In the accompanying manuscript Saul (1998) points
out that a model of within-host malaria population
dynamics (Anderson, May & Gupta, 1989) can
exhibit unrealistically large growth rates. He
suggests that this error can be avoided by replacing
the parameter r, which is the number of merozoites
produced by each parasite at schizogony, with the
value ln (r)+1. This substitution does not, however,
address the true underlying problem with the model,
namely that whilst in reality there is small variation
in the distribution of Plasmodium spp. life-spans, the
use of the constant rate α assumes an exponential,
and hence much more variable, distribution. This
can allow the population to increase, over the time
period of the average life-span (48 h in the case of
Plasmodium falciparum), by factors considerably
larger than r. Saul identifies this assumption as
unrealistic in terms of the biology of malaria, but it
is not remedied in the proposed model.
Within the structure of the original model there
are two ways of addressing the growth rate problem.
Firstly, the ‘growth constant’ r can be replaced by
the value ln (r)+1, so that if all parasites reinvade,
the model increases by a factor r over 1 generation.
As pointed out by Saul, this is an artificial device
since it means that each parasite produces a reduced
number of merozoites. Alternatively, a parasite can
produce the observed number of merozoites, r,
many of which do not reinvade. This is the situation
in the original paper and Gravenor, McLean &
Kwiatkowski (1995). In these papers the model does
grow at a reasonable rate because the parameter β is
estimated directly from observed growth rates.
These points, however, are only an aside. Both the
original and Saul's modified model can grow at the
same rate and they both have the same distributional
assumptions concerning parasite life-span.