Если в общем значении, то лучше расследовать.

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Note added at 35 mins (2022-12-09 12:44:39 GMT)
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... medical investigative duties in the United States are normally the responsibility of the Centers for Disease Control and Prevention (CDC) and local health departments...
https://en.wikipedia.org/wiki/Medical_Investigation

The Epidemic Intelligence Service (EIS) is composed of "boots-on-the-ground disease detectives" who investigate public health problems domestically and globally. When called upon by a governmental body, EIS officers may embark on short-term epidemiological assistance assignments, or "Epi-Aids", to provide technical expertise in containing and investigating disease outbreaks.
https://en.wikipedia.org/wiki/Centers_for_Disease_Control_an...

The EIS is operated by the CDC's Center for Surveillance, Epidemiology, and Laboratory Services (CSELS), in the Office of Public Health Scientific Services (OPHSS).[6]

Persons participating in the program, known colloquially as "disease detectives", are formally called "EIS officers" (or EIS fellows) by the CDC and have been dispatched to investigate hundreds of possible epidemics created by natural and artificial causes. Since 1951, more than 3,000 EIS officers have been involved in domestic and international response efforts, including the anthrax, hantavirus, West Nile virus in the United States, and the 2014–16 Ebola epidemic in West Africa.
https://en.wikipedia.org/wiki/Epidemic_Intelligence_Service

EIS officers attend an annual conference in Atlanta, Georgia, to present components of their work from the preceding year.[15]

During the conference, the Alexander D. Langmuir Prize is awarded "to a current officer or first-year alumnus of the EIS for the best scientific publication...
A complete list of Langmuir Prize winners is...
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1974 Oyster-Associated Hepatitis: Failure of Shellfish Certification Programs To Prevent Outbreaks. B.L. Portnoy, P.A. Mackowiak, C.T. Caraway, J.A. Walker, T.W. McKinley, C.A. Klein Jr.
...
https://en.wikipedia.org/wiki/Epidemic_Intelligence_Service


OYSTER-ASSOCIATED HEPATITIS: LESSONS FROM THE LOUISIANA EXPERIENCE
PHILIP A. MACKOWIAK
An outbreak of oyster-associated hepatitis occurring in Louisiana during the months of October and November 1973 has been shown to be related directly to two similar outbreaks occurring at the same time in Houston. Texas and Calhoun, Georgia. Oysters incriminated in all three outbreaks were traced to approved oyster growing areas along the east Louisiana coast. Although no entirely satisfactory explanation could be found for the mode of contamination of these oysters, the epidemiologic and environmental data are most consistent with the conclusion that Mississippi River flooding provided the necessary source of fecal pollution into the oyster growing areas.
https://academic.oup.com/aje/article-abstract/103/2/181/1056...


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Note added at 46 mins (2022-12-09 12:56:03 GMT)
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если бы health department "изучал случай", то в оригинале было бы study

также, как вариант, - выяснять причины заболевания

Если заболевание опасное, или случаев несколько - выяснять причины вспышки (скажем, гепатита)

Если говорить о лечении пациента, то - изучать клинический случай

Складывается впечатление, что случай гепатита рассматривается здесь в эпидемиологическом контексте, тогда, определенно, "расследовать случай заболевания" в рамках эпидемиологического расследования.

В этом же контексте есть еще вариант, что речь идет об обнаружении причины вспышки, то, наиболее вероятно "выявить" в том смысле, что некий эпидемиологический орган проводит расследование и устанавливает первоначальный источник заражения.

Если, все же, более применим контекст научного исследования по конкретному клиническому случаю, тогда, наиболее подойдет "исследовать" или "изучить".

Наименее вероятный, но вдруг, - это конкретный клинический контекст. Тогда - "диагностировать".

Расследовать больше в криминалистике