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\subsection{Counts}  During the time course of invasion by \textit{Salmonella}, a bacterium can be in 4 different generalized stages: swimming (looking to attach to a host) $B$, attached to a host cell $B_a$, invaded and vacuolar $B_v$, or invaded and cytosolic $B_c$. The counts of bacteria in these stages are constantly changing in time, but ignoring xenophagy and replication, the total number will remain constant:  $ jidofjdf $  $B_[\rm tot} = B(t) + B_a(t) + B_x(t) + B_c(t)$  We classify the relationship between host cells and bacteria in one of three ways: host cells with no bacteria $H$, with attached bacteria $H_a$, with invaded vacuolar bacteria $H_v$, and with invaded cytosolic bacteria $H_c$. A host cell may have a combination of attached, vacuolar and cytosolic bacteria at one time, so the total number of host cells is the union of these sets: