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OMM - Uvod
OMM - Uvod
February 25, 2021
OMM - Uvod
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OMM - Uvod
modellus (lat) - prikazati realnost
Modeliranje?
Matematicko modeliranje?
OMM - Uvod
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OMM - Uvod
"All models are wrong, but some are useful". George E. P. Box
Ptolomejev model (150 gne) Keplerov model (1619)
Njutn (Zakon gravitacije) Obracunava gravitaciju ne samona ZemljiZašto meseci kruže oko svojihplaneta?Zašto se komete ponavljaju?Plima/oseka?
OMM - Uvod
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OMM - Uvod
Uran se ponaša cudno?
(1781)
Urbain Le Verrier (1811-1877)
(1846)OMM - Uvod
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OMM - Uvod
Merkur se ponaša cudno?
Merkur
Vulkan!
OMM - Uvod
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OMM - Uvod
A gde je Vulkan?
(1915)
"A model should be as simple as it can be, but no simpler." AlbertEinstein
OMM - Uvod
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OMM - Uvod
Korona
R0 - osnovna re-produktivna stopavirusa
R0 = cβc - broj osoba sa kojim zaraženi stupa u kontaktβ - verovatnoca infekcije pri kontaktu
p - deo populacije koji je imun/vakcinisanRp = β(c − cp) = βc − βcp = R0 − R0p = R0(1− p)Cilj Rp < 1⇒ R0(1− p) < 1⇒ p > 1− 1
R0
1− 1R0
- prag neophodan za sprecavanje širenja bolesti.
OMM - Uvod
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OMM - Uvod
Modeliranje zaraznih bolesti
OMM - Uvod
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OMM - Uvod
Model SI
→S(t) + I(t) = 1 - konstantan broj jedinkiλ - koeficijent zaraze∆t - vremenski intervalλSI∆t - verovatnoca da dodje do infekcije za ∆tSt+∆t = St − λSI∆t - promena broja jedinki u S za vreme ∆t−λSI = St+∆t−St
∆t∆t → 0⇒ dS
dt = −λSI
It+∆t = It + λSI∆t - promena broja jedinki u I za vreme ∆tλSI = It+∆t−It
∆t∆t → 0⇒ dI
dt = −λSI
Zamenom S(t) + I(t) = 1⇒ dI(t)dt = λI(t)(1− I(t)) (DJ sa razdvojenim
promenljivim)
OMM - Uvod
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OMM - Uvod
Model SIS
→ →
λ - koeficijent zarazeγ - koeficijent oporavka
dS(t)dt = −λS(t)I(t) + γI(t)
dI(t)dt = λS(t)I(t)− γI(t)
(sistem DJ)
OMM - Uvod
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OMM - Uvod
Model za koronu???
OMM - Uvod
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OMM - Uvod
Model za koronu???
OMM - Uvod