A spatial queuing model for the location decision of emergency medical vehicles for pandemic outbreaks: the case of Za'atari refugee camp
Journal of Humanitarian Logistics and Supply Chain Management
ISSN: 2042-6747
Publication date: 21 January 2021
Abstract
Purpose
Refugee camps can be severely struck by pandemics, like potential COVID-19 outbreaks, due to high population densities and often only base-level medical infrastructure. Fast responding medical systems can help to avoid spikes in infections and death rates as they allow the prompt isolation and treatment of patients. At the same time, the normal demand for emergency medical services has to be dealt with as well. The overall goal of this study is the design of an emergency service system that is appropriate for both types of demand.
Design/methodology/approach
A spatial hypercube queuing model (HQM) is developed that uses queuing-theory methods to determine locations for emergency medical vehicles (also called servers). Therefore, a general optimization approach is applied, and subsequently, virus outbreaks at various locations of the study areas are simulated to analyze and evaluate the solution proposed. The derived performance metrics offer insights into the behavior of the proposed emergency service system during pandemic outbreaks. The Za'atari refugee camp in Jordan is used as a case study.
Findings
The derived locations of the emergency medical system (EMS) can handle all non-virus-related emergency demands. If additional demand due to virus outbreaks is considered, the system becomes largely congested. The HQM shows that the actual congestion is highly dependent on the overall amount of outbreaks and the corresponding case numbers per outbreak. Multiple outbreaks are much harder to handle even if their cumulative average case number is lower than for one singular outbreak. Additional servers can mitigate the described effects and lead to enhanced resilience in the case of virus outbreaks and better values in all considered performance metrics.
Research limitations/implications
Some parameters that were assumed for simplification purposes as well as the overall model should be verified in future studies with the relevant designers of EMSs in refugee camps. Moreover, from a practitioners perspective, the application of the model requires, at least some, training and knowledge in the overall field of optimization and queuing theory.
Practical implications
The model can be applied to different data sets, e.g. refugee camps or temporary shelters. The optimization model, as well as the subsequent simulation, can be used collectively or independently. It can support decision-makers in the general location decision as well as for the simulation of stress-tests, like virus outbreaks in the camp area.
Originality/value
The study addresses the research gap in an optimization-based design of emergency service systems for refugee camps. The queuing theory-based approach allows the calculation of precise (expected) performance metrics for both the optimization process and the subsequent analysis of the system. Applied to pandemic outbreaks, it allows for the simulation of the behavior of the system during stress-tests and adds a further tool for designing resilient emergency service systems.
Keywords
Citation
Blank, F. (2021), "A spatial queuing model for the location decision of emergency medical vehicles for pandemic outbreaks: the case of Za'atari refugee camp", Journal of Humanitarian Logistics and Supply Chain Management, Vol. ahead-of-print No. ahead-of-print. https://doi.org/10.1108/JHLSCM-07-2020-0058
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:Emerald Publishing Limited
Copyright © 2021, Emerald Publishing Limited
1. Introduction
Refugee camps can be severely struck by the outbreak of epidemics and pandemics like cholera in Kakuma refugee camp (Mahamud et al., 2010), diphtheria in a Rohingya refugee camp in Bangladesh (Matsuyama et al., 2018) or other respiratory diseases in long-term settlements (Ahmed et al., 2012). Due to basic and oftentimes insufficient medical infrastructure that is provided in most refugee camps, potential outbreaks of COVID-19 may threaten the overall population of the camp because of the vital necessity to treat patients that are seriously affected and contagious at the same time. In addition, the usual strategy of home quarantine for mild courses of the diseases is not practicable within the setting of a refugee camp due to the overall high population density, the sharing of homes by inhabitants in double-digit numbers as well as the inadequate sanitary conditions (Nott, 2020). The use of epidemiologic surveillance and investigation has been proven to be an effective measure against outbreaks of diseases in refugee camps (Elias et al., 1990). For the treatment of potential outbreaks of COVID-19 in refugee camps, the common strategy should be to isolate infected inhabitants in hospitals or other temporary treatment facilities and to retrace their contacts as fast as possible, as noted by Raju and Ayeb-Karlsson (2020). Only then, the containment strategy in densely populated environments like refugee camps can be successful. Hellewell et al. (2020) showed in a simulation study the effects of fast contact tracing after symptom onset and concluded that case isolation can contribute to decreasing the virus spread. Due to the necessary protective measures in the contact with persons potentially affected with a virus, trained medical staff is required. As a result, potential virus outbreaks have to be contained under the existing medical infrastructure. This leads to additional demands for medical and emergency services, which can overstress the already insufficient medical infrastructure. The conditions of a refugee camp and the necessity of isolating COVID-19 infected persons instantly should qualify potential virus outbreaks as emergency medical incidents.
