Convolutional modelling of epidemics

Traditional deterministic modeling of epidemics is usually based on a linear system of differential equations in which compartment transitions are proportional to their population, implicitly assuming an exponential process for leaving a compartment as happens in radioactive decay. Nonetheless, this assumption is quite unrealistic since it permits a class transition such as the passage from illness to recovery that does not depend on the time an individual got infected. This trouble significantly affects the time evolution of epidemy computed by these models. This paper describes a new deterministic epidemic model in which transitions among different population classes are described by a convolutional law connecting the input and output fluxes of each class. The new model guarantees that class changes always take place according to a realistic timing, which is defined by the impulse response function of that transition, avoiding model output fluxes by the exponential decay typical of previous models. The model contains five population compartments and can take into consideration healthy carriers and recovered-to-susceptible transition. The paper provides a complete mathematical description of the convolutional model and presents three sets of simulations that show its performance. A comparison with predictions of the SIR model is given. Outcomes of simulation of the COVID-19 pandemic are discussed which predicts the truly observed time changes of the dynamic case-fatality rate. The new model foresees the possibility of successive epidemic waves as well as the asymptotic instauration of a quasi-stationary regime of lower infection circulation that prevents a definite stopping of the epidemy. We show the existence of a quadrature function that formally solves the system of equations of the convolutive and the SIR models and whose asymptotic limit roughly matches the epidemic basic reproduction number.