Model II: Medical Resource Is Limited
5.5 Conclusions
In this study, we rely on a discrete time-space network to describe medical resources allocation problem when an unexpected influenza is outbreak. We formulate the problem as a multi-stage integer programming model with time-varying demand based on SEIR diffusion rule. The three main differences that distinguish this work from the past literature are presented as follows.
Firstly, the model proposed in this study addresses a time-series demand that is forecasted in match of the course of influenza diffusion. The model couples a multi-stage integer programming for optimal allocation of medical resources with a proactive forecasting mechanism cultivated from influenza diffusion dynamics. The rationale that medical resources allocated in early periods take effect in subduing the spread of influenza and thus impact demand in later periods has been for the first time incorporated into our model.
Secondly, the computational results based on a numerical example show that the proposed model is superior to the general measures in both terms of cost reduction and medical resources control. Our model can reduce the total operation cost of medical resources allocation and may get influenza diffusion in control earlier than general measures.
Last but not least, medical resources allocation problem has always been formu- lated as vehicle routing problem (VRP), or vehicle routing problem with time win- dows (VRPTW) in precious literatures, which includes many sub-tour constraints and is difficult to solve. In this study, we decompose medical resources allocation problem into several mutually correlated sub-problems, and solve them systemati- cally in the same decision scheme subsequently. Therefore, the proposed method is more suitable for an actual decision-making support.
The next research steps of this study incorporate a more realistic influenza dif- fusion model including features such as subdivision of the population by risk group and disease stage. It can also include the cross area diffusion between two or more geographic areas, and the incorporation of purchase lead time of medical resources.
In addition, the utilization of multiple resources in model is another important topic for the further research.
References
1. Mishra BK, Saini DK. SEIRS epidemic model with delay for transmission of malicious objects in computer network. Appl Math Comput. 2007;188(2):1476–82.
2. Sun CJ, Hsieh YH. Global analysis of an SEIR model with varying population size and vacci- nation. Appl Math Model. 2010;34(10):2685–97.
3. Li MY, Graef JR, Wang LC, Karsai J. Global dynamics of a SEIR model with varying total population size. Math Biosci. 1999;160(2):191–213.
4. Zhang J, Li JQ, Ma ZE. Global dynamics of an SEIR epidemic model with immigration of different compartments. Acta Math Scientia. 2006;26(3):551–67.
5. Zhang J, Ma ZE. Global dynamics of an SEIR epidemic model with saturating contact rate.
Math Biosci. 2003;185(1):15–32.
6. Wein LM, Liu YF, Leighton TJ. Evaluation of a HEPA/vaccine plan for indoor remediation after an airborne anthrax attack. Emerg Infect Dis. 2005;11(1):69–76.
7. Craft DL, Wein LM, Alexander HW. Analyzing bioterror response logistics: the case of anthrax.
Manag Sci. 2005;51(5):679–94.
8. Germann TC, Kadau K, Longini IM, Macken CA. Mitigation strategies for pandemic influenza in the United States. Proc Natl Acad Sci. 2006;103(15):5935–40.
9. Stroud P, Del Valle S, Sydoriak S, Riese J, Mniszewski S. Spatial dynamics of pandemic influenza in a massive artificial society. J Artif Soc Soc Simul. 2007;10(4):9.
10. Elizabeth Halloran M, Ferguson NM, Eubank S, et al. Modeling targeted layered containment of an influenza pandemic in the United States. Proc Natl Acad Sci. 2008;105(12):4639–44.
11. Aleman DM, Wibisono TG, Schwartz B. A nonhomogeneous agent-based simulation approach to modeling the spread of disease in a pandemic outbreak. Interfaces. 2011;41(3):301–15.
12. Kim KI, Lin ZG, Zhang L. Avian-human influenza epidemic model with diffusion. Nonlinear Anal Real World Appl. 2010;11(1):313–22.
13. Liu JL, Zhang TL. Epidemic spreading of an SEIRS model in scale-free networks. Commun Nonlinear Sci Numer Simul. 2011;16(8):3375–84.
14. Samsuzzoha M, Singh M, Lucy D. Numerical study of an influenza epidemic model with diffusion. Appl Math Comput. 2010;217(7):3461–79.
15. Samsuzzoha M, Singh M, Lucy D. A numerical study on an influenza epidemic model with vaccination and diffusion. Appl Math Comput. 2012;219(1):122–41.
16. Adida E, Dey D, Mamani H. Operational issues and network effects in vaccine markets. Eur J Oper Res. 2013;231(2):414–27.
17. Deo S, Corbett CJ. Cournot competition under yield uncertainty: the case of the U.S. influenza vaccine market. Manuf Serv Oper Manag. 2009;11(4):563–76.
18. Arifo˘glu K, Deo S, Iravani S. Consumption externality and yield uncertainty in the influenza vaccine supply chain: interventions in demand and supply sides. Manag Sci.
2012;58(6):1072–91.
