Stochastic Methodology for the Study of an Epidemic Decay Phase, Based on a Branching Model

We present a stochastic methodology to study the decay phase of an epidemic. It is based on a general stochastic epidemic process with memory, suitable to model the spread in a large open population with births of any rare transmissible disease with a random incubation period and a Reed-Frost type infection. This model, which belongs to the class of multitype branching processes in discrete time, enables us to predict the incidences of cases and to derive the probability distributions of the extinction time and of the future epidemic size. We also study the epidemic evolution in the worst-case scenario of a very late extinction time, making use of the Q-process. We provide in addition an estimator of the key parameter of the epidemic model quantifying the infection and finally illustrate this methodology with the study of the Bovine Spongiform Encephalopathy epidemic in Great Britain after the 1988 feed ban law. 1. Introduction Outbreaks of infectious diseases of animals or humans are subject, when possible, to control measures aiming at curbing their spread. Effective measures should force the epidemic to enter its decay phase and to reach extinction. The decay phase can then be simply detected by a decrease of the number of cases, when this decrease is obvious. However this is not always the case, and this rough qualitative information might not be sufficient to evaluate accurately the effectiveness of the proposed measures to reduce the final size and duration of the outbreak. The goal of this paper is to present a stochastic methodology in discrete time to study more accurately the decay phase of an epidemic. Our framework is the spread, in a large open population, of a rare transmissible disease such that the infection process may be assumed to follow a Reed-Frost type model, with a probability for a susceptible to become infected by a given dose of pathogens inversely proportional to the total population size. Moreover the latent period (during which an individual is infected but not yet infectious) may be random and long compared to the generation time. Questions about the decay phase include the following: which quantitative criteria can ensure that the disease has entered an extinction phase? What is the probability distribution of the epidemic extinction time, of the epidemic final size, and of the incidence of infected individuals? Finally, what would be the evolution of the epidemic in the event of a very late extinction of the disease? From a practical point of view, it is generally impossible to observe all infections. Susceptible and