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# MBI intro to spatial models

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### MBI intro to spatial models

1. 1. Introduction Deﬁnitions General examples Speciﬁc examples Literatura Overview of spatial models in epidemiology Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology 10 October 2011Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologySpatial models
2. 2. Introduction Deﬁnitions General examples Speciﬁc examples LiteraturaOutline 1 Introduction 2 Deﬁnitions 3 General examples and issues 4 Speciﬁc examplesBen Bolker McMaster University Departments of Mathematics & Statistics and BiologySpatial models
3. 3. Introduction Deﬁnitions General examples Speciﬁc examples LiteraturaOutline 1 Introduction 2 Deﬁnitions 3 General examples and issues 4 Speciﬁc examplesBen Bolker McMaster University Departments of Mathematics & Statistics and BiologySpatial models
4. 4. Introduction Deﬁnitions General examples Speciﬁc examples LiteraturaOverview 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 BiologySpatial models
5. 5. Introduction Deﬁnitions General examples Speciﬁc examples LiteraturaGoals of modeling Why model space, and how? Implicit 7 vs. explicit spatial problems Model-building tradeoﬀs 22;23 : Realism Computational cost Analytical tractability Connections with dataBen Bolker McMaster University Departments of Mathematics & Statistics and BiologySpatial models
6. 6. Introduction Deﬁnitions General examples Speciﬁc examples LiteraturaScope “Look, old boy,” said the machine, “if I could do everything starting with n in every possible language, I’d be a Machine That Could Do Everything in the Whole Alphabet . . . ”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 BiologySpatial models
7. 7. Introduction Deﬁnitions General examples Speciﬁc examples LiteraturaOutline 1 Introduction 2 Deﬁnitions 3 General examples and issues 4 Speciﬁc examplesBen Bolker McMaster University Departments of Mathematics & Statistics and BiologySpatial models
8. 8. Introduction Deﬁnitions General examples Speciﬁc examples LiteraturaModel properties Time discrete vs continuous Space discrete (patch) vs discrete (contiguous) vs continuous 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 BiologySpatial models
9. 9. Introduction Deﬁnitions General examples Speciﬁc examples LiteraturaTrivial models No connections, just (exogenous) variability in the environment “Space is what keeps everything from happening in the same place” Very practical, if exogenous heterogeneity swamps everything elseBen Bolker McMaster University Departments of Mathematics & Statistics and BiologySpatial models
10. 10. Introduction Deﬁnitions General examples Speciﬁc examples LiteraturaNon-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 stochasticity 13;19Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologySpatial models
11. 11. Introduction Deﬁnitions General examples Speciﬁc examples LiteraturaNetwork 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 BiologySpatial models
12. 12. Introduction Deﬁnitions General examples Speciﬁc examples LiteraturaNetworks of patches Collapse local groups of nodes into patches or populations Patch-occupancy: incidence function models Gravity models 9;44 Often matches the scale of data: cases per region Distinguish “truly” spatial models: dimensionality? i.e. (number of neighbors within r ) ∼ power law (∝ r D rather than exp(r )): contiguityBen Bolker McMaster University Departments of Mathematics & Statistics and BiologySpatial models
