f??e? µ??f??
  ?a?????? ?a? ?a?d?a??? pa??se?? ep??e????ta? ap?
  t? ?e?et??? p??d???es? ?a? t?? t??p?? ????
  (??p??sµa, ?s??s?, d?at??f?). ? s?µa?t??? d?af???
  e??a? ?t? de? µp????? ?a ep????????s??? ap?
  p??s?p? µe p??s?p?. ??t?? e??a? ?? e?d?µ????
  as???e?e? ?a? ß??s???ta? ta?t??? se ??a?
  p????sµ?.

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• ?st???a t?? ep?d?µ??????a?
• Hippocrates On the Epidemics (circa 400 p.?.)
• John Graunt's Natural and Political Observations
  madethe Bills of Mortality (1662)
• Louis Pasteur ?a? Robert Koch (µ?sa 1800's)
• ?st???a t?? µa??µat???? ep?d?µ??????a?
• ? Daniel Bernoulli ?de??e e?e??? t?? eµß???asµ?
  e???t?a t?? e??????? ?a? a???se t??
  ?p???????µe?? d????e?a ???? t?? G????? (1760)
• ?p?? ep?d?µ??? µ??t??? t?? Ross (1911)
• ?a? t? ?e???? ep?d?µ??? µ??t??? Kermack
  McKendrick's (1927)

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• ?? e??a? t? SIR model
• ?? SIR model e??a? ??a ep?d?µ???????? µ??t???
  p?? ?p??????e? t? ?e???t??? a???µ? a????p?? p??
  µ??????ta? µe µ?a µetad?t??? as???e?a se ??a?
  ??e?st? p????sµ? ?at? t? d????e?a t?? ??????.
• G?at? ?????µe ?a µe?et?s??µe ta SIR model
• Me?et?µe ta SIR models ??a ?a ?ata??ß??µe
  ?a??te?a t?? e??p??s? t?? ep?d?µ??? st???
  p????sµ???

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??a ap? ta ap???ste?a SIR models e??a? t?
Kermack-McKendrick Model

• ?? µ??t??? Kermack- McKendrick e??a? ??a SIR
  model ??a t?? a???µ? a????p?? p?? µ??????ta? µe
  µ?a µetad?t??? as???e?a se ??a? ??e?st? p????sµ?
  ?at? t? d????e?a t?? ??????.
• ???t????e ??a ?a e?????e? ? ??????? ???d?? ?a?
  ????d?? st?? a???µ? µ???sµ???? as?e?e??? p??
  pa?at??????e st?? ep?d?µ?e? ?p?? ? pa?????a
  (???d??? 1665-1666, ??µß?? 1906) ?a? ? ?????a
  (???d??? 1865).

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SIR model

• ??a????µe t?? p????sµ? se t?e?? ?µ?de?
• -??pa?? ?t?µa S (t)
• -????sµ??a ?t?µa I (t)
• -??a?t?µ??a ?t?µa R (t)

ß I S
a ?
S
I
R
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• ?p???se??
• 1. ?? µ??e??? t?? p????sµ?? e??a? µe???? ?a?
  sta?e??
• S (t) I (t) R (t)
• 2. ?aµ?a ?????s?, ???at?, µeta??ste?s? ? ap?d?µ?a
  
• 3. ?aµ?a ?a??????sa pe???d??, e?? ? pe???d??
  ep?as?? t?? µ???sµat???? p???t??a e??a? st??µ?a?a
• 4. ? d????e?a t?? µ???sµat???t?ta? e??a? ?d?a µe
  t? µ????
• t?? as???e?a?
• 5. ?p???te? ep?s?? ??a? ap???t?? ?µ????e??
  p????sµ? ????? ?????a, ???? ?a? ????????? d?µ?
• 6. ?? p?s?st? µ????s?? e??a? a?????? µe t??
  a???µ? t?? µ???sµ???? p.?. ?rß?
• 7. ?? p?s?st? ap??at?stas?? e??a? sta?e??

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• ??a s?st?µa t???? d?af?????? e??s?se?? pe?????fe?
  a?t? t? µ??t???

?p?? ß e??a? t? p?s?st? µ????s?? ?a? a t?
p?s?st? ap??at?stas?? ?a? r µ?a sta?e??
?
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• ? ßas??? a??a p?? ??ße??? t? ??????? e??????
  a?t?? t?? e??s?se?? e??a? t? ap??a???µe??
  ep?d?µ???????? ?at?tat? ????

ßS a

• ???a? p??a??? ? s?µa?t???te?? e??a?a p?s?t?ta
  st?? ep?d?µ??????a.

R0
R0 o???eta? ?? ? a???µ?? de?te??ß??µ??? µ????se??
p???a???µe??? ap? ??a µeµ???µ??? µ???sµ???
?t?µ? p?? e?s??eta? se ??a? p????sµ?
p?? ap?te?e?ta? ap? e?pa?e?? µ???.
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?ta? R0 lt 1 , ???e p??s?p? p?? s?µß???eta? ap?
t?? as???e?a ?a µ????e? ????te???? ap? ??a ?t?µ?
p??? pe???e? ? a?a?t?se?, ?ts? t? ??spasµa t??
ep?d?µ?a? ?a e?a?t???e? stad?a?? (dI/dt lt 0).
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?ta? R0 gt 1 , ???e p??s?p? p?? pa???e? t??
as???e?a ?a µ????e? pe??ss?te?a ap? ??a p??s?pa,
?ts? ? ep?d?µ?a ?a e?ap???e? (dI/dt gt 0).
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S?µp??asµa

• ?e ded?µ??? ?t? ?? ep?d?µ?e? e??a? ??a s?µa?t???
  p??ß??µa ??e?a? s??e?d?t?p????µe ?t? ta SIR
  models µp????? ?a ß????s??? ?a ?ata??ß??µe
  ?a??te?a t?? e??p??s? t?? µ???sµat???? as?e?e???
  ?a? ?a e?et?s??µe pa??????a ?a? st?at??????
  e?????? ?ste µe????t??? ?a µp???s??µe ?a
  ap?t?????µe t?? e??e?a e??p??s?.

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????S