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Modelling infectious agents in
food webs
Hans Heesterbeek
Small selection of examples from Selakovic, de Ruiter & H, submitted review
• It would be hard to study the ecology of a
natural system without this being
influenced by infectious agents
• Only in recent decades have we started to
explore these explicitly
• Theory to think about these influences is
lagging behind
Changes in the
Serengeti ecosystem:
increased tree cover
since 1980’s
Picture collage: Ricardo Holdo
Photo’s: John Fryxell
Serengeti ecosystem & rinderpest
Cascade: rinderpest
disappears
tree density
increases
in the ecosystem
Via “the effect of
rinderpest on a
herbivore that does
not even consume
trees”
(Holdo et al)
Holdo et al. PLoS Biology, 2009, 7(9), e1000210
Unhealthy herd effect
Daphnia
Predator produces
chemical that induces
larger body size in
its prey, Daphnia.
Larger Daphnia are
more susceptible to a
fungal parasite because
of their increased
feeding rate
Body size and spore yield of
Daphnia in presence of chemical
compared to absence
Chaoborus
Larger infected
Daphnia produce
more fungal spores
Pictures and example from
Duffy et al., Functional Ecol., 2011
Work of Spencer Hall/Meghan Duffy
CDV and Babesia in Serengeti lions
Dynamics lion population ’75-’05
Red bars: outbreaks
of CDV with massive
lion mortality 1994,
2001
Grey bars: ‘silent’
outbreaks of CDV
detected by
serology (retrosp.)
C,D: number of
buffalo carcasses
in lion diet
Extensive herbivore
deaths after extreme
drought in 1993 (S)
and 2000 (N)
From: Munson et al, 2008, PLoS One
Nematomorph parasites in crickets
(community)
Sato et al., Ecol. Lett. 2012
Savanah ecosystem of Kruger National Park, SA
From: Han Olff et al. Phil. Trans. R. Soc. B 2009;364:1755-1779
Infectious agents are species
Pred.
Parasite 1
Pred.
Pathogen 2
Prey
Prey
Suggests effects on
topology, connectivity,
path length, ‘complexity’
Kevin Lafferty
Salt marsh food web
Yellow = parasite species
Red = host species
PNAS, 2008, Ecol. Lett, 2008
Arctic food web
Beckerman & Petchey
J. Anim. Ecol. 2009
Pelagic food web of
sub-arctic lake Takvatn
with only predator-prey
interaction (top)
and including parasite
species and their links
(bottom)
Amundsen et al
J. Anim Ecol. 2009
US-Army General: “it’s dangerous because it creates the illusion
of understanding” (New York Times)
Three approaches to food webs
• Possible ways to think about infectious agents in food web;
I is “pathogen”; II is “parasite”
Type of questions for modelling
ecological questions
epidemiological questions
Study infectious agents as a
biological species to determine
its role in ecosystems.
Study the effects of an infectious
agents on its host species in their
ecosystem (and vice versa).
How do infectious agents influence
(shape, determine?) food-web
topology & ultimately stability? What
is their role in persistence and
evolution of the ecosystem? What is
their contribution to energy flow
through the system? Are there
essential differences between an
agent-host link and a consumerresource link? How are species of
infectious agents distributed over
trophic levels? What are the effects of
loss/gain or increase/decrease of
species (succession?)? How do
infectious agents influence/cause
trophic cascades?
Under what conditions can an
infectious agent invade the
ecosystem? How does the ecosystem
context influence evolution of
virulence, and jumps to new host
species? How are control measures
aimed at a specific host influenced by
the ecosystem context? How is longterm persistence influenced by host
and non-host interaction and
dynamics? How does the prevalence
over host species change with
ecosystem change? What are possible
mechanisms for a positive or negative
“dilution effect”?
