Andrea Rinaldo

!!

Laboratory of Ecohydrology ENAC/IIE/ECHO Ecole Polytechnique Fédérale Lausanne (EPFL) CH Dipartimento ICEA Università di Padova

RIVER NETWORKS

AS ECOLOGICAL CORRIDORS

FOR SPECIES

POPULATIONS AND WATER-BORNE DISEASE

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

explore two critical characteristics (directional dispersal & network structure as environmental matrix)

for spreading of organisms, species & water-borne disease

questions of scientific & societal relevance

(population migrations, loss of biodiversity, hydrologic

controls on the spreading of Cholera, meta-history)

Muneepeerakul et al., JTB, 2007

Rodriguet-Iturbe et al., PNAS, 2012

Carrara et al., PNAS, 2012

Carrara et al., PNAS, 2012

Carrara et al., Am. Nat., 2014

TOOLS - about the progress (recently) made on

how to decode the mathematical language

of the geometry of Nature

DTM - GRID (Planar view)

DTM – GRID format (Perspective – North towards bottom)

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

TOOLS

from O(1) m scales…

the MMRS

random-walk

drainage basin network

(Leopold & Langbein, 1962)

& the resistible

ascent of the

random paradigm

!

Eden growth & self-avoiding random walks !

Rigon et al., WRR, 1998

Huber, J Stat Phys, 1991; Takayasu et al., 1991

Scheidegger’s construction

is exactly solved for

key geometric & topologic features

Rodriguez-Iturbe et al., WRR, 1992 a,b; Rinaldo et al. WRR, 1992

optimal channel networks

Rigon et al., WRR, 1997

Rinaldo et al., PNAS, in press

Peano – exact results & subtleties

(multifractality

binomial multiplicative process & width functions)

Marani et al., WRR, 1991; Colaiori et al., PRE, 2003

TOOLS 1 - comb-like structures, diffusion processes & CTRW framework in terms of density

of particles ρ(x,t)

l

A B

from traditional unbiased random-walks to general cases

!heterogeneous distributions

of spacing, Δx & length of the comb leg, l

AB A B

delay ~ reactions, lifetime distributions

tools - 2

models of reactive transportnetwork → oriented

graph made by nodes & edges

TRANSPORT MODELS BETWEEN

NODES

NODAL REACTIONS

COUPLED MODELS

individuals, species, populations (metacommunities)

TOOLS 2 - reactive continuous time random walk

x

pdf of jump &

waiting time

),( txΨ

)0,(xρ

Φ(t)

reaction)(ρf

∫ ∫∞ +∞

∞−Ψ=

ttxdxdtt )',(')(φ

diffusion

?

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

transport + possibly reactions or interactions

hydrochory

!!

human-range expansion, population migration

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)

!

a little background on Fisher’s fronts

phase plane → the sign of the eigenvalues of an

appropriate Jacobian matrix

determines the nature of the equilibria

!(e.g. Murray, 1993)

a few further mathematical details

the network slows the front! you waste time trapped in the pockets

the Hamilton-Jacobi formalism

Peano’s networkinitial cond

r=1

logistic growth at every node )1()( ρρρ −= af

f

ρ0 1

a

reaction

transport at every timestep each particle moves towards a nearest neighbour

w.p. p= 1 / # nn

SIMULAZIONI

P+=0.5 a=0.5

isotropic migration – Fisher’s model

v = 2√aD Murray, 1988

Peano (exact)

Peano (numerical)

a (logistic growth)

v sp

eed

of fro

nt [

L/T]

Campos et al., Theor. Pop. Biol., 2006; Bertuzzo et al., WRR, 2007

Campos et al., Theor. Pop. Biol., 2006; Bertuzzo et al., WRR, 2007

geometric constraints imposed by the network

(topology & geometry) impose strong corrections

to the speed of propagation of migratory fronts

Rel

ativ

e fr

eque

ncy

(%)

what is a node? !!

strong hydrologic controls

!

Giometto et al., PNAS, 2013

What  about  variability?

