8 Parameter Sensitivities of the Speeds and Duration of Model Mortality Waves
8.1 Wave Speed
We earlier derived an analytical solution of the continuous turnover model based on work by Yachi et al. (1989)
V = √(2D{√[4σβS0)] – (σ + α)}) (8.1)
As we showed earlier, this formula gives a very good description of the wave's speed as generated numerically in this model.
We thus are able to see exactly what parameters this model's predicted wave speed depends upon. Essentially it depends upon the square root of D, and the fourth roots of βS0 and σ the rates of infection of susceptible and of the onset of the infectious period among infected individuals. This indicates that inclusion of the latent period gives a much lower sensitivity to transmission rate and population density than in simple SIR models.
In the previous chapter we derived an analysis method which allows rapid evaluation of the wave speed generated by the model version containing a fixed latent period length.
In this chapter we use this method to show how the wave speed depends upon parameter values. We then compare the results obtained by the analytical method to those obtained numerically to analyse models with a fixed latent and infectious period lengths and show that the analysis works well for the parameter ranges appropriate to the pilchard epidemic.
In chapter 3 we showed that the epidemic wave's observed speed varies from about 10 to 40 km d-1. We wish to find an appropriate parameter space for waves with
velocities in this range based on the three parameters singled out above. The analytical solution depends only upon the latent period length, b, the rate of turnover of infectious individuals, α, infection rate β and susceptible population S0 and upon diffusion.
The wave velocity is largely independent of the rate of turnover of infected individuals (mortality or recovery). We used the analytical solution to find the effect of varying α + µ from 0.125 to 0.5 d-1 that is an infective period of 8 to 2 days, which is the extremes
of what might be consistent with the observed timing of lesion formation. We found the wave speed varied from 31.8 to 31.1, about 2%. Effects of changes in these parameters become larger for small values of b, but at such small values of b produce unrealistic local patterns of mortality (see 8.2).
The epidemic wave's development is independent of the fate of fish post infection.
Whether infected fish eventually recover or die is unimportant at this stage, although this of course determines the longer term impact. We therefore set p equal to 0, so that all infected fish die. This means model mortality results presented here describe mortality not as a fraction of population but of total mortality. In terms of population, the mortality must be multiplied by p, which lies between 0.6 and 0.15.
Because the wave speed shows little variation with the parameterisation of the turnover rate of infected individuals it may also be independent of the formulation of that
turnover. Since we are also interested in models with fixed infection period length, we have also tested the wave speeds generated from that model against the analytical method in the hope that we will be able to use it to analyses this model. As we show in the next three figures the analytical solution does indeed turn out to describe the fixed infection period length model very well although it was not developed for this version of the model.
As demonstrated in the previous chapter, the wave speed depends only upon the rate of diffusion of infected individuals, not uninfected ones. We confirmed this in the
numerical model by turning off diffusion in S and R fish. There was, as predicted, no change in the speed of propagation of the epidemic front. However for this analysis we use a single diffusion coefficient, since behaviour of uninfected fish has no effect on the epidemic wave's properties we do not need to consider these parameters separately and lack the data to do so.
We use a model population that is normalised to the average initial population
N0 (N0 = S0 =1) and also normalise β to the average population density, so these two are essentially a combined parameter. We have no means of evaluating b independently, and thus must evaluate it for current population. Although variation in the population may affect the wave, we do not need to explicitly determine the average population.
Thus the critical parameters to which epidemic wave speed is sensitive are D, β and b.
Using the analytical method we find that the parameter values D = B = 200 km2 d-1, β = 200 d-1, and b = 4 d and α + µ =0.25 d-1 gives a wave speed of 31.5 km d-1. We use this as our initial point for our extended sensitivity analysis and investigation of the applicability of the analytical solution.
Our first test of the model is of the calculated epidemic wave speed against the infection transmission efficiency β. The analytical solution fits both numerical solutions very closely over four order of magnitude of β and a factor of 4 change in wave speed (Fig. 8.1). It is an effective analysis tool. The wave speed is only weakly dependent on β, increasing at a rate slightly less than proportional to the log of β. We are very
uncertain about the range of β, the weakness of the wave's dependency on this parameter means that uncertainty in this parameter does not translate into large
uncertainties in the model results. If transmission per se were very efficient then there
would be a maximum limit imposed by the rate of mixing of schools at some large but unknown value. Because of the saturation in response to change in β for large β the existence of such a maximum value is not a problem that could affect the model results.
