# 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