Fisher-­‐Kolmogorov  Equation

∂ρ

∂t=D∂

2ρ

∂x2+ rρ 1−

ρ

K$

%&'

()

∂ρ

∂t= rρ 1−

ρ

K$

%&'

()+σ ρ η

∂ρ

∂t=D∂

2ρ

∂x2+ rρ 1−

ρ

K$

%&'

()+σ ρ η

ML  estimates  for  r,K,  σ

η  is  a  δ-­‐correlated  gaussian  white  noise  Itô  stochastic  calculus

Transitional  probability  densities  are  computed  by  numerical  integration  of  the  related  Fokker-­‐Planck  equation.

∂ρ

∂t= rρ 1−

ρ

K$

%&'

()+σ ρ η

∂ρ

∂t=D∂

2ρ

∂x2+ rρ 1−

ρ

K$

%&'

()+σ ρ η

Demographic  stochasticity

ML  estimates  for  r,K,  σ

η  is  a  δ-­‐correlated  gaussian  white  noise  Itô  stochastic  calculus

Transitional  probability  densities  are  computed  by  numerical  integration  of  the  related  Fokker-­‐Planck  equation.

Front  variability

Giometto et al., PNAS, 2013

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

Zebra MusselDreissena polymorpha

1989

1990

1991

1992

1994

1995

1988

1993

data: Nonindigenous Aquatic species program USGS

larval stages transported along the

fluvial network

Zebra MusselMari et al., in review, 2007

local age-growth model (4 stages) !larval production

!larval transport

(network)

Zebra Mussel

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

river biogeography

!spatial distribution of

biodiversity within a biota

!riparian vegetation

fluvial fauna

freshwater fish

neutral metacommunity model

metacommunity modelevery link is a community of organisms & internal implicit spatial dynamics

Explicit spatial dynamics among different communities

the neutral assumptionall 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

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

∑=

−= N

kkik

jijij

HK

HKvP

1

)1(

:habitat capacity link ijH

ijK :dispersal kernel

run up to steady state

river biogeography

abundance

# of

spe

cies

20 21 22 24 26 28 21223 25 27 29 210 211

global properties

γ-diversity: total # of species

patterns of abundance

preston plot

river biogeography

α-diversitynumber of

species at local scale

LOCAL PROPERTIES

river biogeography

β-diversity

Jaccard similarity index

a)

b)

abba

abab S

SxJ

−+=

αα)( abS # common species

x distance measured along the network

57.0456

4)( =−+

=xJab

river biogeography

geographic range

area occupied

by a species

ranked species

geog

raph

ic ra

nge

USGS, hydrologic data, NatureServe, Bill Fagan’s ecological data

Mississippi-Missouri freshwater fauna

presence(absence) of 429 species freshwater fish in 421 subbasins

database

α-diversity, β-diversity, γ-diversity, geographic range

fonti

Mississippi-Missouri freshwater fish

α-diversity

runoff

strong correlation

habitat capacity ~ runoff

Muneeperakul, Bertuzzo, Fagan, Rinaldo, Rodriguez-Iturbe, Nature, 2008

distance to outlet

Muneeperakul et al., Nature, May 8 2008

constant

habitat

capacity

per DTA

hydrologic controls

weak interspecific interactions & weak/strong

formulations of the neutral model

model

comparison between geographic ranges of individual species: a) data b) results from the neutral

metacommunity model (after matching procedure)

patterns -- weak or strong impliations of neutrality?

equiprobability map – ratio between the number of common species and the number of species in

the central DTA

Bertuzzo et al., submitted, 2008

environmental resistance R50 for data & the model

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

!

Corridors for pathogens of waterborne disease !!

