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

[1] Boerlijst MC & van Ballegooijen WM, Dec. 2010.PLoS Computational Biology, 6:e1001030. ISSN1553-7358.doi:10.1371/journal.pcbi.1001030.

[2] Brown DH & Bolker BM, 2004. Bulletin ofMathematical Biology, 66:341–371.doi:10.1016/j.bulm.2003.08.006.

[3] Cantrell RS, Cosner C, & Fagan WF, Feb. 2001.Journal of Mathematical Biology, 42:95–119.ISSN 0303-6812, 1432-1416.doi:10.1007/s002850000064.

[4] Chesson P & Lee CT, Jun. 2005. TheoreticalPopulation Biology, 67(4):241–256. ISSN0040-5809. doi:10.1016/j.tpb.2004.12.002.

[5] Dewhirst S & Lutscher F, May 2009. Ecology,90:1338–1345. ISSN 0012-9658.doi:10.1890/08-0115.1. URLhttp://www.esajournals.org/doi/abs/10.

1890/08-0115.1?journalCode=ecol.

[6] Dimitrov NB, Goll S et al., Jan. 2011. PLoSONE, 6:e16094. ISSN 1932-6203.doi:10.1371/journal.pone.0016094.

[7] Durrett R & Levin S, 1994. TheoreticalPopulation Biology, 46(3):363–394.

[8] Durrett R & Neuhauser C, 1991. Annals ofApplied Probability, 1:189–206.

[9] Eggo RM, Cauchemez S, & Ferguson NM, Feb.2011. Journal of The Royal Society Interface,

8(55):233 –243. doi:10.1098/rsif.2010.0216.URL http://rsif.royalsocietypublishing.

org/content/8/55/233.abstract.

[10] Elderd BD, Dukic VM, & Dwyer G, Oct. 2006.Proceedings of the National Academy of Sciences,103(42):15693–15697.doi:10.1073/pnas.0600816103. URLhttp://www.pnas.org/cgi/content/abstract/

103/42/15693.

[11] Ellner SP, Sasaki A et al., 1998. Journal ofMathematical Biology, 36(5):469–484.

[12] Ferguson NM, Donnelly CA, & Anderson RM,May 2001. Science, 292(5519):1155–1160.doi:10.1126/science.1061020. URLhttp://www.sciencemag.org/cgi/content/

abstract/292/5519/1155.

[13] Grenfell BT & Bolker BM, 1998. Ecology Letters,1(1):63–70.

[14] Hu W, Clements A et al., 2010. The AmericanJournal of Tropical Medicine and Hygiene,83(3):722 –728.doi:10.4269/ajtmh.2010.09-0551. URLhttp:

//www.ajtmh.org/content/83/3/722.abstract.

[15] Kamo M & Boots M, 2006. Evolutionary EcologyResearch, 8(7):1333–1347.

[16] Keeling MJ, Sep. 2000. Journal of AnimalEcology, 69(5):725–736.

Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology

Spatial models

Introduction Definitions General examples Specific examples Literatura

[17] Keeling MJ, Jun. 2005. Proceedings: BiologicalSciences, 272(1569):1195–1202. ISSN 0962-8452.URL http://www.jstor.org/stable/30047668.

[18] Keeling MJ, Rand DA, & Morris AJ, aug 221997. Proceedings of the Royal Society B,264(1385):1149–1156.

[19] Keeling MJ, Wilson HB, & Pacala SW, 2002.The American Naturalist, 159(1):57–80.

[20] Keeling MJ, Woolhouse MEJ et al., Oct. 2001.Science, 294(5543):813–817. ISSN 0036-8075.URL http://www.jstor.org/stable/3085067.

[21] Lem S, 1985. The Cyberiad. Harvest/HBJBooks. URL http://english.lem.pl/works/

novels/the-cyberiad/

57-a-look-inside-the-cyberiad. OriginalPolish edition 1965.

[22] Levins R, 1966. American Scientist, 54:421–431.

