The most commonly used epidemiological models are exponential in early phases of an outbreak, because they assume roughly constant infectiousness, up until the

The most commonly used epidemiological models are exponential in early phases of an outbreak, because they assume roughly constant infectiousness, up until the point where some non-negligible portion of the population has recovered and gained immunity. If the nth infected person produces as many new infections as the 1st, you get exponential growth. For example, to take a very simple version of that model: if each infected person infects R other people over some time period, and you have 1 infection at t=0, then you have R^t new infections at time step t.