Optimal Control Prevents Itself from Eradicating Stochastic Disease Epidemics

Abstract

The resources available for managing disease epidemics - whether in animals, plants or humans - are limited by a range of practical and financial constraints. This motivates study of where these resources should be allocated to maximise their impact. Optimal control has been widely explored for optimising management of epidemics. The most common approach assumes a deterministic, continuous model based on differential equations to approximate the epidemic dynamics. However, real systems are stochastic and so a range of outcomes are possible for any given epidemic situation and choice of control. The deterministic models are also known to be poor approximations in cases where the number of infected hosts is low - either globally or within a subset of the population - and these cases are highly relevant in the context of control. Hence, this work explores the effectiveness of disease management strategies derived using optimal control theory when applied to a more realistic, stochastic form of disease model. We demonstrate that solutions to the deterministic optimal control are not optimal in cases where the disease is eradicated or close to eradication. The range of potential outcomes in the stochastic models means that optimising the deterministic case will not reliably eradicate disease, even when it is possible. For effective eradication, the rate of control must be higher than optimal control as applied to the deterministic model would predict. Using Model Predictive Control, in which the optimisation is performed repeatedly as the system evolves to correct for deviations from the optimal control predictions, improves performance but does not fix the underlying issue and the level of control calculated at each repeated optimisation is still insufficient. To demonstrate this, we present several very simple heuristics to identify which sites to control which can outperform the strategies calculated by MPC when the control budget is sufficient for eradication to be possible. Our illustration uses examples based on stochastic simulation of the spatial spread of plant disease but similar issues would be expected in any deterministic model where infection is driven close to zero or targeted to remain below some critical value.