  Weak Kleene logics are three-valued logics characterized by the presence of
an infectious truth-value. In their external versions, as they were originally
introduced by Bochvar and Hallden, these systems are equipped with an
additional connective capable of expressing whether a formula is classically
true. In this paper we further expand the expressive power of external weak
Kleen logics by modalizing them with a unary operator. The addition of an
alethic modality gives rise to the two systems $\B^{\square}$ and $MPWK$, which
have two different readings of the modal operator. We provide these logics with
a complete and decidable Hilbert-style axiomatization w.r.t. a three-valued
possible worlds semantics. The starting point of these calculi are new
axiomatizations for the non-modal bases B and PWKe, which we provide using the
recent algebraization results about these two logics. In particular, we prove
the algebraizability of $\PWKe$. Finally some standard extensions of the basic
modal systems are provided with their completeness results w.r.t. special
classes of frames.
