Ross, macdonald, and a theory for the dynamics and control of mosquito-transmitted pathogens

Abstract

Ronald Ross and George Macdonald are credited with developing a mathematical model of mosquito-borne pathogen transmission. A systematic historical review suggests that several mathematicians and scientists contributed to development of the Ross-Macdonald model over a period of 70 years. Ross developed two different mathematical models, Macdonald a third, and various "Ross-Macdonald" mathematical models exist. Ross-Macdonald models are best defined by a consensus set of assumptions. The mathematical model is just one part of a theory for the dynamics and control of mosquito-transmitted pathogens that also includes epidemiological and entomological concepts and metrics for measuring transmission. All the basic elements of the theory had fallen into place by the end of the Global Malaria Eradication Programme (GMEP, 1955-1969) with the concept of vectorial capacity, methods for measuring key components of transmission by mosquitoes, and a quantitative theory of vector control. The Ross-Macdonald theory has since played a central role in development of research on mosquito-borne pathogen transmission and the development of strategies for mosquito-borne disease prevention.

Figure 1. The Ross-Macdonald theory of transmission dynamics.

(Top left) In a hypothetical location, for a fixed value of R₀ (plotted here for R₀ = 5), the model describes changes in the proportion of infected humans or infectious mosquitoes during an epidemic. (Top right) Alternatively, the models predict the endemic parasite rate or sporozoite rate as a function of R₀. Malaria is not endemic if R₀<1, or after control, if R_C<1, or equivalently, if mosquito density is below a critical threshold. (Bottom left) The model also describes changes in the parasite rate with respect to age (e.g., in a cross-sectional study) in infants or others who were previously unexposed to malaria. (Bottom right) Finally, the models also predict the response timelines and endpoints following the implementation of control (grey).

Figure 2. The Ross-Macdonald theory of control.

(Top left) A relationship exists between the length of a mosquito feeding cycle (2, 3, or 5 days in blue, black, or red), the proportion of parous mosquitoes (denoted O), and the mosquito lifespan (denoted 1/g). (Top right) This relationship can be used to measure predicted changes in the mosquito lifespan (Δg⁻¹) through estimated proportional changes in the proportion parous, which are invariant to the mosquito blood feeding rate (ΔO/O). (Bottom left) These changes can be translated into an effect size on transmission, a proportional change in reproductive numbers (R₀/R_C). (Bottom right) Finally, these can be translated into changes in the endemic parasite rate for a given effect size: R_C = R₀/2.5 (dashed) or R₀/5 (dotted).