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Transcript of ò ± w: g ¶ Ö ´ t A ÿ ¢² s w ® Í ¯ T £ 1 · 2020. 8. 5. · 2020.5.7. ® Ú ¿ Ä h t ò...
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2020.5.7. 1
1 →
1. =
= −
2. SIR
4
= (1− )× × ×
= (1− )× ×
= ×
1
1
→ Liu et al (2020) 12
1.4-6.49 3.29 2.79
1. 1
2. *
*
3.
4.
5.
A. B.
C1.
C2.
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2020.5.7. 2
13 SIR 3
13.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
13.2 SIR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
13.3 R0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
14 SIR 5
15 6
16 7
16.1 . . . . . . . . . . . . . . . . . 7
16.2 x y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
16.3 . . . . . . . . . . . . . . . . . . . . . . . 9
16.4 x y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
16.5 x y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
16.6 y x ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
16.7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
16.8 x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
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32020.5.7.
13 SIR
13.1
1.
2.
3.
3
1. Susceptibles
2. Infected
3. Removed *1
SIR 3 Susceptibles Infected
Removed/Recovered 3
S I R
S(t) ≥ 0, I(t) ≥ 0, R(t) ≥ 0 for 0 ≤ t
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2020.5.7. 4
13.2 SIR
SIR S I R t
dS(t)
dt= −β
S(t)
NI(t) (3)
dI(t)
dt= β
S(t)
NI(t)− γI(t) (4)
dR(t)
dt= γI(t) (5)
β > 0 γ > 0
dS(t) t dt
S difference
dS(t) = S(t+ dt)− S(t) (6)
(3)(4)(5) 0
dS(t)
dt+
dI(t)
dt+
dR(t)
dt= 0 (7)
dS(t) + dI(t) + dR(t) = 0 (8)
N
→
(2) t
(4)(5) I γ R
(5)(3) R S
= *2
→ → (9)
*2 1 =
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2020.5.7. 5
13.3 R0
R0 1
=
R0 =β
γ(10)
*3 (10) ×
14 SIR
(3)(4)(5)
(3)(4)(5) N
x(t) = S(t)/N y(t) = I(t)/N z(t) =
R(t)/N (3)(4)(5)
dx(t)
dt= −βx(t)y(t) (11)
dy(t)
dt= βx(t)y(t)− γy(t) (12)
dz(t)
dt= γy(t) (13)
β
γ
tx(t) + y(t) + z(t) = 1 (14)
*3 Hethcote (2000)
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62020.5.7.
0 ≤ x(t) ≤ 1, 0 ≤ y(t) ≤ 1, 0 ≤ z(t) ≤ 1 for 0 ≤ t
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72020.5.7.
16
16.1
*4
1.
2.
3.
4.
1. →
2.
3.
4. →
SIR
x(t) = S(t)/N y(t) = I(t)/N z(t) = R(t)/N
(11)(12)(13)
dx(t)
dt= −βxy = 0 (16)
dy(t)
dt= (βx− γ)y = 0 (17)
dz(t)
dt= γy = 0 (18)
y = 0 (19)
*4
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2020.5.7. 8
y = 0 (x, y, z)
x, y, z
1. y
2. y
3. y
16.2 x y
(14) x(t) y(t)
z(t) x y
N
t = 0
t = 0
y = 0
(x(0), y(0), z(0)) = (x(0), 0, z(0)) (20)
t = 0
z
y(x(0), y(0), z(0)) = (1, 0, 0) (21)
→
y
y(t) > 0; t > 0 (22)
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2020.5.7. 9
t t = 0
y(t)
limt→0
y(t) = 0 (23)
t > 0 I 1
N 1
y = I/N t = 0
I(0) = y(0) = 0
→
x(0) > 0 (24)
16.3
y(t) (12)
dy(t)
dt= βx(t)y(t)− γy(t) (12)
= (βx(t)− γ)y(t) (25)
t > 0, t→ 0
dy(t)
dt= (βx(0)− γ)y(t) > 0 (26)
(21)
βx(0)− γ > 0 (27)
βx(0) > γ (28)
x(0) >γ
β(29)
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2020.5.7. 10
x(0) >γ
β(30)
16.4 x y
(14) x(t) y(t)
z(t) x y
x(t) (11)
dx(t)
dt= −βx(t)y(t) (11)
β > 0 (15)(22)(24) dx(t)/dt < 0
x(t) t
y(t) (12)
dy(t)
dt= βx(t)y(t)− γy(t) (12)
= (βx(t)− γ)y(t) (25)
(22) y(t) t x(t) (30)
x(0) > γ/β
1. x(t) > γ/β t dy(t)/dt > 0
2. x(t) = γ/β t dy(t)/dt = 0
3. x(t) < γ/β t dy(t)/dt < 0
x(t) y(t)
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2020.5.7. 11
16.5 x y
x(t) y(t)
(15)-(18) y = 0
y(t)
t = t∗
(x(t∗), y(t∗), z(t∗)) = (x(t∗), 0, z(t∗)) (31)
t = t∗
y = 0
t y(t)
limt→∞
y(t) = 0 (32)
y = 0 t = 0 t→∞
16.6 y x ?
