The shape of the epidemic curve tells the story. A propagated epidemic is fuelled by one infected person passing the agent to the next, so cases arrive in successive waves; each wave sits about one incubation period after the previous one, giving a curve with several rising peaks rather than a single spike.
For that pattern to appear, the disease must spread directly between people. Measles fits exactly — an extremely transmissible droplet-spread virus that moves from case to case and generates the classic multi-peaked propagated curve.
The food-poisoning options behave differently. Staphylococcal and Salmonella food poisoning are point-source events: many people eat the same contaminated meal and fall ill together, producing one tight peak with no serial generations. Typhoid is likewise usually a water- or food-borne common-source outbreak, not the prototypical propagated epidemic.
$\text{Peaks} \approx 1 \text{ incubation period apart} \Rightarrow \text{person-to-person spread}$
\[\boxed{\text{Measles}}\]