In a comment on our recent paper (1), the commentators point out that infection curves on small-world (SW) networks are linear only in the vicinity of the critical point (2) and that infection dynamics on complex or SW networks have already been studied, for instance, in ref. 3. In our work (1), we extend the existing literature by showing, in detail, how an explanation of the apparent sustained algebraic growth regimes observed in many countries during the COVID-19 pandemic involves the structure of the underlying contact network of individuals and how a transition from exponential spreading to a power-law behavior happens when the reproduction number takes values in the vicinity of R ≈1. (…)

S. Thurner, P. Klimek, R. Hanel, Reply to Kuśmierz and Toyoizumi: A network-based explanation of why most COVID-19 infection curves are linear, PNAS 118 (10) (2021) e2100906118