Mathematical Models in Population Biology and Epidemiology

In this chapter we look at a population in which all individuals develop independently of one another. For this situation to occur these individuals must live in an unrestricted environment where no form of competition is possible. If the population is small then a stochastic model is more appropriate, as the likelihood that the population becomes extinct due to chance must be considered. However, a deterministic model may provide a useful way of gaining sufficient understanding about the dynamics of a population whenever the population is large enough. Furthermore, perturbations to populations at equilibrium often generate on short time scales independent individual responses, which are appropriately modeled by deterministic models. For example, the propagation of a disease in a large population via the introduction of a single infected individual leads to the generation of secondary cases. The environment is free of “intraspecific” competition, at least at the beginning of the outbreak, when a large population of susceptibles provides a virtually unlimited supply of hosts. In short, the spread of disease in a large population of susceptibles may be thought of as an invasion process via independent contacts with a few infectious individuals.

In this chapter we shall consider populations with a fixed interval between generations or possibly a fixed interval between measurements. Thus, we shall describe population size by a sequence {x _n }, with x ₀ denoting the initial population size, x ₁ the population size at the next generation (at time t ₁), x ₂ the population size at the second generation (at time t ₂), and so on. The underlying assumption will always be that population size at each stage is determined by the population sizes in past generations, but that intermediate population sizes between generations are not needed. Usually the time interval between generations is taken to be a constant.

Up to now in our study of continuous population models we have been assuming that x′(t), the growth rate of population size at time t, depends only on x(t), the population size at the same time t. However, there are situations in which the growth rate does not respond instantaneously to changes in population size. One of the first models incorporating a delay was proposed by Volterra (1926) to take into account the delay in response of a population’s death rate to changes in population density caused by an accumulation of pollutants in the past. Other causes of response delays which have been mentioned in the biological literature include differences in resource consumption with respect to age structure, migration and diffusion of populations, gestation and maturation periods, delays in behavioral response to environmental changes, and dependence of a population on a food supply that requires time to recover from grazing. In deriving a mathematical model to reflect a particular biological delay mechanism one must consider carefully how this mechanism affects the growth rate. One approach to modeling delays that has been used is formulation of a discrete model (or difference equation) and consideration of the delay in the time between steps. While this is appropriate for populations with a discrete reproduction cycle, such as many fish populations, it does not accurately model populations with continuous growth and time lags. The metered models studied in Section 2.5 allow for a continuous death process but involve a discrete reproduction stage.

In the 1920s Vito Volterra was asked if it were possible to explain the fluctuations which had been observed in the fish population of the Adriatic sea—fluctuations which were of great concern to fishermen in times of low fish populations. Volterra (1926) constructed the model which has become known as the Lotka-Volterra model (because A.J. Lotka (1925) constructed a similar model in a different context about the same time), based on the assumptions that fish and sharks were in a predator-prey relationship.

In this chapter we will consider populations of two interacting species with population sizes x(t) and y(t), respectively, modeled by a system of two first order differential equations: .

The topics in this chapter are part of the subject of natural resource management and bioeconomics. This is an important and rapidly-developing subject. The classical reference is the book by Clark (1990), where additional references may be found.

The idea of invisible living creatures as agents of disease goes back at least to the writings of Aristotle (384 BC-322 BC). It developed as a theory in the 16th century. The existence of microorganisms was demonstrated by Leeuwenhoek (1632–1723) with the aid of the first microscopes. The first expression of the germ theory of disease by Jacob Henle (1809–1885) came in 1840 and was developed by Robert Koch (1843–1910), Joseph Lister (1827–1912), and Louis Pasteur (1827–1875) in the latter part of the 19th century and the early part of the 20th century.

This book attempts to bridge the gap between mathematics and population biology. It is intended to show students of biology how to apply mathematics to the study of some questions of importance to population biology and to introduce modeling in the natural sciences to students of mathematics. It may also be used as a reference on mathematical methods for working biological scientists.