Production, Manufacturing and LogisticsModeling the logistics response to a bioterrorist anthrax attack
Introduction
Bacillus anthracis, the bacteria causing Anthrax, is classified by the Centers for Disease Control and Prevention (Atlanta, U.S.A.) as one of the most likely agents to be used for biological attack. It may spread across a large area, and needs a great deal of planning for the protection of the people's health. Already known as an efficient biological warfare weapon, anthrax spores become part of terrorists’ arsenals in the aftermath of the September 11, 2001 terrorist's attacks, when concentrated anthrax spores were delivered by mailing postal letters containing the spores. As a result, 22 were infected and five died. Since then, the use of weaponized anthrax spores has become a major threat for the civil population (Berman & Gavious, 2007). An effective emergency management plan can obviously reduce the number of casualties and the serious consequences of an anthrax attack. Even though several studies have been made about the response to the anthrax attack from different perspectives, some crucial questions are still unanswered. Questions concerning the suitable dispensing capacity of the antibiotics distribution centers (ADC), the utilization of sensors to improve the anthrax detection ability, and the choice of patients who should get the medical help first need, among others, need to be addressed in the design of the most effective post-attack response.
This paper proposes an original modeling approach inspired by Markov decision process. It combines the progress of the anthrax disease, the medical response, and the logistic deployment choices. In particular, the approach allows us to capture dynamically the impact of different medical responses on the infected population. Dynamics are important because the time elapsed since the patient is infected till the moment he/she receives medical treatment has a major impact on the recovery rate, the survival rate and, therefore, on the number of deaths. Crisis managers need to assess how their decisions, in particular these concerning the deployment and operation of an antibiotics’ distribution network, will determine the access of infected people to medical treatment. Our model allows to estimate the number of individuals infected in different periods and the number of patients in each stage of the disease for each period. Based on this information, crisis managers can optimize the resources’ deployment for the best response. The model is flexible and can adapt to specific situations and various resource deployment scenarios. The remainder of this paper is organized as follows: Section 2 briefly reviews the related literature. Section 3 describes the background of the problem. The anthrax response model and the mathematical formulation are presented and discussed in Section 4. Section 5 reports numerical experiments assessing the potential of our approach. Conclusion and future research directions are provided in Section 6.
Section snippets
Literature review
This part contains a brief review of the related papers studying the response to the anthrax attack in terms of methodologies and research content. Doubtless, the three-paper series (Craft et al., 2005, Wein and Craft, 2005, Wein et al., 2003) constitutes one of the first and most important contributions to the field. The first paper in this series (Wein et al., 2003) considered an airborne anthrax attack and proposed a mathematical model to compare five priority policies to determine who
Basis of the model
In order to give the reader a general view of the situation that we intend to address, this section describes firstly the progression of the anthrax and the transitions between the different stages of the disease; secondly it presents the three phases of a response to an anonymous bioterrorist attack with anthrax; finally it describes the decisions concerning the logistics of medical help delivery.
The anthrax response model and the mathematical formulations
This section presents the anthrax response model and the related mathematical formulation. We first present an extended compartmental model and discuss the transitions between its stages. Then, a discrete-time mathematical formulation aiming at minimizing the number of deaths caused by an airborne anthrax bioterrorist attack under limited medical resources is presented.
Numerical experiments
This section illustrates how to use our model to solve the aforementioned logistics questions, but also presents some analysis assessing the sensitivity of the models’ output (number of deaths) with respect to resources’ availability and different system's parameters. Those experiments were conducted using both the exponential and the lognormal distribution to assess the impact of this choice. All the experiments were run on a workstation with 2.67 gigahertz Intel(R) Xeon(R) X5650 and 64
Conclusion
This paper studies the logistic response to an anthrax attack based on a real situation. The model proposed in this paper can predict the different antibiotics requirements on any given day, find the reasonable dispensing capacity of ADC, decide the amount of antibiotics that should be delivered to the ADCs or to the hospitals every day, and show which patients should get the medical help first. Based on numerical experiments, two factors can decrease the number of deaths effectively:
Acknowledgment
We acknowledge the valuable collaboration of the hospital pharmacy of HCL (Hospices Civils de Lyon) and we would like to thank particularly her zonal coordinator for health crisis: Dr. Annick Terrier. Our thanks go also to China Scholarship Council (No. 2011629058 (CSC)) for the financial support for the Ph.D. student who is working on this project under a French-Canadian supervision.
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