Before going any further, it is worthwhile to spend time thinking about how we characterize the transmission of disease between infected and susceptible individuals. This is, as one might imagine, a topic with an immense literature. Here, I provide sufficient information for our needs, but not an overall discussion - see the nice review paper of McCallum et al. (2001) for that.
In the previous section, we modeled the dynamics of disease transmission by bIS. This form might remind you of introductory chemistry and of chemical kinetics. In fact, we call this the mass action model for transmission. Since dS/dt = —bIS, and the units of the derivative are individuals per time, the units of b must be 1/(time)(individuals); even more precisely, we would write 1/(time)(infected individuals). Thus, b is not a rate, but a composite parameter.
The simplest alternative to the mass action model of transmission is called the frequency dependent model of transmission, in which we write dS/dt =—b(I/N)S. Now b becomes a pure rate, because I/Nhas no units. Note that we assume here that the rate at which disease transmission occurs depends upon the frequency, rather than absolute number, of infected individuals. If we were working with an open, rather than closed, population in which infected individuals are removed by death or recovery, instead of N we could use I + S.
A third model, which is phenomenological (that is, based on data rather than theory) is the power model of transmission, in which we write dS/dt =—bSpIq where p and q are parameters, both between 0 and 1. In this case, the units of b could be quite unusual.
A fourth model, to which we will return in a different guise, is the negative binomial model of transmission, for which where k is another parameter - and is intended to be exactly the overdispersion parameter of the negative binomial distribution. This model is due to Charles Godfray (Godfray and Hassell 1989) who reasoned as follows. Over a unit interval of time, let us hold I constant and integrate Eq. (5.4) by separating variables so that we see that in one unit of time, the fraction of susceptibles escaping disease is given by the zeroth term of the negative binomial distribution.
As in Chapter 3, where you explored the negative binomial distribution, it is valuable here to understand the properties of the negative binomial transmission model.
(a) Show that as k n, the negative binomial transmission model approaches the mass action transmission model. (Hint: what is the Taylor expansion of log(1 + x)? Alternatively, set k = 1/x and apply L'Hospital's rule.) (b) Define the relative rate of transmission by and do numerical investigations of its properties as k varies. (c) Note, too, that your answer depends only on the product bl, and not on the individual values of b or I. How do you interpret this? (d) The force of infection is now kSlog(1 + (bI/k)). Holding S and I constant, investigate the level curves of the force of infection in the b — k plane.
In most of what follows, we will use the mass action model for disease transmission. In the literature, mass action and frequency dependent transmission models are commonly used, but rarely tested (for an exception, see Knell et al. 1996). Because of this, one must be careful when reading a paper to know which is the choice of the author and why.
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