We generalize the phenomenological, law of mass action-like, SIR and SEIR epidemiological models to situations with anomalous kinetics. Specifically, the contagion and removal terms, normally linear in the fraction I of infected people, are taken to depend on I-qup and I-qdown, respectively. These dependencies can be understood as highly reduced effective descriptions of contagion via anomalous diffusion of susceptible and infected people in fractal geometries and removal (i.e., recovery or death) via complex mechanisms leading to slowly decaying removal-time distributions. We obtain rather convincing fits to time series for both active cases and mortality with the same values of (q(up), q(down)) for a given country, suggesting that such aspects may in fact be present in the early evolution of the COVID-19 pandemic. We also obtain approximate values for the effective population N-eff, which turns out to be a small percentage of the entire population N for each country.