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

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

9. Carrara et al., Am. Nat., 2014

10. TOOLS - about the progress (recently) made on
how to decode the mathematical language
of the geometry of Nature

11. DTM - GRID
(Planar view)
DTM – GRID format
(Perspective – North towards bottom)

12. 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

13. TOOLS

14. from O(1) m scales…

15. the MMRS

16. random-walk
drainage basin network
(Leopold &
Langbein, 1962)
& the resistible
ascent of the
random paradigm
!

17. Eden growth & self-avoiding random walks
Rigon et al., WRR, 1998
!

18. Scheidegger’s construction
is exactly solved for
key geometric & topologic features
Huber, J Stat Phys, 1991; Takayasu et al., 1991

19. optimal channel networks
Rodriguez-Iturbe et al., WRR, 1992 a,b; Rinaldo et al. WRR, 1992

20. Rigon et al., WRR, 1997

21. Rinaldo et al., PNAS, in press

22. Peano – exact results & subtleties
(multifractality
binomial multiplicative process & width functions)
Marani et al., WRR, 1991; Colaiori et al., PRE, 2003

23. TOOLS 1 - comb-like structures, diffusion processes &
CTRW framework
in terms of density
of particles ρ(x,t)
l
A
B

24. from traditional
unbiased random-walks to general cases
!
heterogeneous distributions
of spacing, Δx
& length of the comb leg, l

25. tools - 2
A
B
A
B
delay ~ reactions, lifetime distributions

26. network → oriented
graph made by nodes
& edges
models of reactive transport
COUPLED MODELS
NODAL REACTIONS
TRANSPORT
MODELS BETWEEN
NODES
individuals, species, populations (metacommunities)

27. 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 ' )

28. 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

29. hydrochory
!
!
human-range expansion, population migration

30. 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)
!
!
!
!

31. a little background on Fisher’s fronts

32. phase plane → the sign of the eigenvalues of an
appropriate Jacobian matrix
determines the nature of the equilibria
!
(e.g. Murray, 1993)

33. a few further mathematical details

35. the Hamilton-Jacobi formalism
the network slows the front!
you waste time trapped in the pockets

36. 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

37. SIMULAZIONI

38. P+=0.5
a=0.5

39. 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)

40. 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

41. what is a node?
strong hydrologic controls
!
!
!

42. Giometto et al., PNAS, 2013

44. Fisher-­‐Kolmogorov
Equation
$ ρ'
∂ρ
∂2 ρ
= D 2 + rρ & 1− )
∂t
∂x
% K(
What
about
variability?

45. $ 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.

46. 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.

47. Front
variability
Giometto et al., PNAS, 2013

48. 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

49. 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

50. Mari et al., in review, 2007
local age-growth model
(4 stages)
!
larval production
!
larval transport
(network)
Zebra Mussel

51. Zebra Mussel

52. Mari et al., WRR, 2011; Mari et al., Ecol. Lett., 2014

53. river biogeography
!
spatial distribution of
biodiversity within a biota
!
riparian vegetation
fluvial fauna
freshwater fish

54. 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

55. 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

56. 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

57. river biogeography
LOCAL PROPERTIES
α-diversity
number of
species at local
scale

58. 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

59. river biogeography
geographic range
geographic range
area occupied
by a species
ranked species

60. 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

61. Mississippi-Missouri freshwater fish
α-diversity
strong correlation
habitat capacity ~
runoff
runoff

62. distance to outlet
Muneeperakul, Bertuzzo, Fagan, Rinaldo, Rodriguez-Iturbe, Nature, 2008

63. Muneeperakul et al., Nature, May 8 2008

64. constant
habitat
capacity
per DTA
hydrologic controls

65. model
weak interspecific interactions & weak/strong
formulations of the neutral model

66. 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)

67. Bertuzzo et al., submitted, 2008
equiprobability map – ratio between the number of
common species and the number of species in
the central DTA

68. environmental resistance R50 for data & the model

69. 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
!

70. Corridors for pathogens of waterborne disease
!
!
Of cholera epidemics & hydrology

71. Haiti (2010-2011)

72. Piarroux et al., Emerging Infectious Diseases, 2011

73. no elementary correlation between population
and cholera cases

75. Mari et al., J Roy Soc Interface, 2011

76. 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)

77. Chao et al., PNAS, 2011

78. the class of SIB models
Capasso et al, 1979; Codeco, JID, 2001

79. 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

80. person
susceptibles
infected
0
50
100
t [days]
150
200

81. nodes are human communities with population H
in which the disease can diffuse & grow

82. Hydrologic Networks
Human-Mobility Network
i
Pij
i
Qij
Rij
j
Qij
j

84. Mari et al., J Roy Soc Interface, 2011

87. uniform population
600
500
I(t)
infected
400
300
200
100
0
0
50
100
150
200
time [days]
250
t
300
350

88. Zipf distribution of population size & self-organization

90. 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

91. 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

94. 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

95. Bertuzzo et et al., GRL, 2011

98. θ
m
σ
l
∝B
φ
D
ρ
β
γ
∝
α
−60%
−40%
−20%
0
20%
40%
60%

99. 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

100. 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

101. 25
Weekly Cases [103]
20
15
10
5
0
Nov 10
Jan 11
Mar 11
May 11
Jul 11
Sep 11

102. 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

104. 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

105. recorded cholera
cases in Haiti
(2010-2013)
(normalized)
normalized maximum
eigenvector
Gatto et al, PNAS, 2012; Gatto et al, Am Nat, 2014

106. river networks & biodiversity
!
tradeoff versus neutral models of
the ecology of riparian vegetation
Muneepeerakul et al., JTB, 2007 --> Mari et al., Ecol Lett., 2014

107. Mari et al., Ecol. Lett., 2014

108. Muneepeerakul, Weitz, Levin, Rinaldo, Rodriguez-Iturbe, JTB, 2007
links are essentially patches within a landscape
cointaining sites that are occupied by individual plants

109. 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

110. power laws matter alot - hotspots & geomorphology
!
indeed a frontier of ecological research
Muneepeerakul et al., WRR, 2007

111. remote sensing & (much) hydrologic research

112. →
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
!

113. 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
!

114. 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