Mean Field Model of Infection

Parameters

Counts

\(N\equiv\) number of host cells

\(N_{I}\equiv\) number of infected cells

\(N_{B}\equiv\) number of bacteria

\(N_{R}\equiv\) number of ruffles

\(N_{r}\equiv\) number or ruffling cells (\(\geq 1\) ruffles)

\(t_{\mathrm{max}}\equiv\) total incubation time

Fractions

\(m\equiv\) multiplicity of infection (MOI) \(=\frac{N_{B}(t=0)}{N}\)

\(c\equiv\) confluency \(=\frac{Na}{L^{2}}\)

\(\quad a\equiv\) mean cellular area

\(\quad L\equiv\) side length of square well

\(x\equiv\) fraction of host cells infected \(=\frac{N_{I}}{N}\)

\(b\equiv\) fraction of bacteria remaining (i.e. not landed on a host) \(=\frac{N_{B}}{N_{B}(0)}\)

\(f\equiv\) fraction of attached bacteria that form ruffles

\(r\equiv\) fraction of host cells with ruffling (\(\geq\) 1 ruffle)

\(\tilde{r}\equiv\) ruffles per cell \(=\frac{N_{R}}{N}\)

\(\tilde{b}_{R}\equiv\) bacteria per ruffle

\(\quad\tilde{b}_{R}(t=0)=1\)

Rates

\(\Gamma_{0}\equiv\) primary attachment rate per bacterial density

\(\Gamma_{1}\equiv\) ruffle recruitment rate per bacterial density

Proofs

Bacterial density

The bacterial density \(\rho_{B}\) is the number of bacteria (available for attachment) per unit area, but is more helpful in terms of MOI and \(b\).

\begin{equation} \rho_{B}=\frac{B_{u}}{L^{2}}=\frac{(1-b)B_{\rm tot}}{HA/c}=\frac{(1-b)mc}{A}\\ \end{equation}

Rate of infectivity

The rate of change of the number of host cells with bacteria (i.e. \(\geq\) 1 bacteria have attached) depends on the number of remaining cells without bacteria attached, the primary attachment rate and the bacterial density.

\begin{equation} \dot{H}_{a}=\dot{a}H=(H-H_{a})\Gamma_{0}\rho_{B}=(H-H_{a})\Gamma_{0}\left(\frac{bmc}{AL^{2}}\right)\nonumber \\ \end{equation}

In our model, we assume limited invasion (i.e. we impose a maximum number of internalized bacteria per cell). The rate of change of infected cells

\begin{equation} \dot{H}_{x}=\dot{x}H=(H-H_{x})\\ \end{equation}