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.

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

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

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 =

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)

62020.5.7.

0 ≤ x(t) ≤ 1, 0 ≤ y(t) ≤ 1, 0 ≤ z(t) ≤ 1 for 0 ≤ t

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

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)

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)

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)

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

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)

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)

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)

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)

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)

2020.5.7. 17

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