May the fourth be with flu

APL (Dyalog 16.0), 54 characters or 60 bytes

Takes enclosed matrix as argument, returns the step number which completes infection, i.e. 1=is already fully infected. 0=does not fully spread, which is just 1 + the OP's numbers.

54 characters (Unicode):

(≢×0=0∊⊃){(⊢≡f←⊢∨2≤{+/,⍵×3 3⍴0 1}⌺3 3)⊃⍵:⍵⋄(⊂f⊃⍵),⍵}⍣≡

60 bytes (Classic):

(≢×0=0∊⊃){(⊢≡f←⊢∨2≤{+/,⍵×3 3⍴0 1}⎕U233A 3 3)⊃⍵:⍵⋄(⊂f⊃⍵),⍵}⍣≡

⌺ is equivalent to ⎕U233A

Examples' run:

      g←(≢×0=0∊⊃){(⊢≡f←⊢∨2≤{+/,⍵×3 3⍴0 1}⌺3 3)⊃⍵:⍵ ⋄ (⊂f⊃⍵),⍵}⍣≡
      ⎕IO←0
      b←⊂⊖⍉~@(⎕JSON'[[0,5],[1,4],[2,3],[2,1],[3,3],[3,0],[4,5],[5,0]]')⊢0⍴⍨2⍴6
      g b
8
      b←⊂⊖⍉~@(⎕JSON'[[1,1],[0,3],[1,0],[3,0],[3,3]]')⊢0⍴⍨2⍴4
      g b
10

The steps are as follows:

┌─────────────┬─────────────┬─────────────┬─────────────┬─────────────┬─────────────┬─────────────┬─────────────┐
│ X       X   │ X X     X   │ X X X   X   │ X X X X X   │ X X X X X   │ X X X X X   │ X X X X X   │ X X X X X X │
│   X         │ X X X       │ X X X X     │ X X X X X   │ X X X X X   │ X X X X X   │ X X X X X X │ X X X X X X │
│     X X     │   X X X     │ X X X X     │ X X X X     │ X X X X X   │ X X X X X X │ X X X X X X │ X X X X X X │
│             │     X       │   X X X     │ X X X X X   │ X X X X X X │ X X X X X X │ X X X X X X │ X X X X X X │
│     X       │     X X     │     X X X   │   X X X X X │ X X X X X X │ X X X X X X │ X X X X X X │ X X X X X X │
│       X   X │     X X X X │     X X X X │     X X X X │   X X X X X │ X X X X X X │ X X X X X X │ X X X X X X │
└─────────────┴─────────────┴─────────────┴─────────────┴─────────────┴─────────────┴─────────────┴─────────────┘
┌─────────┬─────────┬─────────┬─────────┬─────────┬─────────┬─────────┬─────────┬─────────┬─────────┐
│ X     X │ X     X │ X     X │ X     X │ X     X │ X     X │ X   X X │ X X X X │ X X X X │ X X X X │
│         │         │         │         │       X │     X X │   X X X │ X X X X │ X X X X │ X X X X │
│   X     │   X     │   X X   │   X X X │   X X X │   X X X │   X X X │   X X X │ X X X X │ X X X X │
│   X   X │   X X X │   X X X │   X X X │   X X X │   X X X │   X X X │   X X X │   X X X │ X X X X │
└─────────┴─────────┴─────────┴─────────┴─────────┴─────────┴─────────┴─────────┴─────────┴─────────┘

MATL, 29 28 bytes

tn:"tlY6Z+1>Z|t?@.]]Nl=?l_]&

Input is in the form of a 2D matrix of 1's and 0's

Try it at MATL Online

Explanation

        % Implicitly grab user input as a 2D matrix
t       % Duplicate the inputs
n:      % Count the number of elements in the input (N) and create the array [1...N]
"       % Loop this many times (maximum number of steps we analyze)
  t     % Duplicate the top element
  lY6   % Push the 2D array => [0 1 0; 1 0 1; 0 1 0]
  Z+    % Perform 2D convolution (and maintain the size)
  l>    % Find all values that are >= 2
  Z|    % Perform an element-wise OR with the previous state
  t?    % If all elements are 1's
    @.  % Push the current index and break out of the loop
  ]     % End of if 
]       % End of for loop
Nl=?    % If there is only one element on the stack
  l_    % Push a negative one
]       % End of if statement
&       % Display the top stack element

Python, 231 bytes

g=input()
q=lambda r,c:g[r][c]if(0<=r<m)*(0<=c<m)else 0
m=len(g);p=t=0;n=range(m)
while not all([r for k in g for r in k]):h=[[g[r][c]or sum([q(r+1,c),q(r-1,c),q(r,c+1),q(r,c-1)])>1 for c in n] for r in n];t+=1;0/(g!=h);g=h
print t

It raises an error if it isn't possible.

Try it online!