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- 1. RIVER NETWORKS AS ECOLOGICAL CORRIDORS FOR SPECIES POPULATIONS AND WATER-BORNE DISEASE Andrea Rinaldo ! ! Laboratory of Ecohydrology ENAC/IIE/ECHO Ecole Polytechnique Fédérale Lausanne (EPFL) CH Dipartimento ICEA Università di Padova
- 2. PLAN tools: reactive transport on networks nodes (reactions) + branches (transport) metacommunity & individual-based models ! ! modeling migration fronts & human range expansions ! spreading of water-borne disease hydrologic controls on cholera epidemics ! invasion of vegetation or freshwater fish species along fluvial corridors ! hydrochory & biodiversity
- 3. questions of scientific & societal relevance (population migrations, loss of biodiversity, hydrologic controls on the spreading of Cholera, meta-history) explore two critical characteristics (directional dispersal & network structure as environmental matrix) for spreading of organisms, species & water-borne disease
- 4. Muneepeerakul et al., JTB, 2007
- 5. Rodriguet-Iturbe et al., PNAS, 2012
- 6. Carrara et al., PNAS, 2012
- 7. Carrara et al., PNAS, 2012
- 8. Carrara et al., Am. Nat., 2014
- 9. TOOLS - about the progress (recently) made on how to decode the mathematical language of the geometry of Nature
- 10. DTM - GRID (Planar view) DTM – GRID format (Perspective – North towards bottom)
- 11. remarkable capabilities to remotely acquire & objectively manipulate accurate descriptions of natural landforms over several orders of magnitude if I remove the scale bar …consilience… Rodriguez-Iturbe & Rinaldo, Fractal River Basins: Chance and Self-Organization, Cambridge Univ, Press, 2007
- 12. TOOLS
- 13. from O(1) m scales…
- 14. the MMRS
- 15. random-walk drainage basin network (Leopold & Langbein, 1962) & the resistible ascent of the random paradigm !
- 16. Eden growth & self-avoiding random walks Rigon et al., WRR, 1998 !
- 17. Scheidegger’s construction is exactly solved for key geometric & topologic features Huber, J Stat Phys, 1991; Takayasu et al., 1991
- 18. optimal channel networks Rodriguez-Iturbe et al., WRR, 1992 a,b; Rinaldo et al. WRR, 1992
- 19. Rigon et al., WRR, 1997
- 20. Rinaldo et al., PNAS, in press
- 21. Peano – exact results & subtleties (multifractality binomial multiplicative process & width functions) Marani et al., WRR, 1991; Colaiori et al., PRE, 2003
- 22. TOOLS 1 - comb-like structures, diffusion processes & CTRW framework in terms of density of particles ρ(x,t) l A B
- 23. from traditional unbiased random-walks to general cases ! heterogeneous distributions of spacing, Δx & length of the comb leg, l
- 24. tools - 2 A B A B delay ~ reactions, lifetime distributions
- 25. network → oriented graph made by nodes & edges models of reactive transport COUPLED MODELS NODAL REACTIONS TRANSPORT MODELS BETWEEN NODES individuals, species, populations (metacommunities)
- 26. TOOLS 2 - reactive continuous time random walk Φ(t) Ψ ( x, t ) pdf of jump & waiting time x diffusion f (ρ ) reaction ρ (x,0) ? ∞ +∞ t −∞ φ (t ) = ∫ dt ' ∫ dx Ψ ( x, t ' )
- 27. transport + possibly reactions or interactions a master equation – if we consider many realizations of independent processes (large number of noninteracting propagules) ρ(i,t) is proportional to the number of propagules in i at time t
- 28. hydrochory ! ! human-range expansion, population migration
- 29. quantitative model of US colonization 19th century & transport on fractal networks Campos et al., Theor. Pop. Biol., 2006 ! ! the idea that landscape heterogeneities & need for water forced settling about fluvial courses ! ! Ammerman & Cavalli Sforza, The Neolithic transition and the Genetics of population in Europe, Princeton Univ. Press 1984 exact reaction-diffusion model (logistic with rate parameter a for population growth) ! ! ! !