19. Dai TL, Cho SH, Zhang FQ. Contracting for on-time delivery in the U.S. influenza vaccine supply chain. Manuf Serv Oper Manag. 2016;18(3):332–46.
20. Yamin D, Gavious A. Incentives’ effect in influenza vaccination policy. Manag Sci.
2013;59(12):2667–86.
21. Stinnett A, Paltiel A. Mathematical programming for the efficient allocation of health care resources. J Health Econ. 1996;15(5):641–53.
22. van Zon AH, Kommer GJ. Patient flows and optimal health-care resources allocation at the macro-level: a dynamic linear programming approach. Health Care Manag Sci.
1999;2(2):87–96.
23. Zaric GS, Brandeau ML. Resources allocation for epidemic control over short time horizons.
Math Biosci. 2001;171(1):33–58.
24. Zaric GS, Brandeau ML. Dynamic resources allocation for epidemic control in multiple pop- ulations. J Math Appl Med Biol. 2002;19(4):235–55.
25. Brandeau ML, Zaric GS, Richter A. Resources allocation for control of infectious dis- eases in multiple independent populations: beyond cost-effectiveness analysis. J Health Econ.
2003;22(4):575–98.
26. Zaric GS, Bravata DM, Cleophas Holty JE, et al. Modeling the logistics of response to anthrax bioterrorism. Med Decis Making. 2008;28(3):332–50.
27. Duintjer Tebbens RJ, Pallansch MA, Alexander JP, Thompson KM. Optimal vaccine stockpile design for an eradicated disease: application to polio. Vaccine. 2010;28(26):4312–27.
28. Vlah JS, Rui FJ. Multi-objective scheduling and a resources allocation problem in hospitals. J Sched. 2012;15(5):513–35.
29. Savachkin A, Uribe A. Dynamic redistribution of mitigation resources during influenza pan- demics. Socio-Econ Plann Sci. 2012;46(1):33–45.
30. Rottkemper B, Fischer K, Blecken A. A transshipment model for distribution and inven- tory relocation under uncertainty in humanitarian operations. Socio-Econ Plann Sci.
2012;46(1):98–109.
31. Rachaniotis NP, Dasaklis TK, Pappis CP. A deterministic resources scheduling model in epi- demic control: A case study. Eur J Oper Res. 2012;216(1):225–31.
32. Sun L, DePuy GW, Evans GW. Multi-objective optimization models for patient allocation during a pandemic influenza outbreak. Comput Oper Res. 2014;51:350–9.
33. Brandeau M. Allocating resources to control infectious diseases. Oper Res Health Care.
2005;70(3):443–64.
34. Dasaklis TK, Pappis CP, Rachaniotis NP. Epidemics control and logistics operations: a review.
Int J Prod Econ. 2012;139(2):393–410.
35. Liu M, Zhao LD. An integrated and dynamic optimisation model for the multi-level emergency logistics network in anti-bioterrorism system. Int J Syst Sci. 2012;43(8):1464–78.
36. Brauer F, Castillo-Chávez C. Mathematical models in population biology and epidemiology.
NY: Springer; 2012.
37. Kermack WO, McKendrick AG. A Contribution to the mathematical theory of epidemics. Proc R Soc A Contain Pap Math Phys Character. 1927;115(772):700–21.
38. Liu Q, Chen QM. Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence. Physica A. 2015;428:140–53.
39. Harko T, Lobo F, Mak MK. Exact analytical solutions of the susceptible-infected-recovered (SIR) epidemic model and of the SIR model with equal death and birth rates. Appl Math Comput. 2014;236:184–94.
40. Yan SY, Shih YL. Optimal scheduling of emergency roadway repair and subsequent relief distribution. Comput Oper Res. 2009;36(6):2049–65.
41. Yan SY, Lin JR, Lai CW. The planning and real-time adjustment of courier routing and scheduling under stochastic travel times and demands. Transp Res Part E Logist Transp Rev.
2013;53:34–48.
42. Yan SY, Lin CK, Chen SY. Logistical support scheduling under stochastic travel times given an emergency repair work schedule. Comput Ind Eng. 2014;67:20–35.
Integrated Optimization Model for Two-Level Epidemic-Logistics Network
As mentioned in the above chapters, the demand of emergency resource is usually uncertain and varies quickly in anti-bioterrorism system. With the consideration of emergency resources allocated to the epidemic areas in the early rescue cycles will affect the demand in the following periods, we construct an integrated and dynamic optimization model with time-varying demand for the emergency logistics network based on the epidemic diffusion rule. The heuristic algorithm coupled with ‘DDE23 tool’ in MATLAB is adopted to solve the optimization model, and the application of the model as well as a short sensitivity analysis of the key parameters in the time- varying demand forecast model is presented by a numerical example. The win-win emergency rescue effect is achieved by such an optimization model. Thus, it can provides some guidelines for decision makers when coping with emergency rescue problem with uncertain demand, and offers an excellent reference when issues are pertinent to bioterrorism.