The number is 45. This is optimal because the sum (the number of infected cells + the total perimeter of the infected area) never increases when the infection spreads. At the end, we want 144 infected cells with perimeter 36, so the sum is 180. If at the start we have $n$ infected cells, the total perimeter is at most $3n$, so the sum is at most $4n$, and $$4n \geq 180 \Longrightarrow n \geq 45.$$ A construction is easy to motivate --- all we need to make sure is the above inequality is tight and our chosen sum never goes down, which holds if no two initial triangles are adjacent and it is never true during the process that three infected triangles surround an uninfected one. I don't have an extremely neat one, though. Here's a picture: