On the Robust Optimization to the Uncertain Vaccination Strategy Problem.

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On the robust optimization to the uncertain vaccination strategy problem

D. Chaerani, N. Anggriani, and Firdaniza

Citation: AIP Conference Proceedings 1587, 34 (2014); doi: 10.1063/1.4866528 View online: http://dx.doi.org/10.1063/1.4866528

View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1587?ver=pdfcov

Published by the AIP Publishing

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On the Robust Optimization to the Uncertain Vaccination

Strategy Problem

D. Chaerani, N. Anggriani, Firdaniza

Department of Mathematics Faculty of Mathematics and Natural Sciences University of Padjadjaran Indonesia Jalan Raya Bandung Sumedang KM 21 Jatinangor Sumedang 45363 email: d.chaerani@unpad.ac.id

Abstract. In order to prevent an epidemic of infectious diseases, the vaccination coverage needs to be minimized and also the basic reproduction number needs to be maintained below 1. This means that as we get the vaccination coverage as minimum as possible, thus we need to prevent the epidemic to a small number of people who already get infected. In this paper, we discuss the case of vaccination strategy in term of minimizing vaccination coverage, when the basic reproduction number is assumed as an uncertain parameter that lies between 0 and 1. We refer to the linear optimization model for vaccination strategy that propose by Becker and Starrzak (see [2]). Assuming that there is parameter uncertainty involved, we can see Tanner et al (see [9]) who propose the optimal solution of the problem using stochastic programming. In this paper we discuss an alternative way of optimizing the uncertain vaccination strategy using Robust Optimization (see [3]). In this approach we assume that the parameter uncertainty lies within an ellipsoidal uncertainty set such that we can claim that the obtained result will be achieved in a polynomial time algorithm (as it is guaranteed by the RO methodology). The robust counterpart model is presented.

Keywords: robust optimization, robust counterpart, vaccination strategy, conic quadratic optimization

PACS: 87.23 .Cc

INTRODUCTION

To control human infectious disease, vaccination is con-sidered as one of the primary strategy used by public health authorities. In this paper we discuss the vaccina-tion in households. As menvaccina-tioned by Keeling et al.[8] the intensity and frequency of interactions between peo-ple where one initial case can either lead to several more cases within the household or can recover leaving few other household members infected.

Beckeret al.in [2] proposed linear optimization mod-els in order to obtain an optimal vaccination strategy within households. Some assumptions is used in the model such as the disease spreads quickly within individ-ual households and spreads more slowly between them through close contacts between infected and susceptible members of different households. They also assume pro-portionate mixing between households, to ensure that the problem constraints are linear. This allows them to find a closed form equation for the post-vaccination reproduc-tion number. On the other hand, Ballet al.in [1] shows that this linear program does not allow an easy character-ization of the optimal strategy, meaning that the optimal strategy may not be easy to implement. Thus, vaccination policies found for any kind of model should be consid-ered very carefully, especially if the uncertainty of the parameters is not taken into account.

Optimization under uncertainty refers to the branch of optimization where the data vector ζ is uncertain (see [5]). This means that the data vector ζ is not known exactly at the time when its solution has to be

deter-mined. The uncertainty may be due to measurement or modelling errors or simply to the unavailability of the re-quired information at the time of the decision. A recent comprehensive survey on RO can be found in [7].

Some authors, for example Tanner et al. [9] and Clancy et al. [6], have considered the uncertainty pa-rameters on the vaccination strategy problem. Tanneret al.[9] presents a stochastic programming framework for finding the optimal vaccination policy for controlling in-fectious disease epidemics under parameter uncertainty. Clancyet al. [6] propose their work on the optimal inter-vention for an epidemic model under parameter uncer-tainty, which consider the effect upon the optimal pol-icy of changes in parameter estimates, and of explicitly taking into account parameter uncertainty via a Bayesian decision theoretic framework.

This paper discuss a different approach to those re-sults, i.e., we discuss how the robust design model of uncertain vaccination strategy is modeled using Robust Optimization. This optimization methodology incorpo-rates the uncertain data in a so-called uncertainty setU

and replaces the uncertain problem by its so-called ro-bust counterpart. Citing from [3], the main challenge in this RC methodology is how and when we can refor-mulate the robust counterpart of the uncertain vaccina-tion strategy problem as a computavaccina-tionally tractable op-timization problem or at least approximate the model by a tractable problem. Due to its definition the robust coun-terpart highly depends on how we choose the uncertainty setU_{. As a consequence we can meet this challenge only}

if this set is chosen in a suitable way.

Symposium on Biomathematics (Symomath 2013) AIP Conf. Proc. 1587, 34-37 (2014); doi: 10.1063/1.4866528

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