Download Modeling talk Sandland Final

Combining empirical and
theoretical approaches to better
understand the persistence of
waterfowl disease in the Upper
Mississippi River
James Peirce, Gregory Sandland,
Roger Haro and Barbara Bennie
Departments of Biology and Mathematics
University of Wisconsin – La Crosse
Invasive species
•  Over 120 billion dollars spent on invasive species each
year
•  Many species introductions are aquatic
•  Over 50 species introduced in the Great Lakes
Dreissena polymorpha
Hypophthalmichthys molitrix
Bithynia tentaculata
•  Native of Europe
•  Introduced into the Great Lakes
in the 1880s
•  Detected in the Mississippi in 2002
•  Adults reach 12-15 mm in length
•  3 year lifespan
•  Disrupts the integrity of native
aquatic communities
+
And if that wasn t bad enough……
Cyathocotyle bushiensis
Sphaeridiotrema pseudglobulus
Sphaeridiotrema pseudglobulus
Leyogonomus polyoon
Trematode lifecycle
Pathology
•  Found in the lower intestines and
cecae
•  Fluke attachment and penetration
causes severe tissue damage
•  Extreme hemorrhaging and
plaque formation 5-7 days postinfection
•  Death in 5-9 days
Complexities of the system require
numerous approaches
Understanding
the system
Complexities of the system require
numerous approaches
Field
Understanding
the system
Complexities of the system require
numerous approaches
Field
Experiments
Understanding
the system
Complexities of the system require
numerous approaches
Field
Experiments
Understanding
the system
Mathematical
models
Complexities of the system require
numerous approaches
Field
Experiments
Understanding
the system
Mathematical
models
Introduction to Epidemiological Models
Bithynia-Waterfowl Model
A simple idea
A simple model
Analysis of the simple model
Motivation
The primary reason for studying infectious disease is to
improve control and ultimately eradicate the infection from the
population.
Mathematical models allow us to
provide an ideal world in which individual factors can be
examined in isolation
optimize the use of limited resources
The results can target control methods more efficiently.
James Peirce
Epidemiological Models
Introduction to Epidemiological Models
Bithynia-Waterfowl Model
A simple idea
A simple model
Analysis of the simple model
The SIR model
Kermack and McKendrick (1927) developed a model for a
single pathogen that causes illness for a period of time followed
by recovery.
The population is divided into three disjoint categories
S = susceptible - previously unexposed to the pathogen
I = infected - currently colonized by the pathogen
R = recovered - successfully cleared the infection
James Peirce
Epidemiological Models
Introduction to Epidemiological Models
Bithynia-Waterfowl Model
A simple idea
A simple model
Analysis of the simple model
Disease dynamics
Parameter
µ
β
γ
Meaning
natural mortality rate ≈ 1/lifespan
transmission rate from S to I
recovery rate ≈ 1/(length of infection)
James Peirce
Epidemiological Models
Introduction to Epidemiological Models
Bithynia-Waterfowl Model
A simple idea
A simple model
Analysis of the simple model
The mathematical model
dS
dt
dI
dt
dR
dt
= µ(S + I + R) − β
= β
SI
− µS
N
SI
− γI − µI
N
= γI − µR
After we rescale the values s = S/N, i = I/N, and r = R/N
ds
dt
di
dt
dr
dt
= µ − βsi − µs
= βsi − (γ + µ)i
= γi − µr
James Peirce
Epidemiological Models
Introduction to Epidemiological Models
Bithynia-Waterfowl Model
A simple idea
A simple model
Analysis of the simple model
Mathematical analysis
The sir equations have two equilibrium solutions
µ − βsi − µs = 0
βsi − (γ + µ)i = 0
γi − µr
= 0
A disease-free equilibrium at (s∗ , i ∗ , r ∗ ) = (1, 0, 0) and
An endemic equilibrium at
!
"
"
!
γ+µ µ
β
∗ ∗ ∗
∗
∗
(s , i , r ) =
,
− 1 , 1 − (s + i )
β
β γ+µ
James Peirce
Epidemiological Models
Introduction to Epidemiological Models
Bithynia-Waterfowl Model
A simple idea
A simple model
Analysis of the simple model
The basic reproduction number
The basic reproduction number R0 =
as
β
can be thought of
γ+µ
β : the transmission probability of S to I
1
γ+µ
: the average time spent infectious
R0 = the number of new infections resulting from an
infected individual being introduced into an entirely
susceptible population.
The basic reproduction number suggests methods of
controlling the spread of the pathogen.
