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Open Access Technical advance

An optimal control theory approach to non-pharmaceutical interventions

Feng Lin1*, Kumar Muthuraman2 and Mark Lawley1

Author Affiliations

1 School of Biomedical Engineering, Purdue University, 206 S Martin Jischke Drive, West Lafayette, IN 47907-2032, USA

2 CBA 5.234, Dept of Information, Risk, and Operations Management, Austin, TX 78712, USA

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BMC Infectious Diseases 2010, 10:32  doi:10.1186/1471-2334-10-32

Published: 19 February 2010



Non-pharmaceutical interventions (NPI) are the first line of defense against pandemic influenza. These interventions dampen virus spread by reducing contact between infected and susceptible persons. Because they curtail essential societal activities, they must be applied judiciously. Optimal control theory is an approach for modeling and balancing competing objectives such as epidemic spread and NPI cost.


We apply optimal control on an epidemiologic compartmental model to develop triggers for NPI implementation. The objective is to minimize expected person-days lost from influenza related deaths and NPI implementations for the model. We perform a multivariate sensitivity analysis based on Latin Hypercube Sampling to study the effects of input parameters on the optimal control policy. Additional studies investigated the effects of departures from the modeling assumptions, including exponential terminal time and linear NPI implementation cost.


An optimal policy is derived for the control model using a linear NPI implementation cost. Linear cost leads to a "bang-bang" policy in which NPIs are applied at maximum strength when certain state criteria are met. Multivariate sensitivity analyses are presented which indicate that NPI cost, death rate, and recovery rate are influential in determining the policy structure. Further death rate, basic reproductive number and recovery rate are the most influential in determining the expected cumulative death. When applying the NPI policy, the cumulative deaths under exponential and gamma terminal times are close, which implies that the outcome of applying the "bang-bang" policy is insensitive to the exponential assumption. Quadratic cost leads to a multi-level policy in which NPIs are applied at varying strength levels, again based on certain state criteria. Results indicate that linear cost leads to more costly implementation resulting in fewer deaths.


The application of optimal control theory can provide valuable insight to developing effective control strategies for pandemic. Our findings highlight the importance of establishing a sensitive and timely surveillance system for pandemic preparedness.