Reply to: Models for the in-host dynamics of malaria revisited: errors in some basic models lead to large over-estimates of growth rates

Abstract

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.