Two approaches to infectious
agents in food webs
• Direct approach: agent as separate species/node
• Indirect approach: agent only through its effect in splitting
host species split into epidemiological (infected) states
Energy flow soil food web
Phytophagous
Nematodes
Collembolans
Predaceous
Mites
Cryptostigmatic
Mites
Roots
Saprophytic
Fungi
Fungivorous
Nematodes
Enchytraeids
Detritus
Predaceous
Collembolans
Noncryptostigmatic Mites
Nematode
Feeding Mites
Predaceous
Nematodes
Bacteriophagous
Nematodes
Bacteria
Flagellates
Amoebae
Bacteriophagous
Mites
Measurements of feeding, energy flow,
biomass, interaction strength
picture: Peter de Ruiter
Distribution of interaction strengths and biomass within a food
web maintains stability with increasing complexity
resource
top predators
predatory collembola
nematophagous mites
predatory nematodes
predatory nematodes
collembola
cryptostigmatic mites
non-cryptostigmatic mites
fungivorous nematodes
bacteriophagous nematodes
bacteriophagous mites
predatory nematodes
fungivorous nematodes
bacteriophagous nematodes
fungivorous nematodes
bacteriophagous nematodes
phytophagous nematodes
amoebae
fungivorous nematodes
flagellates
bacteriophagous nematodes
phytophagous nematodes
phytophagous nematodes
flagellates
phytophagous nematodes
bacteria
bacteria
fungi
fungi
fungi
fungi
bacteria
bacteria
bacteria
fungi
bacteria
detritus
roots
detritus
detritus
consumer
predatory mites
predatory mites
predatory mites
predatory collembola
predatory mites
predatory mites
predatory mites
predatory mites
predatory mites
predatory mites
nematophagous mites
predatory collembola
predatory collembola
nematophagous mites
nematophagous mites
predatory mites
predatory nematodes
predatory nematodes
predatory nematodes
predatory nematodes
predatory collembola
nematophagous mites
amoebae
predatory nematodes
predatory nematodes
amoebae
collembola
cryptostimatic mites
non-cryptostigmatic mites
fungivorous nematodes
flagellates
bacteriophagous nematodes
bacteriophagous mites
enchytraeids
enchytraeids
enchytraeids
phytophagous nematodes
fungi
bacteria
basal resources
A.M. Neutel, et al., Science (2002) & Nature (2007)
How do infectious agents influence this? Theory based on steady state situation
of biomass distribution over species: “only” the ecological questions can be studied
basal species
Top species
real
1
1
2
4
5
6
0.021
0.021
0.021
0.021
-1.5
-0.085 0.015
0.015
0.015
0.015
-1.5
-0.15
-0.085 0.017
0.017
0.017
0.017
0.017
-2.8
-0.29
-0.32
-0.085 0.019
0.019
0.019
0.019
0.019
-2.8
-0.28
-0.32
-0.19
-0.085 0.022
0.022
0.022
0.022
-1.4
-0.14
-0.16
-0.093
-0.11
-0.085 0.024
0.024
-0.16
-0.093
-0.11
-0.24
0.023
-13
-7.5
-8.5
-7.5
-8.5
-0.085 0.021
3
7
8
9
2
Effects of prey
on their predator
0.015
3
4
5
6
Effects of
predators on
their prey
7
-0.085
8
-11
basal species
-0.085
9
-19
-18
-0.085
Self-limiting
effects
(diagonal)
Direct approach: challenges
• Infectious agent as a species, with links to host species
• Is an agent-host link “the same” as a predator-prey link in
a topological analysis?
– Agent consumes part of resource, but even when agent
kills host, this host is still available as prey for
predators. So how to account for this?
– Some parasite stages and most pathogens inside host
• How to make this precise before studying effects on path
lengths, complexity, nr. of trophic levels, … ?
• Much of the current theory restricted to systems in steady
state (e.g. with respect to biomass distribution)
Intermediate view
Predator
• Structure host species by
epidemiological state
Susc.
• Incorporate effects
Pred.
through interaction
strengths
• Study food-web dynamics with
“weighted” interaction strength
driven by changes
in distribution over epi-states
Infect
Pred.
Prey
Recov.
Pred.
Intermediate approach: challenges
• Similarities to network models on which infection spreads:
– Network is known and fixed
– But: it is the dynamic strength of the link that describes
the system
– This strength changes depending on within-species
dynamics of infectious agent in the species involved in
the link
– The strength itself influences the between-species
dynamics
• How to model (let alone analyse) this feed back?
