Passwords are still the most widespread means for authenticating users, even though they have been shown to create huge security problems. This motivated the use of additional authentication mechanisms used in so-called multi-factor authentication protocols. In this paper we define a detailed threat model for this kind of protocols: while in classical protocol analysis attackers control the communication network, we take into account that many communications are performed over TLS channels, that computers may be infected by different kinds of malwares, that attackers could perform phishing, and that humans may omit some actions. We formalize this model in the applied pi calculus and perform an extensive analysis and comparison of several widely used protocols-variants of Google 2-step and FIDO's U2F. The analysis is completely automated, generating systematically all combinations of threat scenarios for each of the protocols and using the PROVERIF tool for automated protocol analysis. Our analysis highlights weaknesses and strengths of the different protocols, and allows us to suggest several small modifications of the existing protocols which are easy to implement, yet improve their security in several threat scenarios.