Mathematical modelling and optimal control analysis of pandemic dynamics as a hybrid system

Abstract

The study of epidemics using mathematical modelling is critical in understanding its dynamics and proposing potential control measures. We propose a generalised epidemiological model corresponding to a pandemic wherein its dynamics is represented as a novel hybrid system obtained by coupling a deterministic model with a stochastic model. The hybrid system dynamics is established in individualistic (macroscopic) and intra-individualistic (microscopic) scales. The established hybrid system is then considered the basis for an optimal control problem, with the rate of vaccination and velocity of spatial dynamics taken as the control parameters affecting the system’s trajectory. We define the cost functional constituted by the continuous cost corresponding to the deterministic model and discrete costs corresponding to the transitions in the microscopic scale. The objective of the control problem is to find an optimal control pair of vaccination rate and spatial velocity, which minimises the cost functional. We use the Dynamic Programming Principle (DPP) as the optimisation technique, followed by verification of the value function obtained by DPP as a viscosity solution of the appropriate Hamilton–Jacobi–Bellman equation to analyse the existence of an optimal control pair to the hybrid system. We prove the existence of optimal controls to the multi-scale dynamics for pandemic modelling, along with an abstract method to synthesise it.