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Introduction Definitions General examples Specific examples Literatura
Overview of spatial models in epidemiology
Ben Bolker
McMaster UniversityDepartments of Mathematics & Statistics and Biology
10 October 2011
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models
Introduction Definitions General examples Specific examples Literatura
Outline
1 Introduction
2 Definitions
3 General examples and issues
4 Specific examples
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models
Introduction Definitions General examples Specific examples Literatura
Outline
1 Introduction
2 Definitions
3 General examples and issues
4 Specific examples
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models
Introduction Definitions General examples Specific examples Literatura
Overview
Themes:
How can we reduce dimensionality?
Which model properties interact?
Which details are important?
What are the best summary metrics for spatial behavior?How do they differ among model types?
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models
Introduction Definitions General examples Specific examples Literatura
Goals of modeling
Why model space, and how?Implicit7 vs. explicit spatial problemsModel-building tradeoffs22;23:
Realism
Computational cost
Analytical tractability
Connections with data
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models
Introduction Definitions General examples Specific examples Literatura
Scope
“Look, old boy,” said the machine, “if I could doeverything starting with n in every possible language, I’dbe a Machine That Could Do Everything in the WholeAlphabet . . . ”21
Important connections:
biological invasions
epidemics in heterogeneous populations
predator-prey (parasitoid-host) models
graph theory, percolation theory, . . .
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models
Introduction Definitions General examples Specific examples Literatura
Outline
1 Introduction
2 Definitions
3 General examples and issues
4 Specific examples
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models
Introduction Definitions General examples Specific examples Literatura
Model properties
Time discrete vs continuous
Space discrete (patch) vs discrete (contiguous) vscontinuous
State discrete (binary) vs discrete (integer) vs continuous
Dispersal local vs distance-based vs global
Randomness stochastic vs deterministic
Infection dynamics Simple vs complex(e.g. SIR vs age-of-infection models)
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models
Introduction Definitions General examples Specific examples Literatura
Trivial models
No connections, just (exogenous) variabilityin the environment
“Space is what keeps everything from happeningin the same place”
Very practical, if exogenous heterogeneity swampseverything else
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models
Introduction Definitions General examples Specific examples Literatura
Non-contiguous (pseudo-spatial) models
No degree of locality:within- vs between-patch (metapopulation models)Simplest:
Two-patch model
Patch-occupancy model (≡ microparasite model)
More complex: multi-patch models, typically with stochasticity13;19
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models
Introduction Definitions General examples Specific examples Literatura
Network models
Simple (binary state) nodes, interesting contact structure:
Random graphs
Scale-free networks (power-law degree distribution)
Markov models (local neighborhoods)
Small world networks (local structure, global rewiring)
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models
Introduction Definitions General examples Specific examples Literatura
Networks of patches
Collapse local groups of nodes into patches or populations
Patch-occupancy: incidence function models
Gravity models9;44
Often matches the scale of data: cases per region
Distinguish “truly” spatial models: dimensionality?i.e. (number of neighbors within r) ∼ power law(∝ rD rather than exp(r)): contiguity
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models
Introduction Definitions General examples Specific examples Literatura
Contiguous models: bestiary
Space Time Populations Random ModelDisc Disc Disc deterministic cellular automatonDisc Disc Disc stochastic stochastic CADisc Disc Cont either coupled-map latticeDisc Cont Disc stochastic interacting particle system
≈ pair approximationCont Disc Cont either integrodifference equationCont either Disc stochastic spatial point process
≈ spatial moment equationsCont Cont Cont deterministic integrodifferential,
partial differential equation(reaction-diffusion equation)
Cont Cont Cont stochastic stochastic IDE/PDE
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models
Introduction Definitions General examples Specific examples Literatura
Reaction-diffusion equations
∂S
∂t= −βSI + DS∆S
∂I
∂t= βSI + DI∆I − γI
Analyze by finding asymptotic wave speed of traveling-wavesolutions
Details matter: Is ∆S = ∆I?Is contact local or distributed (→ ∆I term in contact rate)27?
