3 The Ross-Macdonald Model
The Ross-Macdonald model, a system of delay differential equations describing malaria. The basic reproductive number as a threshold condition. Vectorial capacity. The classification of transmission.
When I first learned about malaria, I remember reading about the Ross-Macdonald model. I had been reading Ross, but I wanted to find a canonical formulation of a model with George Macdonald’s name attached to it. I was looking for a publication where I could find a system of differential equations. Instead, I found that Macdonald was associated with several new ideas spread across several publications, including a new model of superinfection [57], a synthesis of quantitative medical entomology and analysis of the sporozoite rate [42], and a formula for the basic reproductive number for malaria [43]. His papers introduced a few ideas he attributed (or ought to have attributed) to others, including the concept of a reproductive number (it was Lotka’s idea) [2]. Two useful publications describing George Macdonald’s models by Paul Fine [55] and Norman Bailey [58] about Ross, Macdonald, and the various mathematical versions of malaria models with two or more names attached. Rereading, I found the phrase so-called Ross-Macdonald in a 1982 book chapter by Joan Aron and Bob May [59].
After learning about Ross, I wondered why it was called the Ross-Macdonald model? Macdonald’s papers are conceptually rich discussions of mathematical concept and data from various sources; the formulas that do appear are mostly relegated to footnotes and appendices. Between 1950 and 1956, Macdonald published 14 papers on malaria epidemiology, dynamics, and control and related topics. In 1957, he published a book called The Epidemiology and Control of Malaria, and he was rapporteur for the Sixth Report of the Expert Committee on Malaria, which established the basic plan for the Global Malaria Eradication Programme [60]. In the 1960s, he published some relevant follow-up papers, including a diagram of the basic reproductive number [61]. Macdonald’s core contributions were to provide a synthesis of malaria epidemiology and quantitative medical entomology; to introduce Lotka’s concept of a basic reproductive number into malaria epidemiology (and indeed, to all of mathematical epidemiology); and to introduce some very basic ideas about a theory of vector control. Macdonald’s paper on the sporozoite rate revolutionized quantitative malaria epidemiology.
To understand why Macdonald’s name attaches, it’s worth taking a closer look at the papers from 1950 and 1952. In 1950, he published a synthetic review of malaria malaria epidemiology [62], and a new model for superinfection [57]. That paper describing the model superinfection has a flaw – the model is sound, but it does not match the process described in the paper. (The issue was discussed at length by Paul Fine [55].) In 1952, Macdonald published his analysis of the sporozoite rate [42], and then he wrote a paper about endemic malaria that included a formula for the basic reproductive number, \(R_0\) [43]. Macdonald reports that the mathematical analysis of superinfection and the sporozoite rate had been done by Armitage; the paper appeared in 1953 [63]. Macdonald’s formula for the clearance rate with superinfection, from the model in his 1950 paper, simplifies to Ross’s equation to get the expression for \(R_0.\) In 1955-56, he published papers discussing the measurement of malaria transmission theory for control and for malaria eradication [64–66]. The formulas in Macdonald’s footnotes and appendices from the 1952 papers have enough information to write down the equations he must have been working from, and this is what we will present in the next section.
The codification of the Ross-Macdonald model in the 1980s coincided with a rising interest in disease ecology and mathematical epidemiology in departments of ecology. All this history, spanning the period from roughly 1899 to 1969, has been reviewed before [2], and I have added some historical notes at the end of this chapter. My search for The Ross-Macdonald Model was not in vain – I learned a lot more about the history, and I discovered that no canonical version of the model exists. What is presented in the following is the best candidate for a canonical model. It is system of equations that uses Macdonald’s notation from the 1952 papers, that ignores any discussion of superinfection, and that replicates Macdonald’s footnotes from both papers.
This chapter assumes you understand the material in Basic Malaria Models.