At very low values of β the wave speed becomes much more sensitive to β's value.
0 5 10 15 20 25 30 35 40 45
0.1 1 10 100 1000 10000
β (d-1)
V (km d-1)
Figure 8.1 Analytical solution (solid diamonds) and numerical solution (open squares) of the epidemic wave speed versus infection transmission β for the fixed latent period length model. Also shown is the fixed latent plus fixed infectious period lengths (open triangles) model's wave speed.
It must be emphasised that the transmission efficiency is normalised to the average population density. Because of this, the rate of transmission of the disease depends on population density in the same way that it depends upon the actual transmission
efficiency. Hence the wave's speed will respond significantly to changes in population density only if these are very large and will only cease if these changes are of several orders of magnitude. While changes of three orders of magnitude have occurred in Japanese pilchard populations following recruitment failure (Wada and Jacobson 1998), changes in Australian waters have been much smaller. Changes of the order of 50%
may have occurred off Albany between the two epidemics and not driven by the mass mortality (Fletcher 1992, Cochrane 1999).
The model solutions show a much stronger relationship with the diffusion coefficient.
The relationship is linear with the square root of the diffusion coefficient (Fig. 8.2); this is the same relationship as exhibited by all the other standard epidemic wave models (Murray 1999). In fact, as discussed in the previous chapter, it is the diffusion of infected individuals which control the wave's speed. To test this, we have run the numerical model with D set to zero, while the value of B is maintained. As predicted, this was found to have no effect on wave speed as predicted under the analytical method.
0 10 20 30 40 50
0 5 10 15 20 25
D0.5 (km2 d-1) V (km d-1 )
Figure 8.2 Analytical solution (solid diamonds) and numerical solution (open squares) of the epidemic wave speed versus the square root of diffusion coefficient D (= B) for the fixed latent period length model. Also shown is the fixed latent plus fixed infectious period lengths (open triangles) model's wave speed.
0 10 20 30 40 50
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
b-0.5 (d) V (km d-1 )
Figure 8.3 Analytical solution (solid diamonds) and numerical solution (open squares) of the epidemic wave speed versus the inverse of the square root of the latent period for the fixed latent period length model. Also shown is the fixed latent plus fixed infectious period lengths (open triangles) model's wave speed.
The final parameter that the model wave speed is strongly sensitive to is the length of the latent period (Fig. 8.3). The model wave speed depends linearly on the inverse of its square root. Because of the dispersion of infected fish, large values of b are not consistent with the observed restricted period of mortality at any given location (see section 8.3) so we have a relatively restricted range for this parameter. If the latent period is reduced towards zero (not shown) the wave speed tends towards 400 km d-1. This is the wave speed predicted from a simple SIR model which the model is in this case equivalent to. The numerically calculated speed of an SIR model is, as discussed earlier, 2√(Dβ).
0 10 20 30 40 50 60 70 80
β=10 β=20 β=40 β=80 β=160 β=320
β=640
V km d-1
Figure 8.4 Speed of epidemic waves calculated using β as shown and D of 50, 100, 150, 200, 250, 300, 350 or 400 km2 d-1. Default latent period is 4 days; ranges shown by empty lines indicate the effect of varying latent period from 2 (faster) to 6 (slower) days. Bold lines mark the 10 and 40 km d-1 limits of observed wave speeds.
We bring this information together to show how the wave speed varies with the three parameters acting in combination. As can be seen from Fig. 8.4 even over large changes of β there is little change in wave speed. Change in latent period, b, seems to be not enough to account for the large changes in wave speed on it own. For large D and/or β (i.e. for fast waves) the wave speed becomes more sensitive to b. Only D seems to be able to vary enough and the wave speed is sensitive enough for this parameter to be able to account for the observed variation in wave speed on is own.
We summarise wave speed parameter sensitivity. As Yachi et al. (1989) determined, the presence of a latent period makes wave speed far less sensitive to the local rate of spread of the infection than in simple SIR models. But we also show that by making this latent period of a fixed length the sensitivity is even further reduced. This weak sensitivity applies particularly for large β, as is the case of the pilchard herpesvirus.
Sensitivity to the latent period is greatly increased from the fourth root (Yachi et al.
1989) to the square root of turnover time. All versions of the model are sensitive to the square root of the diffusion coefficient. As demonstrated in the previous chapter, it is the diffusion of infected animals only that affects the wave's speed, so any change in their behaviour may be very important to the spread of an epidemic.