integro-differential--equation models for infectious...
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Integro-differential–EquationModels for Infectious Disease
Jan MedlockYale University
Epidemiology and Public Health
16 March 2005
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Integro-differential–Equation Models for Infectious Disease 1
Introduction
I Many ecological and epidemiological systems are better modeled
in continuous time instead of discrete time.
I There are many continuous-time frameworks: reaction–diffusion
integral equations, integro-differential equations.
I Diffusion is a local operator, i.e., individuals can only influence
their immediate neighbors and may also present some difficulty in
matching with experimental data
I Integrals have been used instead to model non-local spatial
processes giving rise to integral and integro-differential equations.
I These models have all the advantages of the discrete-time
integrodifference equations: more flexibility in using dispersal
data, predicting faster wave speeds, and the possibility for
accelerating waves.
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Integro-differential–Equation Models for Infectious Disease 2
Spatial epidemic models
I Reaction–diffusion: Noble (1974); Bailey (1975); Murray, Staley,
& Brown (1986)
∂tS = −β(I + α∇2I)S + DS∇2S,
∂tI = β(I + α∇2I)S − γI + DI∇2I,
∂tR = γI + DR∇2R
I Integral equations: Thieme (1977, 1979); Diekmann (1978, 1979);
Rass & Radcliffe (1984, 1986, 2003)
S = −λS,
i(t, 0, x) = λS,
i(t, τ, x) = i(t− τ, 0, x),
λ(t, x) =∫ ∞
0
∫Ω
i(t, τ, ξ)A(τ, x, ξ) dξdτ
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Integro-differential–Equation Models for Infectious Disease 3
I Integro-differential equations: Kendall (1957, 1965) and Mollison
(1972); Medlock & Kot (2003)
I Integrodifference equations: Allen & Ernest (2002)
St+1 =∫
Ω
[St(y) + (µ + γ)It(y)− βIt(y)St(y)] k(x− y)dy,
It+1 =∫
Ω
[(1− µ− γ)It(y) + βIt(y)St(y)] k(x− y)dy
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Integro-differential–Equation Models for Infectious Disease 4
Outline
I Two integro-differential–equation models from epidemiology
I The SI model: Brief reviewI Distributed Contacts modelI Distributed Infectives modelI The movement process: Linear integro-differential equationsI Wave speedI Wave shape
I Conclusions
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Integro-differential–Equation Models for Infectious Disease 5
SI model: Brief review
I We divide the population into two groups:
I Susceptible individuals, S(t)I Infective individuals, I(t)
-S IλS
I Assumptions
I Population size is large and constant, S(t) + I(t) = KI No birth, death, immigration, or emigrationI No recovery or latencyI Homogeneous mixingI Infection rate is proportional to the number of infectives, i.e.,
λ = βI
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Integro-differential–Equation Models for Infectious Disease 6
-S IβIS
I A pair of ordinary differential equations describes this model
(Ross, 1911):
S′(t) = −βI(t)S(t),
I ′(t) = βI(t)S(t).
I With constant population size, K = S(t) + I(t), this reduces to
I ′(t) = βI(t)(K − I(t)).
I The solution is the logistic curve,
I(t) =I(0)K
I(0) + (K − I(0))e−βKt.
t0
K
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Integro-differential–Equation Models for Infectious Disease 7
Spatial spread: Two different non-local mechanisms
From Brown, J.K.M. & Hovmøller, M.S., Science, 297 537.
From http://www.hort.uconn.edu/ipm/veg/htms/potblpic.htm.
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Integro-differential–Equation Models for Infectious Disease 8
Distributed Contacts (DC)
I The governing equations (Kendall, 1957, 1965; Mollison, 1972):
∂tS = −β
(∫Ω
k(x− y)I(y, t) dy
)S,
∂tI = β
(∫Ω
k(x− y)I(y, t) dy
)S.
I β is the infection rateI k(u), the kernel, is the contact distribution
I Assumptions: K = S + I is constant and Ω = R, give
∂tI = β
(∫R
k(x− y)I(y, t) dy
)(K − I).
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Integro-differential–Equation Models for Infectious Disease 9
From http://www.phls.wales.nhs.uk/measlgen.htm.
From Cliff, A.D., Haggett, P., Ord, J.K. & Versey, G.R., Spatial Diffusion, 1981.
