2 Basic Malaria Models
Basic malaria models as deterministic or stochastic systems of difference or differential equations
We present several different kinds of mathematical models that implement the dynamics for a mosquito-transmitted pathogen. Two of these look a lot like models that Ross developed. The others introduce concepts that were introduced long after Ross.
Section 2.1 present’s Ross’s quantitative logic.
Section 2.2 presents a basic deterministic discrete-time system. Since it is the first model in the book, we go slowly, walking through the model step-by-step and discussing the basics in detail.
In Section 2.3, we discuss some basic issues about coding and computation.
In Section 2.4, we show that the model presented in Section 2.2 is numerically unstable. To fix the numerical instability, we introduce the Poisson model, a probability distribution function describing a counting process. We use the idea to present a new deterministic discrete-time system of equations that is numerically stable.
Section 2.5 presents a stochastic discrete-time system, where the state of the system is updated each time step using random variables.
Section 2.6 presents a deterministic continuous-time system.
Section 2.7 presents a stochastic continuous-time system with asynchronous updating: events in the system occur one at a time, and the system is updated after each event
In Section 2.8, we compare and contrast discrete vs. continuous, and deterministic vs. stochastic models.
In Section 2.9, we discuss some of the discrete-time systems that played a role in development of theory for infectious diseases.
The goals for this chapter are to introduce the basic elements of models, to present several different different kinds of models, to develop some basic intuition about how they work and their differences, and to emphasize that all models as approximations. In this chapter, we’re trying to develop basic skills that we can use later. There’s a lot of basic material to digest, and since we’ll use it later, we’re not trying to delve too deeply into malaria epidemiology. Each model is implemented as R code2
We assume that the gentle reader has some basic familiarity with these concepts in the next chapter, when we present The Ross-Macdonald Model.
In developing this book, we assumed the reader would be familiar with R and RStudio.↩︎