Cite as: J. Dehning et al., Science

10.1126/science.abb9789 (2020).

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0 .3 5 , 0 .5 1

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0 .2 0 , 0 .3 0

0 .1 2 , 0 .2 0

0 .0 6 , 0 .1 3

λ μ

D0

I

λ μ D

λ 0 .41= μ 0 .1 2= 8 .6D =0

1 9I =

0λ / μR =

03 .4R = 2 .4 4 .7

λ

0I σ

μ D

*λ λ μ= −

λ μ

λ μ

*λ

*λ 2 8%=

λ μ

λ μ

*λ 0= λ μ=

λ

0λ 0 .4 1=

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μ 0 .1 2=

0λ

1λ

1 0λ λ=

1λ 0 .41 μ=

5 0 %1 0

λ λ  / 2=1

λ 0 .21 μ=

D 8 .6D =

*

1 1λ λ μ 0= −

1 0 %1 0

λ λ  /1  0=1

λ 0 .04 μ=

D

*λ 0

0 1λ λ→

D

*λ λ μ= −

*λ 0

0λ

0 1λ λ→

1t

1 2λ λ→

2t

2 3λ λ→

3t

Δ )i

t

( )it

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0λ

0λ 0 .4 1= 0 .3 2 , 0 .5 1 λ

σ

0I

( )λ t0

λ 0 .43=

0 .3 5 , 0 .5 11

λ 0 .2 5=

( ) ( )*

λ λ μt t= −

0λ μ 0 .3− =

1λ μ 0 .12− =

μ 0 .1 3=

( )λ t1

λ 0 .2 5=

2λ 0 .15=

( )*

2λ λ μ 0 .0 2 0t = − =

( )λ t2

λ 0 .15=3

λ 0 .09=

( )*

3λ λ μ 0 .0 3 0t = − = − 0 .0 5, 0 .0 2− −

1 1 .4D = 3

*λ

3 0 % 1 2 % 2 %

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3 %− 5 % , 2 %− −

D

D

1 1 .4D =

( )λ t λi

λi

3 %−

D

D

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λ I

S

R μ

N

dλ

d

dλ μ

d

dμ  .

d

S S I

t N

I S II

t N

RI

t

= −

= −

=

I

R S N I / 1S N

( ) ( ) ( )( )λ μ

λ μ      s           0  .td I

I o lv e d b y I t I ed t

−

= − =

Δ t =

d / d Δ / ΔI t I t

n e w1

1 1

n e w

1 1

n e w n e w1

1 1

λΔ :

       μΔ :      

λ μ Δ  .

t

t t t t

t t t t

t

t t t t t

SS S t I I

N

R R tI R

SI I tI I R

N

−

− −

− −

−

− −

− = − = −

− = =

− = − = −

tI

n e w

tI

Dn e w

tI

tC

n e w.

t t DC I

−=

0t =0

I

ˆt

C

t D

0I

λi

1, ...,i n=

it

1λ

i −λ

i

Δi

t

λ

it Δ

it λ

i

λi

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( )( )

( ) ( )

n e w1  ,              w ith

π 11 1 s in Φ  ,

7 2

t t D

w w

C I f t

f t f t

−= −

= − − −

wf Φ

w

0θ λ , , μ , , σ , , , Φ

i i w wt D I f=

σ

( θˆ | )t

P C

θ

( ) ( ) θ θt

C C=

ˆ ˆt

C C= θ

θ

( )(θ | ) (ˆ ˆ | θ ) θ ,p C p C p ( θˆ | )p C

( )θp

( ) ( )( )ν 4( | θ ) S tu d en tT m ean θ , w id tˆ h σ θ  .

t t tp C C C

= = =

t ( )θtC

ˆt

C

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( )( )μ L ogN o rm a l lo g 1 / 8 , 0 .2

