Finding optimal vaccination strategies under parameter uncertainty using stochastic programming Matthew W. Tanner a, * , Lisa Sattenspiel b , Lewis Ntaimo a a Department of Industrial and Systems Engineering, Texas A&M University, 241 Zachry, 3131 TAMU, College Station, TX 77843-3131, USA b Department of Anthropology, University of Missouri, 105 Swallow Hall, Columbia, MO 65211-1440, USA article info Article history: Received 11 January 2008 Received in revised form 30 April 2008 Accepted 11 July 2008 Available online 24 July 2008 Keywords: Parameter uncertainty Epidemic model Vaccination Stochastic programming Chance constraints abstract We present a stochastic programming framework for finding the optimal vaccination policy for control- ling infectious disease epidemics under parameter uncertainty. Stochastic programming is a popular framework for including the effects of parameter uncertainty in a mathematical optimization model. The problem is initially formulated to find the minimum cost vaccination policy under a chance- constraint. The chance-constraint requires that the probability that R  6 1 be greater than some param- eter a, where R  is the post-vaccination reproduction number. We also show how to formulate the prob- lem in two additional cases: (a) finding the optimal vaccination policy when vaccine supply is limited and (b) a cost–benefit scenario. The class of epidemic models for which this method can be used is described and we present an example formulation for which the resulting problem is a mixed-integer program. A short numerical example based on plausible parameter values and distributions is given to illustrate how including parameter uncertainty improves the robustness of the optimal strategy at the cost of higher coverage of the population. Results derived from a stochastic programming analysis can also help to guide decisions about how much effort and resources to focus on collecting data needed to provide better estimates of key parameters. Ó 2008 Elsevier Inc. All rights reserved. 1. Introduction Vaccination is one of the primary strategies used by public health authorities to control human infectious diseases. Mathe- matical models have long played a major role in identifying and evaluating strategies to allocate resources in order to guarantee maximum effectiveness of vaccination in controlling infectious dis- ease outbreaks. Three primary modeling approaches have been used in this effort – deterministic analytical models, stochastic analytical models, and computer simulations. The determination of optimal vaccination strategies may be sensitive to changes in model parameter values, however, so there is a need for new meth- ods that can take parameter uncertainty into account in order to find more robust vaccination policies. We present here a descrip- tion of one such method, stochastic programming, and illustrate how this method can improve our ability to find optimal vaccina- tion strategies. The goal of most deterministic and stochastic epidemiological models addressing vaccination strategies is to derive appropriate strategies analytically. Deterministic models focused on identify- ing reasonable vaccination strategies for the control of infectious diseases date back to at least the 1960s [early papers include, for example, [11,23,36,40]]. In general, deterministic vaccination models fall into two major groups. The majority of these models are used to evaluate predetermined vaccination strategies to see which of the proposed strategies may be most effective. Analysis of most of these models generally involves exploration of the steady state behavior of the model system and determination of an epidemic threshold. The effectiveness of different proposed vaccination strategies in reducing the susceptible population below the epidemic threshold for the minimum cost is then evaluated. In some of the recent more complex models, com- puter simulation is used to assess the effectiveness of different strategies. Models of this type have been developed for a num- ber of infectious diseases, including tuberculosis [11,40], measles [1,3,20,37], rubella [2,13,19,27], pertussis [18,21,22,24], and respiratory illnesses [34]. The second group of deterministic vaccination models do not start with predetermined strategies; rather, they center on the use of optimization methods in combination with deterministic epidemic models to identify the optimal vaccination strategy. Opti- mization methods have been used both in a theoretical framework [23] and to guide the development of vaccination policies for spe- cific diseases, including tuberculosis [36], influenza [28], and smallpox [16]. 0025-5564/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.mbs.2008.07.006 * Corresponding author. Tel.: +1 573 268 4019. E-mail address: mtanner@tamu.edu (M.W. Tanner). Mathematical Biosciences 215 (2008) 144–151 Contents lists available at ScienceDirect Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs