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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:  $H_{\rm tot} = \left\vert\{ \left{H(t) \right} \left\vert{ \left\{H(t) \right\}  \cup \left{H_a \left\{H_a  (t) \right} \right\}  \cup \left{H_v \left\{H_v  (t) \right} \right\}  \cup \left{H_c \left\{H_c  (t) \right} \}\right\vert \right\} }\right\vert  $ An important and controllable quantity when performing invasion assays is the ratio of bacteria to host cells at inoculation. This is called the multiplicity of infection (MOI) and in terms of previously defined parameters, $m = B_{\rm tot} / H_{\rm tot}$.