Luís M. A. Bettencourt Theoretical Division Los Alamos National Laboratory

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The pace of life in the city:urban population size dependence of the dynamics of disease, crime, wealth and

innovation

Luís M. A. BettencourtTheoretical Division

Los Alamos National Laboratory

ASU - February 4, 2006



Collaboration & Support:

José Lobo : Global Institute of Sustainability, ASUGeoffrey West: Santa Fe Institute

Dirk Helbing & Christian Kuhnert, T.U. Dresden

Support from ISCOM: European Network of Excellence

Special thanks to Sander van der Leeuw: School of Human Evolution and Social Change, ASU



Scaling in Biological Organization

R=R0 Mb b=3/4 Power law solves: R(a N)=ab R(N) Scale Invariance



Cells in organisms are constrained by resource distribution networks:

-Hierarchical Branching-[3d] Space filling-Energy Efficient-Terminate at invariant area units

RBM b , b3

4d

d 1 r R

M BM

1

4

Total metabolic rate metabolic rate/mass

Larger organisms are slower!!



[…] it is evident that the state [polis] is a creation of nature, and that man is by nature a political animal.

The proof that the state is a creation of nature and prior to the individual is that the individual, when isolated, is not self-sufficing; and therefore he is like a part in relation to the whole.

Aristotle: Politics [Book I]

Until philosophers rule as kings or those who are now called kings and leading men genuinely and adequately philosophize, that is, until political power and philosophy entirely coincide, […]

cities will have no rest from evils,... nor, I think, will the human race.

Plato: [Republic 473c-d]

The city as a ‘natural organism’

Raphael's School of Athens (1509-1511)



Is there are analogue betweenbiological and social scaling?

• Metabolic Rates ~ Nd/(d+1)

Energy/resource consumption

• Rates decrease ~N-1/(d+1)

• Times increase ~N1/(d+1)

Is 3> d ~2 ?

We set forth to search for data and estimate power laws:

Y(N)=Y0 Nb



Energy consumption vs. city size

economy of scale

Germany: year 2002

Data source: German ElectricityAssociation [VDEW]

Courtesy of Christian Kuehnert

super-lineargrowth



Structural Infrastructureoptimized global design for economies of scale

Y b 95% CI adj.- R2 observations

Country/year

Gasoline Stations 0.77 [0.74,0.81] 0.93 318 USA/2001

Gasoline Sales 0.79 [0.73,0.80] 0.94 318 USA/2002

Length of electrical

cables0.88 [0.82,0.94] 0.82 387 Germany/

2001

Note that although there are economies of scale in cables the network is still delivering energy at a superlinear rate:

Social rates drive energy consumption rates, not the opposite



Basic Individual needsproportionality to population


Country/year

Total establishmen

ts0.98 [0.95,1.02] 0.95 331 USA/2001

Total employment 1.01 [0.99,1.02] 0.98 331 USA/2001

Total Household electrical

consumption1.00 [0.94,1.06] 0.70 387 Germany/

2001Total Household

electrical consumption

1.05 [0.89,1.22] 0.91 295 China/2002

Total Household water consumption 1.01 [0.89,1.11] 0.96 295 China/2002

Also true for the scaling of number of housing units



The urban economic miracleacross time, space, level of development or

economic system


Country/ year

Total Wages/y

r1.12 [1.09,1.13] 0.96 361 USA/2002

GDP/yr 1.15 [1.06,1.23] 0.96 295 China/2002

GDP/yr 1.13 [1.03,1.23] 0.94 37 Germany/2003

GDP/yr 1.26 [1.03,1.46] 0.64 196 EU/2003



Innovation as the engine


Country/year

New Patents/yr 1.27 [1.25,1.29] 0.72 331 USA/2001

Inventors/yr 1.25 [1.22,1.27] 0.76 331 USA/2001

Private R&D employment 1.34 [1.29,1.39] 0.92 266 USA/2002

“Supercreative” Professionals 1.15 [1.11,1.18] 0.89 287 USA/2003

R&D employment 1.67 [1.54,1.80] 0.64 354 France/1999*

R&D employment 1.26 [1.18,1.43] 0.93 295 China/2002

* France/1999 data courtesy Denise Pumain, Fabien Paulus



Innovation measured by Patents

From “Innovation in the city: Increasing returns to scale in urban patenting”Bettencourt, Lobo and Strumsky Data courtesy of Lee Fleming, Deborah Strumsky

Source data:U.S. patent officeIncludes all patents between 1980-2001



Employment patterns

b=1.15 ( 95% C.I.=[1.11,1.18] ) adjusted R2= 0.89

Supercreative professionals [Florida 2002, pag. 327-329] are “Computer and Mathematical, Architecture and Engineering, Life Physical and Social Sciences Occupations, Education training and Library, Arts, Design, Entertainment, Sports and Media Occupations”.

