dispersal, disturbance and disease spread · dispersal, disturbance and disease control rachel...

Dispersal, Disturbance and Dispersal, Disturbance and Dispersal, Disturbance and Dispersal, Disturbance and

Disease ControlDisease ControlDisease ControlDisease ControlRachel LintottComputing Science and Mathematics

School of Natural Sciences

University of Stirling

Funded by University of Stirling

Horizon Studentship

Overview of the talkOverview of the talkOverview of the talkOverview of the talk

• Introduction

• Biological motivation behind the models

• Building the single species model

• Methods of analysis

• Single species results

• Two host results

IntroductionIntroductionIntroductionIntroduction

• Management of wildlife populations for disease

control e.g. culling

• Heterogeneity of habitat and connectivity through

dispersal

• Perturbation effect: increase in movement

between patches in response to culling

• Effect of this perturbation on disease control

thresholds

Biological Motivations:Biological Motivations:Biological Motivations:Biological Motivations:

Why control wildlife?

• 60% of human pathogens are zoonotic (transmitted from

animals)[1]

• E.g. SARS (severe acute respiratory syndrome) virus killed 750

people in 2003 epidemic

• Virus has been found in populations of Chinese horseshoe bats[2]

[1] L.H. Taylor, S.M. Latham and M.E.J. Woolhouse. Risk Factors for human disease emergence. Philosophical Transactions

of the Royal Society of London. Series B: Biological Sciences, 365.1411 :983-989, (2001):

[2] W. Li. et al. Bats Are Natural Reservoirs of SARS-Like Coronaviruses. Science, 310 (5748), 676-679, (2005)

• 80% of livestock pathogens infect more than one species including

wildlife populations[3]

• Around 30,000 cattle are slaughtered each year as a result of

bovine tuberculosis

• National controversy over proposed cull of badgers to reduce

incidence of bovine TB in cattle

Biological Motivations:Biological Motivations:Biological Motivations:Biological Motivations:

Why control wildlife?

[3] S. Cleaveland, M.K. Laurenson and L.H. Taylor. Diseases of humans and their domestic mammals: pathogen

characteristics, host range and the risk of emergence. Philosophical Transactions of the Royal Society of London. Series

B: Biological Sciences, 356.1411: 991-999, (2001)

Biological Motivations: Biological Motivations: Biological Motivations: Biological Motivations: Dispersal

• Many wildlife species will form distinct groups, either social groups or due to

habitat fragmentation

• Group structure and interactions may have knock on effects on disease

control

• Many models of infectious disease control assume uniform random

distribution throughout an environment

Harbour Seal

Herd

Little Brown Bat

Roost

Badger

Sett

Biological Motivations: Biological Motivations: Biological Motivations: Biological Motivations: Dispersal

• Most populations will have a natural dispersal rate due to

limited resources, or to avoid inbreeding

• Invasive disease control methods such as culling or trapping to

vaccinate will disturb a population and may cause individuals to

leave their natural habitat (e.g. European badger, Rocky

mountain elk, mountain nyala)

• This increased movement may mean that disease spreads to

new areas as a result of control strategies

MathematicalMathematicalMathematicalMathematicalModel: Model: Model: Model: without control

Susceptible:

��

��

Infected:

� �

��

Susceptible Infected

death death

birthinfection

death due to infection


death death

birthinfection


��

��

�� 0

�� 0 ��0 0 ��

0 0 ��

Rare Invader AnalysisRare Invader AnalysisRare Invader AnalysisRare Invader AnalysisIn the absence of disease we can find an equilibrium at �

∗, �∗, 0,0 where

�∗ �

��

�� 1 � ��

Linearising about this equilibrium we are left with a 4 � 4Jacobian matrix:

Rare Invader AnalysisRare Invader AnalysisRare Invader AnalysisRare Invader Analysis

Infection Invasion Matrix:

��∗ � ��

�� ∗ � ��

By Perron-Frobenius Theorem this matrix:

• has real eigenvalues

• has maximum eigenvalue �� bounded above and below

by row and column sums such that

max !�", #!�" $ �� $ min' !() , #!()*


Infection Invasion Matrix: � � 1

+� � ��

�� +� � ��Where +� � ��

∗ � ��

• +�represent the intrinsic properties of the patch, whether it would be able to

support the pathogen in isolation

• +� , 0 patch - could support the pathogen alone and is a RESERVOIR

patch

• +� . 0 patch - unable to support the pathogen and is a NON-RESERVOIR

Infection Invasion Matrix: � � 1

+� � ��

�� +� � ��


With this notation, bounds on the maximum eigenvalue are :

min +�, +� $ �� $ max'+�, +�*

MathematicalMathematicalMathematicalMathematicalModel: Model: Model: Model: with control