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Integro-differential–Equation Models for Infectious Disease 10
Distributed Infectives (DI)
I The governing equations (Fedotov, 2000; Medlock & Kot, 2003;
Lutscher et al., 2004):
∂tS = −βIS −DS + D
∫Ω
k(x− y)S(y, t) dy,
∂tI = βIS −DI + D
∫Ω
k(x− y)I(y, t) dy.
I β is the infection rateI D is the dispersal rateI k(u), the kernel, is the dispersal distribution
I Assumptions: K = S + I is constant and Ω = R, give
∂tI = βI(K − I)−DI + D
∫R
k(x− y)I(y, t) dy.
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Integro-differential–Equation Models for Infectious Disease 11
The movement process:Linear integro-differential equations
I N(x, t) is the density of a population at position x and time t
I At rate D, individuals move to a new location instantaneously
I k(x− y) is the proportion of individuals moving from y to x
∂tN = D
∫R
k(x− y)N(y, t) dy −DN
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Integro-differential–Equation Models for Infectious Disease 12
Position jump process or kangaroo process(Othmer et al., 1988)
I An individual starts at x = 0 at t = 0
I He waits an exponentially-distributed time (with parameter D)...
I ...and then jumps to a new position y that is governed by the
distribution k(x− y)
∂tN = D
∫R
k(x− y)N(y, t) dy −DN
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Integro-differential–Equation Models for Infectious Disease 13
Both models have traveling wave solutions
-100 -80 -60 -40 -20 0x
0
K/2
K
I
-100 -80 -60 -40 -20 0x
0
K/2
K
I
DC
DI
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Integro-differential–Equation Models for Infectious Disease 14
Wave speed
I Distributed Contacts
∂tI = β
(∫R
k(x− y)I(y, t) dy
)(K − I)
Using the traveling-wave coordinate, z = x + ct,
cI ′ = β
(∫R
k(z − y)I(y) dy
)(K − I).
Linearizing about I = 0,
cI ′ = βK
(∫R
k(z − y)I(y) dy
).
Assuming I = Ae−θz,
cθ = −βKM(θ),
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Integro-differential–Equation Models for Infectious Disease 15
where
M(θ) =∫
Rk(u)eθu du.
For a constant speed traveling wave, k(x) ≤ Ae−γ|x|, which
implies M(θ) exists for |θ| < γ.
The critical speeds are
cR = −βK supθ>0
M(θ)θ
, cL = −βK infθ<0
M(θ)θ
.
Parametrically,
c∗ = −βKM ′(θ∗),
M(θ∗) = θ∗M ′(θ∗).
Aronson (1977) showed this is the asymptotic speed.
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Integro-differential–Equation Models for Infectious Disease 16
I Distributed Infectives
∂tI = βI(K − I)−DI + D
∫R
k(x− y)I(y, t) dy
The same procedure gives
c∗ = −DM ′(θ∗),
βK = D [1−M(θ∗) + θ∗M ′(θ∗)] .
Lutscher et al. (2004) showed this is the asymptotic speed.
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Integro-differential–Equation Models for Infectious Disease 17
0 0.5 1 1.5 2β K
0
1
2
3
4
c*
0 0.5 1 1.5 2β K
0
1
2
3
4
c*LaplaceGaussiandiffusion
DC
DI
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Integro-differential–Equation Models for Infectious Disease 18
0 0.5 1 1.5 2β K
0
1
2
3
4
c*
DCDI, LaplaceDI, GaussianDI, diffusion
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Integro-differential–Equation Models for Infectious Disease 19
A perturbation scheme for the wave shape
The key to the perturbation scheme is expanding the convolution
k ∗ I =∫
Rk(y)I(z − y) dz
for large c.
Let ξ = z/c and h(ξ) = I(z). Then
k ∗ I =∫
Rk(y)I(z − y)dy
=∫
Rk(y)h
(ξ − y
c
)dy
=∫
Rk(y)
[h(ξ)− y
ch′(ξ) + O
(1c2
)]dy
=h(ξ)− M ′(0)c
h′(ξ) + O(
1c2
).
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Integro-differential–Equation Models for Infectious Disease 20
Distributed Infectives
cI ′ = βI(K − I)−DI + D
∫R
k(y)I(z − y) dy
with I(−∞) = 0, I(+∞) = K, I(0) = K/2.
