LAUSR

Compartmental models, such as the well-known SEIR model, which divides the population into Susceptible (to the infection), Exposed, Infectious and Recovered persons, and mathematically models their interdependence, are paradigms of mathematical epidemiology. It is therefore natural that epidemiologists attempt to use appropriate variations of such models to forecast quantities of interest and importance to policymakers during the COVID-19 pandemic. Common to all these models and variations is that they are based on the complex mathematics of differential equations. Although differential equations prove appropriate to model a great variety of processes in very different fields, this is not granted in every single case and for every task. To appreciate more easily possible problems or weaknesses in their use for forecasting, we do not argue on a purely abstract level, but in the context of representative models and their evolution equations. Two examples of models that are presently used are described in [1] and called there BT and CZ. They are implemented on icumonitoring.ch, a platform developed for short-term forecasting of intensive care unit (ICU) occupancy in Switzerland during the present pandemic. Currently, the platform provides forecasts derived by yet one further model, termed the MG model, which implements forecasting based on data-driven time series [2]. As far as we know, this model has not been published yet. In the following, we briefly sketch the mathematical content of model BT. Readers not familiar with ordinary differential equations may skip this material and proceed directly to the section “Performance of icumonitoring.ch” where we show that the performance of the models used by icumonitoring.ch has been poor in the past. In the third section, we discuss conceptual shortcomings of models BT and CZ that are likely to contribute to their poor performance. In the next section, we argue on general grounds that differential equations are neither well suited, nor actually needed for short-term forecasting of the impact of infectious diseases. In the last section, we propose a very simple hands-on method for short-term forecasting (in the spirit of recent proposals described in [3]). We now turn to Model BT. This model consists of systems of ordinary differential equations describing the time evolution of the following quantities: The number of susceptible individuals, S(t), the number of exposed individuals, E(t), and the number of infected individuals, I(t), at time t. The relevant equations are: dS/dt = ‒ SβI, dE/dt = SβI ‒ τE, dI/dt = τE ‒ γI. In writing these equations we have replaced S(t) by S, etc. for ease of notation. This system of equations is extended by the following two equations for the numbers of hospitalised patients, H(t), and of ICU patients, U(t): dH/dt = εH γI ‒ γHH, dU/dt = γH εH21 H ‒ γU U. The (posterior) distributions of the parameters β, τ, γ, γH, γU, εH, and εH21 are determined at a cantonal level, using standard techniques and tools; but this is of no importance in the following. For details, as well as for a similar system of deterministic ordinary differential equations, corresponding to the model CZ, and parameters appearing in the equations, we refer the reader to [1]. Forecasting the number of ICU patients is then done as follows. If, on a day t = t0, the true numbers S(t0), E(t0), I(t0), H(t0) and U(t0) are known from direct observation or inferred from other, directly observable quantities, and assuming the distribution of the above parameters is known, we can calculate the distribution of U(t0 + Δ), for Δ > 0, by solving the above system of differential equations with initial values S(t0) = Strue (t0), etc. Among the forecasts provided by icumonitoring.ch are the mean values and the 95% confidence intervals of U(t0 + Δ), for Δ equal to 3 and 7 days. Correspondence: Prof. Daniel Wyler, PhD, Institut für Theoretische Physik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Wyler[at]physik.uzh.ch