Over the past years considerable research effort was made to support the decision making process for emergency medical systems (EMSs) (Bélanger et al., 2019). These systems are characterized by their inherent uncertainty and the spatial distribution of incoming demands for emergency help. One of the most used techniques for the analysis of EMS is the hypercube queuing model (HQM) as debuted by Larson (1974). The HQM is based on queuing theory and can be used to describe the behavior of an already pre-determined set of locations for an emergency system. Moreover, the HQM can be used to calculate probability-based performance metrics to gain further insights. Due to its descriptive nature, it cannot be used to determine optimal or near-optimal locations for emergency vehicles (also called servers) (Galvão and Morabito, 2008). However, as noted by Goldberg (2004) and Takeda et al. (2007) the obtained performance metrics can be embedded into an optimization process. Therefore, for example, Iannoni et al. (2008, 2009), Geroliminis et al. (2009), Toro-Díaz et al. (2013) and Enayati et al. (2019) used HQM calculated performance metrics in the EMS location optimization problem. The approach used in this paper aims to combine the HQM optimization process with a subsequent simulation of virus outbreaks in a refugee camp. For this purpose, the optimization process is done beforehand with the goal of determining optimal or near-optimal server locations. Afterward, random outbreaks of a virus, like for example COVID-19, are simulated and the system behavior is analyzed. Moreover, additional servers supporting the EMS in the treatment of COVID-19 patients are introduced. The objective is to design an EMS that can cope with the regular demands of emergency help as well as with the additional demands imposed by potential virus outbreaks and to support decision-makers during pandemic outbreaks. The real-world case of Za'atari refugee camp in Jordan is applied to illustrate the findings. The remainder of this study is organized as follows: A brief literature review on the design of refugee camps and on the location optimization decision for emergency vehicles is given in section 2. The methodological approach and the corresponding HQM-related formulations are presented in section 3. Section 4 contains the description of the case study as well as the computational results for the optimization and virus simulation studies. Suggestions for future research and concluding remarks are given in section 5.
2. Literature review
In more recent times, an extensive amount of literature on the strategic design of refugee camps has been published. Often refugee camps have been viewed as temporary spaces that are used to isolate refugees from the local community (Jahre et al., 2018). This has shifted toward looking at refugee camps as permanent settlements with the corresponding infrastructure. Additionally, in the planning process additional emphasis has been put on fostering community sense and better accessibility to community services, like health infrastructure (Byler et al., 2015). This also has led to the use of more urban planning principles for the design of refugee camps since there is a strong correlation between camp design and the health of the community (Stevenson and Sutton, 2012). The limited resources of humanitarian organizations and camp planners, the growing populations of refugee camps as well as the uncertain environments of refugee movements require efficient camp design. Facility location models can be a helpful starting point in the decision process. In the context of refugee camps, two recent examples are Smadi et al. (2018) and Karsu et al. (2019). While the first used a facility location model for the location decision of water resources, the latter developed a general facility location model for camp design.
Facility location models for EMS usually place a number of server locations strategically over a study area to provide medical servers to the population. The very early contributions to the location problem of emergency medical vehicles ignored changing inputs or other inherent dynamics of such systems to reduce modeling and computational effort. Some of the most relevant work in this field was stated by Toregas et al. (1971) with the location set covering problem and Church and ReVelle (1974) with the maximal coverage location problem. Daskin and Stern (1981) introduced backup coverages in a hierarchical objective set covering problem, Hogan and ReVelle (1986) developed models to maximize the backup capabilities of emergency vehicles and Gendreau et al. (1997) proposed the double standard model. The HQM was originally debuted by Larson (1974). It is a Markovian state-based probabilistic model that can be used to analyze the performance of an already pre-determined set of server locations to gain insights into performance and behavior of the corresponding system. Moreover, it can be applied to calculate system-specific performance metrics, like the average expected response time, the loss percentages of incoming demand calls or server workloads. To deal with the high computational efforts involved, Larson (1975) later developed the approximate HQM. Atkinson et al. (2006), Budge et al. (2009) and Ansari et al. (2017) also developed approximation techniques to reduce computational burdens. Recent extensions of the basic HQM include the modeling of real-world waiting lines in order to preserve calls that cannot be served immediately (Iannoni et al., 2015; De Souza et al., 2015; Rodrigues et al., 2017). Due to its descriptive nature, the basic HQM, but also its extensions, cannot be used to determine server locations on its own. Therefore, a pre-determined set of server locations that can be subsequently analyzed is necessary (Galvão and Morabito, 2008). The performance metrics that can be calculated with the help of the HQM can be used in an optimization process as noted by Goldberg (2004) and Takeda et al. (2007). Iannoni et al. (2008) and Iannoni et al. (2009) used the HQM in an EMS location study on highways with different call types. Geroliminis et al. (2009) developed the spatial queuing model (SQM) which uses HQM-based metrics in the optimization process. Their model minimizes the average expected response time of the system while simultaneously locating servers and determining areas of responsibility for each server. The SQM differentiates from the basic HQM by automatically determining the areas of responsibility for each server. Geroliminis et al. (2011) extended the SQM by incorporating larger number of servers through reducing the state space. Toro-Diaz et al. (2013) and Enayati et al. (2019) discussed the use of different performance metrics in the optimization process while modeling the dispatch policy as a decision variable. Akdoğan et al. (2018) developed several call-specific formulations of service rates in a SQM-based location study.