- 30. a little background on Fisher’s fronts
- 31. phase plane → the sign of the eigenvalues of an appropriate Jacobian matrix determines the nature of the equilibria ! (e.g. Murray, 1993)
- 32. a few further mathematical details
- 33. the Hamilton-Jacobi formalism the network slows the front! you waste time trapped in the pockets
- 34. Peano’s network initial cond r=1 f a reaction logistic growth at every node transport f ( ρ ) = aρ (1 − ρ ) 0 at every timestep each particle moves towards a nearest neighbour w.p. p= 1 / # nn 1 ρ
- 35. SIMULAZIONI
- 36. P+=0.5 a=0.5
- 37. v speed of front [L/T] Campos et al., Theor. Pop. Biol., 2006; Bertuzzo et al., WRR, 2007 isotropic migration – Fisher’s model v = 2√aD Murray, 1988 Peano (exact) Peano (numerical) a (logistic growth)
- 38. Relative frequency (%) geometric constraints imposed by the network (topology & geometry) impose strong corrections to the speed of propagation of migratory fronts Campos et al., Theor. Pop. Biol., 2006; Bertuzzo et al., WRR, 2007
- 39. what is a node? strong hydrologic controls ! ! !
- 40. Giometto et al., PNAS, 2013
- 41. Fisher-‐Kolmogorov Equation $ ρ' ∂ρ ∂2 ρ = D 2 + rρ & 1− ) ∂t ∂x % K( What about variability?
- 42. $ ρ' ∂ρ ∂2 ρ = D 2 + rρ & 1− ) +σ ρ η ∂t ∂x % K( η is a δ-‐correlated gaussian white noise Itô stochastic calculus ML estimates for r,K, σ $ ρ' ∂ρ = rρ & 1− ) +σ ρ η ∂t % K( Transitional probability densities are computed by numerical integration of the related Fokker-‐Planck equation.
- 43. Demographic stochasticity $ ρ' ∂ρ ∂2 ρ = D 2 + rρ & 1− ) +σ ρ η ∂t ∂x % K( η is a δ-‐correlated gaussian white noise Itô stochastic calculus ML estimates for r,K, σ $ ρ' ∂ρ = rρ & 1− ) +σ ρ η ∂t % K( Transitional probability densities are computed by numerical integration of the related Fokker-‐Planck equation.
- 44. Front variability Giometto et al., PNAS, 2013
- 45. Take-‐home message • Fisher-‐Kolmogorov equation correctly predicts the mean features of dispersal ! • The observed variability is explained by demographic stochasticity Link between scales Giometto et al., PNAS, 2013
- 46. Zebra Mussel Dreissena polymorpha larval stages transported along the fluvial network 1988 1989 1990 1991 data: Nonindigenous Aquatic species program USGS 1992 1993 1994 1995
- 47. Mari et al., in review, 2007 local age-growth model (4 stages) ! larval production ! larval transport (network) Zebra Mussel
- 48. Zebra Mussel
- 49. Mari et al., WRR, 2011; Mari et al., Ecol. Lett., 2014
- 50. river biogeography ! spatial distribution of biodiversity within a biota ! riparian vegetation fluvial fauna freshwater fish
- 51. neutral metacommunity model metacommunity model every link is a community of organisms & internal implicit spatial dynamics Explicit spatial dynamics among different communities the neutral assumption all species are equivalent (equal fertility, mortality, dispersion Kernel) the probability with which a propagule colonizes a site depends only on its relative abundance patterns of biodiversity emerge because of ecological drift Hubbel, 2001
- 52. neutral metacommunity model the model at each timestep an organism is randonly chosen & killed w.p. ν it is substituted by a species non existing (prob of speciation/immigration) w.p. 1-ν the site is colonized by an organism present in the system Pij = (1 − v) K ij H j N ∑K ik Hk k =1 H j :habitat capacity link i K ij :dispersal kernel run up to steady state
- 53. river biogeography global properties γ-diversity: total # of species patterns of abundance # of species preston plot 20 21 22 23 24 25 26 27 28 29 210 211 212 abundance
- 54. river biogeography LOCAL PROPERTIES α-diversity number of species at local scale
