DTM.4:4 - Solution
Specify the surrounding regime, test growth from rarity, find maintained regimes, then examine reachable change and return.
DTM.4:4.1 - Preserve the law and the surrounding conditions
Recover the transmission and uptake construction from DTM.2 and the relevant feedback from DTM.3. Identify the variables being varied and those held fixed, the feasible state region, units of time and the environmental conditions. Keep the counted event and eligible population unchanged across that handoff. A per-recipient probability or hazard, a repeatable-event count and a population flow are different inputs; convert them using the stated receiving event before analyzing regimes.
State the practical question before solving: establishment from a small introduction, persistence of an existing variant, coexistence, reduction to a desired range, or return after a temporary change. “What happens?” is too broad when those questions depend on different initial conditions.
If a capability, permission or resource required for uptake is absent, preserve that absence in the law. A favorable comparison is not itself an event.
DTM.4:4.2 - Test a rare variant in a specified resident regime
Place the proposed variant at a small positive amount in an otherwise specified regime. Determine its initial growth from the modeled receiving events and losses.
For one rare quantity y, a local form y’=r y plus smaller terms gives growth when r>0 and decline when r<0. r concerns this surrounding regime and these assumptions. If r=0 or the neglected terms can dominate at the relevant scale, the first-order test does not decide.
With several linked rare states, use the appropriate coupled linearization or generation-to-generation operator supplied by the mathematical modeling method. Do not assign one scalar “reproduction number” until its construction and threshold apply to that model.
A zero initial amount can remain zero in a deterministic model with no external introduction. The rare-variant test asks about a small positive amount; it does not predict that an introduction will occur.
DTM.4:4.3 - Find what can maintain the variant
Find admissible stationary or recurring regimes and determine their relevant stability. Retain coexistence, oscillation or continued replacement if the model supports those forms; persistence need not mean a constant population share.
Where reinforcement or compatibility matters, examine finite initial amounts as well as rarity. A rare variant can decline while a sufficiently established variant persists. In that case, identify what separates the continuations and whether the proposed action can cross that boundary.
Use analytical reasoning, validated numerical calculation or a justified qualitative argument at the formality needed by the decision. A solver’s last point alone is not evidence of an attracting regime; check the event law, feasible region and behavior under relevant nearby conditions.
DTM.4:4.4 - Distinguish a temporary displacement from a changed law
Model what the contemplated action changes. A one-time alteration of current use changes an initial condition. Continuing support, a compatibility adapter or a revised recognition rule can change a rate or a dependence. A newly available variant changes the candidate set.
Follow the trajectory during the action and after its removal. Ask whether the post-action state remains in the same region of attraction, enters another one, or encounters a different law because capabilities, resources or variants have changed.
For an intended return, identify the route and means needed to realize it. Lowering a parameter is insufficient if a necessary alternative has been lost or an endpoint is invariant under the model. Use C.36.RP and the relevant domain method to obtain an unavailable contribution; do not invent it as a free change in the state variable.
DTM.4:4.5 - Express the result as conditional continuations
Give the conditions and their consequences together:
- the surrounding regime in which a rare variant grows or declines;
- the maintained regimes relevant to the question;
- initial conditions or interventions that distinguish reachable continuations;
- the observations or changed assumptions that would defeat the account.
Use the existing comparison and portfolio methods when several actions remain viable. DTM.1 preserves whose consequences are being compared. DTM.6/.9 reopen the law when another variant or protective response changes the surrounding regime.
Stop with the weakest supported claim that answers the practical question. A useful conditional threshold need not become an unconditional prediction, and a qualitative return may be enough to reject an ineffective intervention.