Dynamic optimization model for allocating medical resources in epidemic controlling
Ming Liu^{1}, Jing Liang^{2}
^{1}Nanjing University of Science and Technology, ^{2}Nanjing College of Chemical Technology (China)
Received August 2012
Accepted February 2013
Liu, M., & Liang, J. (2013). Dynamic optimization model for allocating medical resources in epidemic controlling. Journal of Industrial Engineering and Management, 6(1), 7388. http://dx.doi.org/10.3926/jiem.663

Abstract:
Purpose: The model proposed in this paper addresses a dynamic optimization model for allocating medical resources in epidemic controlling.
Design/methodology/approach: In this work, a threelevel and dynamic linear programming model for allocating medical resources based on epidemic diffusion model is proposed. The epidemic diffusion model is used to construct the forecasting mechanism for dynamic demand of medical resources. Heuristic algorithm coupled with MTLAB mathematical programming solver is adopted to solve the model. A numerical example is presented for testing the model’s practical applicability.
Findings: The main contribution of the present study is that a discrete timespace network model to study the medical resources allocation problem when an epidemic outbreak is formulated. It takes consideration of the time evolution and dynamic nature of the demand, which is different from most existing researches on medical resources allocation.
Practical implications: In our model, the medicine logistics operation problem has been decomposed into several mutually correlated subproblems, and then be solved systematically in the same decision scheme. Thus, the result will be much more suitable for real operations.
Originality/value: In our model, the rationale that the medical resources allocated in early periods will take effect in subduing the spread of the epidemic spread and thus impact the demand in later periods has been for the first time incorporated. A winwin emergency rescue effect is achieved by the integrated and dynamic optimization model. The total rescue cost is controlled effectively, and meanwhile, inventory level in each urban health departments is restored and raised gradually.
Keywords: epidemic diffusion, timevarying demand, timespace network, dynamic optimization

