Group the diseases by whether a person can keep spreading the agent after or without symptoms. Typhoid is the classic example of a chronic carrier (the gallbladder harbours $Salmonella\ Typhi$, as in the historic case of Typhoid Mary). Diphtheria produces nasopharyngeal carriers who shed the bacillus, and poliomyelitis is propagated mostly by silent sub-clinical infections that function exactly like carriers, with paralytic cases being a tiny fraction. Measles behaves differently: it is a self-limiting acute viral illness whose virus is shed only during active clinical disease through respiratory secretions, roughly four days before to four days after the rash. After recovery the virus is eliminated and lifelong immunity follows, so no chronic or convalescent carrier ever forms. Because measles spreads only from frank cases and never from carriers, it is the disease in which the carrier state is unimportant.
\[\boxed{\text{Measles}}\]