Model II: Medical Resource Is Limited
5.1 Introduction
A serious influenza can test the ability of a nation to effectively protect its popula- tion, to reduce human loss and to rapidly recover. Meanwhile, it can also cause a great economic loss. For example, during the period from 1997 to 2002, more than 3,400,000 chickens were killed in Hong Kong, to prevent the avian influenza trans- fers from animals to human. Generally, it is difficult to predict when an unexpected influenza outbreaks, and our security measures to against such problem rest largely on consequence management, i.e., what can be done after the influenza outbreak occurs? How to ensure the supply of medical resources so that the efficiency of medical care can be maximized? Unfortunately, the available medical resources in the control of influenza are usually limited. Therefore, government decision makers must understand how the influenza spreads and then determine how to allocate the limited medical resources.
© Science Press and Springer Nature Singapore Pte Ltd. 2020 M. Liu et al.,Epidemic-logistics Modeling: A New Perspective on Operations Research,https://doi.org/10.1007/978-981-13-9353-2_5
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Most mathematic models for influenza diffusion analysis are compartmental mod- els [1–5]. In these models, the total population is divided into several classes and each class of individuals is closed into a compartment. The mixing of members is homogeneous, meaning these models are constructed with the assumption of both homogeneous infectivity and homogeneous connectivity of each individual. Another stream of research is focused on the development of influenza diffusion models by applying simulation methods, including computer simulation and numerical compu- tation [6–11] proposed an agent-based simulation model that treated each individual as unique, with non-homogeneous transmission and infection rates correlated to demographic information and behavior. Kim et al. [12] described the transmission of avian influenza among birds and humans. Liu and Zhang [13] presented a SEIRS epidemic model based on the scale-free networks, where the active contact number of each vertex was assumed to be either constant or proportional to its degree in their model. Samsuzzoha et al. [14] used a diffusive epidemic model to describe the transmission of influenza. The equations were solved numerically by using the splitting method under different initial distribution of population density. Further, Samsuzzoha et al. [15] presented a vaccinated diffusive compartmental epidemic model to explore the impact of vaccination as well as diffusion on the transmission dynamics of influenza. The above mentioned works provide numerous and signif- icant references to research the influenza diffusion. Although the emphasis of this study is focused on how to allocate the limited medical resources, a basic component of our model, the forecasting mechanism for the dynamic demand, utilizes one of such epidemic diffusion models.
So far, influenza vaccination policy is one of the most effective strategies to prevent a wide spread influenza occurs. However, the level of influenza vaccination coverage in all age groups is suboptimal, even in the majority of developed countries.
There are several reasons for this phenomenon, where mismatch between the vaccine supply and the demand side of is one of them. Recently, some significant studies on the subject are focused on the coordination of the influenza vaccine supply chain.
For example, Adida et al. [16] considered how a central policy-maker can induce socially optimal vaccine coverage through the use of incentives to both consumers and vaccine manufacturer in a monopoly market for an imperfect vaccine. The result shows that a fixed two-part subsidy is unable to coordinate the market. Deo and Corbett [17] examined the interaction between yield uncertainty of influenza vaccine and firms’ strategic behavior and found that yield uncertainty can contribute to a high degree of concentration in an industry and a reduction in the industry output and the expected consumer surplus in equilibrium. Arifo˘glu et al. [18] studied the impact of yield uncertainty (supply side) and self-interested consumers (demand side) on the inefficiency in the influenza vaccine supply chain. The result shows that the equilibrium demand can be greater than the socially optimal demand after accounting for the limited supply due to yield uncertainty and manufacturer’s incentives, which is contrast to the previous economic studies. To break the negative feedback loop between the retailer and the manufacturer in influenza vaccine industry, Dai et al.
[19] introduced two coordinating contracts, the Delivery-time-dependent Quantity Flexibility (D-QF) contract and the Buyback-and-Late-Rebate (BLR) contract, and
connected them to those used in practice. Furthermore, Yamin and Gavious [20]
built a theoretical epidemiological game model to find the optimal incentive for vaccination and the corresponding expected level of vaccination coverage.
