11 Discussion of the Modelling, Impacts and Management of Epidemics
11.2 What Controls Epidemic Behaviour?
Model epidemic wave speed has been found to depend essentially on three parameters:
the infection rate, the diffusion coefficient and the length of the latent period. Two other factors have surprisingly little effect on wave speed. These are the duration of the infectious phase and population density.
11.2.1 Infectious Period
Duration of the infection has little effect on the mature waves speed, although it has significant effects on its initiation. The effect is so small that the analytical solution derived under a continuous turnover of infected individuals works just as well for the
version with a fixed length infectious period. The infectious period length does become a significant constraint when β is small. The model has provided unexpected evidence that the infectious period only applies to a short period towards the end of the infection.
This has significant effect on the pattern of origin of the epidemic.
11.2.2 Population Density
Change in population density actually has as much effect on wave speed as do changes in viral transmission rate in the model. However, changes of an order of magnitude are required to have large effects on wave velocity, and while this may be possible for viral transmission rate such large changes are not consistent with known changes in
population density in Australia. One need only examine the near constant rate of spread of either epidemic in Western Australia to see how constant the speed is over a couple of thousand km, a range in which there must surely be variation in population density.
Populations in Western Australia declined between 1994 and 1997, for reasons other than the epidemics (Cochrane 1999).
It is also possible that transmission is population density independent, in which case there would be no response at all to changes in population density.
Because of non-linear effects of population density it is possible that the degree of dispersal of the pilchard population could impact on the averaged rate of transmission.
A population that was denser in some local areas and less dense in others would be less effective at transmission than would a population that travelled across all areas. The weak response to population density means this effect is likely to be small, but in other epidemics which are more responsive to population density, this factor could be
significant.
11.2.3 Viral Transmission Rate
The model wave speed shows a weak but significant dependence on the viral
transmission rate β. Sensitivity declines as the transmission rate increases. At large values of this parameter wave speed almost ceases to respond to further increases in transmission rate.
The viral transmission rate is only weakly constrained and so may vary by orders of magnitude; indeed there is evidence that it does so vary in that the lesions on infected fish's gills in 1998/9 were much fewer than in 1995. So it may remain a significant control on changes in the epidemic's behaviour provided it is not very large.
The population mixing rate does not affect moderate viral transmission rates but it does impose an ultimate cap on the maximum value of the transmission rate. The value of the cap depends upon fish behaviour and so is unknown. Sensitivity analysis shows that the rate at which the epidemic wave propagates in this model becomes increasingly insensitive to the transmission coefficient as this becomes large. Therefore our ignorance as to the exact value of the maximum transmission is not important even when transmission rate is large, so long as the maximum is large.
Behaviour at the origin is probably a much stronger constraint on the viral transmission parameter. Large values of this parameter are inconsistent with mortality occurring at
intervals near the origin for two-three weeks. Low values produce a single smooth peak, which is similar to the mature wave. Again, this is inconsistent with a long initial period of mortality.
11.2.4 Diffusion Coefficient
The model wave speed responds strongly to the diffusion coefficient, in fact response is controlled by the diffusion of infected fish alone. The control of epidemic speed by the behaviour of infected individuals is an important insight that is applicable epidemic models in general. However, we have generally used a single diffusion coefficient for this model. We have no data to separate the behaviours of infected and uninfected fish and as no detectable effects occur when diffusion is at a different rate for uninfected fish Diffusion coefficient can be constrained with reference the briefness of local mortality in the stabilised epidemic wave. We are able to find a maximum diffusion coefficient because as this becomes large infected fish spread out before dying. The degree to which this occurs is surprisingly insensitive to the latent period over the range of 1 to a few days. We find that the 40 km d-1 eastbound 1995 wave has a maximum D of about 200 km2 d-1, the 20 km d-1 westbound wave a D of 100 km2 d-1 and the 10 km d-1 1998/9 wave has a maximum D of only 30 km2 d-1.
The movement of fish alone can easily generate these values and so no vector is required to explain them. The pattern of infection, killing adults but usually avoiding juveniles, also suggests that it is fish-to- fish contact which spreads the disease.
However, the fact that D can be generated without a vector does not mean that birds or other organisms do not sometimes transport the virus.
11.2.5 The Latent Period
Epidemic wave speed in this model with its fixed length latent period is far more sensitive to the length of the latent period than is the model with a continuous turnover formulation. This model responds to the square root of the latent period, while the traditional model responds to the fourth root (Yachi et al. 1988).
Because wave speed becomes essentially independent of β for large β, we have a maximum latent period that is consistent with the maximum diffusion coefficient and wave speed. For the 40, 20 and 10 km d-1 waves this latent period is four, eight or 12 days. However, we lack direct data for the length of the latent period and infectious period other than observed lesions present for two to 4 days before mortality
(Whittington et al. 1997). Direct data should be obtainable experimentally, and if it were available it would be a highly valuable test of, and constraint on, the model.
The latent period is a major factor in the pattern of recurrent persistent mortality during the initial phase of the epidemic. This recurrence pattern would tend to support latent periods of around the 4 days that can still reproduce the observed wave speed.
The existence of the latent period gives time for infected fish to mix among schools before they become infectious. As a result the details of the mixing of fish populations are of less importance as a control on the epidemic's spread than is fish-to- fish infection.
This result greatly simplifies modelling, since we can dispense with models of the local structure and dynamics of the fish population.
11.2.6 Mortality
The proportion of mortality, as opposed to recovery, of infected individuals does not impact on the epidemic's behaviour in terms of the speed and local persistence of the mortality. It is of great significance for its longer-term impact (see 11.4). The model produces very high levels of infection and so the epidemic's long-term impact is largely controlled by the fraction of those infected individuals that die. We do not have a good handle on this value. The average value in Western Australia varied from 15% in 1995 to 60% in 1999 and showed very large local variation. In 1995 in South Australia mortality was 60% of the population. This figure is the fraction of the population killed, but with infection levels of over 90% the fraction of the population killed is not very different from the fraction of infected fish killed.