  In many-body chaotic systems, the size of an operator generically grows in
Heisenberg evolution, which can be measured by certain out-of-time-ordered
four-point functions. However, these only provide a coarse probe of the full
underlying operator growth structure. In this article we develop a methodology
to derive the full growth structure of fermionic systems, that also naturally
introduces the effect of finite temperature. We then apply our methodology to
the SYK model, which features all-to-all $q$-body interactions. We derive the
full operator growth structure in the large $q$ limit at all temperatures. We
see that its temperature dependence has a remarkably simple form consistent
with the slowing down of scrambling as temperature is decreased. Furthermore,
our finite-temperature scrambling results can be modeled by a modified epidemic
model, where the thermal state serves as a vaccinated population, thereby
slowing the overall rate of infection.
