This is also true under much weaker assumptions. Here's a simple example calculation. Say 50% of a population has R0=1.5, 50% has R0=2.5; the average R0 is 2. T

This is also true under much weaker assumptions. Here's a simple example calculation. Say 50% of a population has R0=1.5, 50% has R0=2.5; the average R0 is 2. Then the required herd immunity is (2-1/1.5-1/2.5)/2=0.47, below the classical 1-1/2=0.5. That this always true for any number of independent subpopulations is a consequence of the harmonic-arithmetic mean inequality.