y
x y
x
y x
y x
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2020.5.7. 12
y x
= x y
y
(11)(12)(13)
dx(t)
dt= −βx(t)y(t) (11)
dy(t)
dt= βx(t)y(t)− γy(t) (12)
dz(t)
dt= γy(t) (13)
(12) (11)
dy(t)
dx(t)= −1 +
γ
β
1
x(t)(33)
= −1 +R−101
x(t)(34)
(34)
y(t)− y(0)
x(t)− x(0)= −1 +R−10
1
x(t)(35)
y(t)− y(0) = x(0)− x(t) +R−10x(t)− x(0)
x(t)(36)
= x(0)− x(t)−R−10x(0)− x(t)
x(t)(37)
= x(0)− x(t)−R−10 logx(0)
x(t)(38)
= x(0)− x(t) +R−10 logx(t)
x(0)(39)
R−10 logx(t)
x(0)= (y(t)− y(0)) + (x(t)− x(0)) (40)
logx(t)
x(0)= R0((y(t)− y(0)) + (x(t)− x(0)) (41)
x(t)
x(0)= exp
{
R0((y(t)− y(0)) + (x(t)− x(0))}
(42)
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2020.5.7. 13
x(t) = x(0) exp{
R0((y(t)− y(0)) + (x(t)− x(0))}
(43)
= x(0) exp{
R0((y(t) + x(t))− (y(0) + x(0))}
(44)
(43) t → ∞ (23)(32) x(t) x(∞) = limt→∞ x(t)
x(∞) = x(0) exp{
R0((y(∞)− y(0)) + (x(∞)− x(0))}
(45)
= x(0) exp{
R0(x(∞)− x(0))}
(46)
= x(0) exp{
−R0(x(0)− x(∞))}
(47)
(47) (24)
x(∞) 6= 0 (48)
(15)
x(∞) > 0 (49)
y x
x y y
(47) x(t)
a. R0 b. x(0)
16.7
(23)(32) (38) x(t) x(∞) = limt→∞ x(t)
x(0)− x(∞)−R−10 logx(0)
x(∞)= 0 (50)
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2020.5.7. 14
f(R0, x(0), x(∞)) = x(0)− x(∞)−R−1
0 logx(0)
x(∞)(51)
R0
dx(∞)
dR0= −
∂f(R0, x(0), x(∞))/∂R0∂f(R0, x(0), x(∞))/∂x(∞)
(52)
= −
−(
−1
R20
)
logx(0)
x(∞)
−1−R−10x(∞)
x(0)
(
−x(0)
(x(∞))2
)
(53)
= −
1
R20log
x(0)
x(∞)
−1 +R−10x(∞)
(54)
(30) (11)
x(0) > R−10
(
=γ
β
)
> x(∞) (55)
dx(∞)
dR0< 0 (56)
R0
=
x
x(0)
dx(∞)
dx(0)= −
∂f(R0, x(0), x(∞))/∂x(0)
∂f(R0, x(0), x(∞))/∂x(∞)(57)
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2020.5.7. 15
= −
1−R−10x(∞)
x(0)
1
x(∞)
−1−R−10x(∞)
x(0)
(
−x(0)
(x(∞))2
)
(58)
= −
1−R−10x(0)
−1 +R−10x(∞)
(59)
(55)
dx(∞)
dx(0)< 0 (60)
=
x
16.8 x
R0 = β/γ x(0)
ȳ
(17) y(t) (11) y ȳ > 0
x̄
dy(t)
dt= (βx̄− γ)ȳ = 0 (17)
βx̄− γ = 0 (61)
x̄ =γ
β(62)
=1
R0(63)
(38) (23)
y(t) = x(0)− x(t)−1
R0log
x(0)
x(t)(38)
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2020.5.7. 16
ȳ = x(0)− x̄−1
R0log
x(0)
x̄(64)
= x(0)−1
R0−
1
R0logR0x(0) (65)
(65) R0
∂ȳ
∂R0= −
(
−1
(R0)2
)
−(
−1
(R0)2
)
logR0x(0)−1
R0
x(0)
R0x(0)(66)
=1
(R0)2−(
−1
(R0)2
)
logR0x(0)−1
(R0)2(67)
= −(
−1
(R0)2
)
logR0x(0) > 0 (68)
→(30) R0x(0) > 1
R0
(65) x(0)
∂ȳ
∂x(0)= 1−
1
R0
R0R0x(0)
(69)
= 1−1
R0x(0)> 0 (70)
→(30) R0x(0) > 1
x(0)
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2020.5.7. 17
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