Faculty of Technology and Society Computer Science and Media Technology

An optimization model for the placement of psychiatric emergency units

The case of Region Skåne

Albin Médoc Daniel Subasic

Degree: Bachelor Degree 180 credits Supervisor: Johan Holmgren Subject: Computer Science Examiner: Saeid Amouzad Mahdiraji Program: Information Architecture Final Seminar Date: 01-06-2021 Sammanfattning

Psykisk sjukdom är ett stort problem i dagens samhälle och många individer har upplevt psykiska hälsoproblem. Allvarlig psykisk sjukdom, såsom schizofreni, bipolär sjukdom och allvarlig depression, är också relativt vanligt, och psykisk sjukdom är en av de främsta anledningarna till att en person bestämmer sig för att ta sitt eget liv. För individer som har sådana destruktiva tankar är snabb tillgång till vård och korrekt utvärdering och behandling avgörande. Därför har värdet av att inskaffa en specialambulans med fokus på psykiatrivård identifierats. För att utnyttja dessa specialambulanser till dess fulla potential är det dock viktigt att de placeras optimalt. Vi föreslår en optimeringsmodell som syftar till att identifiera optimala placeringar för psykiatriska akutenheter i en viss geografisk region. Ett samarbete med Region Skåne tillät oss att använda riktig data, och på så vis utföra en scenariostudie för att utvärdera optimeringsmodellen. I vår scenariostudie använde vi vår modell för att identifiera de optimala placeringarna av en, två och tre ambulanser utifrån befolkning respektive risksannolikhet. Resultaten från vår scenariostudie visar att den optimala platsen för ett visst område kan variera beroende på vilket perspektiv som väljs. Det är därför viktigt att ha tydliga och väl genomtänkta mål för placering av specialambulanser.

Abstract

Mental illness is a major problem in today's society and many individuals have experienced a mental health problem. Severe mental illness, such as schizophrenia, bipolar disorder, and major depression, is also relatively common, and mental illness can even lead to a person taking their own life. For individuals who have such destructive thoughts, quick access to care and proper evaluation and treatment is crucial. Therefore, the value of acquiring a special ambulance with a focus on psychiatric care has been identified. However, to utilize these special ambulances to their full potential, it is important that they are placed at optimal locations. We propose an optimization model that aims to identify optimal locations for psychiatric emergency units in a specific geographical region. A collaboration with Region Skåne allowed us to use real data, and thus perform a scenario study to evaluate the optimization model. In our scenario study, we used our model to identify the optimal placements of one, two, and three psychiatric ambulances based on population and risk probability, respectively. The results from the scenario study show that the optimal location for a certain area can vary depending on which perspective is chosen. It is therefore important to have clear and well-thought-out goals for the placement of special ambulances.

Table of contents

1. Introduction 1 1.1 Purpose and objectives 3 1.2 Research Question 3 1.3 Related work 4 1.4 Scope 5 1.5 Thesis overview 5 2. Methodology 6 2.1 Design science 6 2.2 Method motivation and implementation 7 2.3 Optimization process 9 2.4 Evaluation method 10 2.5 Literature study 10 3. Data collection and processing 11 4. Optimal Psychiatric Emergency Unit Placement Model 14 5. Scenario study 16 5.1 Scenario description 16 5.1.1 Scenario cases 18 5.1.2 Solution generation 19 5.2 Conditions 19 5.3 Results 20 5.3.1 Placement of one PAP unit 20 5.3.2 Optimal placement of two PAP units 21 5.3.3 Optimal placement of three PAP units 23 5.4 Analysis 25 6. Discussion 28 6.1 Validity threats 30 7. Conclusions and future research 31 References 32 Appendix A (data) 34 Appendix B (literature study) 37