James Peirce
Epidemiological Models
Concept Map
The basic reproduction number R0
Model Analysis
Introduction to Epidemiological Models
Bithynia-Waterfowl Model
Bithynia-Waterfowl Model
χ1
χ2
IB1
ρ
SB
IB2
1−ρ
β
J1
S1
A
γ
IA
James Peirce
α
Epidemiological Models
I1
β
Introduction to Epidemiological Models
Bithynia-Waterfowl Model
Concept Map
The basic reproduction number R0
Model Analysis
The basic reproduction number R0
For an epidemiological model, the basic reproduction number
R0 =
the average number of new infections created by a single infected snail or waterfowl introduced to an entirely
susceptible population.
R0 > 1 means that the parasite infection will persist.
In the bithynia-waterfowl model
!
"1/3
R0 = R0,A R0,S R0,B
,
where each of the three values on the right-side of the equation
represents a stage in the parasite life cycle.
James Peirce
Epidemiological Models
Introduction to Epidemiological Models
Bithynia-Waterfowl Model
Concept Map
The basic reproduction number R0
Model Analysis
Stage 1
R0,1 =
!
m + µJ
m + µJ + b
"!
γ
γ + µ + da
"!
α
µ + dA
"
During this period, the parasite develops from the miracidial
stage to the free-swimming cercarial form. R0,1 is the average
number of new infections in the snail host produced a single IA
after it has survived the amplification stage.
James Peirce
Epidemiological Models
Introduction to Epidemiological Models
Bithynia-Waterfowl Model
Concept Map
The basic reproduction number R0
Model Analysis
Stage 2
R0,2 =
βωK
µ + d + τ ωSB∗
is the average number of new infections produced by a single I1
in the waterfowl populations during its infectious period.
Stage 3
R0,B =
τ (1 − ρ)χ2
τ ρχ1
+
µB + e1 + k1 µB + e2 + k2
Given that exposed waterfowl can fall into two infectious
classes, R0,B is the average number of new low- and
high-intensity infections produced by a single I1 in the
susceptible bird sub-population during its infectious period.
James Peirce
Epidemiological Models
Introduction to Epidemiological Models
Bithynia-Waterfowl Model
Concept Map
The basic reproduction number R0
Model Analysis
The value and sensitivity of R0
Parasite
C. bushiensis
S. globulus
R0
2.820
5.413
Parameter name, p
α, transmission rate IA → S1
β, transmission rate I1 → SB
K , carrying capacity of adult S1
χ1 , transmission rate IB1 → S1
ω, rate of foraging of SB
τ , hours per day foraging
dA , mortality rate of IA from infection
James Peirce
A 1% change in p results in
a % change in R0 of
1/3
1/3
1/3
0.3144
0.3068
0.3068
-0.3030
Epidemiological Models
Introduction to Epidemiological Models
Bithynia-Waterfowl Model
Concept Map
The basic reproduction number R0
Model Analysis
Other Sensitivity Analysis
We write the host-parasite system in the form
d !x !
= f (t, !x (t, !q ), !q )
dt
where !q is the vector of parameter values. The sensitivity of the
solution !x with respect the !q is a solution to the differential
equation
∂!f ∂!x
∂!f
d ∂!x
=
+
dt ∂ !q
∂!x ∂ !q
∂ !q
James Peirce
Epidemiological Models
Introduction to Epidemiological Models
Bithynia-Waterfowl Model
James Peirce
Concept Map
The basic reproduction number R0
Model Analysis
Epidemiological Models
Introduction to Epidemiological Models
Bithynia-Waterfowl Model
Concept Map
The basic reproduction number R0
Model Analysis
Control methods
Strategies to reduce R0 :
prevent hosts from feeding in areas where infected snails
are abundant.
introduce physical barriers
the size of Pool 7 makes this difficult
reduce population size of lesser scaup.
other waterfowl species could replace as hosts
target and reduce Bithynia population
chemicals are difficult to deliver and have the potential to
have adverse effect on non-target species
SO . . . no one thing will work. . .
The ability to control parasite persistence in the UMR will
require multi-faceted approaches.
James Peirce
Epidemiological Models
Concept Map
The basic reproduction number R0
Model Analysis
Introduction to Epidemiological Models
Bithynia-Waterfowl Model
Bithynia-Waterfowl Model
χ1
χ2
IB1
ρ
SB
IB2
1−ρ
β
J1
S1
A
γ
IA
James Peirce
α
Epidemiological Models
I1
β