3.9 History
In 1975 [55], Paul Fine discusses various versions of mathematical models developed by Ross and Macdonald, discussing various formulations of models with superinfection. In 1982, Norman Bailey discusses the history of development of the models in The Biomathematics of Malaria [58]; and Joan Aron and Robert May present a “Ross-Macdonald” model and several additional variants [59]. Later, we argued that the name Ross-Macdonald model was, in fact, a
- The history of development of mathematical models for malaria is covered in depth by Norman T.J. Bailey in The Biomathematics of Malaria.
3.9.1 Lotka
In retrospect, it became clear that Alfred Lotka played an important yet overlooked role in development of basic theory for malaria. Ross had actually published three mathematical models. Two of these described malaria transmission. Lotka analyzed both transmission models in 1923, and in a collaboration with Sharpe, he had extended the models to consider the effects of a delay. Lotka had also introduced the idea of a basic reproductive number for human demography. Macdonald adapted Lotka’s ideas to malaria, largely without attribution. Arguably, it could be called the Ross-Lotka-Macdonald model.
Ross’s first model described malaria transmission dynamics using difference equations [17]. That model was reviewed, analyzed, and critiqued first by H. Waite in 1910 [69].
Aron and May [59] present a “Ross-Macdonald” model that is close to what we have Ross model, and then a v. The history of mathematical models for malaria is covered in depth by Norman T.J. Bailey in The Biomathematics of Malaria.
Ross’s second model.
The Ross-Macdonald model and its development have been discussed elsewhere [2,58,59]; it was a model developed by Macdonald that was based on Ross’s earlier work and that was supported by the mathematical talents of Armitage. There is not a single paper where Macdonald described the system of differential equations. Instead, the model appeared in parts of several different publications that presented equations and formulas describing malaria transmission, and that reviewed existing data.
The model that Maconald started with was first developed by Ronald Ross (who published two models of malaria transmission), but we also owe a lot to Alfred Lotka, who analyzed both models.
The formulation is a bit hard to follow, so we will present a system of difference equations in place of the equations Ross actually wrote down.
Lotka
Ross’s second model was thoroughly analyzed by Lotka, who had taken an active interest in Ross’s malaria models. In 1912, he published a set of solutions to Ross’s equations [54]. In 1923, Lotka published an analysis of both of Ross’s models in five parts. The first two parts reformulate Ross’s models [38,39]. The third part tackles numerical issues, which includes a photograph of a clay model of the phase plane as a surface [40], and the fifth part is a concise summary [41]. In the \(4^{th}\), which was led by Sharpe, a new model was introduced that included delays for the latent periods [68].
### Alfred J. Lotka
While Alfred J. Lotka is more famous for his work in demography and ecology, he took an interest in Ross’s work on malaria and he made some important contributions to mathematical malaria epidemiology:
Most importantly, Lotka developed the concept of the basic reproductive number in his work in human demography, which was defined as the expected number of females that would be born to a newborn female.