Motivated by the supply chain coordination concept, we study an interactive coor- dination problem between influenza diffusion and medical resources allocation. This paramount life-saving and costly logistics problem opens up a wide range of appli- cations of OR/MS techniques and has motivated much research works in the past decades [21–25]. These models, however, are not applicable to epidemics with dis- crete rates of growth and are restricted by several assumptions like the number of interventions or independence of populations. Recent mathematical approaches for healthcare resources allocation, on the other hand, suggest advanced models of dis- ease prevalence among several populations, and consider more general forms of cost function for the prevention programs [26–30] designed a mixed-integer programming model for distribution and inventory relocation under uncertainty in humanitarian operations. Rachaniotis et al. [31] presented a resources scheduling model in epi- demic control with limited resources. The objective is to minimize the total infected people in a certain time horizon under consideration by effectively relocating the available resources over several regions. Sun et al. [32] built a mathematical model to optimize the patient allocation considering two objectives: to minimize the total travel distance by patients to hospitals, and the maximum distance a patient travels to a hospital. In addition, it is worth mentioning that a concise survey of OR/MS contri- bution to epidemics control can be found in Brandeau [33]. The popular techniques that have been used for resources allocation in epidemics control are linear and inte- ger programming models, numerical analysis procedures, cost-effectiveness anal- ysis, simulation, non-linear optimization and control theory techniques. Recently, Dasaklis et al. [34] focused on defining the role of logistics operations and their management that may assist the control of epidemic outbreaks. They reviewed the literature and pointed out the research gaps critically.
In summary, this section does not aim to be an exhaustive review of the litera- ture; rather, we introduce an illustrative subset of existing models. In our previous work [35], we divided influenza diffusion process into three stages. The first stage is the inception of influenza in very limited population. If the infectious disease is noticed in time and treated properly, the epidemic can be controlled without causing a wide spread. Otherwise, influenza diffusion develops into the second widespread diffusion stage. The third stage is the recovery stage that influenza diffusion is under controlled. In this study, we attempt to model the interactive coordination process between influenza diffusion and medical resources allocation in the second response stage. The model couples a forecasting mechanism for dynamic demand of medical resources based on the classical SEIR epidemic model [36]. As shown in Fig.5.1, we decompose the whole interactive coordination process intoncorrelated sub-problems (ndecision-making cycles). Each sub-problem includes three phases, which are influenza diffusion analysis, demand forecasting and medical resources allocation. We briefly introduce the connections among these three phases as below.
No
Initialize, decompose the problem into nsub-problems
Demand forecasting t=0
Medical resources allocation t>n
Output the result
t=t+1
Yes
Epidemic diffusion analysis
Fig. 5.1 The dynamic operational procedure of medical resources allocation
(i) Initially, we employ a SEIR model to depict the dynamic epidemic diffusion process. The model will give us a forecast of the growing (or decreasing) number of the infected population in the course of the epidemic diffusion, which will be embedded in the following demand forecasting model.
(ii) Secondly, we define a difference factor to illustrate the change in the number of infected population. Coupling with this factor and medical resources allocation result in the current decision cycle, we can get the demand of medical resources for the next decision cycle.
(iii) Based on the forecasting demand for the next decision cycle, we solve an integer programming problem for the optimized allocation of medical resources in a supportive logistics system to meet the dynamic demand.
The latter two phases are executed iteratively. The details of the demand forecast- ing model are presented in Sect.5.2.2. It is worth mentioning that medical resources allocated in current period will take effect in subduing the spread of influenza and thus impact the demand in the next period. To the best of our knowledge, such an operational procedure is different from any existing influenza response operations, which have always been carried out under the assumption that demand is deter- ministic or stochastic. While the proposed method is adopted, we can take a fixed time interval (i.e. one day) as the decision-making cycle and then update the alloca- tion result for each epidemic area periodically. Moreover, we believe the proposed model should serve for the benefit of a centralized decision maker, usually a local or regional governmental agent, in control of the influenza diffusion, who needs an analytic model to plan for the logistics and to revise and update such plan in the actual implementation.