## Info

Time

200-600 km 6-12 Years

100-300 km 3-6 Years

100-300 km 3-6 Years

40-100 km Distance 1-3 Years

Figure 13.6. Fluctuations in the susceptible fox population density as a function of the passage of the rabies epizootic obtained from data from Centre National d'Etudes sur la Rage 1977. In dimensional terms, S0 is the uninfected susceptible population ahead of the epidemic wave. Note the periodic, but decreasing, fluctuations in S, which follow the main wavefront, as S tends to its steady state. (Redrawn from Macdonald 1980).

Figure 13.6. Fluctuations in the susceptible fox population density as a function of the passage of the rabies epizootic obtained from data from Centre National d'Etudes sur la Rage 1977. In dimensional terms, S0 is the uninfected susceptible population ahead of the epidemic wave. Note the periodic, but decreasing, fluctuations in S, which follow the main wavefront, as S tends to its steady state. (Redrawn from Macdonald 1980).

trated in Figure 13.6. There is a clear schematic difference in the behaviour behind the front in the two figures. The model (13.13) is only intended to cover the passage of an epidemic front. Clearly after the passage of the wavefront the susceptible population will start to increase again since the foxes find themselves in an environment which admits a larger carrying capacity. In other words, the timescale of the model (13.13) is considerably shorter than that associated with the oscillations in Figure 13.6. To include in our model the situation which obtains after the front has passed we must include a term for the fox reproduction. If we model this by a simple logistic growth, the equation for the susceptibles in place of the first of (13.13) becomes d S / S \

where B is the linear growth rate. With the same nondimensionalisation (13.2) as before, the model now becomes

91 d 21

where b = B/rSo, that is, the ratio of linear birth rate to the basic rate of infection per infective. Figure 13.7 shows an example of the resulting epidemic wave of susceptibles