On the Robust Optimization to the Uncertain Vaccination Strategy Problem.
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l-lidetaka Arimura and NuninE itlunaini
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 considered as one of the primary strategy used by public
health authorities. In this paper we discuss the vaccination in households. As mentioned by Keeling et al. [8]
the intensity and frequency of interactions between people where one initial case can either lead to several more
cases within the household or can recover leaving few
other household members infected.
Becker et al. in [2] proposed linear optimization models in order to obtain an optimal vaccination strategy
within households. Some assumptions is used in the
model such as the disease spreads quickly within individual households and spreads more slowly between them
through close contacts between infected and susceptible
members of different households. They also assume proportionate mixing between households, to ensure that the
problem constraints are linear. This allows them to find a
closed form equation for the post-vaccination reproduction number. On the other hand, Ball et al. in [1] shows
that this linear program does not allow an easy characterization 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 considered 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 required 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 parameters on the vaccination strategy problem. Tanner et
al. [9] presents a stochastic programming framework for
finding the optimal vaccination policy for controlling infectious disease epidemics under parameter uncertainty.
Clancy et al. [6] propose their work on the optimal intervention for an epidemic model under parameter uncertainty, which consider the effect upon the optimal policy 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 results, i.e., we discuss how the robust design model of
uncertain vaccination strategy is modeled using Robust
Optimization. This optimization methodology incorporates the uncertain data in a so-called uncertainty set U
and replaces the uncertain problem by its so-called robust counterpart. Citing from [3], the main challenge in
this RC methodology is how and when we can reformulate the robust counterpart of the uncertain vaccination strategy problem as a computationally tractable optimization problem or at least approximate the model by
a tractable problem. Due to its definition the robust counterpart highly depends on how we choose the uncertainty
set U . 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
© 2014 AIP Publishing LLC 978-0-7354-1219-4/$30.00
34
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
111.223.255.4 On: Tue, 04 Mar 2014 05:11:20
( y & mthzffitm)
est
ava,
Em ummsim
27'Ifl [ntrbnr Zft3
!tmrs
l-lidetaka Arimura and NuninE itlunaini
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
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
111.223.255.4 On: Tue, 04 Mar 2014 05:11:20
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 considered as one of the primary strategy used by public
health authorities. In this paper we discuss the vaccination in households. As mentioned by Keeling et al. [8]
the intensity and frequency of interactions between people where one initial case can either lead to several more
cases within the household or can recover leaving few
other household members infected.
Becker et al. in [2] proposed linear optimization models in order to obtain an optimal vaccination strategy
within households. Some assumptions is used in the
model such as the disease spreads quickly within individual households and spreads more slowly between them
through close contacts between infected and susceptible
members of different households. They also assume proportionate mixing between households, to ensure that the
problem constraints are linear. This allows them to find a
closed form equation for the post-vaccination reproduction number. On the other hand, Ball et al. in [1] shows
that this linear program does not allow an easy characterization 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 considered 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 required 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 parameters on the vaccination strategy problem. Tanner et
al. [9] presents a stochastic programming framework for
finding the optimal vaccination policy for controlling infectious disease epidemics under parameter uncertainty.
Clancy et al. [6] propose their work on the optimal intervention for an epidemic model under parameter uncertainty, which consider the effect upon the optimal policy 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 results, i.e., we discuss how the robust design model of
uncertain vaccination strategy is modeled using Robust
Optimization. This optimization methodology incorporates the uncertain data in a so-called uncertainty set U
and replaces the uncertain problem by its so-called robust counterpart. Citing from [3], the main challenge in
this RC methodology is how and when we can reformulate the robust counterpart of the uncertain vaccination strategy problem as a computationally tractable optimization problem or at least approximate the model by
a tractable problem. Due to its definition the robust counterpart highly depends on how we choose the uncertainty
set U . 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
© 2014 AIP Publishing LLC 978-0-7354-1219-4/$30.00
34
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
111.223.255.4 On: Tue, 04 Mar 2014 05:11:20