Table 1.

Summary of the state variables and parameters.

symbolmeaningdimensionless analogue
Sdensity of susceptible hostsŜ = S/κ
Idensity of infected hostsÎ = I/κ
Rdensity of removed hosts = bR/
Xdensity of soil-borne inoculum = bX/
ttime = bt
g(S)production of susceptible hostsn.a.
hrate 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) − hSsee below
Model Mf(S) = b(κS)(Ŝ) = 1 − Ŝ
Model Lf(S) = bS(1 − S/κ)(Ŝ) = Ŝ(1 − Ŝ)
bbirth rate1
κcarrying capacity1
βprate of primary infectionβ̂p = βpaκ/b2
βsrate of secondary infectionβ̂s = βsaκ/b
mrate of disease-induced mortality for infected hosts1
arate of production of inoculum by infected hosts1
crate of decay of inoculumĉ = c/b
μtotal rate at which individuals leave class I, with μ = m + hμ̂ = μ/b