Complete observation of the epidemic process

Abstract

In this chapter we assume that the standard SIR epidemic process E _n,m (λ, I) is observed completely. By complete observation is meant that the infection times τ _i and removal times ρ _i (hence also the length of the infectious period I _i =ρi-τi for formulas to be consistent. From the data it is obviously possible to deduce how many individuals are susceptible, infectious (and removed) for each time t implying that (X,Y)= {(X)(t),Y(t);t≥0} as well as the final size Z=n-X(∞) are observed. Based on the observed data we want to draw inferences on the transmission parameter λ and the distribution of the infectious periodI.We do this by means of Maximum Likelihood (ML) theory. First, we have to make the meaning of ‘likelihood’clear for such epidemic processes. Hence, in Section 9.1 we derive the log-likelihood for a vector of counting processes, and also state some properties of martingales which will turn out useful in the statistical analysis.