In this chapter we consider a non-linear transmission of infection model. This model has major implications for the origin of the epidemic since it requires a large initial dose of infection to induce the epidemic, whereas under the standard model any dose may induce an epidemic provided the environment is suitable.
Viral transmission may depend on a threshold concentration of viruses that is required to induce infection. The immune systems of hosts that are exposed to small numbers of viruses may be able to fight this off, but a massive dose is another matter. This appears to be the case for IPNV for which a minimum challenge dose of virus may be required to trigger infection (Hill 1982). Confinement of Pacific herring in cages allowed VHSV (viral hemorrhagic septicemia virus) to build up in the water and thus an epidemic to take off (Hershberger et al. 1999). Free herring, from which the caged stock was taken, did not develop disease at this time. This may be an effect of linearly higher infection probability caused by the local retention of the fish in the vicinity of accumulating virus numbers, but it is suggestive of the minimum viral dose effect.
If there is such a dependence of infection on high viral density then infection may depend upon a higher power of I, perhaps βSI2(Liu et al. 1986). Infectivity, the rate at which new infections are produced per infected host, is low both when I is low and when it is high because then S is low as a proportion of the population. This situation means that there is a minimum product of both susceptible and infective host
populations required for an epidemic to take off (Fig. 10.1). Under the standard linear model the number of infections produced per infected individual is only low when I is
high and hence S is small. The actual number of infections produced when I is low is also low, but the infections produced per infected host are then maximal.
[SI]t = α/β (10.1)
This relationship of infection rate to I has particular significance for the origin of the epidemics. Under the quadratic (I2) model a locally stressed host population may have an infection induced by this stress that can then spread freely to adjacent unstressed populations. A localised environmental shock could thus be the trigger of a wide- ranging epidemic. However, the presence of a small amount of virus in an unstressed host population will not trigger disease. This is in contrast to the standard linear model, under which any presence of virus will trigger the disease, provided the initial host population exceeds the threshold.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.2 0.4 0.6 0.8 1
I
Infectivity
Figure 10.1 Infectivity (normalised to maximum infectivity) under βIS and βI2S models (while R and E = 0). The dashed line gives an example threshold population for net infection. The maximum infectivity = βS0 under βIS and 0.25βS0 under βI2S model.
There is no evidence of any environmental shock that could have initiated an infection of the non- linear transmission type. There was a cold upwelling prior to the 1995 epidemic, but events on a similar scale occur every three-four years (Griffin et al.
1997). No upwelling occurred prior to the 1998/9 epidemic (Ward et al. 1999).
The transmission of the epidemic to caged pilchards, physically isolated from the wild stock, is evidence supporting the linear transmission model since it is unlikely that very large number of viruses would be transmitted to this isolated population. Similarly, the coincidental outbreak in New Zealand in 1995 supports the linear model. It would be unsurprising if the disease were transported to New Zealand given its then prevalence in Australia, but it would be less likely that a similar environmental shock occurred
coincidentally in the same year and not in some other year.
The linear model is the form of transmission that is almost invariably used to model epidemics (e.g. Anderson and May 1979). However, because of the significant
differences between the two models, both linear and quadratic transmission we examine the latter formulation here as well
We can apply the quadratic version of non-linear infection (i.e. infection dependent on I2) to the spatially resolved model. We can thus generate waves of infection which are similar to those generated under the linear model. However, with the initial intrusion of the wave of infection I is small and I2 is very small. This means, in the quadratic model the forward front of the infection is an area of low production of infection. The speed of the wave is driven by the area of high production, when I becomes large and where S is still not small. But by this point the relative gradient of the distribution of I has declined and so diffusive flux is weakened. Hence the epidemic wave progresses more slowly. The dispersal of E and I still limits maximum diffusion rates consistent with observations. Hence, for a given latent period, the value of β selected must be much larger than under the linear model.
We lack an analytical solution of the quadratic model, but we have been able to find curves for various latent periods under the two speeds by fitting the value of β required to produce the appropriate wave speed (Fig. 10.2). These waves are of constant speed.
However, longer latent periods result in the need for even higher infection rates. The high infection rates results in waves that mature very rapidly, hence these waves depart from the origin only a few days after infection. Indeed the wave forms only a couple of days after the first mortality, which occurs at 4 days after the infection is introduced.
Mortality does continue at the origin for a short while.
0 200 400 600 800
0 5 10 15 20
time (days)
position (km)
b=1 b=2 b=1 b=2 b=4
Figure 10.2 Development of 10 and 40 km d-1 waves under quadratic infection. For the 10 km d-1 waves D = 20 km2 d-1 and b = 1 β = 100, b = 2 β = 1000, b = 4 β = 40000. For the 40 km d waves D = 200 and b = 1 β = 1000, b = 2 β = 20000.
The quadratic model also produces a similar mature wave shape (Fig. 12.3) to that obtained under the linear mortality model. Mortality is spread over a few days, as observed. There are similar peaks for the two diffusion coefficients to the values obtained under the linear infection model.
0 0.05
0.1 0.15 0.2 0.25 0.3 0.35 0.4
0 10 20 30 40 50
time (days)
mortality (/day)
b=1 b=2 b=4 b=1 b=2
Figure 10.3 Shape of stable 10 and 40 km d-1 waves under quadratic infection. For the 10 km d-1 waves D = 20 km2 d-1 and b = 1 β = 100, b = 2 β = 1000, b = 4 β = 40000. For the 40 km d waves D = 200 and b = 1 β = 1000, b = 2 β = 20000.
The quadratic infection model can also produce oscillations similar to those obtained under the linear model. However, due to the relatively short latent periods that in this model are compatible with the observed wave speeds, given limits to diffusion imposed by the short-lived pattern of mortality in the mature waves, only brief periods of
oscillation occur (Figs. 10.4, 10.5). For latent periods of 1 day, these peaks are spaced at daily intervals and so would be undetectable given the necessary limits to the resolution of observational data, except possibly as initially high mortality. Only the slower waves can be made compatible with a significant gap between first and secondary mortality events.
0 0.5 1 1.5 2 2.5
0 5 10 15 20 25
time (days)
mortality (/day)
b=1 b=2 b=4
Figure 10.4 Initial development of the 10 km d-1 quadratic epidemic wave under three latent period lengths.
0 0.5 1 1.5 2 2.5
0 5 10 15 20
time (days)
mortality (/day)
b=1 b=2
Figure 10.5 Initial development of the 40 km d-1 quadratic epidemic wave under two latent period lengths.
The quadratic model appears less compatible with the evidence than is the standard linear model. It can be made to produce stable epidemic waves whose behaviour is identical to those obtained under linear infection models, although higher transmission coefficients are required. However, the initial persistence at the origin under this model is too short lived. This occurs because of the requirement for rapid viral transmission rates. Initial gaps or oscillations in the observed mortality also cannot be reproduced, except for the slower wave speeds because long incubation periods are not consistent with observed wave speeds. The quadratic model would be entirely ruled out in its current form if it were shown that latent periods do exceed two to 3 days.
Because quadratic infection spreads more rapidly as the number of infected individuals increases it gives an inherent tendency for epidemics to take off suddenly and so not to persist near the origin as was observed.
The difference between the linear and quadratic models is largely in their implications for the origin of the epidemic. Under the linear model any initial dose of viable virus can lead to the epidemic taking root. The quadratic model requires a large initial viral dose and so is less easy to introduce. Since it is precisely the behaviour at the origin that the quadratic model fails to reproduce the linear model would appear to be a better choice.