Table 4.

Snail population-transmission dynamics

Dynamic variables for the snail model are population densities (per unit habitat) x: susceptible; y: prepatent; z: patent; N = x + y + z- total. $\stackrel{\beta }{\to }\underset{\begin{array}{l}\downarrow \\ \nu \end{array}}{x}\underset{\left(1-c\right)r}{\overset{\Lambda }{\rightleftarrows }}\underset{\begin{array}{l}\downarrow \\ \nu \end{array}}{y}\stackrel{cr}{\to }\underset{\begin{array}{l}\downarrow \\ \nu \end{array}}{z}$ Basic processes and parameters include (i) snail reproduction (logistic growth) β = β 0(x + y)(1 − N/K), with maximal reproduction rate β 0 and carrying capacity K; (ii) snail mortality v; (iii) snail FOI Λ(determined by human host egg outputs) ; (iv) recovery rate r (prepatency period 1/r) (v) patency conversion fraction c. In population growth term β, only susceptible and prepatent snails (x + y) reproduce. Combined growth-SEI dynamics consists of 3 differential equations $\begin{array}{l}\frac{dx}{dt}=\beta -\Lambda x-\nu x+r\left(1-c\right)y\\ \frac{\mathit{dy}}{dt}=\Lambda x-\left(r+\nu \right)y\\ \frac{dz}{dt}=cry-\nu z\end{array}$ Parameter values and ranges for the snail system are given in Table 5. Short-lived larval stages (M, C) equilibrate rapidly at levels proportion to human/snail (H, N) multiplied by their respective infectivity. Specifically, C* = α C  N z; M* = β M ω H E (6) where ${\alpha }_C=\frac{{\pi }_C}{{\nu }_C}$ (“C-production /patent snail” over “C-mortality”). For miracidia the relevant inputs include environmental egg-release by host population ω H E, ω = human-snail contact rate, H - population size, E - mean host infectivity - egg release (Eq. 5), coefficient ${\beta }_M=\frac{{\sigma }_M}{{\nu }_M}$ (“survival fraction of eggs” over “M - mortality”)