13. 13. Introduction Deﬁnitions General examples Speciﬁc examples LiteraturaContiguous models: bestiary Space Time Populations Random Model Disc Disc Disc deterministic cellular automaton Disc Disc Disc stochastic stochastic CA Disc Disc Cont either coupled-map lattice Disc Cont Disc stochastic interacting particle system ≈ pair approximation Cont Disc Cont either integrodiﬀerence equation Cont either Disc stochastic spatial point process ≈ spatial moment equations Cont Cont Cont deterministic integrodiﬀerential, partial diﬀerential equation (reaction-diﬀusion equation) Cont Cont Cont stochastic stochastic IDE/PDEBen Bolker McMaster University Departments of Mathematics & Statistics and BiologySpatial models
14. 14. Introduction Deﬁnitions General examples Speciﬁc examples LiteraturaReaction-diﬀusion equations ∂S = −βSI + DS ∆S ∂t ∂I = βSI + DI ∆I − γI ∂t Analyze by ﬁnding asymptotic wave speed of traveling-wave solutions Details matter: Is ∆S = ∆I ? Is contact local or distributed (→ ∆I term in contact rate) 27 ? Simplest model → criticisms [e.g. “atto-fox” problems 1 , eﬀects of long-distance dispersal] Limit of many other modelsBen Bolker McMaster University Departments of Mathematics & Statistics and BiologySpatial models
15. 15. Introduction Deﬁnitions General examples Speciﬁc examples LiteraturaOutline 1 Introduction 2 Deﬁnitions 3 General examples and issues 4 Speciﬁc examplesBen Bolker McMaster University Departments of Mathematics & Statistics and BiologySpatial models
16. 16. Introduction Deﬁnitions General examples Speciﬁc examples LiteraturaReaction-diﬀusion equations II Linear conjecture: as long as nonlinearity in local growth rate is decelerating (f (log N) ≤ 0), asymptotic wave speed is the same as in the linear case 43 Allee eﬀects (cf. backward bifurcation), interaction with heterogeneity: pinning interactions among stochasticity and nonlinearity 24;25 heterogeneity 31 Boundary/edge eﬀects 3Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologySpatial models
17. 17. Introduction Deﬁnitions General examples Speciﬁc examples LiteraturaIntegro-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) = βS(x) K (x, y)I (y) dy − γI (x) ∂t Ω 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 dispersalBen Bolker McMaster University Departments of Mathematics & Statistics and BiologySpatial models
18. 18. Introduction Deﬁnitions General examples Speciﬁc examples LiteraturaLattice models discrete (but contiguous) space, usually stochastic and local cellular automata/interacting particle system square/hexagonal lattice incorporate discreteness, stochasticity computationally straightforward probability theory 8 physics/percolation literature, self-organized criticality etc. closed-form quantitative solutions diﬃcult nonlocality with realistic neighbourhoods? 4 alternative: irregular lattice connecting neighboring patches 26Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologySpatial models
19. 19. Introduction Deﬁnitions General examples Speciﬁc examples LiteraturaApproximation techniques: Correlation/moment equations approximate via local neighbourhood conﬁguration on patches 16 on square lattices: pair approximation 15;41 on networks: triples vs. triangles 18;32;33 in continuous space: correlation models 2;30;32 Challenges boundaries/ﬁnite domains 11 maintaining discreteness (extinction dynamics) rigorBen Bolker McMaster University Departments of Mathematics & Statistics and BiologySpatial models
20. 20. Introduction Deﬁnitions General examples Speciﬁc examples LiteraturaHeterogeneity endogenous sampling variability in discrete/stochastic models spatial (static) vs temporal (global) vs spatiotemporal eﬀects on rate of invasion in (R-D models, spatial 38 ); (integrodiﬀerence equations, temporal 29 ): geometric mean. more complex interactions in other models 5 eﬀects on diﬀerent parameters (density of hosts; contact rates; susceptibility; movement rate or distance . . . )Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologySpatial models