None of the mentioned SQM studies combines optimization procedures and subsequent simulation studies of additional incoming demands for emergency help, like virus outbreaks. Theoretically, it would be possible to include the outbreaks into the optimization process with, for example, stochastic programming principles. The assumption here is that the virus outbreaks are a temporary factor that should not play a role in the strategic facility location decision. To account for the outbreaks, the approach taken in this paper simulates them subsequently to the location decision. In order to deal with the increased risk of congestion, additional servers supporting the EMS in the treatment of virus patients are introduced. For the present study, an exact HQM is used to derive performance metrics as precise as possible. The contribution of this study can then be summarized as follows:
Application of the SQM for the real-world case of a refugee camp
Randomized simulation of virus outbreaks in the refugee camp
Analysis of system behavior during virus outbreaks
3. Methodological approach
A number of J demand points is located in the study area. In order to serve incoming demands for emergency help, a number of N servers need to be located within the study area. All servers are assumed to be identical. Furthermore, it is assumed that not every server can be dispatched to every demand point, which is called partial backup. The level to which downstream servers can be dispatched to demand calls is called districting. So, for example, if third-level districting is considered, the three nearest servers for every demand point can be dispatched in the case of an incident. As noted by Geroliminis et al. (2009) and Enayati et al. (2019) dispatching more distant servers can lead to sub-optimal performance in some key metrics, like the average expected response time of the system. However, dispatching more distant servers also can lead to lower loss percentages of incoming demands. To cope with this trade-off, a level of districting, D, has to be defined, which is considered a strategic decision.
The HQM is a binary-coded model, in which each server only has two states, available or unavailable for incoming demands for emergency calls. For a five-server system in which the second and fourth servers are unavailable, the state description can be denoted as {0, 1, 0, 1, 0}. As can be seen state specification, the total number of states is dependent on the number of servers N and can be calculated by
The demand at each demand point is assumed to be independent of the demand at other demand points and not per definition identical but known over the study period. Each demand point is assigned the D-nearest servers and the nearest-available server is always dispatched. The assignment of servers forms a preference list for the particular demand point. Whenever a demand call for emergency help enters the system, the preference list is checked and the nearest-available server is dispatched. After the server has completed its service, it returns to its base location and is available for future incoming demands. If no server on the preference list is available, the incoming demand call is assumed to be lost to the system. In reality this call would be served later or by another EMS.
3.1 Formulation of upward and downward transition rates
The upward and downward transition rates are the key inputs into the SQM and describe the potential congestion of the analyzed system. The calculation of each transition rate is done with respect to the current state of the system and the preference lists of the demand points. For the construction of the latter an ordered vector
The upward transition rate
Equation (3) denotes that, given the state of the system a, server k responds to any demand of its primary area of responsibility
Generally, the rules of districting are also relevant for the calculation of the downward transition rates. There are several different approaches in the academic literature to calculate the downward transition rate. Most of them are formulated without regard to the origin of the demand call. Iannoni et al. (2008), Iannoni et al. (2009), Geroliminis et al. (2009), Geroliminis et al. (2011), Toro-Diaz et al. (2013) and Enayati et al. (2019) used fixed values for the service that is provided by the system. Akdoğan et al. (2018) debuted a formulation of the service rate that incorporates the origin of the call for emergency help as well as the time that is spent at the scene of the incident. The service rates are slightly altered for the approach used in this paper: Since emergency vehicles in refugee camps are oftentimes not stationed at a hospital or other medical facilities, the travel time from the server location to the demand point, from the demand point to the hospital and, if valid, from the hospital to the server location are calculated and used separately for the service rate.
When the dispatch of a single emergency vehicle to a demand point is considered, the state descriptions between state a and state b differ at only one position. For example, if state a is described as {1, 0, 1, 0, 1} and state b as {1, 0, 0, 0, 1}, they differ at the third position which means that the third server is unavailable in state a, but available in state b. This server is denoted by k with
This formulation describes the maximum deployments per time period T. The denominator consists of the time spent at the incident as well as the relevant travel times of server k to the demand point, to the hospital and to its original location. It can then be used to calculate the overall downward transition rate between state a and state b based on Akdoğan et al. (2018). It represents the weighted sum of the incident handling rates of all demand points j (
3.2 Model formulation
The notation used for the model is given in Table 1.
The objective function (OF) minimizes the average expected response time of the system and can be formulated as follows:
Subject to
The term Hamming-distance refers to the difference between the state descriptions of two states. Applied to the HQM, the definition of the Hamming-distance controls whether one-step or multiple-step transitions are allowed. A multiple-step transition would occur if multiple vehicles are dispatched to one demand point.
The core structure of the model is defined by the constraints (6)–(8). Constraint (6) ensures that
Constraint (9) constructs the linear equations of the equation system that needs to be solved to derive the probabilities of each state. Each equation specifies the flows of one state for the N-dimensional hypercube. Constraint (10) ensures that the sum of all probabilities is equal to one.
