DTM.2:4.3 - Choose a contact law that matches the mechanism

For independent opportunities with per-contact success probability p and n contacts, the chance of at least one success is 1−(1−p)^n. This construction requires comparable independent trials. It is not justified merely by counting contacts.

When uptake needs reinforcement, use the relevant history: distinct supporting neighbours, repeated successful demonstrations, trust, resource availability or a shared commitment. Give repeated and independent sources the effects supported by the case. Repetition may cease to add information, and another contact can discourage uptake.

In continuous time, define the hazard for a specified recipient who is still eligible for the event. The conditional probability of its first event over a small interval Δt is approximately hΔt; h has units of inverse time. Under a constant hazard, the interval probability is 1−exp(−hΔt), not h itself.

To construct the population flow, count the eligible recipients once. In an illustrative fixed population of N participants, with fraction f already using the variant, homogeneous mixing may support an eligible recipient’s hazard h=βf. Then:

eligible recipients S = N(1−f)
new-uptake flow J = S h = N(1−f)βf          [events per unit time]
flow of population share j = J/N = βf(1−f) [share per unit time].

Here β combines the supported encounter and conditional-uptake rates. The factor 1−f belongs to the eligible population, not again inside its per-recipient hazard. This is one two-state construction, not a universal law. Network, institutional or broadcast exposure may require recipient-specific hazards; sum those over the actual eligible set. New participants, departures and return to eligibility require their own transitions.

If competing variants or mechanisms could explain the same uptake, retain those alternatives and use MMP.16 to find a distinguishing observation. A fitted contact coefficient does not establish the missing mechanism.