21. 21. Introduction Deﬁnitions General examples Speciﬁc examples LiteraturaLarge-scale simulation 6;39 Abandon analytical tractability for realism Restricted by computational cost Many parameters 10 Fill in contact structures from census data, transport networks, etc. 37 Validation? Propagation of uncertainty?Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologySpatial models
22. 22. Introduction Deﬁnitions General examples Speciﬁc examples LiteraturaStatistical approaches Data extremely heterogeneous; rarely have direct information about contact Dynamic spatial point processes Hierarchical Bayesian models 14 : blurring the boundary (but still mostly static, or correlation-based) various MCMC-based approaches 9Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologySpatial models
23. 23. Introduction Deﬁnitions General examples Speciﬁc examples LiteraturaOutline 1 Introduction 2 Deﬁnitions 3 General examples and issues 4 Speciﬁc examplesBen Bolker McMaster University Departments of Mathematics & Statistics and BiologySpatial models
24. 24. Introduction Deﬁnitions General examples Speciﬁc examples LiteraturaExample: raccoon rabies 26;35;36 Spread of raccoon rabies in northeastern US Data on ﬁrst reported date of rabies per county Discrete space (county network), 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 control 28 )Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologySpatial models
25. 25. Introduction Deﬁnitions General examples Speciﬁc examples LiteraturaExample: UK 2001 Foot and mouth disease virus 12;17;20;40 UK FMDV epidemic: decisions about optimal (spatial) control policies Three models 17 : non-spatial, integrodiﬀerence (day-by-day), complex simulation later development of moment approximations for deeper understanding 32Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologySpatial models
26. 26. Introduction Deﬁnitions General examples Speciﬁc examples LiteraturaChallenges When do diﬀerences in microscopic assumptions have macroscopic 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 genetics 34 ?Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologySpatial models
27. 27. Introduction Deﬁnitions General examples Speciﬁc examples Literatura [1] Boerlijst MC & van Ballegooijen WM, Dec. 2010. 8(55):233 –243. doi:10.1098/rsif.2010.0216. PLoS Computational Biology, 6:e1001030. ISSN URL http://rsif.royalsocietypublishing. 1553-7358. org/content/8/55/233.abstract. doi:10.1371/journal.pcbi.1001030. [10] Elderd BD, Dukic VM, & Dwyer G, Oct. 2006. [2] Brown DH & Bolker BM, 2004. Bulletin of Proceedings of the National Academy of Sciences, Mathematical Biology, 66:341–371. 103(42):15693–15697. doi:10.1016/j.bulm.2003.08.006. doi:10.1073/pnas.0600816103. URL [3] Cantrell RS, Cosner C, & Fagan WF, Feb. 2001. http://www.pnas.org/cgi/content/abstract/ Journal of Mathematical Biology, 42:95–119. 103/42/15693. ISSN 0303-6812, 1432-1416. [11] Ellner SP, Sasaki A et al., 1998. Journal of doi:10.1007/s002850000064. Mathematical Biology, 36(5):469–484. [4] Chesson P & Lee CT, Jun. 2005. Theoretical [12] Ferguson NM, Donnelly CA, & Anderson RM, Population Biology, 67(4):241–256. ISSN May 2001. Science, 292(5519):1155–1160. 0040-5809. doi:10.1016/j.tpb.2004.12.002. doi:10.1126/science.1061020. URL [5] Dewhirst S & Lutscher F, May 2009. Ecology, http://www.sciencemag.org/cgi/content/ 90:1338–1345. ISSN 0012-9658. abstract/292/5519/1155. doi:10.1890/08-0115.1. URL [13] Grenfell BT & Bolker BM, 1998. Ecology Letters, http://www.esajournals.org/doi/abs/10. 1(1):63–70. 1890/08-0115.1?journalCode=ecol. [14] Hu W, Clements A et al., 2010. The American [6] Dimitrov NB, Goll S et al., Jan. 2011. PLoS Journal of Tropical Medicine and Hygiene, ONE, 6:e16094. ISSN 1932-6203. 