Noise can lead to exponential epidemic spreading despite $R_0$ below one

Abstract

Branching processes are widely used to model evolutionary and population dynamics as well as the spread of infectious diseases. To characterize the dynamics of their growth or spread, the basic reproduction number $R_0$ has received considerable attention. In the context of infectious diseases, it is usually defined as the expected number of secondary cases produced by an infectious case in a completely susceptible population. Typically $R_0>1$ indicates that an outbreak is expected to continue and to grow exponentially, while $R_0<1$ usually indicates that an outbreak is expected to terminate after some time. In this work, we show that fluctuations of the dynamics in time can lead to a continuation of outbreaks even when the expected number of secondary cases from a single case is below $1$. Such fluctuations are usually neglected in modelling of infectious diseases by a set of ordinary differential equations, such as the classic SIR model. We showcase three examples: 1) extinction following an Ornstein-Uhlenbeck process, 2) extinction switching randomly between two values and 3) mixing of two populations with different $R_0$ values. We corroborate our analytical findings with computer simulations.