Of cholera epidemics & hydrology

Haiti (2010-2011)

Piarroux et al., Emerging Infectious Diseases, 2011

no elementary correlation between population and cholera cases

Mari et al., J Roy Soc Interface, 2011

continuous SIR model

susceptibles S

infected I

vibrios/m3 B

recovered R

persons

vibriosSBK

B+

βIγ

BBµ

IWp

Iµ Iα

Rµ Sµ Hµ

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)

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)

Codeco, JID, 2001; Pascual et al, PLOS, 2002; Chao et al, PNAS, 2011

Chao et al., PNAS, 2011

Capasso et al, 1979; Codeco, JID, 2001

the class of SIB models

0 50 100 150 200t [days]

pers

on

susceptiblesinfected

I(t) S(t)

!SIR model for the temporal &

spatial evolution of water-transmitted disease revisited → network

!a few assumptions

!!

total population of humans is 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

0 50 100 150 200t [days]

pers

on

susceptiblesinfected

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

Hydrologic Networks Human-Mobility Network

i

j

i

jRij

Qij

Pij Qij

Mari et al., J Roy Soc Interface, 2011

0 50 100 150 200 250 300 3500

100

200

300

400

500

600

time [days]

infe

cted

uniform population

I(t)

t

Zipf distribution of population size & self-organization

0 50 100 150 200 250 300 3500

200

400

600

800

1000

1200

1400

time [days]

infe

cted

Zipf’s distribution of population & secondary peaks of infection

I(t)

t

spatio-temporal dynamics

initial conditions

the higher the transport rate, the better the system is approximated by a well-mixed reactor (spatially implicit scheme)

refelecting boundary condition at all the leaves

and at the outlet

Bertuzzo et al., J Roy Soc Interface, 2010

0

5

10

15

20

25

Wee

kly

Cas

es [1

03 ]

0

10

20R

ainf

all [

mm

/day

]

prediction

calibration

Nov 10

Jan 11 Mar 11 May 11 Jul 11

Sep 11

100

300

500

prediction

Cum

ulat

ive

Cas

es [1

03 ]

Nov 10 Sep 11

Bertuzzo et al. GRL 2011

Bertuzzo et et al., GRL, 2011

−60% −40% −20% 0 20% 40% 60%

α

∝

γ

β

ρD

φ

∝B

lσ

mθ

Nov 10 Jan 11 Mar 11 May 11 Jul 11 Sep 110

5

10

15

20

25W

eekl

y C

ases

[103 ]

calibration hindcast

0

10

20R

ainf

all [

mm

/day

]

Rinaldo et al., PNAS, in press

0

5 Nord−Ouest

0

5 Nord

0

5 Nord−Est

0

5

10Artibonite

0

5 Centre

0

5 Grande Anse

0

5 Nippes

0

5

10

15 Ouest

0

5 Sud

0

5 Sud−Est

15

Wee

kly

Cas

es [1

03 ]

Jan 11 May 11 Sep 11 Jan 11 May 11 Sep 11

0

5

10

15

20

25

Wee

kly

Cas

es [1

03 ]

Nov 10 Jan 11 Mar 11 May 11 Jul 11 Sep 11

Wee

kly

Cas

es [1

03 ]

0

10

20

30

40

Jan 11 Jul 11 Jan 12 Jul 12 Jan 13 Jul 13 Jan 14

effects of rates of loss of acquired immunity (1-5 years)

Rinaldo et al., PNAS, in press

Jan 11 Jul 11 Jan 12 Jul 12 Jan 13 Jul 130

10

20

30

Wee

kly

Cas

es [1

03 ]0

5

10

15

20

Rai

nfal

l [m

m/d

ay]

Jan 14

Rinaldo et al., PNAS, in press

recorded cholera cases in Haiti (2010-2013) (normalized)

normalized maximum eigenvector

Gatto et al, PNAS, 2012; Gatto et al, Am Nat, 2014

river networks & biodiversity

!tradeoff versus neutral models of the ecology of riparian vegetation

Muneepeerakul et al., JTB, 2007 --> Mari et al., Ecol Lett., 2014

Mari et al., Ecol. Lett., 2014

Muneepeerakul, Weitz, Levin, Rinaldo, Rodriguez-Iturbe, JTB, 2007

links are essentially patches within a landscape cointaining sites that are occupied by individual plants

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

power laws matter alot - hotspots & geomorphology

!indeed a frontier of ecological research

Muneepeerakul et al., WRR, 2007

remote sensing & (much) hydrologic research

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

!!

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 !!

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