1. Introduction
As mentioned in (Rachaniotis, Dasaklis & Pappis, 2012), a serious epidemic is a problem that tests the ability of a nation to effectively protect its population, to reduce human loss and to rapidly recover. Sometime such a problem may acquire worldwide dimensions. For example, during the period from November 2002 to August 2003, 8422 people in 29 countries were infected with SARS, 916 of them were dead at last for the effective medical resources appeared late. Other diseases, such as HIV, H1N1 also cause significant numbers of direct infectious disease deaths.
Actually, many recent research efforts have been devoted to understanding the prevention and control of epidemics, such as Wein, Liu and Leighton (2005), Craft, Wein and Alexander (2005) and Kaplan, Craft and Wein (2002). The major purpose of these articles is to compare the performance of the following two strategies, the traced vaccination (TV) strategy and the mass vaccination (MV) strategy, but not address how to optimize the allocation of medical resources.
Another stream of research is on the development of epidemic diffusion models by applying complex network theory to traditional compartment models. For example, Saramäki & Kaski (2005) proposed a susceptibleinfectedrecovered (SIR) model for the spreading of randomly contagious diseases, such as influenza, based on a dynamic smallworld network. Xu, Peng, Wang and Wang (2006) presented a modified susceptibleinfected susceptible (SIS) model based on complex networks, smallworld and scalefree, to study the spread of an epidemic by considering the effect of time delay. Based on twodimensional smallworld networks, a susceptibleinfected (SI) model with epidemic alert is proposed in (Han, 2007). This model indicates that to broadcast an epidemic alert timely is helpful and necessary in the control of epidemic spreading. Jung, Iwami and Takeuchi (2009) extended the previous studies on the prevention of the pandemic influenza to evaluate the timedependent optimal prevention policies, and they found that the quarantine policy is very important, and more effective than the elimination policy. After determining the epidemiologic features of an Escherichia coli O157:H7 outbreak in Xuzhou, Jiangsu Province, China, Zhu, Gu, Yu et al. (2009) provided a scientific approach for establishing prevention and control strategies in local areas. These above mentioned works represent some of the research on various differential equation models for epidemic diffusion and control.
However, after an epidemic outbreak, public officials are faced with many critical issues, one of the most important of which being how to ensure the availability and supply of medical resources so that the loss of life may be minimized and the rescue operation efficiency maximized. Sheu (2007) presented a hybrid fuzzy clusteringoptimization approach to the operation of medical resources allocation in response to the timevarying demand during the crucial rescue period. Yan and Shih (2009) considered how to minimize the length of time required for emergency roadway repair and relief distribution, as well as the related operating constraints. The weighting method is adopted, and a heuristic algorithm is developed to solve a real emergency relief problem, the ChiChi earthquake in Taiwan. To optimize the process of materials distribution in an epidemic diffusion system and to improve the distribution timeliness, Liu and Zhao (2009) modeled the emergency materials distribution problem as a multiple traveling salesman problem with time window. Wang, Wang and Zeng (2009) constructed a multiobjective stochastic programming model with timevarying demand for the emergency logistics network based on the epidemic diffusion rule. A genetic algorithm coupled with Monte Carlo simulation is adopted to solve the optimization model. Recently, Qiang and Nagurney (2011) proposed a humanitarian logistic model for supply/distribution of critical needs in a disruption caused by a nature disaster. They consider a general network structure and disruptions that may have an impact to both network link capacities and product demand. The problem is studied in a bicriteria system optimization framework for network performance.
In this article, a threelevel and dynamic linear programming model for allocating medical resources based on epidemic diffusion model is proposed. The epidemic diffusion model introduced here is to construct the forecasting mechanism for the dynamic demand of medical resources. Heuristic algorithm coupled with MTLAB mathematical programming solver is adopted to solve the model. The remainder of the article is organized as follows: Section 2 is the problem description, including the timevarying demand forecasting model based on the epidemic diffusion rule for the infected areas and the timevarying demand forecasting model for the local health departments. The threelevel and dynamic optimization model and the solution procedure are introduced in Section 3. A numerical example is presented in Section 4. Finally, the limitations of the proposed model and future research directions are proposed in Section 5.
2. Problem description
2.1. The research ideas and way to achieve
As work in Liu and Zhao (2011), this article focus on the recovered stage of epidemic rescue. In such a stage, epidemic diffusion tends to be stable. Thus, optimization goal in such stage is to construct an integrated, dynamic and multilevel emergency logistics network, which includes the national strategic storages (NSS), the urban health departments (UHD), the area disease prevention and control centers (ADPC), and the emergency designated hospitals (EDH). Herein, we introduce a timespace network to depict the network structure relationship of these elements, which is shown as Figure 1. In such figure, the vertical axis represents the time duration, and the horizontal axis represents different emergency departments.
Figure 1. Timespace network of medical resources allocation
The entire recovered stage of epidemic rescue process is decomposed into several mutually correlated subproblems (i.e. n decisionmaking cycles). To each decisionmaking cycle, there exist two subproblems. In the upper level, we consider the problem how to replenish medical resources to the UHDs. Besides, we adjust the replenishment arcs among these NSSs by a heuristic algorithm, and construct a mixedcollaborative delivery system. Thus, the total rescue cost of the upper level subproblem would be minimized. In the lower level, we present the problem how to distribute medical resources to the ADPCs and then allocate medical resources to EDHs. We propose a forecasting model for the timevarying demand in EDHs based on a SEIR epidemic diffusion model. Such two phases are executed iteratively. Besides, at the end of each rescue cycle, effect of medical resources allocated is analyzed and the number of infected people is updated. The research idea of such rescue stage is shown as Figure 2.
