When does the Eucharist stop being Jesus?

This is actually a classic objection, and the Church has addressed it clearly through her sacramental theology.

The short answer: Christ's presence in the Eucharist endures only as long as the species (the appearances of bread and wine) remain. Once the accidents of bread and wine are corrupted or cease to exist, the Real Presence ceases. The Eucharist does not become waste; Christ's presence simply withdraws when the sacramental sign is destroyed.

The Council of Trent taught that Christ is present "truly, really, and substantially" under the appearances of bread and wine (Session XIII, Canon 1). St. Thomas Aquinas clarifies that this presence is sacramental, not natural or physical in the ordinary sense. Christ's Body is present per modum substantiae (by way of substance), not per modum quantitatis (by way of physical dimension). This means His presence is not subject to digestion, division, or corruption the way ordinary matter is (Summa Theologiae III, q. 77, a. 4–6).

Aquinas directly addresses this: once the species are altered to the point they would no longer be recognized as bread or wine — through digestion, decomposition, or corruption — the sacramental presence ceases. What remains is simply the matter into which the species have been converted, not the Body of Christ (ST III, q. 77, a. 4). The sacrament is tied to the sign; when the sign is destroyed, the sacramental reality is no longer present.

So the faithful need not worry about irreverence in digestion. The Church has long understood this, which is why we also take care with particles and the purification of vessels — to honor Christ's presence while the species endure.

This is actually a question the Church has considered carefully, and the answer lies in understanding what "the species" of bread and wine means.

Catholic teaching holds that Christ is present "truly, really, and substantially" under the Eucharistic species — that is, under the appearances (or "accidents") of bread and wine (CCC §1374). The key principle is this: Christ remains present as long as the species of bread and wine remain.

St. Thomas Aquinas addressed this directly in the Summa Theologica (III, q. 77, a. 4). He explains that once the species are corrupted — meaning they no longer retain the properties of bread or wine through digestion, decomposition, or other substantial change — Christ's sacramental presence ceases. This isn't because Christ "leaves," but because the sacramental sign itself no longer exists.

So to your specific question: if someone were to vomit shortly after receiving and the host were still recognizably intact as bread, the Church would treat that with reverence, as the species may still be present. Traditional practice is to collect it reverently and allow it to dissolve in water, which is then disposed of properly (poured into a sacrarium or the ground). Once the species is substantially corrupted — no longer bread in its accidents — Christ is no longer sacramentally present there.

The Eucharist is not magic; it's a sacrament, and sacraments work through signs. When the sign ceases, the unique mode of presence proper to that sacrament ceases — though Christ himself, of course, remains eternal and unchanged.