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However, as of version 11.1 specifying common CA rules has become a lot more convenient. The possibility to specify a CA rule via an association with allows for rather high-level classifications. In fact, Mathematica now knows about various neighbourhoods:

However, as of version 11.1 specifying common CA rules has become a lot more convenient. The possibility to specify a CA rule via an association allows for rather high-level classifications. In fact, Mathematica now knows about various neighbourhoods:

If it is at all an option to represent the grid as a 2D list instead of a list of infected coordinates, I would model this is a cellular automaton. What you've essentially got is an outer totalistic cellular automaton with a von Neumann neighbourhood. The rule in Game-of-Life notation is B234/S01234, i.e. a cell comes to life if it has two or more live neighbours and it always survives. Implementing simple CAs is quite straight-forward with Mathematica's CellularAutomaton, and I've written another answer here about how to figure out the rule number of the CA.

If it is at all an option to represent the grid as a 2D list instead of a list of infected coordinates, I would model this is a cellular automaton. What you've essentially got is an outer totalistic cellular automaton with a von Neumann neighbourhood. The rule in Game-of-Life notation is B234/S01234, i.e. a cell comes to life if it has two or more live neighbours and it always survives. Implementing simple CAs is quite straight-forward with Mathematica's CellularAutomaton, and I've written another answer here about how to figure out the rule number of the CA.

To simulate the infection to convergence, I'd recommend FixedPointList instead of NestWhileList. It simply applies a function over and over until the value stops changing, and then gives you all the intermediate values.

Module[{a, b, d = 25}, a = RandomChoice[{0, 0, 0, 0, 0, 0, 0, 1}, {d, d}]; b = Most@ FixedPointList[ CellularAutomaton[{1018, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2,    0}}}, {1, 1}}, {#, 0}][[1, 2 ;; -2, 2 ;; -2]] &, a]; ListAnimate[ArrayPlot /@ b] ] 

However, as of version 11.1 specifying common CA rules has become a lot more convenient. The possibility to specify a CA rule via an association with allows for rather high-level classifications. In fact, Mathematica now knows about various neighbourhoods:

CellularAutomaton[<| "OuterTotalisticCode" -> 1018, "Neighborhood" -> "VonNeumann", |>, {#, 0} ][[1, 2 ;; -2, 2 ;; -2]] & 

And we don't even need to compute that rule code, because we can specify the rule directly via a set of growth cases:

CellularAutomaton[<| "GrowthCases" -> {2, 3, 4}, "Neighborhood" -> "VonNeumann", |>, {#, 0} ][[1, 2 ;; -2, 2 ;; -2]] & 

This says "when a dead cell has 2, 3 or 4 live neighbours, the cell comes alive", which is exactly what we're looking for.

To simulate the infection to convergence, I'd recommend FixedPointList instead of NestWhileList. It simply applies a function over and over until the value stops changing, and then gives you all the intermediate values.

Module[{a, b, d = 25},   a = RandomChoice[{0, 0, 0, 0, 0, 0, 0, 1}, {d, d}];   b = Most @ FixedPointList[  CellularAutomaton[<| "GrowthCases" -> {2, 3, 4},    "Neighborhood" -> "VonNeumann",     |>,  {#, 0} ][[1, 2 ;; -2, 2 ;; -2]] &, a];   ListAnimate[ArrayPlot /@ b] ]