Constraint (11) calculates the fraction of dispatches server n sends to demand point j. Therefore, it sums up all state probabilities in which server n is the nearest-available for demand point j. Since the loss probabilities of incoming demand calls are usually greater than zero, the denominator normalizes the sum of the fraction of dispatches per demand point j to “one” which is also ensured by constraint (12). The loss probabilities that are necessary for the normalization per demand point j are calculated by constraint (13) by summing up all probabilities in which no server on the preference list is available for demand point j.
3.3 Performance metrics
The application of the HQM allows for the calculation of (server-specific) performance metrics. The degree to which the first server is available, the longest response time to a demand point as well as the overall calculation of the loss percentages are used as additional performance metrics here.
The average degree to which the first server of the preference lists is available to serve demands can be calculated by equation (14):
The longest response time to a demand point can be calculated by equation (15):
The overall loss percentages of the system can be calculated by equation (16):
To gain further insights into the behavior of the EMS in the case of a virus outbreak, several other performance metrics are calculated and used. First, the Gini-Index is used to capture the impact of the outbreaks on the overall medical care as a measure of equity. Usually, it is used to measure the amount by which the value of an individual differs when compared to a scenario in which all individuals receive the same value (Toro-Díaz et al., 2013; Enayati et al., 2019). Applied to the virus outbreak simulations, two different versions of the Gini-Index are introduced to capture the difference that non-virus-related patients experience in (1) the loss percentages of incoming demands and (2) their expected average response time when compared to the base scenario. The Gini-Index for both versions can be calculated by the following formulas (equations 17 and 18):
As additional performance metrics, the loss percentages and the server workloads are analyzed. The loss percentages are calculated by equation (13) and the server workloads can be calculated by equation (19):
3.4 Genetic algorithm
The OF presented in section 3 has no closed-form expression. Therefore, a solution technique has to be applied to solve the applied model. So far the genetic algorithm (GA) (Iannoni et al., 2008, 2009; Geroliminis et al., 2011; Toro-Díaz et al., 2013; Enayati et al., 2019) and the steepest-descent method (Geroliminis et al., 2009) have been used in the location optimization process for the exact HQM. The first is a population-based metaheuristic that mimics evolutionary behavior through gradually eliminating inferior solution components from the population. The second uses a search process that is focused on the direction in which the OF decreases faster. For the approach presented in this paper, the GA is applied. It can be adapted to the respective problem by calibrating certain parameters like the population size, the crossover point as well as the crossover and mutation probabilities (Akdoğan et al., 2018). The utilization of a population ensures genetic diversity and helps to avoid local minima. The GA uses a chromosome structure to represent the different decision variables of the model. In order to evolve toward better solutions, a fitness function, in this case the value of the OF, is used to compare the different individuals of each population per generation. The main components relevant for the application within the SQM are described hereafter:
3.4.1 Initial population
At the start of the GA, a pre-defined number of individuals, the population size, is randomly initialized with N (the number of servers) chromosomes as server locations.
3.4.2 Calculation of reproduction probabilities
After each individual of the current generation
Due to the minimizing character of the OF, smaller fitness values are considered superior. Afterward, individuals are selected in pairs according to their probability to create the next generation.
3.4.3 Crossing over
Crossing over is used to transfer genes from the parents to the offspring. Here, it is assumed that two parents produce two offspring. A random crossover point
Afterward, the offspring are added to the solution pool.
3.4.4 Mutation procedures
After the crossover procedures, each chromosome of each individual is subject to mutation to create larger genetic diversity and to avoid local minima. The rule of thumb for the mutation probability is to choose values between 1/(number of chromosomes) and 1/(population size) (Geroliminis et al., 2011).
3.4.5 Solution pool and reproduction conditions
After the crossover and mutation procedures, the model is performed for each individual candidate solution. Afterward, the solutions are compared to each other and only the superior individuals are selected to start the reproduction process for the next generation.
3.4.6 Termination criteria
The GA is terminated after ten iterations in which the best solution of the current iteration has not improved. Afterward, the best solution found is returned.
4. Pandemic simulations and computational results
In this section, the procedure for the pandemic simulations as well as the computational results for the general optimization process for emergency medical vehicles and, subsequently, for the virus outbreak simulations are described. All computations are performed on an i7-8650U.
Za'atari is a refugee camp close to Jordan´s northern border to Syria. Since its establishment in 2012, it has grown from a small cluster of tents to an urban-like settlement with about 76,000 inhabitants. The camp is operated under the joint administration of the Syrian Refugee Camp Directorate and the UNHCR. Due to its size, the camp has most of the common infrastructure that could be found in other like-sized towns (Schön et al., 2018). Four clinics are situated in the camp territory that provide 24/7 health services (UNHCR, 2020). To account for the spatial distribution of incoming demands for emergency helps as well as for the potential origins of pandemic outbreaks, the Za'atari refugee camp was modeled on a virtual 100 × 100 grid. The population density in the camp as well as the population in the individual districts was reconstructed using maps provided by UNICEF (2015) and the UNHCR (2017). The overall layout of the camp with respect to the population densities is presented in Figure 1. Two-hundred demand points were distributed according to the population in the respective districts. A total of thirty potential candidate server locations were placed in the study area with all four clinics as potential locations. The remaining locations were generated by a random procedure. Please note that servers do not necessarily have to be located at a clinic which may lead to the location of a server elsewhere in the study area.