Letting ξ = z/c and h(ξ) = I(z) gives
h′ = βh(K − h)−D
[M ′(0)
ch′ + O
(1c2
)].
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Integro-differential–Equation Models for Infectious Disease 21
Expanding h = h0 + 1ch1 + O
(1c2
),
h′0 = βh0(K − h0),
h′1 = βh1(K − 2h0)−DM ′(0)h′0,
with h0(−∞) = 0, h0(+∞) = K, h0(0) = K/2,
and h1(−∞) = h1(+∞) = h1(0) = 0.
Then
h0(ξ) =KeβKξ
1 + eβKξ,
h1(ξ) = DβK2M ′(0)ξeβKξ
(1 + eβKξ)2.
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Integro-differential–Equation Models for Infectious Disease 22
-30 -15 0 15 30z
0
0.5
1
N
Exponential kernel
-30 -15 0 15 30z
0
0.5
1
N
Laplace kernel
-30 -15 0 15 30z
-0.1
0
0.1
Abs.
Erro
r
-30 -15 0 15 30z
-0.02
-0.01
0
0.01
0.02
Abs.
Erro
r
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Integro-differential–Equation Models for Infectious Disease 23
Asymptotic speed
For an accelerating invasion k(x) > Ae−γ|x|, we can compute the
wave asymptotically.
I Starting with the linearized DC equation
∂tI = βK
∫R
k(x− y)I(y, t)dy, I(x, 0) = I0δ(x),
taking the Fourier transform
∂tI = βKk(ω)I , I(ω) = I0,
gives
I(ω, t) = I0eβKk(ω)t.
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Integro-differential–Equation Models for Infectious Disease 24
Inverting gives
I(x, t) =I0
2π
∫R
e−ıωxeβKk(ω)tdω ≈ I0eβKtk(x) as |x| → ∞,
subject to some technical conditions.
I Likewise for the DI model
I(x, t) ≈ I0eβKtk(x) as |x| → ∞.
Setting some threshold value, I, gives
x ≈ k−1
(I
I0e−βKt
),
the position of the threshold as a function of time.
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Integro-differential–Equation Models for Infectious Disease 25
0 1 2 3t
0
250
500
750
1000
x
asymptoticDCDI
front position vs. time
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Integro-differential–Equation Models for Infectious Disease 26
Time-periodic coefficients
∂tI = β(t)I(K(t)− I)−D(t)I + D(t)∫
Rk(x− y, t)I(y, t) dy
Using the same linearization procedure, the speed is given by
βK = D −DM(θ) + θD∂θM(θ),
c = −D∂θM(θ),
where
f =1T
∫ T
0
f(t) dt.
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Integro-differential–Equation Models for Infectious Disease 27
0 10 20 30 40 50t
-2
0
2
4
6
spee
dspeed vs. time
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Integro-differential–Equation Models for Infectious Disease 28
Numerics: IDEs are better than PDEs!
I Integral dispersal is much more stable than diffusion, which allows
much larger time steps
I In general,
∂t
u1...
un
= f
u1
...
un
,
k1 ∗ u1 . . . km ∗ u1... ...
k1 ∗ un . . . km ∗ un
.
I Trapezoid rule and Simpson’s rule are O(N2).
I FFT is O(N log N) (Andersen, 1991)
F(ki ∗ uj) = F(ki)F(uj).
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Integro-differential–Equation Models for Infectious Disease 29
I Algorithm
I Compute F(ki) at the beginning.I Each time step, compute F(uj) and F−1 [F(ki)F(uj)].I Step in time with Runge–Kutta scheme.
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Integro-differential–Equation Models for Infectious Disease 30
Conclusions
I There are more flexible frameworks than reaction–diffusion for
continuous-time models.
I Integro-differential equations can be used with empirical dispersal
data, give faster speeds than r–d, and can give accelerating waves.
I Traveling wave speed is sensitive to the form of the model; care
should be used.
I Perturbation techniques can be used to compute approximate
solutions to the IDEs.
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Integro-differential–Equation Models for Infectious Disease 31
Problems
I Instead of space, x represents some other characteristic.
I Estimating kernels from dispersal data.
I The inverse problem: Given a traveling wave, what is the kernel?