Herd immunity
Herd immunity refers to population-level reduction in transmission when enough individuals are immune or otherwise protected that sustained spread becomes difficult.
Simple threshold
In an idealised homogeneous model with perfect, permanent protection, the critical immune proportion is
\[p_c=1-\frac{1}{R_0}.\]This follows from reducing the susceptible fraction until the effective reproduction number is below one.
Why the real threshold can differ
Imperfect vaccines, waning immunity, age structure, heterogeneous contact patterns and pathogen evolution can change the required level of protection.
Key idea. The familiar formula \(1-1/R_0\) belongs to a simplified model; herd-immunity thresholds depend on model assumptions.