1. Introduction

Mental illness is a major problem in today's society. Mental health conditions cause 1 in 5 years lived with disability, leading to one trillion dollars being lost annually in the global economy [1]. Approximately 13% of Swedes from the age 10 and older have experienced suicidal thoughts [2]. If not treated, these extreme symptoms of mental illness will affect a person's life negatively and can even lead to them taking their own life. According to WHO, around 800 000 suicides are estimated to be committed around the world annually [3]. For individuals having such destructive thoughts, a quick access to care, including proper evaluation and treatment, is critical. Psychological conditions requiring emergency care are often highly time sensitive, and they require medical personnel that possess the knowledge and are properly trained to make the correct decisions for treatment of the patient. This includes intense and dangerous circumstances where the police need to be called in. Acute psychiatric calls are often dangerous for not only the individuals who are in need of psychiatric care but also for the people in their vicinity. The short-term consequences can be costly, ranging from property damage and injured people to cost in human life, without personnel with proper training in psychiatric treatment on-site. By giving individuals in need of care an environment created by mental health specialists on-site, it is typically possible to create a safer situation for all involved. The personnel on standard issue ambulances possess a basic level of knowledge of mental illnesses, but their knowledge is often insufficient for making treatment decisions, which impacts the quality of treatment [4]. Therefore, the value of acquiring mobile psychiatric emergency units with a focus on psychiatric care has been identified. A mobile psychiatric emergency unit is an ambulance that responds to SOS calls to improve the treatment of patients with psychiatric illnesses. A psychiatric emergency unit is operated by specialists in the psychiatric field to improve the on-site treatment of patients in need for psychiatric care. The specialists can give patients better care and assessment than ordinary ambulance staff, which is expected to decrease the time of the visit. To get the most out of these units, it is crucial to identify the optimal placement for them. In November 2019, the Region of Skåne in Sweden started to use a mobile psychiatric emergency unit which the region named PAP (Prehospital Acute Psychiatry). The PAP vehicle looks like a standard ambulance, but the stretcher has been replaced with chairs for a more chat-friendly environment. The municipalities within Skåne that are currently covered

1 by the PAP are Malmö, Svedala, Skurup, Vellinge, Trelleborg, Lund, Staffanstorp, Lomma, and Burlöv, as shown in Figure 1. The emergency calls that are assigned to the PAP include acute psychiatric states, such as psychosis and suicide situations, for people of all ages. In 2020, a total of 6623 emergency units were dispatched (see Appendix A) to help persons with acute psychiatric symptoms in the region of Skåne. Our work is conducted as part of the ongoing PAP-project by Region Skåne.

Figure 1: The Skåne regions, where the green area is covered by the PAP.

A similar unit, called PAM, is located in the region of Stockholm [5]. It has been active since 2017 and is taking 15-20 suicide prevention calls daily. Patients are taken care of by psychiatric personnel, who are capable of making the right treatment decisions, instead of police officers, which is typically the case for this type of patient. Observations show signs that the load at the acute psychiatric clinics have decreased but no data was found to prove it [5]. Another similar type of unit, called PPR, has been active in Västra Götaland since 2017 [5]. The PPR unit design separates it from the PAP and PAM as rather than being a standard issue ambulance, it is a passenger car with special equipment and blue lights. The study on mobile psychiatric emergency units is further justified by several, recently published, studies related to both emergency mental illness treatment and placement of ambulances, both regular ones and units that are dedicated to providing special expertise and tools for persons who need psychiatric, stroke and other specialized care. A study conducted in Sweden by Carlborg et al. [6] shows that public health and society in general would strongly benefit from having a dedicated psychiatric unit. While it might not be a direct economic benefit from the alternate interventions, the use of mobile psychiatric units makes

2 other resources such as the police and ambulances available for other missions. Furthermore, as a result of a positive impact on the patients' health, long term savings in human value and decreased societal costs are expected. Toderova et al. [4] conduct a study on conventional ambulance nurses’ own perceptions on judgement on their own competence regarding psychiatric mental illness. The results show that they considered their basic knowledge is not enough for assessing treatment options. The nurses that participated in the study suggest combining pre-hospital and psychiatric expertise in the psychiatric unit. Furthermore, Lindström et al. [7] conduct a study on patients that have been treated by the mobile psychiatric unit PAM (Psykiatrisk Akut Mobilitet) in Stockholm, showing that their experience of being more involved in their treatment was positive. The patient had a safe space created for them, in which they could take part in the decision making without being afraid of getting dismissed or judged.

