der Wirt
innkeeper
A man who owns or runs a pub, inn, or restaurant.
Der Wirt brachte uns persönlich die Speisekarte.
The innkeeper personally brought us the menu.

innkeeper / host
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innkeeper
A man who owns or runs a pub, inn, or restaurant.
Der Wirt brachte uns persönlich die Speisekarte.
The innkeeper personally brought us the menu.

The word 'Wirt' is related to the verb 'bewirten' (to serve guests food and drink) and the noun 'Wirtschaft' (another word for a pub or inn).
Wir wurden vom Wirt freundlich bewirtet.
We were served warmly by the innkeeper.
The term 'Wirt' for the operator of a pub or inn is traditional and especially common in southern Germany, Austria, and Switzerland. It often has a cozy, personal feel.
Der Wirt kennt alle seine Stammgäste mit Namen.
The innkeeper knows all his regular customers by name.
You can recognize the biological meaning from the context. If 'Parasiten' (parasites), 'Viren' (viruses), 'Bakterien' (bacteria), or 'Organismen' (organisms) are being discussed, the biological host is meant. In the context of 'Gaststätte' (pub/restaurant), 'Bier' (beer), or 'Essen' (food), it refers to the restaurant or pub owner.
Mücken können als Wirt für verschiedene Krankheitserreger dienen.
Mosquitoes can serve as hosts for various pathogens.
sponsor / carrier
'der Träger' and 'der Wirt' can both refer, in medicine and biology, to a living being in which something else is present. 'Der Träger' has a pathogen or genetic trait inside it and can pass it on, often without being ill itself. 'Der Wirt' is the living organism in or on which another organism lives and from which it gets nourishment. The words can be confusing because both have to do with living organisms and disease. 'Träger' emphasizes carrying something and possibly passing it on, while 'Wirt' emphasizes the habitat or source of nourishment for a parasite.

Bei manchen Parasiten ist ein Tier der Wirt, in dem sie leben und wachsen.
With some parasites, an animal is the host in which they live and grow.
Der Patient ist Träger des Virus, obwohl er keine Beschwerden hat.
The patient is a carrier of the virus, even though he has no symptoms.