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 2011
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models

2. 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

3. 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

4. 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

5. Introduction Definitions General examples Specific examples Literatura
Goals 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 data
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models

6. Introduction Definitions General examples Specific examples Literatura
Scope
“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 Biology
Spatial models

7. 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

8. Introduction Definitions General examples Specific examples Literatura
Model 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 Biology
Spatial models

9. Introduction Definitions General examples Specific examples Literatura
Trivial 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 else
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models

10. 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 stochasticity 13;19
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models

11. 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

12. 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 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 )): contiguity
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models

13. Introduction Definitions General examples Specific examples Literatura
Contiguous 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/PDE
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models

14. Introduction Definitions General examples Specific examples Literatura
Reaction-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 models
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models

15. 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

16. Introduction Definitions General examples Specific examples Literatura
Reaction-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 3
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models

17. 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)
= 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 dispersal
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models

18. 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 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 26
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models

19. Introduction Definitions General examples Specific examples Literatura
Approximation 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)
rigor
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models

20. 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, 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 Biology
Spatial models

21. Introduction Definitions General examples Specific examples Literatura
Large-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 Biology
Spatial models

22. Introduction Definitions General examples Specific examples Literatura
Statistical 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 9
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models

23. 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

24. Introduction Definitions General examples Specific examples Literatura
Example: 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 Biology
Spatial models

25. Introduction Definitions General examples Specific examples Literatura
Example: 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 32
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Spatial models

26. Introduction Definitions General examples Specific examples Literatura
Challenges
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 Biology
Spatial models

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 Biology
Spatial models

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 Biology
Spatial models

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 Biology
Spatial models