A nonlocal reaction-diffusion system modeling the Schistosomiasis transmission with multiple hosts and periodic delays.

In this paper, we investigate the effects of seasonality, spatial heterogeneity and multiple hosts on the transmission dynamics of schistosomiasis. The seasonal environment imposes a temporary maturation period, parasites' extrinsic incubation phase, and periodic developments within the hosts. This results in multiple periodic time delays in stage transforms. Incorporating both the movements of parasites and their hosts, the transmission becomes periodic, time-delayed, and spatially non-local. We first show the well-posedness of the model and introduce its basic reproduction number R 0 . Accordingly, we confirm the threshold-type global dynamics where the disease is uniformly persistent with at least one positive periodic solution when R 0 > 1 , while the disease-free periodic solution is globally attractive when R 0 < 1 . In addition to the well-known results indicating that varied delays and spatial heterogeneity can affect the threshold value, our numerical simulations reveal some interesting findings: (1) The value of R 0 decreases in environments with more complex fragmentation, while increases with higher spatial variation of transmission rates. (2) The optimal control approach is to initiate control of transmission in intermediate and definitive hosts at different timings. (3) Employing a space-dependent resource distribution is more effective than applying a spatially uniform resource distribution in reducing the spread of the disease.