Relatively Robust Timing of Medical Tests under Ambiguous Prevalence Dynamics

Medical tests are investments in information whose value depends on the prior probability of disease, on how that belief evolves over time, and on actions taken after evidence is observed. Deterministic prevalence dynamics are tractable, but policies calibrated under a single intensity can be unreliable when epidemiological speeds are misspecified. We model belief as evolving deterministically under an autonomous differential equation whose drift is scaled by an unknown multiplicative speed factor. The decision maker may wait, test at a cost, and treat (or not treat) after outcomes, with retesting possible after negative results. For stationary policies defined by two belief thresholds, we obtain a closed-form expression for the expected discounted value, yielding a fast evaluator for computing optimal thresholds. Relative robustness is measured by a performance ratio that compares the value of a candidate threshold pair with the ex-post optimal value under the realized speed factor; relatively robust thresholds maximize the worst-case ratio over an ambiguity interval. We provide a grid-search procedure and a numerical illustration motivated by post-exposure tuberculosis infection screening, where timing decisions unfold over weeks to years and intensity ambiguity is clinically consequential.