Figure 2. Operational procedure of the dynamic medicine logistics network
2.2. Timevarying forecasting method for the dynamic demand
In this article, we divide people into four classes: the susceptible people (S), the exposed people (E), the infected people (I), and the recovered people (R). The following SEIR epidemic diffusion model in Liu and Zhao (2012) is adopted to depict the epidemic diffusion rule.
Herein, <k> is the average degree distribution of the smallworld network; b is the propagation coefficient; d is the recovered rate of the infected people; d is the death rate caused by the disease; stands for the incubation period. Furthermore, <k>,b,d,d,t>0. Traditionally, a simple linear function is used to formulated the demand for medical resources as follows:
Herein, I(t) is the number of infected people in the epidemic area at time t. a is the proportionality coefficient. Note that emergency demand for medical resources is closely related to the number of infected people, and medical resources allocated in the early rescue cycle will affect the demand later, here we propose a timevarying forecasting method for the demand in each EDH as follows:
Herein, . q is the effective rescue rate; G is the treatment cycle for each infected person. To facilitate the calculation process in the following sections, we assume that G to be an integral multiple of rescue cycle. Equation (3) is used to calculate the linear scale factor of the change in demand. Equation (4) is the initial demand in the epidemic area, and I(0) represents the initial number of infected people in the epidemic area. Equations (5)(7) represent demand for medical resources in rescue cycle 1,2,…,n, respectively.
2.3. Dynamic demand and inventory for the UHD
To facilitate the calculation process in the following sections, we assume that initial inventory of medical resources in each UHD is zero. Besides, we suppose that capacity of each UHD is V_{cap }· d_{t}^{v }represents demand for medical resources in UHD in rescue cycle t. P_{t }represents the total output of medical resources in UHD in rescue cycle t. Thus, the forecasting model for dynamic demand in UHD can be formulated as follows:
Correspondingly, suppose that V_{t} is inventory of medical resources in UHD in rescue cycle t, we can get the following equation:
3. Optimization model and solution procedure
3.1. Optimization model
The following assumptions are needed to facilitate the model formulation in the following sections:
(1) Once an epidemic outbreak, each EDH can be isolated from other areas to avoid the spread of the disease.
(2) It is reasonable to assume that the government can ensure the adequate supply of the needed medicines either from domestic pharmaceutical companies or imported. Hence, there are enough medical resources during the entire operation process.
(3) Holding cost of medical resources is not considered in this paper for two reasons.
(4) Medical resources in this section are an assembled product, which may includes water, vaccine, antibiotic, etc.
Notations used in the following optimization model are specified as follows.
nc_{ij}: Unit replenishment cost of medical resources from NSS i to UHD j.
ce_{jk}: Unit distribution cost of medical resources from UHD j to ADPC k.
eh_{kl}: Unit distribution cost of medical resources from ADPC k to EDH l.
ns_{i}: Amount of medical resources supplied by NSS i in each rescue cycle.
V_{cap}: Capacity of UHD.
d_{lt}: Demand for medical resources in EDH l in rescue cycle t.
d_{jt}^{v}: Demand for medical resources in UHD j in rescue cycle t.
P_{jt}: Total output of medical resources in UHD j in rescue cycle t.
V_{jt}: Inventory of medical resources in UHD j in rescue cycle t.
x_{jt}: Amount of medical resources that will be transported from NSS i to UHD j in rescue cycle t.
y_{jkt}: Amount of medical resources that will be transported from UHD j to ADPC k in rescue cycle t.
z_{klt}: Amount of medical resources that will be transported from ADPC k to EDH l in rescue cycle t.
TC: Total rescue cost of the threelevel medical logistics network.
N: Set of NSSs.
C: Set of UHDs.
E: Set of ADPCs.
H: Set of EDHs.
T: Set of decisionmaking cycles.
According to the above explanations and assumptions, the threelevel and dynamic linear programming model for allocating medical resources based on epidemic diffusion model can be formulated as follows:
Herein, the objective function in Equation (10) is to minimize the total rescue cost of the threelevel medical distribution network. Equations (11) and (12) are constraints for flow conservation in the upper level subproblem. Equations (13)(15) are the dynamic demand models in the upper level subproblem. Equations (16)(18) are constraints for flow conservation in the lower level subproblem. Equations (19)(21) are the timevarying demand models in the lower level subproblem. At last, Equations (21)(23) ensure all the arc flows in the timespace network within their bounds.
3.2. Solution procedure
As Figure 2 shows, the solution procedure for the proposed optimization model is presented as follows:
Step 1. Decompose the entire recovered stage of epidemic rescue process into n decisionmaking cycles.
Step 2. Let t=0, and initialize parameters in the SEIR epidemic diffusion model.
Step 3. Analyze the epidemic diffusion rule, and calculate the initial demand for medical resources in each EDH according to Equations (3)(7).
Step 4. Solve the programming model in rescue cycle t=0 and obtain the initial solution.
Step 5. Improve the initial solution by heuristic algorithm. Detail about the heuristic algorithm, please go to Liu, Li and Chan (2003).
Step 6. Get the final solution for medical resources allocation in this rescue cycle.
Step 7. Let t=t+1, if the termination condition for the rescue cycle has not been satisfied, update the demand in each EDH, and update the inventory level of medical resources in each UDH, go back to Step 3. Else, go to the next step.
Step 8. End the program and output the final result.
4. Numerical example
To test how well the proposed model may be applied in an actual event, we present a numerical example to illustrate its efficiency. Assume there is a smallpox outbreak in a city, which has 3 NSSs, 4 UHDs, 6 ADPCs and 8 EDHs. The values of parameters in SEIR epidemic diffusion model are given in Table 1.
Figure 3 depicts a numerical simulation of the epidemic model at EDH 1 in this effected region. The four curves respectively represent the number of four groups of people (S, E, I, R) over time. As mentioned in [18], the process of epidemic diffusion is divided into three stages and our research focus on the recovered one. According to the above figure, we assume that such stage range from the 45^{th }day (decisionmaking cycle t=0) to the 55^{th} day (decisionmaking cycle t=10). For simplicity, the decisionmaking cycle is assumed to be one day. A total of 138240 variations are generated in the experiment. During an actual epidemic activity, decision makers can adjust these parameters according to the actual situation.

