doi:10.1016/S0025-5564(03)00041-5 
Copyright © 2003 Elsevier Science Inc. All rights reserved.
Spreading disease: integro-differential equations old and new
Jan Medlock
, 
and Mark Kot
Department of Applied Mathematics, University of Washington, P.O. Box 35240, Seattle, WA 98195-2420, USA
Received 20 June 2002;
revised 3 December 2002;
accepted 25 February 2003. ;
Available online 16 April 2003.
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Abstract
We investigate an integro-differential equation for a disease spread by the dispersal of infectious individuals and compare this to Mollison’s [Adv. Appl. Probab. 4 (1972) 233; D. Mollison, The rate of spatial propagation of simple epidemics, in: Proc. 6th Berkeley Symp. on Math. Statist. and Prob., vol. 3, University of California Press, Berkeley, 1972, p. 579; J. R. Statist. Soc. B 39 (3) (1977) 283] model of a disease spread by non-local contacts. For symmetric kernels with moment generating functions, spreading infectives leads to faster traveling waves for low rates of transmission, but to slower traveling waves for high rates of transmission. We approximate the shape of the traveling waves for the two models using both piecewise linearization and a regular-perturbation scheme.
Author Keywords: Biological invasions; Epidemics; Dispersal; Traveling waves; Integro-differential equations
Fig. 1. Traveling DC and DI waves. Waves were computed numerically for both the DC and DI models with the Laplace kernel. The initial condition is the step function on the left. Successive curves are separated by 5 time units. Numerical computations were done using a fourth-order Runge–Kutta scheme for the time derivative and fast Fourier transforms for the spatial convolution integral. The computations were for β=1, D=1, N=1, and . The computations were done on a 10 000-point spatial domain with a spatial step of 0.05 and a time step of 0.05. The initial condition is 1 on the 200 spatial points in the middle of the domain and 0 on the remainder. Only the right half of the spatial domain is shown.
Fig. 2. Wave speeds for various symmetric kernels. The wave speed,
c*, is plotted as a function of β
N with
D=1 for some symmetric kernels. The kernel parameters were chosen so that the DC wave-speed is the same for each kernel and so that each plot has a slope of 2: for both models with the Laplace kernel, , and with the Gaussian kernel, . Note that the DI model is faster for β
N<
D and the DC model is faster for β
N>
D for each kernel.
Fig. 3. Traveling-wave shapes. The numerically computed wave shapes are shown for both the DC and the DI model with the Laplace kernel. The top row shows the shapes of traveling waves computed both numerically and by the piecewise-linear and perturbation approximations. The middle row shows the absolute error between the approximations and the numerical result. The bottom row contains phase portraits,
I′ versus
I, for the numerical solutions and the approximations. In the figures, the solid line is the wave shape computed by numerically solving the full models, the dotted line is the piecewise-linear approximation, the dashed-dotted line is the perturbation expansion to O(1) and the dashed line, which is barely distinguishable from the solid line, is the perturbation expansion to O(1/
c2). The parameters match those of
Fig. 1. Here,
c=2. Even though
c is not large, the perturbation approximation is quite good. The data is taken from the same computation as in
Fig. 1. The wave shapes are for
t=100.
Fig. 4. Position of a threshold value for an accelerating wave. Position of the threshold value for the kernel , which has all moments but no moment generating function and satisfies condition (
B.5). The solid line is the asymptotic position of the threshold (
B.12), the pluses are the numerical values from solving the full DC model and the circles are the numerical values from solving the full DI model. The asymptotic result closely agrees with the numerical results. The initial condition for the numerics is a discrete approximation to the Dirac delta. The numerical results were shifted on the time axis to remove the transient effects of this approximation to the Dirac delta. The computations were for β=10,
D=1,
N=1, α=1, and
I0=1. The computations were done using an adaptive fourth–fifth-order Runge–Kutta scheme in time and fast Fourier transforms for the spatial convolution integral. The computations were done on a spatial domain of 100 000 points with spatial step 0.05. The initial condition is 20 at the midpoint of the domain and 0 elsewhere. The plot here is for the rightward traveling front relative to its starting position.
Table 1. Table of kernels
Some kernels, moment generating functions (MGFs), and wave speeds for the DC and DI models. The positive value for c* is cR, the right-wave speed, and the negative value of c* is cL, the left-wave speed. For the DI wave speed, cR corresponds to θ>0 and cL corresponds to θ<0. Note that the Laplace and Gaussian kernels are symmetric and that the exponential kernel is asymmetric. In each case the parameter α is positive. H(u) is the Heaviside function.
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