Malaria in Northwest India: Data Analysis via Partially Observed Stochastic Differential Equation Models Driven by Lévy Noise

Many biological systems are appropriately described by partially observed Markov process (POMP) models, also known as state space models. Such models also arise throughout the physical and social sciences, in engineering, and in finance. Statistical challenges arise in carrying out inference on nonlinear, nonstationary, vector-valued POMP models. Methodologies that depend on the Markov process model only through numerical solution of sample paths are said to have the plug-and-play property. This property enables consideration of models for which the evaluation of transition densities is problematic. Our case study employs plug-and-play methodology to investigate malaria transmission in Northwest India. We address the scientific question of the respective roles of environmental factors, immunity, and nonlinear disease transmission dynamics in epidemic malaria. Previous debates on this question have been hindered by the lack of a statistical investigation that gives simultaneous consideration to the roles of human immunity and the fluctations in mosquito abundance associated with environmental or ecological covariates. We present the first time series analysis integrating these various components into a single vector-valued dynamic model. We are led to investigate a POMP involving a system of stochastic differential equations driven by Levy noise. We find a clear role for rainfall and evidence to support models featuring the possibility of clinical immunity. An online supplement presents details of the methodology implemented and two additional figures.