Simple Model for the Spatial Spread of an Epidemic

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We consider here a simpler version of the epidemic model discussed in detail in Chapter 10, Volume I, Section 10.2. We assume the population consists of only two populations, infectives I (x, t ) and susceptibles S(x, t ) which interact. Now, however, I and S are functions of the space variable x as well as time. We model the spatial dispersal of I and S by simple diffusion and initially consider the infectives and susceptibles to have the same diffusion coefficient D. As before we consider the transition from susceptibles to infectives to be proportional to rSI, where r is a constant parameter. This form means that rS is the number of susceptibles who catch the disease from each infective. The parameter r is a measure of the transmission efficiency of the disease from infectives to susceptibles. We assume that the infectives have a disease-induced mortality rate al; 1/a is the life expectancy of an infective. With these assumptions the basic model mechanism for the development and spatial spread of the disease is then d S

— = rIS - al + DV21, d t where a, r and D are positive constants. These equations are simply (10.1) and (10.2) in Chapter 10, Section 10.2, Volume I with the addition of diffusion terms. The problem we are now interested in consists of introducing a number of infectives into a uniform population with initial homogeneous susceptible density S0 and determining the geotemporal spread of the disease.

Here we consider only the one-dimensional problem; later we present results of a two-dimensional study. We nondimensionalise the system by writing i *=-, S*=-, x *=( s y2 x,

where S0 is a representative population and the model (13.1) becomes, on dropping the asterisks for notational simplicity, dS = -iS+d2S

The three parameters r, a and D in the dimensional model (13.1) have been reduced to only one dimensionless grouping, k. The basic reproduction rate (cf. Chapter 10, Volume I, Section 10.2) of the infection is 1/k; it has several equivalent meanings. For example, 1/k is the number of secondary infections produced by one primary infective in a susceptible population. It is also a measure of the two relevant timescales, namely, that associated with the contagious time of the disease, 1 /(rS0), and the life expectancy, 1 /a, of an infective.

The specific problem we investigate here is the spatial spread of an epidemic wave of infectiousness into a uniform population of susceptibles. We want to determine the conditions for the existence of such a travelling wave and, when it exists, its speed of propagation.

We look for travelling wave solutions in the usual way (cf. Chapter 1) by setting

I (x, t) = I (z), S(x, t) = S(z), z = x - ct, (13.4)

where c is the wavespeed, which we have to determine. This represents a wave of constant shape travelling in the positive x-direction. Substituting these into (13.3) gives the ordinary differential equation system

I" + cI' + I(S - k) = 0, S" + cS'- IS = 0, (13.5)

where the prime denotes differentiation with respect to z. The eigenvalue problem consists of finding the range of values of k such that a solution exists with positive wave-speed c and nonnegative I and S such that

I (-&>) = I(rn) = 0, 0 < S(-<x>) < S(<x>) = 1. (13.6)

Figure 13.1. Travelling epidemic wave of constant shape, calculated from the partial differential equation system (13.3) with X = 0.75 and initial conditions (that is, with compact support) compatible with (13.6). Here a pulse of infectives (I) moves into a population of susceptibles (S) with speed c = 1 which in dimensional terms from (13.2) is (rSoD)1/2 which agrees with the analytical wavespeed (13.11) with X = a/rS0 = 0.75.

Figure 13.1. Travelling epidemic wave of constant shape, calculated from the partial differential equation system (13.3) with X = 0.75 and initial conditions (that is, with compact support) compatible with (13.6). Here a pulse of infectives (I) moves into a population of susceptibles (S) with speed c = 1 which in dimensional terms from (13.2) is (rSoD)1/2 which agrees with the analytical wavespeed (13.11) with X = a/rS0 = 0.75.

The conditions on I imply a pulse wave of infectives which propagates into the uninfected population. Figure 13.1 shows such a wave; Figure 13.5 below, which is associated with the spread of a rabies epidemic wave, is another example, although there, only the infectious population I diffuses.

The system (13.5) is a fourth-order phase space system. We can determine the lower bound on allowable wavespeeds c by using the same technique we employed in Chapter 13, Volume I, Section 13.2 in connection with wave solutions of the Fisher-Kolmogoroff equation. Here we linearise the first of (13.5) near the leading edge of the wave where S ^ 1 and I ^ 0 to get

solutions of which are

Since we require I (z) ^ 0 with I (z) > 0 this solution cannot oscillate about I = 0; otherwise I (z) < 0 for some z. So, if a travelling wave solution exists, the wavespeed c and X must satisfy c > 2(1 - X)1/2, X< 1. (13.9)

If X > 1 no wave solution exists so this is the necessary threshold condition for the propagation of an epidemic wave. From (13.2), in dimensional terms the threshold condition is a

This is the same threshold condition found in Chapter 10, Volume I, Section 10.2 for an epidemic to exist in the spatially homogeneous situation.

With our experience with the Fisher-Kolmogoroff equation we expect such travelling waves computed from the full nonlinear system will, except in exceptional conditions, evolve into a travelling waveform with the minimum wavespeed c = 2(1 — X)l/2. In dimensional terms, using (13.2), the wave velocity, V say, is then given by

The travelling wave solution S(z) cannot have a local maximum, since there S' = 0 and the second of (13.5) shows that S" = IS > 0, which implies a local minimum. So S(z) is a monotonic increasing function of z. By linearising the second equation of (13.5) as z ^œ, where S = 1 — s, with s small, we have s" + cs' — I = 0, a a which, with I(z) from (13.8), shows that

S(z) - 1 — O (exp[{—c ± [c2 — 4(1 — X)]1/2}z/2]J

The threshold result (13.10) has some important implications. For example, we see that there is a minimum critical population density Sc = a/r for an epidemic wave to occur. On the other hand for a given population S0 and mortality rate a, there is a critical transmission coefficient rc = a/S0 which, if not exceeded, prevents the spread of the infection. With a given transmission coefficient and susceptible population we also get a threshold mortality rate, ac = rS0, which, if exceeded, prevents an epidemic. So, the more rapidly fatal the disease is, the less chance there is of an epidemic wave moving through a population. All of these have implications for control strategies. The susceptible population can be reduced through vaccination or culling; we discuss this and immunity effects below. For a given mortality and population density S0, if we can, by isolation, medical intervention and so on, reduce the transmission factor r of the disease, it may be possible to violate condition (13.10) and hence again prevent the spread of the epidemic. Finally with a/(rS0) < 1 as the threshold criterion we note that a sudden influx of susceptible population can raise S0 above Sc and hence initiate an epidemic.

Here we have considered only a simple two-species epidemic model. We can extend the analysis to a three-species SIR system. It becomes, of course, more complicated. In Sections 13.5-13.9 we discuss in some detail such a model for the current European epidemic of rabies.

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