μ

( )( )λ L o gN o rm a l lo g 0 .4 , 0 .5

0 .4 λ μ− 0 .2 7 5

0λ / μR =

3 .2

( )( )L o g N o rm a l lo g 8 , 0 .2D

( ) (2

H a lfC a u c h y , β 2 / π β[1 / β )x x = +

( )βO β

( )0H alfC au ch y 1 0 0I

( )1 0 0O

( )σ H alfC au chy 10

λ

5 0 %

λ

( )( )0λ L o gN o rm a l lo g 0 .4 , 0 .5

( )( )1λ L o gN o rm a l lo g 0 .2 , 0 .5

5 0 %

0λ 0 .4

( )( )2λ L ogN o rm a l lo g 1 / 8 , 0 .5

λ 5 0 %

( )( )3λ L o gN o rm a l lo g 1 / 1 6 , 0 .5

5 0 %3

λ

μ

( )1N o rm al 2 0 2 0 / 0 3 / 0 9 , 3t

( )2N o rm al 2 0 2 0 / 0 3 / 1 6 ,1t

( )3N o rm al 2 0 2 0 / 0 3 / 2 3,1t

Δi

t ( )( )Δ LogN o rm a l lo g 3 , 0 .3i

t

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( )B eta m ean 0 .7 , s td 0 .1 7w

f = =

( )Φ vonM ises m ean 0 , κ 0 .0 1w = =

( )o s

p o s t

1

lp p d = lo g ( | θ ) (θ ) θ

N

i

i

p y p d

=

o s o s

1 1, ,y y

N ( )p o s t p o s tθ (θ | , )

jp p y M=

yj

M

( )p o s tθp

2−

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λ

σ

λ λ λ

λ

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λ

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λ

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9 2 7 9 8 .3 1

8 1 9 1 6 1 3 .4 6

7 9 6 1 7 1 2 .5 3

7 8 7 1 7 1 3 .4 2

( )( )L o g N o rm al lo g 0 .4 , 0 .5

( )( )L o g N o rm al lo g 1 / 8 , 0 .2

D ( )( )L o g N o rm a l lo g 8 , 0 .2

0I ( )H alfC au ch y 1 0 0

( )H alfC au ch y 1 0

1t ( )N o rm al 2 0 2 0 / 0 3 / 0 9 , 3

2t ( )N o rm al 2 0 2 0 / 0 3 / 1 6 ,1

3t ( )N o rm al 2 0 2 0 / 0 3 / 2 3,1

Δi

t ( )( )L o g N o rm a l lo g 3 , 0 .3

0 ( )( )L o g N o rm al lo g 0 .4 , 0 .5

1 ( )( )L o g N o rm al lo g 0 .2 , 0 .5

2 ( )( )L o g N o rm al lo g 1 / 8 , 0 .5

3 ( )( )L o g N o rm a l lo g 1 / 1 6 , 0 .5

( )( )L o g N o rm al lo g 1 / 8 , 0 .2

D ( )( )L o g N o rm a l lo g 8 , 0 .2

wf ( )B eta m ean 0 .7 , s td 0 .1 7= =

Φw ( )vo n M ises m ean 0 , 0 .0 1= =

0I ( )H alfC au ch y 1 0 0

( )H alfC au ch y 1 0

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Inferring change points in the spread of COVID-19 reveals the effectiveness of interventionsJonas Dehning, Johannes Zierenberg, F. Paul Spitzner, Michael Wibral, Joao Pinheiro Neto, Michael Wilczek and Viola Priesemann

published online May 15, 2020

ARTICLE TOOLS http://science.sciencemag.org/content/early/2020/05/14/science.abb9789

MATERIALSSUPPLEMENTARY http://science.sciencemag.org/content/suppl/2020/05/14/science.abb9789.DC1

CONTENTRELATED

http://stm.sciencemag.org/content/scitransmed/9/396/eaal3653.fullhttp://stm.sciencemag.org/content/scitransmed/11/499/eaat0360.fullhttp://stm.sciencemag.org/content/scitransmed/12/534/eabb1469.fullhttp://stm.sciencemag.org/content/scitransmed/12/541/eabb5883.full

REFERENCES

http://science.sciencemag.org/content/early/2020/05/14/science.abb9789#BIBLThis article cites 36 articles, 10 of which you can access for free

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