106

■Appendices

The model described in Equation 5.B.1 is an advance compared to some early mod-

els [40], [95], [203] since it illustrates how the reduction in transmission can make things

worst for adults by diminishing naturally acquired immunity. Nevertheless, it gives a sim-

plistic description of the interaction between immune responses and the population biology

of malaria. This is because, the classification of humans into discrete categories does not

reveal the fact that malaria parasite populations can exist at various densities within the hu-

man hosts. Aron [181] has shown how such false negatives, which result in underestimation

of the prevalence among older age groups, can also confound the statistics of prevalence,

rates of infection and apparent recovery. This suggests that an accurate model will need to

abandon the approach of dividing people into discrete classes of infected and uninfected,

and move toward a more detailed description in which the clinical symptoms and the im-

mune response depend upon the magnitude of the parasite burden in the individual host.

In view of the above, Aron [43] developed age-specific density-dependent model as a

different characterization of immunity that is boosted by exposure. The model character-

izes the experience of a birth cohort based on: the asexual parasites density at age α, p(α);

the immunity level geared against asexual parasite at age α, r(α); and gametocyte density

at age α, g(α). The interaction between immunity and infection is modeled by assuming

that the immunity against asexual parasitemia, r increases at a rate proportional to the asex-

ual parasitemia whereas in the absence of parasitemia, it decreases. The density model is

given as

dp

dα ^{= υ −(r +rb)p,}

dr

dα ^{= ap−βr,}

(5.B.2)

dg

dα ^{= γpe−(r+rb)T −δg,}

where υ is the influx rate of the asexual parasites (also called the force of infection) which

declines at a clearance rate r + rb (see [146] for a robust formula for υ). The clearance

rate is the sum of the rate that denotes the effect of acquired immunity r and the back-

ground rate rb. T is the development period after which the gametocytes are produced by

the asexual parasite, δ is the clearance rate of the gametocytes. Thus, the density of asexual

parasites and gametocytes as a function of age, simulated from the above model is given in

Figure 11. It can be seen from Figure 11 that the patterns for asexual parasites are different

from that of gametocytes. It appears that compartmental model (Figure 5.2 a) characterize

different levels of endemicity better than the density model (Figure 11 a).

However, the density model captures the repression of malaria as a result of reduced

parasite densities, based on the levels of usage of antimalarial drug reflected in rates of

recovery. In comparison with the compartmental model, the density model did not capture

the potential of reduced transmission to increase malaria prevalence in older age groups,

as predicted in the former model (see Equation 5.B.1). Thus, it is not wrong to expect

that both models might have been created for different cases. Perhaps, the compartmen-

tal model was intended to be used for the study of prevalence whereas the density model