On the Numerical Simulation of Exponential Decay and Outbreak Data Sets Involving Uncertainties

Abstract

Measurement data sets collected when observing epidemiological outbreaks of various diseases often have specific shapes, thereby the data may contain uncertainties. A number of epidemiological mathematical models formulated in terms of ODE’s (or reaction networks) offer solutions that have the potential to simulate and fit well the observed measurement data sets. These solutions are usually smooth functions of time depending on one or more rate parameters. In this work we are especially interested in solutions whose graphs are either of “decay” shape or of a specific wave-like shape briefly denoted as “outbreak” shape. Furthermore we are concerned with the numerical simulation of measurement data sets involving uncertainties, possibly coming from one of the simplest epidemiological models, namely the two-step exponential decay process (Bateman chain). To this end we define a basic exponential outbreak function and study its properties as far as they are needed for the numerical simulations. Stepping on the properties of the basic exponential decay-outbreak functions, we propose numerical algorithms for the estimation of the rate parameters whenever the measurement data sets are available in numeric or interval-valued form.