Dynamical and time-series approach to understanding compartmental stock and flow models: a case study in malaria...

Insecticide-treated nets (ITNs) are one of the most cost-effective malaria control measures widely adopted today. The construction of mechanistic models to describe the structural distribution, ownership and attrition of ITNs is crucial to quantify historical impact and burden, and perform predictive estimates of intervention effect. Compartmental stock-and-flow (SNF) models have been the traditional approach to modelling ITN inventories and have remained the most commonly employed approach owing to their simplicity and ease of implementation. However, insight into the mathematical justification for commonly adopted modelling decisions is sparse. The calibration of SNF to observed data is also challenging owing to the fact that data across disparate sampling frequencies and sparsity need to be reconciled. In this paper, we present a mathematical analysis of compartmental SNF models from both a time-series analysis and a dynamical systems approach to provide more insight into their dynamical behaviours. Using a reduced form of an SNF model, we show its equivalence to a linear time-invariant system and demonstrate the criticality of attrition functions in the design of SNF models. In addition, we propose an iterative adapted expectation-maximization (EM) algorithm to address SNF calibration challenges arising from disparate sampling frequencies alongside a list of required assumptions. Statistical analyses are presented to verify the validity of these assumptions. To demonstrate its application, the subsequent EM method is applied to collected delivery, distribution and household survey data across 44 countries spanning 24 years to provide robust and statistically rigorous estimates of net distribution and volumes. Results for numerical convergence and uniqueness of output are also given.