- 55. river biogeography a) β-diversity b) Jaccard similarity index S ab S ab # common species J ab ( x) = x distance measured along the α a + α b − S ab 4 J ab ( x) = = 0.57 6+5−4 network
- 56. river biogeography geographic range geographic range area occupied by a species ranked species
- 57. Mississippi-Missouri freshwater fauna database presence(absence) of 429 species freshwater fish in 421 subbasins α-diversity, β-diversity, γ-diversity, geographic range fonti USGS, hydrologic data, NatureServe, Bill Fagan’s ecological data
- 58. Mississippi-Missouri freshwater fish α-diversity strong correlation habitat capacity ~ runoff runoff
- 59. distance to outlet Muneeperakul, Bertuzzo, Fagan, Rinaldo, Rodriguez-Iturbe, Nature, 2008
- 60. Muneeperakul et al., Nature, May 8 2008
- 61. constant habitat capacity per DTA hydrologic controls
- 62. model weak interspecific interactions & weak/strong formulations of the neutral model
- 63. patterns -- weak or strong impliations of neutrality? comparison between geographic ranges of individual species: a) data b) results from the neutral metacommunity model (after matching procedure)
- 64. Bertuzzo et al., submitted, 2008 equiprobability map – ratio between the number of common species and the number of species in the central DTA
- 65. environmental resistance R50 for data & the model
- 66. is topology reflected in the spatial organization of the species? ! species range & maximum drainage area – the max area experienced by a species is that in blue color, range is cross-hatched red ! containment effect favors colonization !
- 67. Corridors for pathogens of waterborne disease ! ! Of cholera epidemics & hydrology
- 68. Haiti (2010-2011)
- 69. Piarroux et al., Emerging Infectious Diseases, 2011
- 70. no elementary correlation between population and cholera cases
- 71. Mari et al., J Roy Soc Interface, 2011
- 72. Codeco, JID, 2001; Pascual et al, PLOS, 2002; Chao et al, PNAS, 2011 continuous SIR model αI µI infected I γI persons p I W β B S K+B vibrios vibrios/m3 B recovered R µB B µR H: total human population at disease free equilibrium µ: natality and mortality rate (day-1) β: rate of exposure to contaminated water (day-1) K: concentration of V. cholerae in water that yields 50% chance of catching cholera (cells/m3) susceptibles S µS µH α: mortality rate due to cholera(day-1) γ : rate at which people recover from cholera (day-1) µB:death rate of V. cholerae in the aquatic environment (day-1) p : infected rate of production of V. cholerae (cells day-1 person-1) W: volume of water reservoir (m3)
- 73. Chao et al., PNAS, 2011
- 74. the class of SIB models Capasso et al, 1979; Codeco, JID, 2001
- 75. person ! SIR model for the temporal & spatial evolution of water-transmitted disease revisited → network susceptibles I(t) infected ! S(t) a few assumptions ! ! 0 50 100 150 200 total population of humans is t [days] unaffected by the disease ! diffusion of infective humans is small w.r. to that of bacteria thus set to zero ! density-dependent reaction terms depend on local susceptibles ! ! Capasso et al, 1979; Codeco, JID, 2001; Pascual et al, PLOS, 2002; Hartley et al, PNAS, 2006
- 76. person susceptibles infected 0 50 100 t [days] 150 200
- 77. nodes are human communities with population H in which the disease can diffuse & grow
- 78. Hydrologic Networks Human-Mobility Network i Pij i Qij Rij j Qij j
- 79. Mari et al., J Roy Soc Interface, 2011
- 80. uniform population 600 500 I(t) infected 400 300 200 100 0 0 50 100 150 200 time [days] 250 t 300 350
- 81. Zipf distribution of population size & self-organization