Derived from Standard Occupation Classification System of the U.S. Bureau of Labor Statistics

Data courtesy of Richard Florida and Kevin Stolarick. Plot by Jose Lobo



Social Side Effects


Country/year

Total elect. consumptio

n1.09 [1.03,1.15] 0.72 387 Germany/

2001

Cost of housing 0.09 [0.07,1.27] 0.21 240 USA/2003

New AIDS cases 1.23 [1.18,1.29] 0.76 93 USA/2002

Serious Crime 1.16 [1.11,1.18] 0.89 287 USA/2003

Walking Speed 0.09 [0.07,0.11] 0.79 21 Several/1979

Disease transmission is a social contact process:

dT

dtcSI Standard Incidence



The Pace of Life walking speed vs. population

Borstein & BornsteinNature 1976Bornstein, IJP 1979

bCI [0.071,0.115]



But cities to exist at all must also satisfy:

Basic individual needs (house, job, basic necessities)Require city-wide infrastructure:

- larger population Optimization of system level - higher density distribution networks

Result: 3 categories:

Social - interpersonal interactions - grow with # effective relations Individual - no interactions - proportional to population Structural - global urban optimization - economies of scale

Y Y0Nb

b 1 Social

b1 Individual

b 1 Structural

Scaling Law:



Scale, Pace and Growth

RNRc E0

dN

dt

dN

dtRaE0

N b RcE0

N

N(t) RaRc

N1 b (0) RaRc

exp[

RcE0

(1 b)t]

1

1 b

Consider the energy balance equation:

availableresources

costs growth

General Solution:



b<1 implies limited carrying capacity

biological population dynamics

N RaRc

1

1 b



N(t) RaRc

N1 b (0) RaRc

exp[

RcE0

(1 b)t]

1

1 b

tcrit E0

( 1)RaN1 b (0) 50

T

nb 1years.

b>1 : Finite time Boom and Collapse



Escaping the singularity with b>1:cycles of successive growth & innovation

tcrit E0

( 1)RaN1 b (0) 50

T

nb 1years. tcrit shortens with N



Consequences for epidemiology

Consider:

SIR :N

S,I

Epidemic dynamics over quasi-static background

S

BN b SNI S

I S

N ( )

I

is a small parameter

Disease free fixed point:

SE 1

0

N, IE

0BNb 1 1 N

Endemic fixed point:

SF BN b , IF 0



dynamics at disease free fixed point

I( ) 0BN

b 1 1 I, 0

Unstable if:

0BNb 1 1 b=1

1

b<1

b>1

0BNb 1

N

Even if initially stable b>1 eventuallyleads to endemic state.



Dynamics at endemic steady state

IEN

0BNb 1 1

Infected as a fraction of the population:

N

IEN

b<1

b=1

b>1



Oscillations and decayat endemic state

1

2 0BN

b 1 (0BNb 1)2 4 BN b 10 1

0

Eigenvalues:

Solution:

e (N )t cos (N)t

N

b>1

b=1

b<1

N

b>1

b=1

b<1



General pictureScaling

ExponentDriving Force Organization Growth

b<1Optimization, Efficiency

Stable equilibriumBiological Sigmoidal

Long term finite attractor

b>1Creation of Information, Wealth

and Resourcesnonequilibrium, constant adaptation

SocialBoom / Collapse

Finite time singularityIncreasing acceleration / discontinuities

b=1 Single Individual Maintenance Trivial, Free ExponentialInfinite time divergence

Human social organization is a compromise over many social activitiesEpidemiological dynamics is affected by large-scale

human organization and behavior



An argument for the smallness of the superlinearity in social scaling

exponentsIt is expected that a social process scales with the number of contacts.Naively for a homogeneous population N:

Or per capita ncpc=(N-1)/2

Clearly in a large population, N>1000, not all contacts can be realized.

This naïve estimate is the wrong result, an unattainable upper bound.

nc N(N 1) /2



Now, still assume that the number of effective contacts increases with N but is constrained (time, cognition, energy) to be much smaller:

between largestand smallest city

Now equate the change in productivity per capita R= R1Nwith this increase in effective contacts:

Ancpc (N)

ncpc (N0)~ 101 102

AN

N0

1

1log(A)

log(N /N0)~

1 2

70.14 0.29

Note that ncpc(N) may itself scale, but with a very small exponent

Luís M. A. Bettencourt Theoretical Division Los Alamos National Laboratory

Documents

metabolic rates

social rates

energy consumption rates

social scaling

political power

creation of nature

housing units yb95

opposite yb95