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

Insecticide treated nets (ITNs) represent one of the most cost-effective malaria control measures that is widely adopted today. The construction of mechanistic models to describe structural distribution, ownership and attrition of ITNs are crucial in order to quantify historical impact and burden, and perform predictive estimates of intervention impact. 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 are sparse. The calibration of SNF to observed data is also challenging due as 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 dynamical systems approach to provide more insight on 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 SNF models. Additionally, we propose an iterative adapted expectation-maximisation (EM) algorithm to address SNF calibration challenges arising from disparate sampling frequencies alongside a list of required assumptions. Statistical analyses to verify the validity of these assumptions are presented. To the 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 outputs are also given.