The tipping effect of delayed interventions on the evolution of COVID-19 incidence

Abstract: We combine infectious disease transmission and the non-pharmaceutical intervention (NPI) response to disease incidence into one closed model consisting of two coupled delay differential equations for the incidence rate and the time-dependent reproduction number. The model contains three free parameters, the initial reproduction number, the intervention strength, and the response delay relative to the time of infection. The NPI response is modeled by assuming that the rate of change of the reproduction number is proportional to the negative deviation of the incidence rate from an intervention threshold. This delay dynamical system exhibits damped oscillations in one part of the parameter space, and growing oscillations in another, and these are separated by a surface where the solution is a strictly periodic nonlinear oscillation. For parameters relevant for the COVID-19 pandemic, the tipping transition from damped to growing oscillations occurs for response delays of the order of one week, and suggests that effective control and mitigation of successive epidemic waves cannot be achieved unless NPIs are implemented in a precautionary manner, rather than merely as a response to the present incidence rate.