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

Picture collage: Ricardo HoldoPhoto’s: John Fryxell

Changes in the Serengeti ecosystem: increased tree cover since 1980’s

Serengeti ecosystem & rinderpest

Holdo et al. PLoS Biology, 2009, 7(9), e1000210

Cascade: rinderpestdisappears tree density increasesin the ecosystem

Via “the effect of rinderpest on a herbivore that doesnot even consumetrees”(Holdo et al)

Unhealthy herd effect

Work of Spencer Hall/Meghan DuffyPictures and example fromDuffy et al., Functional Ecol., 2011

Daphnia

Predator produceschemical that induceslarger body size inits prey, Daphnia.

Larger Daphnia are more susceptible to a fungal parasite becauseof their increasedfeeding rate

Larger infected Daphnia producemore fungal sporesBody size and spore yield of

Daphnia in presence of chemicalcompared to absence

Chaoborus

CDV and Babesia in Serengeti lions

Dynamics lion population ’75-’05

From: Munson et al, 2008, PLoS One

C,D: number of buffalo carcasses in lion diet

Extensive herbivoredeaths after extremedrought in 1993 (S)and 2000 (N)

Red bars: outbreaksof CDV with massivelion mortality 1994,2001

Grey bars: ‘silent’outbreaks of CDVdetected by serology (retrosp.)

Nematomorph parasites in crickets(community)

Sato et al., Ecol. Lett. 2012

Savanah ecosystem of Kruger National Park, SAFrom: Han Olff et al. Phil. Trans. R. Soc. B 2009;364:1755-1779

Infectious agents are species

PreyPrey

Pred. Pred. Pred. Pred. Para- site 1Para- site 1

PreyPrey

Patho-gen 2Patho-gen 2

Suggests effects ontopology, connectivity, path length, ‘complexity’

Arctic food webBeckerman & PetcheyJ. Anim. Ecol. 2009

Kevin LaffertySalt marsh food web

Yellow = parasite speciesRed = host species

PNAS, 2008, Ecol. Lett, 2008

Pelagic food web ofsub-arctic lake Takvatn

with only predator-preyinteraction (top)and including parasitespecies and their links(bottom)

Amundsen et alJ. Anim Ecol. 2009

US-Army General: “it’s dangerous because it creates the illusionof 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 modellingecological questions epidemiological questions

Study infectious agents as a biological species to determine its role in ecosystems.

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 consumer-resource 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?

Study the effects of an infectious agents on its host species in their ecosystem (and vice versa).

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 long-term 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

Roots

Detritus

PhytophagousNematodes

SaprophyticFungi

Bacteria

Collembolans

Noncrypto-stigmatic Mites

CryptostigmaticMites

FungivorousNematodes

Bacteriophagous

NematodesBacteriophagous

Enchytraeids

Mites

Flagellates

Amoebae

PredaceousNematodes

PredaceousMites

PredaceousCollembolans

NematodeFeeding Mites

Energy flow soil food web

picture: Peter de RuiterMeasurements of feeding, energy flow,biomass, interaction strength

top predatorsresource

basal resources

predatory collembolanematophagous mitespredatory nematodespredatory nematodes

collembolacryptostigmatic mites

non-cryptostigmatic mitesfungivorous nematodes

bacteriophagous nematodesbacteriophagous mitespredatory nematodes

fungivorous nematodesbacteriophagous nematodes

fungivorous nematodesbacteriophagous nematodes

phytophagous nematodesamoebae

fungivorous nematodesflagellates

bacteriophagous nematodesphytophagous nematodesphytophagous nematodes

flagellatesphytophagous nematodes

bacteriabacteria

fungifungifungifungi

bacteriabacteriabacteria

fungibacteriadetritus

rootsdetritusdetritus

consumerpredatory mitespredatory mitespredatory mites

predatory collembolapredatory mitespredatory mitespredatory mitespredatory mitespredatory mitespredatory mites

nematophagous mitespredatory collembolapredatory collembolanematophagous mitesnematophagous mites

predatory mitespredatory nematodespredatory nematodespredatory nematodespredatory nematodespredatory collembolanematophagous mites

amoebaepredatory nematodespredatory nematodes

amoebaecollembola

cryptostimatic mitesnon-cryptostigmatic mites

fungivorous nematodesflagellates

bacteriophagous nematodesbacteriophagous mites

enchytraeidsenchytraeidsenchytraeids

phytophagous nematodesfungi

bacteria

Distribution of interaction strengths and biomass within a food web maintains stability with increasing complexity

A.M. Neutel, et al., Science (2002) & Nature (2007)

How do infectious agents influence this? Theory based on steady state situationof biomass distribution over species: “only” the ecological questions can be studied

Effects of prey on their predator

Effects of predators on their prey

Self-limiting effects (diagonal)

0.0170.017 0.017 0.017 0.017

0.019 0.0190.019

0.021 0.021 0.021

0.0150.015

-0.085-0.15-1.5

0.0210.021-0.085

0.015-0.085-1.5 0.015 0.015

0.022 0.0220.022

0.024

-0.32-0.29-2.8

-0.32-0.28-2.8

-0.16-0.14-1.4

-0.085 0.019 0.019

-0.19 -0.085 0.022

-0.093 -0.11 -0.085 0.024

-0.16

-13-11

-0.085-18

0.023-0.085

-0.085

-19-8.5-7.5

-0.24-0.11-0.093

-8.5-7.5

31 2 4 5 6 7 8 9

1

2

3

4

5

7

8

9

6

realTop species basal species

basal species

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

• Structure host species by epidemiological state • Incorporate effects

through interactionstrengths

• Study food-web dynamics with “weighted” interaction strength driven by changes in distribution over epi-states

PreyPrey

Susc.Pred. Susc.Pred.

InfectPred. InfectPred.

Recov. Pred. Recov. Pred.

Predator

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 • Jacobian matrix C, community matrix:

Adding an infectious agent (SI)

Stability in combined system

• Jacobian matrix J is, for a particular steady state, given by

• 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)

• 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

• 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:

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

H is a 1 x 1 ‘matrix’(only one infected state)

R0 = β/(μ2 + α)Epidemiological stability does not depend on ϕin this example

Stability is balance between ecologyand epidemiology

Consumer extinct due to infection

Serengeti ecosystem & rinderpest

Data from Holdo et al. PLoS Biology, 2009, 7(9), e1000210

Rinderpest regulatedwildebeest to a low steady state level

Vaccination of cattlearound the parkloweredinfection success in wildebeest

R0 decreased to below 1

Wildebeest settled inhigh steady state;grass in low state

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 non-trivial 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?