[23] Levins R, 1993. Quarterly Review of Biology,68(4):547–555.

[24] Lewis MA, Nov. 2000. Journal of MathematicalBiology, 41(5):430–454.

[25] Lewis MA & Pacala S, Nov. 2000. Journal ofMathematical Biology, 41(5):387–429.

[26] Lucey BT, Russell CA et al., 2002. Vector Borneand Zoonotic Diseases, 2(2):77–86.

[27] Medlock J & Kot M, Aug. 2003. MathematicalBiosciences, 184(2):201–222. ISSN 0025-5564.doi:10.1016/S0025-5564(03)00041-5. URLhttp://www.sciencedirect.com/science/

article/pii/S0025556403000415.

[28] Miller Neilan R & Lenhart S, Jun. 2011. Journalof Mathematical Analysis and Applications,378(2):603–619. ISSN 0022-247X.doi:10.1016/j.jmaa.2010.12.035. URLhttp://www.sciencedirect.com/science/

article/pii/S0022247X10010528.

[29] Neubert MG, Kot M, & Lewis MA, Aug. 2000.Proceedings of the Royal Society B: BiologicalSciences, 267(1453):1603–1610. ISSN 0962-8452.PMID: 11467422 PMCID: 1690727.

[30] Ovaskainen O & Cornell SJ, Aug. 2006.Proceedings of the National Academy of Sciencesof the USA, 103(34):12781–12786. ISSN0027-8424.

[31] Pachepsky E & Levine JM, Jan. 2011. TheAmerican Naturalist, 177(1):18–28. ISSN1537-5323. doi:10.1086/657438. URL http:

//www.ncbi.nlm.nih.gov/pubmed/21117949.PMID: 21117949.

[32] Parham PE, Singh BK, & Ferguson NM, May2008. Theoretical Population Biology,73(3):349–368. ISSN 0040-5809.doi:10.1016/j.tpb.2007.12.010.

[33] Rand DA, Keeling M, & Wilson HB, jan 23 1995.Proceedings of the Royal Society B, 259(1354):9.

[34] Real LA, Russell C et al., 2005. Journal ofHeredity, 96(3):253–260.

[35] Russell CA, Smith DL et al., 2004. Proceedingsof the Royal Society B: Biological Sciences,271(1534):21–25.

Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology

Spatial models

Introduction Definitions General examples Specific examples Literatura

[36] Russell CA, Smith DL et al., 2005. PLoS Biology,3(3):382–388.

[37] Rvachev LA & Longini Jr IM, 1985.Mathematical Biosciences, 75:3–22.

[38] Shigesada N & Kawasaki K, 1997. Biologicalinvasions : theory and practice. Oxford UniversityPress, New York.

[39] Smieszek T, Balmer M et al., 2011. BMCInfectious Diseases, 11(1):115. ISSN 1471-2334.doi:10.1186/1471-2334-11-115.

[40] Tildesley MJ, Deardon R et al., Jun. 2008.Proceedings of the Royal Society B: BiologicalSciences, 275(1641):1459 –1468.doi:10.1098/rspb.2008.0006. URLhttp://rspb.royalsocietypublishing.org/

content/275/1641/1459.abstract.

[41] van Baalen M, 2000. In U Dieckmann, R Law, &JAJ Metz, eds., The Geometry of EcologicalInteractions: Simplifying Spatial Complexity,Cambridge Studies in Adaptive Dynamics,chap. 19, pp. 359–387. Cambridge UniversityPress, Cambridge, UK.

[42] van den Bosch F, Metz JAJ, & Diekmann O,1990. J. Math. Biol., 28:529–565.

[43] Weinberger HF, Lewis MA, & Li B, 2002. J.Math. Biol., 45:183–218.

[44] Xia Y, Bjørnstad ON, & Grenfell BT, 2004.American Naturalist, 164(2):267–281.doi:10.1086/422341.

Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology

Spatial models