*S* | density of susceptible hosts | *Ŝ* = *S*/*κ* |

*I* | density of infected hosts | *Î* = *I*/*κ* |

*R* | density of removed hosts | *R̂* = *bR*/*mκ* |

*X* | density of soil-borne inoculum | *X̂* = *bX*/*aκ* |

*t* | time | *t̂* = *bt* |

*g*(*S*) | production of susceptible hosts | n.a. |

*h* | rate of removal (all classes of host) | *ĥ* = *h*/*b* |

*f*(*S*) | overall dynamics of susceptible hosts in the absence of infection, where *f*(*S*) = *g*(*S*) − *hS* | see below |

Model M | *f*(*S*) = *b*(*κ* − *S*) | *f̂*(*Ŝ*) = 1 − *Ŝ* |

Model L | *f*(*S*) = *bS*(1 − *S*/*κ*) | *f̂*(*Ŝ*) = *Ŝ*(1 − *Ŝ*) |

*b* | birth rate | 1 |

*κ* | carrying capacity | 1 |

*β*_{p} | rate of primary infection | *β̂*_{p} = *β*_{p}*a**κ*/*b*^{2} |

*β*_{s} | rate of secondary infection | *β̂*_{s} = *β*_{s}*a**κ*/*b* |

*m* | rate of disease-induced mortality for infected hosts | 1 |

*a* | rate of production of inoculum by infected hosts | 1 |

*c* | rate of decay of inoculum | *ĉ* = *c*/*b* |

*μ* | total rate at which individuals leave class *I*, with *μ* = *m* + *h* | *μ̂* = *μ*/*b* |