# 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 $$b$$.

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

### 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.

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

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

$$\dot{H}_{x}=\dot{x}H=(H-H_{x})\\$$
$$\dot{x}=(1-x)\Gamma_{0}\left(\frac{bmc}{a}\right)\\$$

### Rate of ruffle formation

The rate of change of the number of total ruffles over all cells depends on the number of total host cells, the primary attachment rate, the bacterial density and the probability of ruffle formation.

$$\dot{N}_{R}=N\Gamma_{0}\rho_{B}f=Nf\Gamma_{0}\left(\frac{bmc}{a}\right)\\$$
$$\dot{\tilde{r}}=\frac{\dot{N}_{R}}{N}=f\Gamma_{0}\left(\frac{bmc}{a}\right)\\$$

### Rate of ruffling

The rate of change of the number or ruffling cells (i.e. $$\geq$$ 1 ruffle) is given by the number of cells without ruffles, the primary attachment rate, the bacterial density and the probability of ruffle formation.

$$\dot{N}_{r}=\dot{r}N=(N-N_{r})\Gamma_{0}\rho_{B}f=(N-N_{r})f\Gamma_{0}\left(\frac{bmc}{a}\right)\nonumber \\$$
$$\dot{r}=(1-r)f\Gamma_{0}\left(\frac{bmc}{a}\right)\\$$

### Rate of bacteria capture

Bacteria are captured through one of two ways: primary attachment to host cells, or secondary recruitment to host cell ruffles.

\begin{aligned} \dot{N}_{B}= & \ \dot{b}N_{B}(0)=\dot{b}mN\nonumber \\ = & -N\Gamma_{0}\rho_{B}-N_{R}\Gamma_{1}\rho_{B}=-N\rho_{B}(\Gamma_{0}+\tilde{r}\Gamma_{1})\nonumber \\ \end{aligned}
$$\dot{b}=-\frac{bc}{a}(\Gamma_{0}+\tilde{r}\Gamma_{1})\\$$
$$\dot{\tilde{b}}=\frac{-\dot{N}_{B}}{N}=-\dot{b}m\\$$