EDH1 
EDH2 
EDH3 
EDH4 
EDH5 
EDH6 
EDH7 
EDH8 
S(0) 
5x10^{3} 
4.5x10^{3} 
5.5x10^{3} 
5x10^{3} 
6x10^{3} 
4.8x10^{3} 
5.2x10^{3} 
4x10^{3} 
E(0) 
30 
35 
30 
40 
25 
40 
50 
45 
I(0) 
5 
6 
7 
8 
4 
7 
9 
10 
R(0) 
0 

b 
4x10^{5} 

<k> 
6 

d 
0.3 

d 
1x10^{3} 

t 
5 
Table 1. Values of parameters in SEIR epidemic diffusion model
Figure 3. Solution of the SEIRS epidemic diffusion model (EDH 1)
Let a=1, q=90% and G=15, the MATLAB mathematic solver coupled with Equations (2)(7) is adopted to forecast the timevarying demand for each EDH. Taking the EDH 1 as an example, the demand for medical resources in each rescue cycle is shown in Figure 4. One can observe in Figure 4 that timevarying demand is way below traditional demand, suggesting that the allocation of medical resources in the early periods will significantly reduce the demand in the following periods. The second observation is that both curves exhibit similar trends, namely, the demand will decrease after the epidemic is brought under control.
After getting the demand for medical resources in each rescue cycle, in what follows, we are going to discuss how to allocate these medical resources to the epidemic areas, and at the same time, how to replenish medical resources to each UHD, with the objective of minimizing the total rescue cost. Table 2 shows the unit cost from the supply points to the demand points (Here, we suppose that NSS 1 has been preset as the HUB location).
Figure 4. Demand in EDH 1 by the two different methods
cost 
N1 
C1 
C2 
C3 
C4 

N1 
 
2 
2 
1 
3 

N2 
3 
7 
2 
10 
9 

N3 
4 
8 
9 
2 
10 


E1 
E2 
E3 
E4 
E5 
E6 

C1 
1 
3 
4 
2 
5 
5 

C2 
2 
1 
2 
3 
5 
4 

C3 
2 
3 
5 
4 
1 
3 

C4 
4 
4 
1 
2 
3 
2 


H1 
H2 
H3 
H4 
H5 
H6 
H7 
H8 

E1 
6 
2 
6 
7 
4 
2 
5 
9 

E2 
4 
9 
5 
3 
8 
5 
8 
2 

E3 
5 
2 
1 
9 
7 
4 
3 
3 

E4 
7 
6 
7 
3 
9 
2 
7 
1 

E5 
2 
3 
9 
5 
7 
2 
6 
5 

E6 
5 
5 
2 
2 
8 
1 
4 
3 

Table 2. Unit cost of medical resources between two different departments
Assume that three NSS can supply 400, 420 and 450 unit of medical resources in each rescue cycle, and suppose that capacity of each UHD is 320. Let us take the allocation result at cycle t=0 as example, we can solve the programming model according to the solution procedure. The initial solution is reported in Table 3.
Amount 

