dispersal, disturbance and disease spread · dispersal, disturbance and disease control rachel...
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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
Susceptible Infected
death death
birthinfection
death due to infection
���� � ����
����� � �����
��� � �� ��� ��� 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' !() , #!()*
Rare Invader AnalysisRare Invader AnalysisRare Invader AnalysisRare Invader Analysis
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
+� � �� ��
�� +� � ��
Rare Invader AnalysisRare Invader AnalysisRare Invader AnalysisRare Invader Analysis
With this notation, bounds on the maximum eigenvalue are :
min +�, +� $ �� $ max'+�, +�*
MathematicalMathematicalMathematicalMathematicalModel: Model: Model: Model: with control
Susceptible:
���
��� �� � ���� � ���
Infected:
� �
��� ���� � ���� � ����
Susceptible Infected
death death
birthinfection
death due to infection
Susceptible Infected
death death
birthinfection
death due to infection
�#�� � /� #� � � �/� #� �
�#�� � /� #� �� � �/� #� ��
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
Control Dependent EquilibriumControl Dependent EquilibriumControl Dependent EquilibriumControl Dependent Equilibrium
Controlled Patch Non-controlled Patch
Constant dispersal
Rate of Control
Disease Free Equilibrium
Control Dependent EquilibriumControl Dependent EquilibriumControl Dependent EquilibriumControl Dependent Equilibrium
Controlled Patch Non-controlled Patch
Saturating dispersal
Rate of Control
Disease Free Equilibrium
Control Dependent EquilibriumControl Dependent EquilibriumControl Dependent EquilibriumControl Dependent Equilibrium
Controlled Patch Non-controlled Patch
Linear dispersal
Rate of Control
Disease Free Equilibrium
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
Control Maps: Control Maps: Control Maps: Control Maps: 3 4
Pathogen
Excluded
Pathogen
Persistent
Saturating dispersal
Control Maps: Control Maps: Control Maps: Control Maps: 3 4
Pathogen
Excluded
Pathogen
Persistent
Linear dispersal
Control Maps: Control Maps: Control Maps: Control Maps: 3 4
Pathogen
Excluded
Pa
tho
ge
n
Pe
rsis
ten
t
Constant dispersal
Control Maps: Control Maps: Control Maps: Control Maps: 3 4
Pathogen
Excluded
Pathogen
Persistent
Saturating dispersal
Control Maps: Control Maps: Control Maps: Control Maps: 3 4
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
Parameter Dependence:Parameter Dependence:Parameter Dependence:Parameter Dependence:
increasing disease transmission
• As � increases so does the amount of control
Parameter Dependence:Parameter Dependence:Parameter Dependence:Parameter Dependence:
increasing death due to infection
• As γ7increases, less control is needed
Parameter Dependence:Parameter Dependence:Parameter Dependence:Parameter Dependence:
Switch in thresholds
Parameter Dependence:Parameter Dependence:Parameter Dependence:Parameter Dependence:
Switch in thresholds
Parameter Dependence:Parameter Dependence:Parameter Dependence:Parameter Dependence:
Switch in thresholds
Parameter Dependence:Parameter Dependence:Parameter Dependence:Parameter Dependence:
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
WildlifeWildlifeWildlifeWildlife----Livestock Model:Livestock Model:Livestock Model:Livestock Model:
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
WildlifeWildlifeWildlifeWildlife----Livestock Model:Livestock Model:Livestock Model:Livestock Model:
Livestock →wildlife
Pathogen
Excluded
�� � 0
• Control of wildlife
alone does not work
• Necessary to control
the livestock species
• As cross species
transmission increases,
control threshold
increases
WildlifeWildlifeWildlifeWildlife----Livestock Model:Livestock Model:Livestock Model:Livestock Model:
Livestock →wildlife
Pathogen
Excluded
�� � 0.2
• Control of wildlife
alone does not work
• Necessary to control
the livestock species
• As cross species
transmission increases,
control threshold
increases
WildlifeWildlifeWildlifeWildlife----Livestock Model:Livestock Model:Livestock Model:Livestock Model:
Livestock →wildlife
Pathogen
Excluded
�� � 0.5
• Increase in connectivity leads to smoothing of threshold
• Control of both reduces threshold further
WildlifeWildlifeWildlifeWildlife----Livestock Model:Livestock Model:Livestock Model:Livestock Model:
Livestock →wildlife
No dispersal of wildlife:
Pathogen
Excluded
• Increase in connectivity leads to smoothing of threshold
• Control of both reduces threshold further
WildlifeWildlifeWildlifeWildlife----Livestock Model:Livestock Model:Livestock Model:Livestock Model:
Livestock →wildlife
Increasing dispersal
Pathogen
Excluded
• Increase in connectivity leads to smoothing of threshold
• Control of both reduces threshold further
WildlifeWildlifeWildlifeWildlife----Livestock Model:Livestock Model:Livestock Model:Livestock Model:
Livestock →wildlife
Pathogen
Excluded
Control of both species
WildlifeWildlifeWildlifeWildlife----Livestock Model:Livestock Model:Livestock Model:Livestock Model:
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
WildlifeWildlifeWildlifeWildlife----Livestock Model:Livestock Model:Livestock Model:Livestock Model:
Wildlife → livestock
�� � 0.2
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
• Increasing between species
transmission leads to increased
threshold
• Reduction in contact between
species goes a long way to easing
the burden of control
WildlifeWildlifeWildlifeWildlife----Livestock Model:Livestock Model:Livestock Model:Livestock Model:
Wildlife → livestock
�� � 0.2
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
• 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.
WildlifeWildlifeWildlifeWildlife----Livestock Model:Livestock Model:Livestock Model:Livestock Model:
Interaction supports the pathogen
Pathogen
Excluded
WildlifeWildlifeWildlifeWildlife----Livestock Model:Livestock Model:Livestock Model:Livestock Model:
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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