Many countries have manifested COVID-19 trajectories where extended periods of constant and low daily case rate suddenly transition to epidemic waves of considerable severity with no correspondingly drastic relaxation in preventive measures. Such solutions are outside the scope of classical epidemiological models. Here, we construct a deterministic, discrete-time, discrete-population mathematical model called cluster seeding and transmission model, which can explain these non-classical phenomena. Our key hypothesis is that with partial preventive measures in place, viral transmission occurs primarily within small, closed groups of family members and friends, which we label as clusters. Inter-cluster transmission is infrequent compared with intra-cluster transmission but it is the key to determining the course of the epidemic. If inter-cluster transmission is low enough, we see stable plateau solutions. Above a cutoff level, however, such transmission can destabilize a plateau into a huge wave even though its contribution to the population-averaged spreading rate still remains small. We call this the cryptogenic instability. We also find that stochastic effects when case counts are very low may result in a temporary and artificial suppression of an instability; we call this the critical mass effect. Both these phenomena are absent from conventional infectious disease models and militate against the successful management of the epidemic.