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How he got tuberculosis:
When he went on the mission for Herr Strauss to get the debt from Thomas Downes.
Thomas Downes is sick, his wife pointed it out plus he was coughing on Arthur's face. He even coughed up blood onto Arthur's face.
How tuberculosis spreads:
By airborne respiratory droplets (coughs or sneezes).
So, in other words, Arthur wouldn't have gotten it, if he didn't go on this mission for Herr Strauss.