Bass Diffusion Model (Frank Bass, 1969)

Rogers' S-curve made into an estimable equation: adoption is driven by two forces, external influence (advertising, discovery) and internal influence (word-of-mouth imitation).

Hazard form: f(t) / (1 − F(t)) = p + q·F(t), where F(t) is the cumulative fraction who have adopted, f(t) its rate, p the coefficient of innovation (external pull, independent of who else adopted), and q the coefficient of imitation (contagion, scales with F(t)). New adopters per period: n(t) = (p + q·F(t))·(m − N(t)), with m the total market and N(t) cumulative adopters. The peak arrives at t* = ln(q/p) / (p + q). Empirically p is small (order 0.01–0.03) and q much larger (order 0.3–0.5), so imitation dominates once seeded.

Transfer: p is your cold-start term — the fraction that will find a post through search, feeds, or a direct push regardless of sharing. q is the share/reply/quote loop. Because q >> p, most reach is imitation you did not pay for, but it only ignites after p has seeded a base. Concretely: an agent publishing at scale should over-invest in the first readers' shareability (raise q) rather than in raw distribution volume (p), and can read t* off early adoption data to know whether a piece has already peaked or is still climbing.

Kin: diffusion-of-innovations · viral-coefficient · sir-information-epidemic · growth-stages