Throughout human history diseases have played an important role. The black death in Europe in the middle ages is one example, and more recent examples include the flu pandemic of 1918 and 1919 and diseases such as AIDS and various childhood diseases. Additionally, the study of population dynamics in ecology has been greatly advanced by the study of diseases for several reasons. First, the data on the incidence of childhood diseases are both extensive and accurate. Second, the processes involved in the dynamics of childhood diseases (essentially, infection and either recovery or death) are relatively simple and straightforward and well understood.
In particular, many questions of scientific or practical interest can be answered using relatively simple models. A fundamental question is why does a disease die out before everyone has the disease? And, what fraction of the population needs to be vaccinated so a disease can be controlled or eliminated? Answering this question can explain why smallpox was more easily eradicated than other so-called childhood diseases.
Perhaps the simplest epidemic model, which also introduces many of the ideas, is the case of a single epidemic of a disease, first studied in detail by Kermack and McKendrick (1927). We focus here on diseases that are caused by microparasites, so individuals either have the disease or do not, as opposed to diseases caused by macroparasites (such as tapeworms) where the number of infectious agents in individuals needs to be considered explicitly in order to understand the disease dynamics. Different models are needed to understand epidemics and diseases which are endemic (Kermack & McKendrick, 1932).
We assume a constant population size and divide the population into three classes: susceptible, infective, and removed. Since the time scale of an epidemic is much shorter than the time scale of changes in human population sizes, we can simplify the system by ignoring any demographic influences in the simplest model for an epidemic, which corresponds to the assumption of a constant population size. In other words, the population size is assumed constant, and births and deaths are ignored (although there are models that do take this into account) because the time scale of an epidemic is short relative to the time scale of human population dynamics.
We assume that the rate at which susceptibles become infected is simply proportional to the product of the number of susceptible and infective individuals, corresponding to an assumption of random encounters. The rate at which infective individuals recover is assumed to be a constant. Then, with S the number of susceptibles, I the number of infectives, and R the number of removed individuals, the dynamics are given by
(1)
(2)
(3)
Under the assumption that the total population size N remains constant, we can use the relationship N=S+I+R, and reduce the system (1)–(3) to one based on just the first two equations. The phase plane analysis of the resulting system is facilitated by the fact that the formula
(4)
is so simple. As first discussed by Kermack and McKendrick (1927), one can solve this system explicitly by integration and using approximations. From this one can see that the solution curves which start with I arbitrarily small return to I=0 before all individuals in the population become infected, or in other words the number of susceptibles at t=∞ is positive.
The qualitative behavior of the system (1)–(3) essentially depends on a single nondimensional parameter, the reproductive number for the disease. The reproductive number is defined as the mean number of infective individuals produced by a single infective individual, which can clearly be calculated by multiplying the rate at which a single infective individual produces new infections by the mean period of infectivity for a single individual. Under our assumptions, the mean rate of infection is simply βS and the mean infective period is 1/γ. Thus the reproductive number is simply
(5)
The dynamics are governed by the observation that the number of infectives will increase if R0>1 and will decrease if R0<1. For the case of a single infective introduced into a population of susceptibles, we can use the total population N instead of S in the formula for R0.
The import of the observation about the importance of R0 (and results from integrating (4)) is typically summarized in the threshold theorem which states that there will be an epidemic if the population initially satisfies R0>1, which may be a reflection of the population size. From integrating (4) one finds that the total number of individuals who get the disease is dramatically larger if the population is initially above the threshold.
Kermack & McKendrick (1932) further studied the cases of endemic diseases, where it is necessary to take into account the demography of the population, since the time scale is long enough that births and deaths need to be explicitly included. Other more complex systems were also studied (Kermack & McKendrick, 1933).
Modifications of the basic equations have been used to study the dynamics of sexually transmitted diseases, including AIDS. Here, an important modification has been to break up the population into classes based on different encounter rates.
More recent work on disease dynamics has played a central role in population biology, as the data sets for the incidence of childhood diseases in the United Kingdom and many large U.S. cities are among the longest, most accurate, and most detailed records of populations available (Bjornstad & Gren-fell, 2001). In particular, the prevaccination dynamics of measles has been carefully analyzed using a variety of approaches, based essentially on extensions of the basic model (Equations (1)–(3)), modified to include a seasonally varying contact rate and, in some cases, stochasticity. Studies of this data has led to important substantial advances in the analysis of the kinds of time series available: relatively short with substantial stochastic influences. Analysis of these time series using nonlinear methods have demonstrated that at least some of the dynamics may be chaotic. Further and more recent efforts have focussed on spatiotemporal dynamics (Rohani et al., 1999) and on applied questions (Keeling et al., 2002) like the recent hoof and mouth epidemic in the UK.
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