83(3):722 –728. doi:10.1371/journal.pone.0016094. doi:10.4269/ajtmh.2010.09-0551. URL [7] Durrett R & Levin S, 1994. Theoretical http: Population Biology, 46(3):363–394. //www.ajtmh.org/content/83/3/722.abstract. [8] Durrett R & Neuhauser C, 1991. Annals of [15] Kamo M & Boots M, 2006. Evolutionary Ecology Applied Probability, 1:189–206. Research, 8(7):1333–1347. [9] Eggo RM, Cauchemez S, & Ferguson NM, Feb. [16] Keeling MJ, Sep. 2000. Journal of Animal 2011. Journal of The Royal Society Interface, Ecology, 69(5):725–736.Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologySpatial models
28. 28. Introduction Deﬁnitions General examples Speciﬁc examples Literatura [17] Keeling MJ, Jun. 2005. Proceedings: Biological [28] Miller Neilan R & Lenhart S, Jun. 2011. Journal Sciences, 272(1569):1195–1202. ISSN 0962-8452. of Mathematical Analysis and Applications, URL http://www.jstor.org/stable/30047668. 378(2):603–619. ISSN 0022-247X. [18] Keeling MJ, Rand DA, & Morris AJ, aug 22 doi:10.1016/j.jmaa.2010.12.035. URL 1997. Proceedings of the Royal Society B, http://www.sciencedirect.com/science/ 264(1385):1149–1156. article/pii/S0022247X10010528. [19] Keeling MJ, Wilson HB, & Pacala SW, 2002. [29] Neubert MG, Kot M, & Lewis MA, Aug. 2000. The American Naturalist, 159(1):57–80. Proceedings of the Royal Society B: Biological [20] Keeling MJ, Woolhouse MEJ et al., Oct. 2001. Sciences, 267(1453):1603–1610. ISSN 0962-8452. Science, 294(5543):813–817. ISSN 0036-8075. PMID: 11467422 PMCID: 1690727. URL http://www.jstor.org/stable/3085067. [30] Ovaskainen O & Cornell SJ, Aug. 2006. [21] Lem S, 1985. The Cyberiad. Harvest/HBJ Proceedings of the National Academy of Sciences Books. URL http://english.lem.pl/works/ of the USA, 103(34):12781–12786. ISSN novels/the-cyberiad/ 0027-8424. 57-a-look-inside-the-cyberiad. Original [31] Pachepsky E & Levine JM, Jan. 2011. The Polish edition 1965. American Naturalist, 177(1):18–28. ISSN [22] Levins R, 1966. American Scientist, 54:421–431. 1537-5323. doi:10.1086/657438. URL http: [23] Levins R, 1993. Quarterly Review of Biology, //www.ncbi.nlm.nih.gov/pubmed/21117949. 68(4):547–555. PMID: 21117949. [24] Lewis MA, Nov. 2000. Journal of Mathematical [32] Parham PE, Singh BK, & Ferguson NM, May Biology, 41(5):430–454. 2008. Theoretical Population Biology, [25] Lewis MA & Pacala S, Nov. 2000. Journal of 73(3):349–368. ISSN 0040-5809. Mathematical Biology, 41(5):387–429. doi:10.1016/j.tpb.2007.12.010. [26] Lucey BT, Russell CA et al., 2002. Vector Borne [33] Rand DA, Keeling M, & Wilson HB, jan 23 1995. and Zoonotic Diseases, 2(2):77–86. Proceedings of the Royal Society B, 259(1354):9. [27] Medlock J & Kot M, Aug. 2003. Mathematical [34] Real LA, Russell C et al., 2005. Journal of Biosciences, 184(2):201–222. ISSN 0025-5564. Heredity, 96(3):253–260. doi:10.1016/S0025-5564(03)00041-5. URL [35] Russell CA, Smith DL et al., 2004. Proceedings http://www.sciencedirect.com/science/ of the Royal Society B: Biological Sciences, article/pii/S0025556403000415. 271(1534):21–25.Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologySpatial models
29. 29. 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. Biological invasions : theory and practice. Oxford University Press, New York. [39] Smieszek T, Balmer M et al., 2011. BMC Infectious 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: Biological Sciences, 275(1641):1459 –1468. doi:10.1098/rspb.2008.0006. URL http://rspb.royalsocietypublishing.org/ content/275/1641/1459.abstract. [41] van Baalen M, 2000. In U Dieckmann, R Law, & JAJ Metz, eds., The Geometry of Ecological Interactions: Simplifying Spatial Complexity, Cambridge Studies in Adaptive Dynamics, chap. 19, pp. 359–387. Cambridge University Press, 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 BiologySpatial models