The HQM, the GA as well as the virus simulations were coded in C++. The locations of the demand points as well as the server locations were used as input variables into the model. The linear equation system necessary for the HQM was solved by using the Eigen library. Section 4.1 contains the description of the virus outbreak simulation procedure. In section 4.2 the overall EMS is optimized and subsequently analyzed. The virus simulations are performed in section 4.3 which also includes the incorporation of two additional servers to cope more adequately with the congestion caused by the virus outbreaks.
4.1 Virus outbreak simulations
To simulate the impact of COVID-19 in the Za'atari refugee camp, random outbreaks in the camp territory are simulated. Due to the high population density, it is assumed that the people affected by the outbreaks need to be isolated and treated separately from their usual homes to prevent further spreading of the virus. Therefore, the outbreaks consist of patients that need immediate medical treatment as well as patients that need to be quarantined. In order to account for the additional demand for emergency medical services, the demand for virus-related dispatches of emergency vehicles is assumed to add to the usual demand and has to be served as soon as possible by the existing medical infrastructure in the camp.
The simulation of cases per outbreak is randomized and follows a normal distribution with varying mean values and a fixed SD of 2. Moreover, it is assumed that more than one outbreak can happen simultaneously. If there is more than one outbreak considered per simulation, the individual outbreaks are then assumed to happen independently from each other as a simplification and for the ease of calculations. The number of outbreaks per simulation also follows a normal distribution with varying mean values and a fixed SD of 0.75. The different mean values for the cases per outbreak and the number of outbreaks per simulation are given in section 4.3. The locations of the cases per outbreak are distributed around the primary origin of the outbreak. This is done to account for spatial hotspots of virus outbreaks. Additionally, it is assumed that the outbreaks are more likely to occur in the parts of the camp with a higher population density due to the facilitated spreading of the virus. As mentioned, all of the incoming pandemic-related demand for emergency help has to be served as soon as possible. All assumptions for the calculation of upward and downward transition rates, as described in section 3.1, are also assumed for the pandemic-related transition rates.
4.2 Optimization and analysis of the EMS
For the following optimization, the population size of the GA is set to 50 and the crossover and mutation probabilities to 0.8 and 0.1, respectively. In order to gain basic insights into the behavior of the proposed EMS, the optimization is performed without the virus outbreaks. A five-server system is considered to reconstruct the medical infrastructure of Za'atari refugee camp. Note that common coverage restrictions to the EMS do not apply fully in the case of refugee camps because of the high population density and overall small size of the camp. Therefore, it is necessary to analyze more performance metrics than the average response time. For this analysis, the percentage to which the first server of the preference list for each demand zone is available, the longest expected response time to a demand zone as well as the loss percentages of the system are introduced as performance metrics and calculated as described in section 3.3. The optimization procedure is performed with
As can be seen from Table 2, all derived performance metrics decrease if the overall demand increases. Due to the normalized calculations for the OF and for all response time-based metrics, the highest and average expected response time only increase moderately. The degree to which the first server of the preference list is available (FSA metric) does decrease significantly for higher λ´s. The expected loss probabilities increase for higher demands. Since there is no modeling of real-world waiting lines, it is important to note that the loss probability only represents the degree to which incoming demands cannot be served immediately. The derived performance metrics for the proposed system show appropriate values for at least λ
4.3 Results for the virus outbreak simulations
For the outbreak simulations, the described simulation process of section 4.1 is used. The simulations are performed with a mean number of cases per outbreak of 1, 5 and 10 and with means of individual outbreaks per simulation of 1, 2 and 3 as well as with all combinations of the parameters. As a result, nine different combinations for the simulations are considered. Each simulation is repeated 2,500 times to yield a sufficient amount of data. The demand for non-virus-related emergency demands is set to two.
The cases per outbreak and the outbreaks per simulation both follow a normal distribution. Figure 2 shows the aggregated behavior of the overall cases with regard to both parameters. “C” denotes the mean values of case per outbreak while “O” denotes the mean value of outbreaks per simulation. So, for example, C5.0-O1.0 describes the parameter combination of a mean of 5 cases per outbreak and a mean of 1 outbreak per simulation.
Figure 2 shows peaks in the case distribution at the mean values of the outbreaks per simulation. For all parameter combinations, Figure 2 shows a peak at the amount of cases per simulation which corresponds to the number of cases per simulation times the number of outbreaks per simulation. Moreover, it can be observed that the overall curve flattens for higher mean values of cases and outbreaks per simulation, see C5.0-O3.0, C10.0-C2.0 and C10.0-O3.0, but still has (multiple) peaks at the respective mean values of the number of cases per outbreak. Higher values of O, in combination with C defined as a positive integer, lead to a more even distribution of the absolute amount of cases per simulation. Since an extended discussion and presentation of nine parameter combinations would be confusing, three parameter combinations, C1.0-O3.0, C5.0-O2.0, C10.0-O1.0, were chosen randomly for further discussion. The values for the median as well as the upper whisker for the remaining parameter combinations can be found in Table 3, the remaining boxplots in Figure A1. The performance metrics derived by the simulations are illustrated in the boxplots in Figure 3.