- 82. Zipf’s distribution of population & secondary peaks of infection 1400 1200 I(t) infected 1000 800 600 400 200 0 0 50 100 150 200 time [days] 250 t 300 350
- 83. spatio-temporal dynamics initial conditions refelecting boundary condition at all the leaves and at the outlet the higher the transport rate, the better the system is approximated by a well-mixed reactor (spatially implicit scheme) Bertuzzo et al., J Roy Soc Interface, 2010
- 84. 10 20 Weekly Cases [103] 25 calibration 20 Cumulative Cases [103] Rainfall [mm/day] 0 500 300 100 prediction Nov 10 Sep 11 prediction 15 10 5 0 Nov 10 Jan 11 Mar 11 May 11 Jul 11 Sep 11 Bertuzzo et al. GRL 2011
- 85. Bertuzzo et et al., GRL, 2011
- 86. θ m σ l ∝B φ D ρ β γ ∝ α −60% −40% −20% 0 20% 40% 60%
- 87. Rainfall [mm/day] 0 10 20 Weekly Cases [103] 25 calibration hindcast 20 15 10 5 0 Nov 10 Jan 11 Mar 11 May 11 Jul 11 Sep 11 Rinaldo et al., PNAS, in press
- 88. 5 Sud−Est 0 Sud 0 5 Nord−Est 5 Nippes 0 0 Weekly Cases [103] 5 5 Nord 0 5 Grande Anse 0 5 Nord−Ouest 5 Centre 0 0 15 Artibonite 15 10 10 5 Ouest 5 0 Jan 11 May 11 Sep 11 0 Jan 11 May 11 Sep 11
- 89. 25 Weekly Cases [103] 20 15 10 5 0 Nov 10 Jan 11 Mar 11 May 11 Jul 11 Sep 11
- 90. effects of rates of loss of acquired immunity (1-5 years) Weekly Cases [103] 40 30 20 10 0 Jan 11 Jul 11 Jan 12 Jul 12 Jan 13 Jul 13 Jan 14 Rinaldo et al., PNAS, in press
- 91. Rainfall [mm/day] 0 5 10 15 Weekly Cases [103] 20 30 20 10 0 Jan 11 Jul 11 Jan 12 Jul 12 Jan 13 Jul 13 Jan 14 Rinaldo et al., PNAS, in press
- 92. recorded cholera cases in Haiti (2010-2013) (normalized) normalized maximum eigenvector Gatto et al, PNAS, 2012; Gatto et al, Am Nat, 2014
- 93. river networks & biodiversity ! tradeoff versus neutral models of the ecology of riparian vegetation Muneepeerakul et al., JTB, 2007 --> Mari et al., Ecol Lett., 2014
- 94. Mari et al., Ecol. Lett., 2014
- 95. Muneepeerakul, Weitz, Levin, Rinaldo, Rodriguez-Iturbe, JTB, 2007 links are essentially patches within a landscape cointaining sites that are occupied by individual plants
- 96. the containment effect: the network structure significantly hinders the dispersal of propagules across subbasins – less sharing of species ! fragmentation increases species richness (both neutral & trade-off communities) (diameters ~ species’ link-scale abundance
- 97. power laws matter alot - hotspots & geomorphology ! indeed a frontier of ecological research Muneepeerakul et al., WRR, 2007
- 98. remote sensing & (much) hydrologic research
- 99. → CONCLUSIONS ! rivers as ecological corridors → containment effects (hydrochory migrations & spreading of epidemics) ! network structure provides strong controls & susceptibility ! e.g. secondary peaks of ‘infections’ or biodiversity hotspots ~ geometric constraints rather than dynamics ! river networks are possibly templates of biodiversity impacts of climate change scenarios on local and regional biodiversity !
- 100. CONCLUSIONS -- 2 ! ecohydrological footprints from rivers as ecological corridors & human mobility for the spreading of epidemic cholera ! network structure(s) provides controls & susceptibility ! from secondary peaks of infections to rainfall prediction ~ it’s all in the water ! rainfall drivers – seasonality, endemicity & impacts of climate change scenarios, water management, sanitation !
- 101. collaborations IGNACIO RODRIGUEZ-ITURBE MARINO GATTO AMOS MARITAN RICCARDO RIGON the ECHO/IIE/ENAC/EPFL Laboratory ENRICO BERTUZZO, LORENZO MARI, SAMIR SUWEIS LORENZO RIGHETTO, FRANCESCO CARRARA SERENA CEOLA, ANDREA GIOMETTO PIERRE QUELOZ, CARA TOBIN, BETTINA SCHAEFLI

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