Simplest model → criticisms [e.g. “atto-fox”problems1, effectsof long-distance dispersal]
Limit of many other models
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models
Introduction Definitions General examples Specific examples Literatura
Outline
1 Introduction
2 Definitions
3 General examples and issues
4 Specific examples
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models
Introduction Definitions General examples Specific examples Literatura
Reaction-diffusion equations II
Linear conjecture: as long as nonlinearity in local growthrate is decelerating (f ′(logN) ≤ 0), asymptotic wave speed isthe same as in the linear case43
Allee effects (cf. backward bifurcation), interaction withheterogeneity: pinninginteractions among stochasticity and nonlinearity24;25
heterogeneity31
Boundary/edge effects3
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models
Introduction Definitions General examples Specific examples Literatura
Integro-diff* models
Nonlocal deterministic models in continuous space
Relax assumption of local dispersal
Dispersal kernel K (x , y) (usually via jumps)27;42
e.g.∂I (x)
∂t= βS(x)
∫ΩK (x, y)I (y) dy − γI (x)
stable wave speed ↔ K has exponentially bounded tails(moment-generating function exists); otherwise accelerates
discrete (integrodifference) or continuous (integrodifferential)time
simpler: small fraction of global dispersal
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models
Introduction Definitions General examples Specific examples Literatura
Lattice models
discrete (but contiguous) space, usually stochastic and local
cellular automata/interacting particle system
square/hexagonal lattice
incorporate discreteness, stochasticity computationallystraightforward
probability theory8
physics/percolation literature, self-organized criticality etc.
closed-form quantitative solutions difficult
nonlocality with realistic neighbourhoods?4
alternative: irregular lattice connecting neighboring patches26
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models
Introduction Definitions General examples Specific examples Literatura
Approximation techniques: Correlation/moment equations
approximate via local neighbourhood configuration
on patches16
on square lattices: pair approximation15;41
on networks: triples vs. triangles18;32;33
in continuous space: correlation models2;30;32
Challenges
boundaries/finite domains11
maintaining discreteness (extinction dynamics)rigor
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models
Introduction Definitions General examples Specific examples Literatura
Heterogeneity
endogenous sampling variability in discrete/stochastic models
spatial (static) vs temporal (global) vs spatiotemporal
effects on rate of invasion in (R-D models, spatial38);(integrodifference equations, temporal29): geometric mean.more complex interactions in other models5
effects on different parameters(density of hosts; contact rates; susceptibility; movement rateor distance . . . )
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models
Introduction Definitions General examples Specific examples Literatura
Large-scale simulation6;39
Abandon analytical tractability for realism
Restricted by computational cost
Many parameters10
Fill in contact structures from census data, transportnetworks, etc.37
Validation?
Propagation of uncertainty?
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models
Introduction Definitions General examples Specific examples Literatura
Statistical approaches
Data extremely heterogeneous;rarely have direct information about contact
Dynamic spatial point processes
Hierarchical Bayesian models14: blurring the boundary (butstill mostly static, or correlation-based)
various MCMC-based approaches9
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models
Introduction Definitions General examples Specific examples Literatura
Outline
1 Introduction
2 Definitions
3 General examples and issues
4 Specific examples
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models
Introduction Definitions General examples Specific examples Literatura
Example: raccoon rabies26;35;36
Spread of raccoon rabies innortheastern US
Data on first reported date ofrabies per county
Discrete space (countynetwork), discrete time,stochastic, binary state
Local (diffusion to neighbours)plus long-distance dispersal
Incorporation of boundaries,barriers (rivers, forests)
Practical rather than analytical(but: optimal control28)
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models
Introduction Definitions General examples Specific examples Literatura
Example: UK 2001 Foot and mouth disease virus12;17;20;40
UK FMDV epidemic: decisionsabout optimal (spatial) controlpolicies
Three models17: non-spatial,integrodifference (day-by-day),complex simulation
later development of momentapproximations for deeperunderstanding32
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models
Introduction Definitions General examples Specific examples Literatura
Challenges
When do differences in microscopic assumptions havemacroscopic consequences?
Separation of space/time scales: what is “local”?
Wave (spread/invasion) vs mosaic (endemic) processes
R0 in a spatial context: exponential vs quadratic growth
Bridging the gap between analytical and realistic models:what else should we be doing?
What about genetics34?
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models
Introduction Definitions General examples Specific examples Literatura
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Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
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Introduction Definitions General examples Specific examples Literatura
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Introduction Definitions General examples Specific examples Literatura
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Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models
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