Strong biological correlations as a cause of autonomous oscillations in epidemics

Abstract

We consider the impact of strong biological correlations on the epidemic process. The biological correlations mean the influence of the environment on the individual state of immunity of the infected person. Accounting for the correlations turns the traditional SIRS model into the 3D Lotka-Volterra model, the parameters of which are uniquely determined by the parameters of the original SIRS model. The measure of the biological correlations in the epidemic is the correlation strength parameter [Formula: see text], where [Formula: see text] and [Formula: see text] are the duration of the infectious period of the disease and the duration of immunity, respectively, both measured in years. If the epidemic is highly correlated ([Formula: see text]), then after the first epidemic outbreak, subsequent oscillations occur, the period [Formula: see text] of which is less than one year. The example is the COVID-19 pandemic. If the epidemic is weakly correlated ([Formula: see text]), the period [Formula: see text] of the oscillations is more than one year. Then in the presence of regular annual outbreaks the oscillations do not have time to manifest themselves. The examples are the ordinary flu annual epidemics. In the absence of the annual epidemic factor, the oscillations can exist and persist regardless of the [Formula: see text] value. The examples are the measles epidemics. The 3D Lotka-Volterra model makes it possible to predict the period [Formula: see text] of the future oscillations based on two known clinical parameters-[Formula: see text] and [Formula: see text]. This period turns out to be equal to 2π multiplied by the geometric mean of the duration of the infectious period of the disease [Formula: see text] and the duration of immunity [Formula: see text]. The adequacy of the 3D Lotka-Volterra model is supported by examples of the 2017-2019 annual flu epidemics and the 2020-2022 COVID-19 epidemic in Israel, as well as the succession of measles epidemics in the UK between 1940 and 1970.

Keywords: Autonomous oscillations; Epidemic; Lotka–Volterra model.