The boxplot of the “Gini Type a” and “Gini Type b” performance metrics shows that the non-virus-related patients receive significantly less value owing to the additional consideration of virus-related demand in the EMS. From the “Gini Type a” metric, it can be seen that the Gini-Index takes the highest median value (0.1530) for the C5.0-O2.0 parameter combination, if compared to C1.0-O3.0 (0.1137) and even in comparison to C10.0-O1.0 (0.1484). This can be explained by the additional occupation of the EMS capacity if more than one outbreak happens at once, the necessity of responding to more than one virus outbreak and the higher average case number as seen in Figure 2. From a virus containment viewpoint, this finding accentuates the need to contain virus hotspots quickly since in reality individual outbreaks are probably correlated to the overall virus spread. From an EMS management viewpoint, this shows that rapid containment and a resulting lower number of outbreaks can lead to a less constrained EMS, even if the initial case number is high. This can also be seen by the other boxplots provided in Figure A1. The outliers can be explained by the probability distribution of outbreaks leading to a very high loss percentage if there are high case numbers and many outbreaks at the same time. The “Gini Type b” metric shows an increase in average response times for non-virus-related patients corresponding to higher mean values of cases per outbreak. If a high number of cases during a singular virus outbreak is considered, the responsible emergency vehicles for this area are able to respond to normal emergency demands only to a relatively low percentage. As a result, more distant emergency vehicles are dispatched and the overall response time increases. Due to the more even spread of incoming demands over multiple servers, multiple outbreaks with lower case numbers at once lead to lower response times in comparison with a singular outbreak with the same case number. For the three randomly picked parameter combinations, the median of “Gini Type b” increases from 0.8939 (C1.0-O3.0), to 1.0561 (C5.0-O2.0) and 1.1405 (C10.0-O1.0). If the other parameter combinations are considered, it can be seen that “Gini Type b” metric increases significantly for a higher number of outbreaks per simulation, for all mean values of cases per outbreak. The “loss percentages” show a relationship between overall case number and outbreak that is similar to the “Gini Type b” metric. Their median for the described three parameter combinations increases from 0.1418 (C1.0-O3.0), to 0.1968 (C5.0-O2.0) and 0.2016 (C10.0-O1.0). In combination with the described effects for the “Gini Type a” metric, the overall loss percentages do increase with higher mean values of cases per outbreak. Moreover, the analysis of all parameter combinations further illustrates the necessity to limit the outbreaks in the camp to secure sufficient emergency capacities for non-virus-related patients and, more relevantly, to avoid unnoticed spreads of outbreaks which ultimately could lead to higher case numbers per outbreak. This can also be seen by the comparison of, for example, C1.0-C3.0 and C5.0-O1.0 “Gini Type a” and “Loss Percentages” values in Table 3 and the plots in Figure A1. The boxplots of the server workloads show that the virus outbreaks are mostly handled by the servers 2, 3 and 4 which are located close to areas with a high population density. Especially server 5 is available to a relatively high degree. Note that this server location cannot be considered to be inferior because of its responsibility to provide medical coverage for the camp. The median of the workloads for servers 2 to 4 increases from about up to 70% (C1.0-O3.0), to up to 80% (C5.0-O2.0) and to about 80% (C10.0-O1.0). Interestingly, if C5.0-O2.0 is compared to C10.0-O1.0, it can be seen that the boxes of the server workloads are about the same, but with significantly shorter whiskers in the boxplot for the C10.0-O1.0 case. This can be explained by the higher average number of outbreaks for C5.0-O2.0. Server 2 has the highest expected workloads for all considered parameter combinations. For all parameter combinations, the high number of outliers can be explained by low numbers of virus outbreaks per simulation and therefore low server workloads due to only mild effects on the overall behavior of the EMS.
One strategy to increase the performance of the EMS could be to consider additional servers that support the existing system in the treatment of virus patients. The two additional servers are located at the upper left of the study area for spatial proximity to the area of high population density in the corresponding district of Za'atari refugee camp. The locations of the original five servers (black symbols) and the two additional servers (red symbols) can be seen in Figure 4 below.
For easier comparison, the same parameter combinations as in the five-server case are chosen for detailed analysis. The values for the median and the upper whisker can be found in Table 3. The boxplots for the remaining parameter combinations can be found in Figure A2 (see Figure 5).
