Small worlds and giant epidemics
Denis Mollison
http://www.ma.hw.ac.uk/∼denis
1
Introduction
Interests
• Invasion – threshold? R0?
• Spread – velocity? diameter? final size?
• Persistence? – duration? control?
Example Foot and mouth disease outbreaks
1967-8: spatial with jumps
2001: two phases
• global
2
Building models
Types• individual or mass-action?
• stochastic or deterministic?
[For more on structures, and pitfalls (hidden assump-tions, unfriendly parameters), see Mollison 1995]
Today shall stick to network models
• mean-field
• metapopulations
• small-world
Simple random graph
Giant component exists iff R0 > 1.
Diameter of giant, T ∼ logN.
Final size (and probability of a large outbreak) are both given by the largest solution of
Structural choices for network models
• Directed or undirected?
• Degree – fixed? Poisson? ‘scale-free’ ?
• Large-scale structure (mean-field to spa-tial)
Note Should in and out links be independent? For undirected graphs, they are only independent in the SRG case. For any other degree distribution, the effec-tive mean number of contacts is ‘size-biased’.
Example: the ‘scale-free’ case, withpn∼n−3, has
Metapopulation models (Ballet al. 1997)
Threshold: RT = R0µ > 1
Spatial
Nearest-neighbour
Spatial stochastic models
• velocity finite iff finite variance
• threshold value of R0 is > 1.
‘Great circle’ / ‘small world’ (Watts & Strogatz 1998)
Threshold: RT = R0µ > 1 (as for
metapopu-lation model)
T reduces from ∼N to∼ logN as the number of global links increases
3
Approximations
• Spatial DEs
• Submultiplicative dynamics ˙
I = βIαSγ
Growth I ∼t1/(1−a) suggests α = 1− D1 ?
• Pair approximations: does local correla-tion capture spatial structure?
Spatial DEs
R&D kernel approach (see Mollison 1991) shows how linear theory can find velocity for a wide range of models . . .
• exactly for deterministic models
Pair approximations
Reconsider the deterministic SIR:
˙ S = −βSI ˙ I = βSI −γI ˙ R = γI
More accurately ˙ S = −β[SI] ˙ I = β[SI]−γI ˙ R = γI ˙ [SS] = −2β[SSI] ˙ [SI] = β([SSI]−[ISI]−[SI]−γ[SI] ˙ [SR] = · · · ˙ [II] = · · ·
For closure, use [ABC] ≈ (1− 1 n) [AB][BC] [B] ×(1−φ+φ [AC] [A][C]) where φ = P(bc|ab & ac)
Example hexagonal lattices (HBFs)
φ = 6/15 = 0.4
So does G(6,0.4) have T ∼ √N ?
whereG(?, φ) is the random graph with degree distribution ? and clustering parameter φ
SIR (dashed line) and its pair approximation (solid line), forφ = 0, 0.2, 0.4.
Also, spatial SIR (‘S’) and ordinary deterministic SIR (‘?’).
4
Discussion
Pair approx is good for ‘typical’ graph G(n, φ) but T ∼logN (mean-field)
Yet there are spatial examples of G(n, φ) – HBFs – with T ∼ √N
Worse, at one extreme we can find a 1-D net-work with the same n and φ
. . . while at the other, with a metapopulation model . . .
. . . we can find an essentially mean-field model with any φ, R0.
Internal 12(k −1)(k −2)p3
The resolution of this paradox is that the non-mean-field cases are of negligible probability – HBFs, even as small as N = 150,
‘We are now cruising at a level of 225,000 to 1 against and falling, and we will be restoring normality just as soon as we are sure what is normal anyway.’
• Local structure is a poor guide to global structure
Small worlds good ,
5
References
Adams, D (1979)The Hitchhiker’s Guide to the Galaxy, Pan Books.
Structure
Mollison, D (1995) ‘The structure of epidemic models’, in Epidemic Models: their Structure and Relation to Data(ed. Denis Mollison), Cambridge UP.
Random graphs
Bollob´as, B (1985) Random Graphs, Academic Press, London.
Newman, MEJ (2002) ‘Random graphs as models of net-works’,cond-mat archive 0202208.
Metapopulations
Ball, FG, Mollison, D and Scalia-Tomba, G-P (1997)
‘Epidemics in populations with two levels of mixing’,
Ann. Appl. Prob.,7, 46-89.
Spatial models
Mollison, D (1972) ‘The rate of spatial propagation of simple epidemics’,Proc 6th Berkeley Symp on Math Statist and Prob3, 579-614.
Cox, JT, and Durrett, R (1988) ‘Limit theorems for the spread of epidemics and forest fires’,Stoch Procs Applics
30, 171-191.
Mollison, D (1991) ‘The dependence of epidemic and population velocities on basic parameters’, Math Bio-sciences107, 255-287.
Durrett, R and Levin, SA (1994) ‘The importance of being discrete (and spatial)’, Theor Pop Biol 46, 363-394.
Pair approximations
Morris, AJ (1997) Representing spatial interactions in simple epidemiological models, PhD Thesis, Warwick Uni-versity.
Keeling, MJ (1999) ‘The effects of local spatial structure on epidemiological invasions’, Proc R Soc LondB 266, 859-867.
Rand, DA (1999) ‘Correlation equations and pair ap-proximations for spatial ecologies’, in Advanced Ecolog-ical Theory(ed. Jacqueline McGlade), 100-142.
Ferguson, N, Donnelly, C, and Anderson, R (2001) ‘The Foot-and-Mouth epidemic in Great Britain: pattern of spread and impact of interventions’,Science,292, 1155-1160.
Small world graphs
Watts, DJ, and Strogatz, SH (1998) ‘Collective dynam-ics of ‘small-world’ networks’, Nature, 393, 440-442. Newman, MEJ (2000) ‘Models of the Small World: A Review’, cond-mat archive 0001118.
Scale-free networks
Albert, R, and Barabasi, A-L (2001) ‘Statistical me-chanics of complex networks’,cond-mat archive 0106096.
