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

1. The pace of life in the city:urban population size dependence of the dynamics of disease, crime, wealth and innovation Luís M. A. Bettencourt Theoretical Division Los Alamos National Laboratory ASU - February 4, 2006

2. Collaboration & Support: José Lobo : Global Institute of Sustainability, ASU Geoffrey 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

3. Scaling in Biological Organization R=R0 Mb b=3/4 Power law solves: R(a N)=ab R(N) Scale Invariance

4. Cells in organisms are constrained by resource distribution networks: • Hierarchical Branching • [3d] Space filling • Energy Efficient • Terminate at invariant area units Total metabolic rate metabolic rate/mass Larger organisms are slower!!

5. The city as a ‘natural organism’ 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] Raphael's School of Athens (1509-1511) […] 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]

6. 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

7. Energy consumption vs. city size Germany: year 2002 Data source: German Electricity Association [VDEW] Courtesy of Christian Kuehnert super-linear growth economy of scale

8. Structural Infrastructureoptimized global design for economies of scale 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

9. Basic Individual needsproportionality to population Also true for the scaling of number of housing units

10. The urban economic miracleacross time, space, level of development or economic system

11. Innovation as the engine * France/1999 data courtesy Denise Pumain, Fabien Paulus

12. Innovation measured by Patents Source data: U.S. patent office Includes all patents between 1980-2001 From “Innovation in the city: Increasing returns to scale in urban patenting” Bettencourt, Lobo and StrumskyData courtesy of Lee Fleming, Deborah Strumsky

13. Employment patterns b=1.15 ( 95% C.I.=[1.11,1.18] ) adjusted R2= 0.89 Data courtesy of Richard Florida and Kevin Stolarick. Plot by Jose Lobo 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

14. Social Side Effects Disease transmission is a social contact process: Standard Incidence

15. The Pace of Life walking speed vs. population Borstein & Bornstein Nature 1976 Bornstein, IJP 1979 b CI [0.071,0.115]

16. 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 Scaling Law:

17. Scale, Pace and Growth Consider the energy balance equation: growth costs available resources General Solution:

18. b<1 implies limited carrying capacitybiological population dynamics

19. b>1 : Finite time Boom and Collapse

20. Escaping the singularity with b>1:cycles of successive growth & innovation tcrit shortens with N

21. Consequences for epidemiology Epidemic dynamics over quasi-static background Consider: Disease free fixed point: Endemic fixed point: m is a small parameter

22. b>1 1 b<1 N dynamics at disease free fixed point Unstable if: b=1 Even if initially stable b>1 eventually leads to endemic state.

23. b=1 b<1 N Dynamics at endemic steady state Infected as a fraction of the population: b>1

24. Oscillations and decayat endemic state Eigenvalues: Solution: w(N) h(N) b>1 b>1 b=1 b=1 b<1 b<1 N N

25. General picture Human social organization is a compromise over many social activities Epidemiological dynamics is affected by large-scale human organization and behavior

26. An argument for the smallness of the superlinearity in social scaling exponents It 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.

27. Now, still assume that the number of effective contacts increases with N but is constrained (time, cognition, energy) to be much smaller: between largest and smallest city Now equate the change in productivity per capita R= R1Nb-1 with this increase in effective contacts: Note that ncpc(N) may itself scale, but with a very small exponent