Population Dynamics and Epidemiology
We now turn to models of population dynamics and epidemics. These typically involve difference or differential equations. We will start with one-species discrete models, discuss the occurrence of chaos, and consider populations with different age groups. We will then move on to continuous models, involving one or two species. The latter are too low-dimensional to exhibit any chaotic behavior, and we will concern ourselves with the appearance of oscillatory dynamics.
Contrarily to classical mechanics, we are still in the process of understanding the laws of biological evolution. Mathematical biology has recently become a topic of broad interest, and it is generally accepted that progress in comprehending the dynamics of populations, diseases, epidemics, etc can only be achieved through interdisciplinary activities involving both mathematicians and life scientists. Modelers should be able to know the significance of any term in their model equations and assess, by looking at experimental results, whether such terms are relevant to the problem in question. Successful modeling thus involves going back and forth between the model, its simulation and/or analysis, and experimental results.
Reliable biological data are difficult to obtain for a variety of reasons. First, in the case of population dynamics, data must be gathered over time periods much longer than the lifetime of an individual. At the human scale, this takes centuries. Fortunately, other systems, such as for instance bacterial systems, have a much shorter intrinsic time scale. Second, different studies are typically performed under different conditions, and the results may be affected by unknown confounding variables. It is also very rare to find very large-scale studies involving primates, in particular human subjects. As a consequence, trends may only become statistically significant when a variety of studies are combined. Finally, many biomedical studies are performed on mice or monkeys, and it is not always clear how the results may be transported to humans. More than ever, it is thus essential for a modeler to understand the nature and the limitations of the available data, as well as those of the models.
The examples discussed in this section are fairly simple, but they should be sufficient to help the reader develop an intuition for the basic generic modeling of population dynamics and epidemics in terms of difference or ordinary differential equations.
