= Solution
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 $s(a)=1-p(a)$ obeys
$$
\frac{ds}{da}=-\lambda(a)s(a),\qquad s(0)=1.
$$
Hence
$$
\boxed{p(a)=1-\exp\left[-\int_0^a\lambda(u)\,du\right]}.
$$
Equivalently, where differentiation is possible, $\lambda(a)=p'(a)/[1-p(a)]$. This identifies the <force of infection> as a <hazard function> and $1-p(a)$ as the <survival function> for remaining uninfected. For constant exposure $\lambda$, \b[$p(a)=1-e^{-\lambda a}$]. 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.
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