The tempered pullback dynamics of the 3D Brinkman–Forchheimer equation with variable delay has been studied in this paper. With the different universes which has some topology property, the existence of minimal and unique family of pullback attractors were obtained. Moreover, the convergence of pullback attractors for the 3D Brinkman–Forchheimer equation as delay term vanishes is also been proved, i.e., the upper semi-continuity of attractors.

Visceral leishmaniasis (VL) is a deadly neglected tropical disease caused by a parasite Leishmania donovani and spread by female sand flies Phlebotomus argentipes . There is conflicting evidence regarding the role of insecticide-treated nets (ITNs) on the prevention of VL. Numerous studies demonstrated the effectiveness of ITNs. However, KalaNet, a large trial in Nepal and India did not support those findings. The purpose of this paper is to gain insight into the situation by mathematical modelling. We expand a mathematical model of VL transmission based on the KalaNet trial and incorporate the use of ITNs explicitly into the model. One of the major contributions of this work is that we calibrate the model based on the available epidemiological data, generally independent of the KalaNet trial. We validate the model on data collected during the KalaNet trial. We conclude that in order to eliminate VL, the ITN usage would have to stay above 96%. This is higher than the 91% ITNs use at the end of the trial which may explain why the trial did not show a positive effect from ITNs. At the same time, our model indicates that asymptomatic individuals play a crucial role in VL transmission.