Recovering the time-dependent transmission rate from infection data via solution of an inverse ODE problem

Background. Mathematical models provide epidemiologists with powerful tools for quantitatively assessing the effectiveness of control methods and uncovering underlying mechanisms of observed infection data. One of the key parameters is the transmission rate. The transmission rate of many acute infectious diseases varies significantly in time, but the underlying mechanisms are usually uncertain. They may include seasonal changes in the environment, contact rate, immune system response, etc. The transmission rate has been thought impossible to measure directly. We derive an algorithm to recover the time-dependent transmission rate directly from infection data. Methodology/Principal Findings. The algorithm is derived from the complete and explicit solution of a mathematical inverse problem for SIR-type transmission models. We illustrate the algorithm with historic UK measles data and discover that the two dominant spectral peaks of the transmission rate have 2and 1/3-year periods, respectively. In contrast, previous measles transmission models assumed a one-year periodic transmission rate function to account for mixing of children in school. Conclusions/Significance. The main objective of this work is to provide a new algorithm to recover the transmission rate function from collectable infection data. Our algorithm also yields that almost any infection profile can be perfectly fitted by an SIR-type model with variable transmissibility. This clearly illustrates the danger of overfitting an SIR transmission model. † To whom correspondence should be addressed. Electronic address: [email protected] The UK measles data and birth data was obtained from [23] and [24]. The authors very much appreciate the efforts of David Earn and Ben Bolker for maintaining these public databases of infectious disease data.