1.1 Purpose and objectives

The purpose of this study is to provide a useful contribution to the mental health response field, which aims to improve the quality of treatment for patients in need of psychiatric care provided by mobile emergency units. The objective is to create an optimization model, which can be used to identify the optimal location of psychiatric emergency units in a particular region such that they are expected to be the most efficient and help the most people that are in acute need of mental health care.

1.2 Research Question

To fulfill the objective of our study, we have formulated the following research question, which we aim to answer in this work: ● RQ: What is an appropriate optimization model for identifying the optimal location of psychiatric emergency units in a geographical region?

3 1.3 Related work

The problem of how to appropriately place ambulances has been studied for decades, and there is plenty of published research on the topic. Models for placing and moving emergency units have been created with the purpose of improving both availability and quality, for example, maximizing survivability for stroke patients [8]-[10]. For example, Amouzad Mahdiraji et al. [8] estimate the expected time to treatment for scenarios where multiple mobile stroke units (MSU), which carry special equipment for treatment on-site, are placed in a region. Their results show that the MSU placements significantly influence what benefits can be expected. Work has been conducted on how the number of total available ambulances and their stations’ locations affect their ability to provide a timely response [11]. Studies on which models perform best for different objectives such as survivability, coverage maximization and more have also been published [12], [13]. “Recent work has shifted the objective from maximizing coverage to improving patient survivability. Our findings show that the maximum survivability objective performs better in both survivability and coverage metrics. Further, the results also support using the survivability objective for resource constrained ambulance operators.” - Zaffar et al. [13]. Bélanger et al. review and discuss modern modeling approaches in regard to ambulance fleet management, specifically vehicle location, reallocation and dispatchment [11]. The paper concludes that although there has been a considerable amount of research on allocation and reallocation, the unpredictability that arises needs to be studied further. Brotcorne et al. [14] study the evolution of ambulance allocation and reallocation models, noting that the early models were deterministic, ignoring factors suchf33 as unavailability of ambulances when dispatched. They observe that models have since evolved and have started to become more dynamic, and can be used to reallocate ambulances as they operate. Brotcorne et al. [14] also anticipate a growing interest in dynamic location models [16] although one drawback is that real-time data needs improved tools to handle it. In summary, there are several studies focusing on placement of special ambulances, but to the best of our knowledge, there exists no published work on the placement of mobile psychiatric units, which creates a research gap for us to explore. Starting with one psychiatric emergency unit, we will suggest a new model for finding the optimal placement of both one and multiple units in a specific region of interest.

4 1.4 Scope

The focus of this study is to construct a model for finding optimal placement of psychiatric emergency units in a geographical region. Due to limitations in data, a limited scope of factors to consider were defined when developing the optimization model, including estimated data. The model should, in its final form, be able to take real data as input. It seemed reasonable to look at the placement of not just one, but multiple psychiatric emergency vehicles from the standpoint of efficiency in the region of Skåne.

1.5 Thesis overview

In the current Section, we have presented the background of the problem, the purpose of the study and the research question. Section 2 presents the methods used and describes how they were applied in the study. The data collection and processing are described in Section 3, after which we present our optimization model for optimal ambulance placement in Section 4. Section 5 contains our scenario study, where the optimization model has been applied, with results and an analysis, followed by a discussion in Section 6. Finally, Section 7 presents the conclusions drawn from the study and some ideas for future work are communicated.

5 2. Methodology

The methods used in this study are design science with an optimization process as well as a literature study and a scenario study. For the process of conducting the study itself, the design science approach is used [17], [18]. A literature study is conducted to find grounded information and knowledge about the problem and methods that could be used to solve it. Furthermore, conducting a literature study strengthens the validity and reliability of the study in regard to the process and results. The scenario study is made to evaluate the optimization model.

