Pathogens carry antigens — molecular structures recognised by specific immune receptors. Phagocytes engulf and digest pathogens, while lymphocytes make antibodies, proteins shaped to bind one specific antigen. The first response is slow because the matching lymphocytes must be selected and multiplied.
antigen: A molecular structure recognised by a specific antibody or immune receptor; antigens can come from the body itself or from outside it; antibody: A protein that binds one specific antigen; memory cell: A long-lived lymphocyte that enables a fast secondary response; vaccination: Giving a vaccine to prepare a specific immune response and memory; herd immunity: Indirect population protection when immunity reduces transmission.
Memory cells make the second response faster and larger; with R₀ = 4, herd immunity needs 75% immune
Known: a pathogen has a basic reproduction number R₀ = 4, meaning four secondary cases per case on average in a fully susceptible population under this model. The threshold is 1 − 1/R₀ = 1 − 1/4 = 0.75. With uniform mixing and complete protection, exactly 75% immune gives R = 4 × (1 − 0.75) = 1. The immune proportion must exceed 75% to give R < 1 and declining transmission in the model.
The threshold model assumes uniform mixing, equal susceptibility and complete protection against transmission among immune people; real populations cluster and protection can wane. If vaccination gives incomplete protection against transmission, the required vaccination proportion is higher than the immune threshold in this simplified model.
Reproduce the reference's R₀ = 4 threshold of 75% completely immune and its R₀ = 2.5 threshold of 60%. Compare immunity below, at and above the boundary to distinguish replacement from declining transmission under uniform mixing and complete protection.