Statics and dynamics of malaria infection in Anopheles mosquitoes

Abstract

The classic formulae in malaria epidemiology are reviewed that relate entomological parameters to malaria transmission, including mosquito survivorship and age-at-infection, the stability index (S), the human blood index (HBI), proportion of infected mosquitoes, the sporozoite rate, the entomological inoculation rate (EIR), vectorial capacity (C) and the basic reproductive number (R0). The synthesis emphasizes the relationships among classic formulae and reformulates a simple dynamic model for the proportion of infected humans. The classic formulae are related to formulae from cyclical feeding models, and some inconsistencies are noted. The classic formulae are used to to illustrate how malaria control reduces malaria transmission and show that increased mosquito mortality has an effect even larger than was proposed by Macdonald in the 1950's.

Figure 2

Classic relationships between EIR or R₀ and the proportion of humans with parasitemia. The relationship between EIR and the proportion of infectious humans, , is plotted to illustrate the shape of the curve, assuming a human infectious period of 100 days (r = 0.01) and transmission efficiency to humans of 50% (b = 0.5). At this rate, 50% of the human population is infectious with a yearly EIR of 7.3 infectious bites. The relationship between R₀ and the proportion of infectious humans, , is also plotted assuming a human feeding rate of 3 human bites every ten days (a = 0.3), a transmission efficiency to mosquitoes of 50% (c = 0.5), and an average mosquito lifespan of about 10 days (g = 0.1). Note that = 0 unless R₀ > 1.

Figure 1

Mosquito survivorship and age-at-infection. Using parameter values from Killeen, et al., (1999), mosquito survivorship λ(A) is plotted from Equation 1 (squares), the proportion of surviving mosquitoes that are infectious, μ(A) from Equation 2 (diamonds), and the proportion of the original cohort that is alive and infectious λ(A)μ(A) ((triangles) note the units are given on a separate axis). The corresponding values from a cyclical feeding model are plotted, assuming that a mosquito with parity τ has chronological age A/f. The discrepancies between this algorithm and the one used by Killeen, et al. are explained in Appendix 2.

Figure 3

Revisiting Kaduna. The statics for a cohort of mosquitoes are plotted using parameter values from Kaduna and the algorithm described by Killeen, et al., with δ rounded to the next lowest integer (using + for the proportion of a cohort that survives and the downward pointing triangles for the proportion of an original cohort that is alive and infectious). Also plotted are the lower values where δ was rounded to the next highest integer (using – or an upward pointing triangle). Killeen et al. (1999) use a linear approximation for the proportion of infectious mosquitoes; the approximation is bad in some circumstances. In this case, it predicts that > 100% of mosquitoes are infected, after about 50 days. For comparison, the age-specific survivorship and the probability infection for Kaduna are replotted, as in Figure 1 (lines).