1.10 Quantification of the Algorithm’s Efficiency
2.1.5 Final Size of the Epidemic and The Basic Reproduction
Before presenting any statistical issues which refer to the general stochastic epi- demic model, we concentrate to the most important measures in stochastic epi- demic modelling; thefinal sizeof epidemic and thebasic reproduction number R0. In this section we will briefly describe these useful epidemiological quantities.
2.1.5.1 Final Size Distribution
The final size the epidemic, say Z, is simply defined as the number of initially susceptible individuals that ultimately become infected. For θ ≥ 0, let φ(θ) =
E[exp{−θD}] be the moment generating function of the infectious period D and letpkbe the probability that the final size of the epidemic is equal tok, 0≤k ≤n. Ball (1986) proved that
l X i=1 N −k l−k pk φλ(N −N l)k+m = N l , 0≤k ≤n (2.2)
Note that an alternative definition of GSE assumes that during their infectious period, an individual makes contacts with each of the susceptibles at times given by the points of a homogeneous Poisson process with intensity λ/N.
The system of equations in 2.2 is triangular in thepk’s and hence, in principle, it is easy to calculate the final size probabilities recursively, i.e. p0,thenp1, p2 and so on. Nevertheless, problems often occur in some specific circumstances such as ex- treme values for the parameters, even for small populations. When an Exponential
infectious period is assumed, Bailey (1975) has derived a different set of equations for the final size of probability that has better numerical stability. However, the Laplace transformation methods which are applied to the forward equations of the Markovian epidemic process do not generalize for a non-Markovian setup. Recently, Demiris and O’Neill (2006) employed multiple precision arithmetic to surmount this numerical problems. They also concluded that the branching pro- cess approximations as used to calculate the probability of an epidemic taking off was found to be effective, even for small numbers of initial susceptibles.
2.1.5.2 R0 and the Threshold Result
The following definition is taken from Heesterbeek and Dietz (1996):
R0 is the expected number of secondary infections produced by a typical infected individual during its entire infectious period in a population consisting of susceptibles only.
In the GSE model, atypical individual can be any of the infectives since the model assumes homogeneous mixing and will, on average, be infectious for time 1/γ. Then, the number of susceptibles infected by one infective per unit time is βN. Hence the total number of infections produced by one infective, is equal toβN/γ. In the case of the deterministic SIR model, the parameterγ can be interred as the reciprocal of the infectious period. In general, for an arbitrary infectious number,
D, the basic reproduction ratio is defined as follows:
R0 =βN ·E[D].
In more complicated models, the definition of R0 is not straightforward and care is required to define an appropriate measure.
We shall describe whyR0 is such a significantly important measure in epidemics.
the number of infectives Yt increases as long as the number of initial susceptibles in the population x0 = N is greater than the quantity γ/β (Kermack and McK- endrick, 1927). In other words, this is equivalent to the inequality that R0 > 1. This reveals the significance of R0. If R0 ≤ 1 then the latter condition cannot be met and therefore only a minor outbreak can result and R0 is considered as
threshold parameter . With an infection for which R0 > 1 a population will be protected from epidemic outbreaks as long as the number of susceptibles is kept below the threshold by vaccination.
The threshold behavior of the stochastic SIR model for large populations (Whit- tle, 1955, Williams, 1971, Bailey, 1975, Andersson and Britton, 2000) is generally speaking analogous to that of the deterministic model. Intuitively, if the initial number of infectives, α, is small then during the early stages of the epidemic in a large population, essentially all the contacts of infectives are with susceptibles and a branching process approximation is appropriate (see also for example, Ball, 1983). We should make clear that such results are exactly valid only asymptot- ically, typically as the population size becomes infinite. Although the branching approximation idea has a long history, Ball and Donnelly (1995) used a coupling argument to investigate how the approximation improves as the population tends to infinity.
Specifically, in a population of infinitely many susceptibles, if R0 ≤ 1 then, with probability one, only a finite number of susceptibles will become infected (i.e. minor outbreak). If R0 > 1 there is a positive probability that infinitely many
susceptibles will become infected (i.e. major outbreak). We should bring to at- tention that, for finite populations, corresponding definitions of major and minor outbreaks are more difficult to define. Nevertheless, it is broadly true, an epidemic is either very likely will die out with minor impact or else might end up with a large proportion of susceptibles getting infected. Depending on the value ofR0and whether is greater or smaller than one, then its value, approximately, will indicate which of the two situations is more likely. Therefore, it now becomes clear whyR0
is so important in epidemic theory since also implies the amount of effort needed to prevent an epidemic.