Production and inventory management often suffers from emergency events, which always lead to the phenomenon of ``stratification'' in market demand. By the mixed producing and outsourcing strategy, we propose a new production and inventory model with monitoring window from the perspective of manufacturers when an emergency event occurs during a fixed production period. With help of the monitoring strategy, the manufacturer determines a sudden change of demand rate. By building and solving the proposed production and inventory model defined by a system of differential equations, the analytical expressions of the productivity adjustment, the outsourcing time and the outsourcing quantity so that the supply matches the demand again. In case study, we apply the developed model and its theoretical properties to analyze impacts of COVID-19 on the production and inventory of small and medium enterprises, including the impacts of the demand fluctuation intensity, the size of monitoring window, and the producing and outsourcing structure. The results from the numerical simulation reveal a series of valuable practical managerial insights underlying the proposed model in this paper.