The two additional servers lead to significantly lower values in the “Gini Type a” metric for the seven-server case. This can be stated for all parameter combinations (see Table 3). The described relationships for the five-server case remain valid which can be seen from the comparison of C5.0-O2.0 (0.1072) and C10.0-O1.0 (0.0987). Moreover, the values of the median for the “Gini Type b” metric 0.35 (C1.0-O3.0), 0.3635 (C5.0-O2.0), 0.3274 (C10.0-O1.0) heavily decrease, which can be explained by the calculation of the average response time of the OF that is also used to calculate the response time in the case of a virus outbreak. If compared to the values of the five-server-case, it can be seen that the median values for C10.0-O1.0 parameter combination are lower than for C5.0-O2.0, which can be explained by more singular outbreaks being served by the additional servers introduced. Due to the proximity of both additional servers to areas of the camp with high outbreak probabilities, the response time to those incidents is very low and the overall response time of the system decreases significantly because of their high proportion of the overall demand. This can be shown by the comparison of the median and size of the box of C5.0-O2.0 and C10.0-O1.0, which are significantly more alike than in the five-server case. Additionally, the boxes and whiskers of the “Gini Type b” metric for all parameter combinations do not increase similarly, but stay more even if compared to the five-server case. Interestingly, the “Loss Percentages” metric, in comparison to the other metrics, does not decrease in a similar manner but even increases for certain parameter combinations. This effect can be explained by the difference in the preference lists compared to the normal situation if additional servers are considered. So, if, for example, an outbreak happens at a more remote place of the camp, it is unlikely that one of the two additional servers will be dispatched. If an outbreak happens in an area of high population density, it will be handled by servers that are already handling common emergencies and other virus-related demands. This argument is strengthened by the lower medians, boxes and whiskers of, for example, C5.0-O3.0 and C10.0-O3.0 in comparison to the five-server case. For those parameter combinations, it is more likely that outbreaks will occur at different parts of the camp, which will then benefit more from the use of additional servers. Additionally, the still existing possibility of system congestion in larger outbreaks as well as the stochastic nature of their location and quantity further add to this effect, which could easily be bypassed by extending the preference lists for all demands to account for the additional servers in future studies. If all “Loss Percentages” boxplots of Figure A1 are compared to Figure A2, it can be seen that the whiskers and the boxes are smaller for almost all parameter combinations, but with only minor differences regarding high mean values of cases and outbreaks. Applied to the management of virus outbreaks, this means that large case numbers in multiple areas of the camp are still hard to contain and therefore have to be avoided, even if additional servers are considered.
If the workloads of the servers are considered, it can be stated that the boxes, whiskers and medians over all parameter combinations show lower values for the already existing five servers of the system. This also explains the lower values of the “Gini Type a” metric since the existing servers are more often able to respond to normal emergency demands. Especially the first of the two new servers has high workloads, while the second new server has slightly lower workloads over all parameter combinations. It is also worth noting that the general distribution of workloads for the first five servers is lower but with mostly the same relations as before. Over all parameter combinations, the rather large boxes and whiskers of the boxplots lead to the impression that the availability and ability to respond to incidents is highly dependent on the overall number of outbreaks and case numbers, even if additional servers are used. Therefore, and also in combination with the results for the “Loss Percentages”, it can be stated that effective and quick management of virus outbreaks still remains relevant but with a slightly larger cushion. From a EMS planning viewpoint, the location of servers that help in the containment and treatment of virus patients has rather large benefits for the whole camp population. As a managerial implication, the number one goal for the containment of virus outbreaks in refugee camps should be the fast containment and contract tracing of infected persons. This strategy will benefit both common emergency and virus-related demands.
5. Concluding remarks
The present study optimizes an EMS in a refugee camp. Due to the threat of potential virus outbreaks, a subsequent simulation study was carried out to analyze the congestion risk and influence of the necessary isolation and treatment of virus patients on the overall EMS, as well as on the normal emergency patients. For the simulation of different intensities of virus outbreaks, several simulations with different parameters are conducted to gain insights on the EMS.
The combination of optimization and subsequent simulation studies allows the decision-makers to (1) derive server locations that can deal appropriately with common demands for emergency help and (2) to simulate potential virus outbreaks and their effect on the EMS. Thus, it can contribute to the goal of public health and containment of virus outbreaks. Furthermore, the embedded queuing model can be used to analyze existing EMS and to support the location optimization decision. The location of the additional servers can be easily adjusted and the simulation approach can be helpful in the analysis of potential additional server locations. The study showed that the proposed system can handle all existing demands for emergency help to a sufficient degree. The derived performance metrics show low loss percentages and response times as well as high coverage values. Moreover, the simulation study showed that the system can be congested to a rather large degree by even modest virus outbreaks that need to be isolated and treated. The incorporation of different equity measures illustrated the differences in value non-virus-related patients receive. In order to mitigate those effects, additional servers were introduced that support the existing system in the treatment of virus patients. Thereby, the performance metrics were significantly improved and the workload of the existing servers could be lowered, which proved to be beneficiary for the common demands for emergency help.
Nevertheless, the present study has some drawbacks. Since there is no incorporation of waiting lines, all demands that cannot be served immediately are assumed to be lost. In a real-world context, these demands would be served by another EMS or later during the study period. Therefore, following studies could include several subsequent time periods. As stated by Geroliminis et al. (2011), the HQM-based approach requires high computational efforts and therefore large-scale study areas with more than 15 servers could be prohibitive due to the computational burdens. Since the simulation study also builds on the proposed SQM, a high number of simulations for a large-scale study area could also lead to high computational times. Existing approaches to lower the computational efforts mostly concentrate on the approximation of the embedded queuing model and therefore could potentially compromise its expressiveness. As Galvão and Morabito (2008) mentioned, the construction effort of the equations for the linear equation system can exceed its solving time. Therefore, future approaches could focus on more efficient procedures to derive the coefficients for the equation system. Moreover, some of the parameter values that were chosen for both the optimization and subsequent simulation study require further validation. This should be done in future studies in close corporation with the relevant EMS authorities that are also responsible for the design and operation of the system. From a practitioners perspective, the application of the proposed HQM requires at least some understanding in the fields of queuing and optimization theory. Therefore, the real-world application could be limited. In this context, the case study should be understood as the demonstration of a viable solution that allows for further insights into the congestion of an EMS in a refugee camp in the case of a virus outbreak.
