TimeLocal Reproduction Number Counting Processes, Martingales, Log-Likelihood MLE’s and Variances

Currently, reproduction numbers of pandemic models (SIR, SEIR, Lotka-Euler and others), basic or not, and their constituent parameters like transmission rate or duration of infectiousness, mainly are estimated as one average value for either a moving period in time [0,t], or for the whole duration [0,T] of the pandemic. This book statistically investigates these parameters, and thereby the reproduction number, as time-dependent: hence the name 'Timelocal Reproduction Number'. For instance, the contact rate as part of the transmisson rate is influenced by government measures like social distancing. In this situation it is not really helpful to include data from, say, the beginning of the epidemic, but, data of exactly how bad it was at the beginning of the measure, will do fine. This way, the direct effects of the 'social distancing'-measure can be validated, evaluated and thereby monitored as they happen in real time. Sudden spectacular rises in daily infections (locally in time !) accompanied by a slight decline of the reproduction number, as happened in the Netherlands, are not exactly helpful to breed trust from the public in the government's handling of the pandemic.