ABSTRACT

The disease models introduced in Chapter 7 assume that the population is “well-mixed”. That is, all individuals in the community interact in such a way that any infected individual is equally likely to infect every other member of the community. For very many diseases this is plainly not the case. In Section 7.4 we saw that such models could only give an approximate description of transmission of SARS in Hong Kong, and only if the infection rate was allowed to vary. However, since SARS only affected a small proportion of the population, the actual contact between infected individuals and the general community was limited to a particular sub-community. In Section 10.2 we will introduce a better model which accounts for actual contact between individuals in the wider community. In this section we consider another special type of disease transmission process. In this case, individuals are subject to some infectious agent, but the individuals are arranged spatially — that is, they can only infect others in the same, or neighbouring geographical region. This model is particularly useful for modelling the spatial spread of diseases

over large regions, and, in particular the spread of diseases such as bubonic plague. Figure 8.1 depicts historical spread of bubonic plague through Europe in the fourteenth century. The model is essentially an extension of the ones presented in Chapter 7 for disease transmission. For simplicity we consider the S and I equations of the SIR model and add a term D ∂

∂x2 for spatial diffusion,

∂S

∂t = −SI +D∇2S (8.1)

∂I

∂t = SI − λI +D∇2I (8.2)

λ = a

rS0 , (8.3)

where D is the rate of diffusion, and the parameter λ is introduced to simplify the above expression. As we saw in the previous chapter, the SIR model maintains a constant total population N , and hence, here we ignore the R population as it can be deduced directly from Eqns. (8.1) and (8.2). Moreover,

Figure 8.1: Plague in Europe. Spread of bubonic plague in fourteenth century Europe can clearly be seen to propagate as a wave across the continent, originating in a port in northern Italy and spreading throughout continental Europe and Britain. In medieval Europe, bubonic plague was fatal within 2 or 3 days in roughly 80% of cases. Hence, during the period 1347-1351, plague spread across Europe and killed at least one quarter of the entire population (conservative estimates tend to exceed 20 million). (Image obtained from https://commons.wikimedia.org/wiki/File:Bubonic plague map 2.png and freely distributed under the GNU Free Documentation License (GFDL)).)

Figure 8.2: Approximate diffusitivity across fourteenth century Europe. Terrain roughness is used as an approximate estimate of diffusitivity. The second derivative of altitude is used to estimate the difficulty of travel in a given location. Flat terrain is easier to travel over and rough terrain is more difficult.