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Pandemics, like the 1918 Spanish Influenza and COVID-19, spread through regions of the World in subsequent waves. There is, however, no consensus on the origin of this pattern, which may originate from human behaviour rather than from the virus diffusion itself. Time-honoured models of the SIR type or others based on complex networks describe well the exponential spread of the disease, but cannot naturally accommodate the wave pattern. Nevertheless, understanding this time-structure is of paramount importance in designing effective prevention measures. Here we propose a consistent picture of the wave pattern based on the epidemic Renormalisation Group (eRG) framework, which is guided by the global symmetries of the system under time rescaling. We show that the rate of spreading of the disease can be interpreted as a time-dilation symmetry, while the final stage of an epidemic episode corresponds to reaching a time scale-invariant state. We find that the endemic period between two waves is a sign of instability in the system, associated to near-breaking of the time scale-invariance. This phenomenon can be described in terms of an eRG model featuring complex fixed points. Our results demonstrate that the key to control the arrival of the next wave of a pandemic is in the strolling period in between waves, i.e. when the number of infections grow linearly. Thus, limiting the virus diffusion in this period is the most effective way to prevent or delay the arrival of the next wave. In this work we establish a new guiding principle for the formulation of mid-term governmental strategies to curb pandemics and avoid recurrent waves of infections, deleterious in terms of human life loss and economic damage.