Open Access
Table I.
Model parameter definitions, initial values and model estimates.
| Symbol | Definition | Initial value | Estimated value | Units |
|---|---|---|---|---|
| Parameters with the same value for all cows | ||||
| gcp | Baseline growth rate of pro-inflammatory cytokines in absence of anti-inflammatory cytokines. Stimulated by contact between E. coli and macrophages | Unknown | 0.0024 | day−1 |
| icp | Inhibitory effect of anti-inflammatory cytokines on growth of pro-inflammatory cytokines | Unknown | 0.035 | none |
| gca | Rate of growth of anti-inflammatory cytokines (dependent on concentration of pro-inflammatory cytokines) | Unknown | 9.8e-5 | |
| ge | Growth rate of E. coli in milk | 25.8–46.1 | 34 | day−1 |
| dp | Rate of loss of PMN from milk | 0.1–1.8 | 0.10 | day−1 |
| dc | Rate of loss of cytokines | 0.7–17 | 10 | day−1 |
| M | Baseline number of macrophages in milk | 5 000 (fixed) | Cells/mL | |
| Vmax | Maximum volume of milk in a quarter | 2.5 (fixed) | Litre | |
| Dmilk | Duration of milking | 5/(60 × 24) (fixed) | Day | |
| pmilk | Proportion of milk remaining after milking | 0.1 (fixed) | None | |
| k | Killing rate by macrophages (M and P) | 3.6e-6–4.8e-6 | 1.0e-5 | Cell/day |
| σE | Mean standard deviation of the model fit from the data for log E. coli concentration | Unknown | 0.54 | log cells/mL |
| σS | Mean standard deviation of the model fit from the data for log somatic cell concentration | Unknown | 0.75 | log cells/mL |
| Parameters that have a different value for each type of infection (transient, persistent) | ||||
| a | Rate of movement of E. coli from milk to intra-cellular reservoirs | Unknown | 0.51 (transient) 5.1 (persistent) | day−1 |
| b | Rate of movement of E. coli from intra-cellular reservoirs to milk | Unknown | 0.55 (transient) 7.5e-3 (persistent) | day−1 |
| Parameters that have a different value for each individual cow | ||||
| di | Death rate of E. coli in intra-cellular reservoirs | ~ 1 (transient) ~ 0.1 (persistent) | 2.1, 0.9, 1.4, 1.2, 1.4, 1.8, 1.9 (transient)0.1 (persistent) | day−1 |
| gp | Rate of influx of PMN into the milk in presence of pro-inflammatory cytokines | Unknown | 0.02, 0.04, 0.02, 0.03, 0.03, 0.50, 2.0, 0.8, 6.0, 2.0, 0.3 | day−1 |