1.9 Historical Notes
1.9.2 Mathematical Epidemiology Before Ross
We note that contemporary scholars of Ross, including Brownlee and Lotka, were also applying mathematics in human demography and health. Ross’s mathematical models of malaria transmission were also not the first models for the transmission of an infectious disease. First was Bernoulli’s model of smallpox, and P’Enko had published a model for flu a few years before. Ross was
Our philosophy has been to design a framework for model building that can be used by programs. In this context, model building means designing ensembles of models. The next step involves applying a set of tools to computational tasks that are beyond what our brains could do. To accomplish our goals, we need more than the mathematical framework. We need to be able to implement and compute models. This requires new software.
The software is structured around three major dynamical components and two interfaces. The dynamical components are: 1) the humans and malaria epidemiology, including the effects of treating malaria with drugs; 2) adult mosquito ecology, behavior and infection dynamics; and 3) aquatic mosquito ecology. Malaria transmission by mosquito populations, including the \(2^{nd}\) and \(3^{rd}\) dynamical component are set up to consider the effects of weather and vector control. The first interface links humans, adult mosquitoes, and parasites to describe parasite transmission through mosquito blood feeding and human exposure to infective mosquito populations. The second interface links adult and mosquito populations through egg laying and emergence. Within each component and interface, there are multiple sub-domains, and there are built in design features to deal with heterogeneity and other features for malaria control. After a 140 years of studying malaria, there’s a lot of detail that could be important in some way.
The software we have developed is meant to lower the costs of building and using models. We want programs to be focused on the decisions, the data, the concepts, and the analysis. As a metaphor, some students learn a numerical method for approximating \(\sqrt{2}\) in school, but after learning it once, they stop worrying about how it is computed and they punch buttons into a calculator. Knowing how to compute something is sometimes useful, but worrying about how to compute it each time would interrupt the process that called for computing it. Instead, we punch the formula into a scientific calculator or any software that does computation confident that the machine knows how to do it. In applying models, the same kind of logic applies. People need to understand the concepts, but like a calculator, the tools should hide the technical details that don’t add to a discussion. The software we have developed is a reliable interface for calculations designed to support policy.
To learn how to use that software, we need to get through a lot of material. The background material in the following presentation is fairly sparse. We are trying to introduce just enough mathematics to teach users the critical concepts so they know what the software can do. We assume that the work will be done by teams that include a few people who understand the mathematics, who can guide others through the process.
All of this sounds very complex, but we must start with something simple and layer on complexity in an ordered way. The first model we present is a Ross-Macdonald model.