Indirect approach
• More pragmatic and close to the ecological
and epidemiological modelling we know
• Basically: take a predator-prey model and
add allow different infected states for each
host species
Developments in math. biology
• Hadeler & Freedman, 1989: parasite mediates coexistence
between predator and prey
• Chattopadhyay & Arino, 1999: similar with disease in prey,
probably coined “eco-epidemiology”
• Venturino, 1994, 1995, 2002: Lotka-Volterra with infection
• Han & Hethcote, 2001: one predator/one prey with infection
• Hsieh & Hsiao, 2008: similar
• Haque & Venturino, 2006: similar
• Han & Pugliese, 2009: similar
• Malchow and others 2005-2008 (papers + book): spatial
predator-prey with infection
• Hilker and others, 2006-2010 (5 papers): Allee effect and
infection, stabilizing predator-prey oscillations, bio-control
• Morozov, 2012: one predator/ one prey and infection
Pathogen can mediate coexistence
between consumer and resource when
feeding rate too high
Consumer-resource dynamics
• n species, population sizes Ni
• Pi set of consumers species for which species i is a resource
– Consumption rate ΦijNj
– Positive effect on species j: eji Φji Ni
• Qi set of species that are consumed by species i
• Density dependent birth and death
From: Roberts & Heesterbeek,
J. Math. Biol. March 2013
Ecological stability
• Steady state solutions N i
• Jacobian matrix C, community matrix:
Cij = -fij N i
Cik = eikfik N i
Cil = 0
j Î Pi , k Î Qi , l Ï Pi ÈQi
Adding an infectious agent (SI)
Stability in combined system
• Jacobian matrix J is, for a particular steady state, given by
æ C D ö
J =ç
÷
è B H ø
• Order by total population sizes Ni, followed by the sizes of all
infected states in the system
• C is the community matrix, as given before
• H is the epidemiological matrix; this matrix is related to the
next-generation matrix (NGM)
æ C D ö
J =ç
÷
è B H ø
• D gives influence of changes in the ecology of individuals
due to epidemiology (i.e. their infected state)
– E.g. changes in feeding behaviour, fecundity, …
• B gives the influence of changes in the epidemiology of
infected individuals due to ecology (e.g. population size Ni)
– E.g. changes in the influence of density dependence for
infected individuals, compared to uninfected
• In the infection-free steady state (invasion problem),
matrix B = 0, the zero matrix
• For endemic states, B is typically not the zero matrix
Stability: spectral bound of J
æ C D ö
J =ç
÷
è 0 H ø
• Regard J for the infection-free steady state:
– Consequence: B = O = zero matrix
• Stability problem decouples in product of ecological stability
(governed by C) x epidemiological stability (governed by H)
• H describes the influence of any infected state on each
infected state
– H=T+Σ
– T the transmission matrix, Σ the transition matrix
– Next-generation matrix with large domain:
-1
K L = -TS
In SI-example: KL = K, next-gen. matrix; in all cases: R0 = spectral radius of KL
Matrix H for the ‘general’ model
Epidemiological stability H = T + Σ
for pred.-prey with infection in both
Epidemiological stability depends on feeding rate ϕ
Wildebeest-grass-rinderpest
Epidemiological stability does not depend on ϕ
in this example
H is a 1 x 1 ‘matrix’
(only one infected state)
R0 = β/(μ2 + α)
Stability is balance
between ecology
and epidemiology
Consumer extinct due to infection
Serengeti ecosystem & rinderpest
Rinderpest regulated
wildebeest to a low
steady state level
Vaccination of cattle
around the park
lowered
infection success in
wildebeest
R0 decreased to
below 1
Wildebeest settled in
high steady state;
grass in low state
Data from Holdo et al. PLoS Biology, 2009, 7(9), e1000210
Afterthought: more tree and shrub cover could lead to increase of tsetse flies
which could lead to more sleeping sickness in cattle and humans
Eco-epi approach: agenda
• Deriving useful analytical results for the stability of nontrivial states (B not equal zero matrix)
• What happens in periodic environments?
• Stability related to adding one host or non-host species?
Exploring the dilution effect (Pete’s lecture!)
• How is overall system stability related to relevant indicators
related to matrix C, D, B, H
• Parasites with i life stages (Andy’s question and conjecture:
hope to deal with that in the coming weeks)
Summary
• On your wish list of future extensions for
your work: add multiple species and
community dynamics!
The web of interactions between microparasite species within a community of infectious
agents in one rodent host species (bank vole), showing the magnitude of effects.
Sandra Telfer et al. Science 2010;330:243-246
How and what to model?
• Within host: using ideas from metabolic/gene regulatory
networks? Relevant questions? Relevant experiments or
empirical set ups?
• Individual level: how is susceptibility and infectivity for agent
A mediated by agents B, C, D, …? Does influence remain
when an agent has been cleared? Immune response.
• Population level: how does dynamics of agents B, C, D, …
and their distribution over the host species influence the
invasion and spread of A into that community?
• How to model the above? Stratify by history of infection?