  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.