Figures
Model notation
| Sets | |
| Set of candidate server locations | |
| Set of demand points | |
| Set of servers to be located | |
| Set of servers covering demand point j | |
| Set of states in which server n is the nearest-available for demand point j | |
| Set of states in which no server is available for demand point j | |
| Variables and HQM notations | |
| Binary variable, that shows whether demand point j is covered | |
| Vertices of the N-dimensional hypercube | |
| Fraction of demand point j on the overall demand | |
| Fraction of dispatches server n sends to demand point j | |
| Travel time of server n to demand point j | |
| Upward transition rate between state a and state b | |
| Downward transition rate between state a and state b | |
| State a and state b of the N-dimensional hypercube | |
| Loss percentages of incoming demand calls for demand point j | |
| Upward and downward Hamming-distances between states a and b | |
| Decision variable | |
| Binary variable, that shows whether candidate server location i is chosen | |
Performance metrics
| FSA in percent | Highest expected response time in minutes | Average expected response time in minutes | Average expected loss probability in percent | |
|---|---|---|---|---|
| λ = 1 | 82.17 | 6.66 | 4.99 | 0.063 |
| λ = 2 | 71.60 | 6.78 | 5.26 | 0.562 |
| λ = 3 | 63.79 | 6.89 | 5.48 | 1,728 |
| λ = 4 | 57.53 | 6.99 | 5.65 | 3,536 |
| λ = 5 | 52.32 | 7.08 | 5.80 | 5,851 |
Median and upper whisker for 5 and 7 server
| 5 Server | 7 Server | ||||
|---|---|---|---|---|---|
| Median | Upper Whisker | Median | Upper Whisker | ||
| C1.0-O1.0 | Gini Type a | 0.0359 | 0.1199 | 0.0203 | 0.06698 |
| Gini Type b | 0.5476 | 1.2243 | 0.1552 | 0.5641 | |
| Loss percentages | 0.0493 | 0.1591 | 0.0369 | 0.1312 | |
| C1.0-O2.0 | Gini Type a | 0.0659 | 0.2044 | 0.03591 | 0.1245 |
| Gini Type b | 0.6973 | 1.3291 | 0.2336 | 0.9122 | |
| Loss percentages | 0.08349 | 0.2492 | 0.0628 | 0.218 | |
| C1.0-O3.0 | Gini Type a | 0.1137 | 0.3068 | 0.0699 | 0.1914 |
| Gini Type b | 0.8939 | 1.3918 | 0.35 | 1.0715 | |
| Loss percentages | 0.1418 | 0.3611 | 0.1135 | 0.3183 | |
| C5.0-O1.0 | Gini Type a | 0.0850 | 0.2345 | 0.0506 | 0.1672 |
| Gini Type b | 0.8732 | 1.429 | 0.2466 | 1.0415 | |
| Loss percentages | 0.1144 | 0.3143 | 0.1116 | 0.3355 | |
| C5.0-O2.0 | Gini Type a | 0.1530 | 0.4855 | 0.1072 | 0.3126 |
| Gini Type b | 1.0561 | 1.6328 | 0.3635 | 1.4658 | |
| Loss percentages | 0.1968 | 0.5726 | 0.2051 | 0.5371 | |
| C5.0-O3.0 | Gini Type a | 0.2761 | 0.6004 | 0.1832 | 0.4362 |
| Gini Type b | 1.2237 | 1.553 | 0.5685 | 1.6078 | |
| Loss percentages | 0.3413 | 0.6139 | 0.3115 | 0.6065 | |
| C10.0-O1.0 | Gini Type a | 0.1484 | 0.4087 | 0.0987 | 0.3129 |
| Gini Type b | 1.1405 | 1.6005 | 0.3274 | 1.4395 | |
| Loss percentages | 0.2016 | 0.5581 | 0.2449 | 0.6255 | |
| C10.0-O2.0 | Gini Type a | 0.2953 | 0.7059 | 0.2037 | 0.5911 |
| Gini Type b | 1.3197 | 1.7008 | 0.4996 | 1.8256 | |
| Loss percentages | 0.4036 | 0.7382 | 0.4056 | 0.7063 | |
| C10.0-O3.0 | Gini Type a | 0.4453 | 0.7694 | 0.3266 | 0.6616 |
| Gini Type b | 1.4263 | 1.6905 | 0.7174 | 1.806 | |
| Loss percentages | 0.5672 | 0.7806 | 0.5336 | 0.7882 | |
Appendix
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Acknowledgements
The author wants to thank the two anonymous reviewers for their comments that helped to greatly improve the paper.