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Let $p(a)$ be the true seroprevalence at age $a$, and $\lambda (a)$ the force of infection, meaning the instantaneous rate of a first infection among people still uninfected. In an irreversible serocatalytic model, people begin seronegative and retain detectable antibodies after infection. Their seronegative fraction $s(a)=1-p(a)$ obeysHenceEquivalently, where differentiation is possible, $\lambda (a)=p^{\prime }(a)/[1-p(a)]$. This identifies the force of infection as a hazard function and $1-p(a)$ as the survival function for remaining uninfected. For constant exposure $\lambda$, $p(a)=1-e^{-\lambda a}$. A single calendar-time survey reflects each birth cohort's accumulated exposure; interpreting the same age curve as a timeless age effect requires assumptions about historical transmission.

An irreversible serocatalytic model describes the transition from seronegative to seropositive after infection. If the force of infection is $\lambda (a)$ and detectable antibodies persist, the seronegative fraction satisfies $s^{\prime }(a)=-\lambda (a)s(a)$, hence seroprevalence is $1-\exp [-{\int }_0^a\lambda (u)du]$ when $s(0)=1$. Antibody loss needs a reverse transition, and imperfect testing needs a separate observation model.