Most measles models are based around the SEIR equations (see section (1.4.3)) because the life- history of a measles infection in an individual is well classied by an exposed (but not infectious) stage followed by the fully infectious stage and then life-long recovery. Typical durations are mea- sured in a few days for each, and the remainder of this chapter will use the values given by Olsen and Schaer (1990) of 10.2 days for the average length of the exposed stage and 3.65 days for the infectious period. With time measured in years these correspond to = 35:78 and = 100. Life expectancy for humans has naturally varied from time to time, and from place to place through history, but an average value of 50 years ( = 0:02) is a reasonable estimate. Estimating the trans- mission rate is more dicult, as previously mentioned because it represents the combined eect of rate of contact formation (assuming the mass-action principle) with eective disease transmission. On their own, with either the parameter values given here or indeed with any others, the SEIR equations like the SIR equations display only long-term equilibrium dynamics, sometimes through damped oscillations. The basic equations therefore dramatically fail to capture the epidemic be- haviour observed in reality, but there are some modications have helped to rectify this discrepancy. These are now discussed.
6.4.1.1 SeasonalForcing
Measles is predominantly a childhood disease with over 90% of a pre-vaccination era population typically having been infected before the age of twenty (Keeling, 1995). Because of the importance of schooling and its consequences for social mixing in this age group, many authors have consid- ered seasonally forcing the basic SEIR model to simulate the changes in population contact rates between school term times and holidays. The simplest approach, and a crude rst step, is to vary sinusoidally over a period of one year, with low during the summer (holiday) and high during the winter term time (Olsen and Schaer, 1990 Rand and Wilson, 1991 Bolker and Grenfell, 1993). With appropriately chosen parameters, previous studies of the forced SEIR equations have shown that they can display much more dynamic (occasionally chaotic) trajectories than the unforced ver- sions (see, for example, Rand and Wilson, 1991). In this respect the behaviour is closer to that observed in the measles data. For realistic parameters, the time series of the number of infected in- dividuals also displays occasional epidemic outbursts (spikes) but there is an accompanying problem in that the troughs in between are typically very low indeed. The deterministic equations can recover from these to produce another epidemic, but in a stochastic model (and certainly in the real world)
the infection would be in great danger of disappearing. In a simulated population of around half a million, the troughs commonly represent the equivalent of way less than one infected individual, and an accurate prediction of the population size for persistence is therefore not available. Another consequence of the very low troughs (through which the orbits are `stretched' by the attractor) is that the spacing between subsequent spikes is very irregular - anything from one to ve years is common - in contrast to the much more regular peaks observed in England and Wales (gure (6.9)) and other data sets.
6.4.1.2 AgeStructure
Because measles primarily aects children, age-structure is an important consideration. Rather than assuming all individuals in a population are indistinguishable, some models split up the population according to the individuals' age distribution. The presence of schools means that individuals of dif- ferent ages will typically have dierent mixing rates both with other individuals of a similar age and with individuals of other ages (they will also have potentially dierent mortality rates and recovery rates etc. too). Tayloring the (increased number of) model parameters to t more specically ob-
served age-related data gives opportunity of rening the model to more accurately reect real events. One approach is to convert the SEIR ODEs into a system of PDEs (see Anderson and May, 1992 for general techniques) that gives age-specic parameter responses to a continuum of ages from zero up to the maximum required age. Such a system then models not only the size but also the complete age distribution of all individuals in each class (S,E,I or R) of the model. A further advantage is that the questionable assumption of exponentially distributed expected life times can then be replaced with age-dependent mortality. The cost, of course, is a resulting PDE system which is very dicult to analyse and relatively dicult to investigate numerically due to the vastly increased set of pa- rameters (each single parameter in the ODE corresponds to an arbitrary distribution with respect to age in the PDE). For reasons of clarity and manageability, many authors prefer to consider a second approach, where the population is divided into a few distinct age categories. A popular choice is the four class system consisting of pre-school children (aged 0 to 5 years), primary school children (aged 6 to 10), secondary school children (aged 11 to 15) and adults (aged 16+).
This latter classication is used by the Realistic Age Structure (RAS) model of measles dynam- ics (Bolker and Grenfell, 1993, 1995). This is another SEIR-based model that also incorporates school-year motivated seasonal forcing. Each of the four age categories is split into the four disease categories resulting in a 16;1 = 15 dimensional model (the population has a xed total size).
To date, the RAS model is one of the most successful measles models in terms of replicating the observed dynamics. It signicantly improves on the behaviour of the basic forced SEIR model, most noticeably by reducing the persistence population size to around two million. This is a step in the
right direction if not yet quite perfect.
One consequence of increasing the number of population categories in models like the RAS model is the diculties associated with the corresponding increase in the number of model parameters. A single transmission parameter, , for example, is replaced by an entire age-structured `who acquires infection from whom' (WAIFW) matrix (Anderson and May 1992). Furthermore, this matrix is given even more exibility with the introduction of seasonal variation. The rst diculty this presents is how to measure realistic values for each of these parameters in the eld. Edmunds, O'Callaghan and Nokes (1997) have taken an experimental rst step and measured the age-structured contact rates (based on the incidence of conversations) amongst a sample of adults that may be important in the transmission of infectious disease. Information of this kind, combined with possible age-dependent physiological factors, is essential for estimating the model parameters. The second diculty is that numerical analysis of the equations becomes a much more laborious task, with vast parameter spaces to search. Given a judicious choice of enough parameters, there is also the increased likelihood of being able to reproduce a whole spectrum of dierent behaviour with a single model. This can lead to the problem of justifying why any one behaviour is more important or relevant than any other.
6.4.1.3 SpatialStructure
Perhaps because of the many possibilities presented by seasonal and age-related modications, and also because of the absence of an obvious practical approach, spatial structure has received com- paratively little attention in measles modelling (but see Grenfell, Bolker and Kleczkowski, 1995 for a metapopulation discussion). This is despite the signicant contributions it has made to other ecological model systems, and the clear prospect of space being an important factor in explaining measles dynamics. A pair model analysis provides a possible solution to tractably incorporating spatial eects and this is now explored. The pair induced limit cycling found in the SIR pair equa- tions is particularly exciting now because of the observed oscillations in the measles data.