Susceptible:

��

��

Infected:

� �

��


death death

birthinfection



death death

birthinfection


�#�� /� #� � � �/� #� �

�#�� /� #� �� /� #� ��

Control dependent dispersalControl dependent dispersalControl dependent dispersalControl dependent dispersal

1. Constant Dispersal

/� #� � ��

2. Saturating, Increasing Dispersal

/� #� � �� 0�#�1� � ��2

1 � #�

3. Linearly Increasing Dispersal

/� #� � �� 0�#�1� � ��2

Introducing control into the model shifts the disease free equilibrium to

�∗ �

/� #� �� /� #� �� #� ��

/� #� �� #� � /� #� �� #� � �� #� �� #� � 1 � �� /� #� /� #�

Control Dependent EquilibriumControl Dependent EquilibriumControl Dependent EquilibriumControl Dependent Equilibrium


Controlled Patch Non-controlled Patch

Constant dispersal

Rate of Control

Disease Free Equilibrium



Saturating dispersal

Rate of Control




Linear dispersal

Rate of Control


ControlControlControlControl ThresholdsThresholdsThresholdsThresholds

Infection invasion matrix with control:

+� � #� � /�1#�2 /�1#�2

/�1#�2 +� � #� � /� #�

With constant, per capita control, bounds on the maximum

eigenvalue become:

min +� � #�, +� � #� $ �� $ max'+� � #�, +� � #�*

Reservoirs of InfectionReservoirs of InfectionReservoirs of InfectionReservoirs of Infection

• If +�, +� . 0both patches are non-reservoir and will not support the pathogen.

• If +� , 0, +� . 0patch 1 is reservoir and is a source of infection to patch 2 – in order to remove infection patch 1 must be controlled.

• If +�, +� , 0 both patches are reservoirs and will support the pathogen in isolation. Culling must be sufficient to reduce population in both patches to remove pathogen.

Control Maps: Control Maps: Control Maps: Control Maps: 3 4

Pathogen

Excluded

Pathogen

Persistent

Constant dispersal


Pathogen

Excluded

Pathogen

Persistent



Pathogen

Excluded

Pathogen

Persistent

Linear dispersal


Pathogen

Excluded

Pa

tho

ge

n

Pe

rsis

ten

t

Constant dispersal


Pathogen

Excluded

Pathogen

Persistent



Pathogen

Excluded

Pathogen

Persistent

Linear dispersal

Parameter Dependence Parameter Dependence Parameter Dependence Parameter Dependence

• Increasing �� is equivalent to increasing culling parameters #�

• Increasing dispersal parameter �� causes a reduction in all thresholds

• Increasing �� causes an increase in the disease free equilibrium, and

therefore increases all thresholds

• What happens when � and �� are varied?

Parameter LabelParameter LabelParameter LabelParameter Label DescriptionDescriptionDescriptionDescription

�� Constant recruitment

�� Per capita death

�� Per capita dispersal from - to 5

#� Per capita culling rate

�� Per capita death due to infection

� Mass action transmission of infection

Parameter Dependence:Parameter Dependence:Parameter Dependence:Parameter Dependence:

increasing natural dispersal

• As �� increases, all thresholds are reduced.

• Constant dispersal is more sensitive to changes in natural

dispersal


increasing disease transmission

• As � increases so does the amount of control


increasing death due to infection

• As γ7increases, less control is needed


Switch in thresholds

Parameter DependenceParameter DependenceParameter DependenceParameter Dependence

• Absence of control –

infection dies out.

• Begin to cull and

disturbance causes change

in population size.

• This may cause a larger

population to support the

infection.

Parameter DependenceParameter DependenceParameter DependenceParameter Dependence

• Absence of control –

infection dies out.

• Begin to cull and

disturbance causes change

in population size.

• This may cause a larger

population to support the

infection.

• E.g. Trophy hunting or

game shooting.

