A phrase to describe the external spreading phase of some infection?

Metastasize can explain the spreading, although no one word completes your example sentence.

to spread injuriously to transform, especially into a dangerous form

There isn't a single word solution for your sentence, but something like

It was like a cancer metastasizing.

comes close.

Consider,

pustulation

The formation or appearance of pustules. The American Heritage® Medical Dictionary

carbunculosis

A condition marked by the formation of numerous carbuncles. Miller-Keane Encyclopedia and Dictionary of Medicine, Nursing, and Allied Health

Contagion — M-W

a : rapid communication of an influence (as a doctrine or emotional state)
b : an influence that spreads rapidly

"It was like a contagion."
"the contagion of disgrace"

Simply put, it means the spreading of a harmful idea or theory.