C1 
C2 
C3 
C4 

Before adjustment 
N1 
100 
0 
0 
320 

N2 
110 
320 
0 
0 

N3 
110 
0 
320 
0 



N1 
C1 
C2 
C3 
C4 
After adjustment 
N1 
 
320 
0 
0 
320 
N2 
110 
0 
320 
0 
0 

N3 
110 
0 
0 
320 
0 
Table 3. Solution of the optimization model at cycle t=0
Then, we can improve the initial solution by heuristic algorithm in Liu, Li and Chan (2003). As Table 3 shows, when replenishment arcs are transferred from the direct shipment delivery system to the hubandspoke delivery system, such as N_{2 }è N_{1}è C_{1 }and N_{3 }è N_{1}è C_{1}, the total rescue cost will be reduced 10.75% when compared with the cost before adjustment (4090/3650). Similarly, we can complete the whole operations according to the solution procedure. Replenishment arcs that need to be transferred in each cycle are shown in Table 4. Total rescue cost at each cycle is presented in Figure 5.
Cycle 
Arcs need to be transferred 
Cycle 
Arcs need to be transferred 

Before 
After 
Before 
After 

t=0 
N2gC1 N3gC1 
N2gN1gC1 N3g N1gC1 
t=1 
N2gC1 N3gC1 
N2gN1gC1 N3gN1gC1 
t=2 
N2gC1 
N2gN1gC1 
t=3 
N2gC1 
N2gN1gC1 
*There is no adjustment when t=4,5,6,7,8,9,10 
Table 4. Transferred arcs in each rescue cycle
Figure 5 shows the change in total rescue cost over time. From this figure, we can get the following two conclusions: (1) Coupled with Figure 4, we can see that demand for medical resources is decreasing, which implies that epidemic diffusion is on the recovered stage. (2) Coupled with Table 4, we can see that the total rescue cost can be reduced in a certain degree by using the proposed heuristic algorithm. It is worth mentioning that there is no adjustment after rescue cycle t=4, for that NSSs and UHDs which are adjacent to the epidemic areas will have stored enough resources at that time.
Figure 5. Total rescue cost at each rescue cycle
Figure 6. Inventory level in different UHD over time
Figure 6 shows the inventory level in different UHDs. It shows that inventory level in each UHD has been improved and raised as time goes by. Therefore, with the application of the threelevel and dynamic optimization model, the total rescue cost can be controlled effectively, and meanwhile, inventory level in each UHD can be restored and raised gradually. Thus, such optimization model achieves a winwin rescue effect.
5. Conclusions
In this article, we develop a discrete timespace network model to study the medical resources allocation problem in the recovered stage when an epidemic outbreak. In each decisionmaking cycle, the allocation of medical resources across the region from NSSs through UHDs and ADPCs to EDHs is determined by a linear programming model with the dynamic demand that is forecasted by an epidemic diffusion rule. The novelty of our model against the existing works in literature is characterized by the following three aspects:
(1) While most research on medical resources allocation studies a static problem taking no consideration of the time evolution and dynamic nature of the demand, the model proposed in this paper addresses a timeseries demand that is forecasted in match of the course of an epidemic diffusion. The model couples a multistage linear programming for optimal allocation of medical resources with a proactive forecasting mechanism cultivated from the epidemic diffusion dynamics. The rationale that the medical resources allocated in early periods will take effect in subduing the spread of the epidemic spread and thus impact the demand in later periods has been for the first time incorporated into our model.
(2) A winwin emergency rescue effect is achieved by the integrated and dynamic optimization model. The total rescue cost is controlled effectively, and meanwhile, inventory level in each UHD is restored and raised gradually.
(3) In this article, the medicine logistics operation problem has been decomposed into several mutually correlated subproblems, and then be solved systematically in the same decision scheme. Thus, the result will be much more suitable for real operations.
As the limitation of the model, it is developed for the medical resources allocation in a geographic area where an epidemic disease has been spreading and it does not consider possible cross area diffusion between two or more geographic areas. We assume that once an epidemic outbreak, the government has effective means to separate the epidemic areas so that crossarea spread can be basically prevented. However, this cannot always be guaranteed in reality.
Acknowledgments
This work has been partially supported by the MOE (Ministry of Education in China) Project of Humanities and Social Sciences (No. 11YJCZH109), and by the Fundamental Research Funds for the Central Universities，No. NUST (2011YBXM96), (2011XQTR10), (AE88072) and (JGQN1102). The authors wish to thank Prof. Ding Zhang from State University of New York (USA), who read earlier versions of this paper. His helpful suggestions improved the presentation significantly.
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Journal of Industrial Engineering and Management, 20082021
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