Single Species Model:Single Species Model:Single Species Model:Single Species Model:

Summary of results

• If one patch is source to another, perturbation means

control must be applied to both patches

• Increasing connectivity between patches can help to ease

the control burden- although this benefit is reduced if

control causes disturbance

• If natural dispersal is high, for a range of potential

pathogens, perturbation predicts a higher control

threshold than constant dispersal

Wildlife Wildlife Wildlife Wildlife ----Livestock ModelLivestock ModelLivestock ModelLivestock Model

Livestock Livestock

Wildlife Wildlife

Transmission Transmission

Dispersal

Wildlife:

89�

8:� �� 9��;� � ��9��9� � ��9� � #9�9�

�/� #9� 9� � �/� #9� 9�

8�9�

8:� ��9��;� � ��9��9� � �� 9� � #9��9�

�/� #9� �9� � �/� #9� �9�

Wildlife Wildlife Wildlife Wildlife ----Livestock ModelLivestock ModelLivestock ModelLivestock Model

Livestock:8;�

8:� <� � ��;��;� � ��;��9� �=�;� � #;�;�

8�;�

8:� ��;��;� � ��;��9� � =� � >� �;� � #;��;�

WildlifeWildlifeWildlifeWildlife----Livestock Model:Livestock Model:Livestock Model:Livestock Model:

Reservoirs of disease

Two host model:Two host model:Two host model:Two host model:

Patch -is a reservoir if ��;� �=� � >� ��;�

��9� ��9� � ��

has maximum eigenvalue �? , 0

Wildlife

Transmission

Livestock

Single host modelSingle host modelSingle host modelSingle host model::::Patch - was a reservoir if +� , 0


Reservoirs of disease

Within a patch a pathogen may be

supported by:

• A single species alone - a source

species

• The interaction between species –

either species in isolation would

have no disease

• Both species in isolation

If any of these occur then the patch is

a reservoir of infection.

Wildlife

Transmission

Livestock

• Control of wildlife

alone does not work

• Necessary to control

the livestock species

• As cross species

transmission increases,

control threshold

increases


Livestock →wildlife

Pathogen

Excluded

�� 0


alone does not work





control threshold

increases



Pathogen

Excluded

�� 0.2


alone does not work





control threshold

increases



Pathogen

Excluded

�� 0.5

• Increase in connectivity leads to smoothing of threshold

• Control of both reduces threshold further



No dispersal of wildlife:

Pathogen

Excluded





Increasing dispersal

Pathogen

Excluded





Pathogen

Excluded

Control of both species


Wildlife → livestock

�� 0

Pathogen

Excluded

• Control of livestock alone does

not work

• Necessary to control wildlife

• If between species transmission

is0then we’re back to single

species model



�� 0.2

Pathogen

Excluded

• Control of livestock alone does

not work


• If between species transmission

is0then we’re back to single

species model

• Increasing between species

transmission leads to increased

threshold

• Reduction in contact between

species goes a long way to easing

the burden of control



�� 0.2

Pathogen

Excluded

• Control of livestock alone does not work


• If between species transmission is0then we’re back to single species model

• Increasing between species transmission leads to increased threshold

• Reduction in contact between species goes a long way to easing the burden of control

• Mixed strategy can reduce control burden

• If pathogen persists due to the

interaction of both species then

increased culling of one species

will ease the required control on

the other.

• Here reduction in livestock density

(modelled by culling) causes

reduction in wildlife threshold

• Reduction in intensity of farmed

animals may be an effective way to

reduce the burden of control on

wildlife populations such as

badgers.


Interaction supports the pathogen

Pathogen

Excluded


Summary of results

• Control of non-reservoir species will not result in

eradication of disease

• Mixed strategy may reduce control burden on a single

species

• Reduction in contact between species is key to reduction

of control

• Reduction in density of livestock through less intense

farming practices would also reduce burden of control on

wildlife

Further Research

• Closer look at the perturbation effect in the two host

model

• Logistic growth of species, and limits on control

parameters

• Different dispersal functions e.g. diffusion, density

dependent dispersal

• Communal resources and their impact on disease

persistence

ReferencesReferencesReferencesReferences

J.V. Greenman and A.S.Hoyle. Exclusion of generalist pathogens in multihost

communities. The American Naturalist, 172(4):576-584, (2008)

R.A. Lintott, R.A. Norman and A.S. Hoyle. The impact of increased dispersal

in response to disease control in patchy environments. Journal of Theoretical

Biology, in press.

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