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Table 4 Snail population-transmission dynamics

From: Refined stratified-worm-burden models that incorporate specific biological features of human and snail hosts provide better estimates of Schistosoma diagnosis, transmission, and control

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”)