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.

\( \overset{\beta }{\to}\underset{\begin{array}{l}\downarrow \\ {}\nu \end{array}}{x}\underset{\left(1-c\right)r}{\overset{\varLambda }{\rightleftarrows }}\kern0.24em \underset{\begin{array}{l}\downarrow \\ {}\nu \end{array}}{y}\overset{c\;r}{\to}\underset{\begin{array}{l}\downarrow \\ {}\nu \end{array}}{z} \)

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

\( \begin{array}{l}\frac{dx}{dt}=\beta -\varLambda\;x-\nu\;x+r\left(1-c\right)y\\ {}\frac{dy}{dt}=\varLambda\;x-\left(r+\nu \right)\;y\\ {}\frac{dz}{dt}=c\;r\;y-\nu\;z\end{array} \)

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”)