Early detection for sepsis, a high-mortality clinical condition, is important for improving patient outcomes. The performance of conventional deep learning methods degrades quickly as predictions are made several hours prior to the clinical definition. We adopt recurrent neural networks (RNNs) to improve early prediction of the onset of sepsis using times series of physiological measurements. Furthermore, physiological data is often missing and imputation is necessary. Absence of data might arise due to decisions made by clinical professionals which carries information. Using the missing data patterns into the learning process can further guide how much trust to place on imputed values. A new multi-task LSTM model is proposed that takes informative missingness into account during training that effectively attributes trust to temporal measurements. Experimental results demonstrate our method outperforms conventional CNN and LSTM models on the PhysioNet-2019 CiC early sepsis prediction challenge in terms of area under receiver-operating curve and precision-recall curve, and further improves upon calibration of prediction scores.