RIVER NETWORKS AS ECOLOGICAL CORRIDORS FOR SPECIES POPULATIONS AND WATER-BORNE DISEASE

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This is the presentation given by Andrea Rinaldo in Trento for the opening day of the 2014 Doctoral School.

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RIVER NETWORKS AS ECOLOGICAL CORRIDORS FOR SPECIES POPULATIONS AND WATER-BORNE DISEASE

  1. 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. 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. 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. 4. Muneepeerakul et al., JTB, 2007
  5. 5. Rodriguet-Iturbe et al., PNAS, 2012
  6. 6. Carrara et al., PNAS, 2012
  7. 7. Carrara et al., PNAS, 2012
  8. 8. Carrara et al., Am. Nat., 2014
  9. 9. TOOLS - about the progress (recently) made on how to decode the mathematical language of the geometry of Nature
  10. 10. DTM - GRID (Planar view) DTM – GRID format (Perspective – North towards bottom)
  11. 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. 12. TOOLS
  13. 13. from O(1) m scales…
  14. 14. the MMRS
  15. 15. random-walk drainage basin network (Leopold & Langbein, 1962) & the resistible ascent of the random paradigm !
  16. 16. Eden growth & self-avoiding random walks Rigon et al., WRR, 1998 !
  17. 17. Scheidegger’s construction is exactly solved for key geometric & topologic features Huber, J Stat Phys, 1991; Takayasu et al., 1991
  18. 18. optimal channel networks Rodriguez-Iturbe et al., WRR, 1992 a,b; Rinaldo et al. WRR, 1992
  19. 19. Rigon et al., WRR, 1997
  20. 20. Rinaldo et al., PNAS, in press
  21. 21. Peano – exact results & subtleties (multifractality binomial multiplicative process & width functions) Marani et al., WRR, 1991; Colaiori et al., PRE, 2003
  22. 22. TOOLS 1 - comb-like structures, diffusion processes & CTRW framework in terms of density of particles ρ(x,t) l A B
  23. 23. from traditional unbiased random-walks to general cases ! heterogeneous distributions of spacing, Δx & length of the comb leg, l
  24. 24. tools - 2 A B A B delay ~ reactions, lifetime distributions
  25. 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. 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. 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. 28. hydrochory ! ! human-range expansion, population migration
  29. 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. 30. a little background on Fisher’s fronts
  31. 31. phase plane → the sign of the eigenvalues of an appropriate Jacobian matrix determines the nature of the equilibria ! (e.g. Murray, 1993)
  32. 32. a few further mathematical details
  33. 33. the Hamilton-Jacobi formalism the network slows the front! you waste time trapped in the pockets
  34. 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. 35. SIMULAZIONI
  36. 36. P+=0.5 a=0.5
  37. 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. 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. 39. what is a node? strong hydrologic controls ! ! !
  40. 40. Giometto et al., PNAS, 2013
  41. 41. Fisher-­‐Kolmogorov  Equation $ ρ' ∂ρ ∂2 ρ = D 2 + rρ & 1− ) ∂t ∂x % K( What  about  variability?
  42. 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. 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. 44. Front  variability Giometto et al., PNAS, 2013
  45. 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. 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. 47. Mari et al., in review, 2007 local age-growth model (4 stages) ! larval production ! larval transport (network) Zebra Mussel
  48. 48. Zebra Mussel
  49. 49. Mari et al., WRR, 2011; Mari et al., Ecol. Lett., 2014
  50. 50. river biogeography ! spatial distribution of biodiversity within a biota ! riparian vegetation fluvial fauna freshwater fish
  51. 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. 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. 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. 54. river biogeography LOCAL PROPERTIES α-diversity number of species at local scale
  55. 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. 56. river biogeography geographic range geographic range area occupied by a species ranked species
  57. 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. 58. Mississippi-Missouri freshwater fish α-diversity strong correlation habitat capacity ~ runoff runoff
  59. 59. distance to outlet Muneeperakul, Bertuzzo, Fagan, Rinaldo, Rodriguez-Iturbe, Nature, 2008
  60. 60. Muneeperakul et al., Nature, May 8 2008
  61. 61. constant habitat capacity per DTA hydrologic controls
  62. 62. model weak interspecific interactions & weak/strong formulations of the neutral model
  63. 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. 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. 65. environmental resistance R50 for data & the model
  66. 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. 67. Corridors for pathogens of waterborne disease ! ! Of cholera epidemics & hydrology
  68. 68. Haiti (2010-2011)
  69. 69. Piarroux et al., Emerging Infectious Diseases, 2011
  70. 70. no elementary correlation between population and cholera cases
  71. 71. Mari et al., J Roy Soc Interface, 2011
  72. 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. 73. Chao et al., PNAS, 2011
  74. 74. the class of SIB models Capasso et al, 1979; Codeco, JID, 2001
  75. 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. 76. person susceptibles infected 0 50 100 t [days] 150 200
  77. 77. nodes are human communities with population H in which the disease can diffuse & grow
  78. 78. Hydrologic Networks Human-Mobility Network i Pij i Qij Rij j Qij j
  79. 79. Mari et al., J Roy Soc Interface, 2011
  80. 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. 81. Zipf distribution of population size & self-organization
  82. 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. 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. 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. 85. Bertuzzo et et al., GRL, 2011
  86. 86. θ m σ l ∝B φ D ρ β γ ∝ α −60% −40% −20% 0 20% 40% 60%
  87. 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. 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. 89. 25 Weekly Cases [103] 20 15 10 5 0 Nov 10 Jan 11 Mar 11 May 11 Jul 11 Sep 11
  90. 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. 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. 92. recorded cholera cases in Haiti (2010-2013) (normalized) normalized maximum eigenvector Gatto et al, PNAS, 2012; Gatto et al, Am Nat, 2014
  93. 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. 94. Mari et al., Ecol. Lett., 2014
  95. 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. 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. 97. power laws matter alot - hotspots & geomorphology ! indeed a frontier of ecological research Muneepeerakul et al., WRR, 2007
  98. 98. remote sensing & (much) hydrologic research
  99. 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. 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. 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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