2.1 Design science

In a design science process, information and knowledge are gathered to understand a problem, and IT-artifacts are built and evaluated for a specific purpose. The produced artifacts can be used by the study conductor to better understand the problem and solve it. Iterative construction and evaluation of the artifact and its utility improves the quality of the process, which means an increase of the quality of the artifact itself. Requirements can be set as inputs. Acceptance criterias can be set so the process of creating and evaluating the artifact iterates until it has the desired effectiveness. To understand the process of design science, Hevner et al. [18] propose 7 guidelines for researchers to follow. These guidelines are listed below and in section 2.2, where we explain how we have connected them to our study.

1) Design as an Artifact: The result must be an artifact that addresses the problem and is well-described in order to be implemented and applied in an appropriate environment. Artifacts can be methods, models, and software. They are rarely a final product, but artifacts define the problem and its solution, ideas and practices that can be used for implementation in a fitting domain. 2) Problem Relevance: The research conducted must have a level of relevance in the field. 3) Design Evaluation: The design evaluation must be based on rigorous evaluation methods, to ensure the quality and efficiency of the design artifact. 4) Research Contributions: The research must be a justifiable and verifiable contribution in the areas of the design artifact, design foundations, and/or design methodologies.

6 5) Research Rigor: Research based on the design science approach must rely on rigorous methods when it comes to the creation and evaluation of the design artifact. 6) Design as a Search Process: The process of finding an effective artifact means using available approaches to reach the desired goal while obeying the laws of the problem environment. 7) Communication of Research: Design science research must be presented in a way that it is understanded by both technology-oriented and management-oriented audiences.

2.2 Method motivation and implementation

The major focus of this thesis is to improve the quality of treatment for patients in need of psychiatric care provided by mobile emergency units. To accomplish this, an optimization model is created, which means a design process must be followed. As mentioned above, the objective of this study is that the model, being an artifact, is created and evaluated. Hence, we argue that we can use the design science approach. Below, we explicitly outline how our study connects to each of the seven guidelines proposed by Hevner et al. [18]

Design as an Artifact An artifact is created, which is an optimization model. Using available ambulance data, the model aims to identify the optimal locations of a set of mobile psychiatric emergency units, where the possible locations are a set of ambulance stations.

Problem Relevance The large interest and the number of previous related works in the ambulance placement field makes it evident that there is a level of relevance that justifies further work on the subject. In addition, related work convincingly suggests that psychiatric emergency units do increase the quality of care for the individuals that are in need for psychiatric emergency care. Optimizing the placement of psychiatric emergency units will increase the number of patients that have access to a psychiatric emergency unit as well. As mentioned in Section 3, there is no published work, to our knowledge, focusing on the placement of psychiatric emergency units. Additionally, there is a growing interest in such emergency units, with multiple units already operating in several parts of Sweden.

7 Design Evaluation In order to evaluate our artefact, a scenario study in the Skåne region was conducted. In the scenario study, the placement of one, two and three psychiatric units is considered, respectively.

Research Contributions The main contribution of our work is the optimization model for identifying the optimal psychiatric emergency unit locations. We demonstrate that it is possible to use aggregated data on ambulances, mental health cases, and population to create an artifact that can be used to identify the placement where psychiatric emergency units are expected to be most helpful. We also provide knowledge on where psychiatric emergency units can be optimally placed in Skåne.

Research Rigor Previous research on ambulance placement approaches using expected value optimization and estimation is used as the foundation of the artifact design in this study. As we explored how to solve the problem of optimal placement of psychiatric emergency units, we decided to use a mathematical approach where we, based on the data available to us, look at the distribution of cases in the region from our specific scenario and calculate the likelihood of cases on a 1x1-km2 level. There are certain limitations due to the data, which are discussed in Section 5.1.

Design as a Search Process In order to create the optimization model, we follow the optimization process approach presented by Lundgren et al. [19]. This process includes a number of model construction steps and elements of iterative evaluation and verification. Section 2.3 explains how the optimization process works.

Communication of Research The results are disseminated through this bachelor thesis report.

8 2.3 Optimization process

In order to build and evaluate our optimization model, we followed the optimization problem by Lundgren et al. [19], see also Figure 2. There are multiple steps involved and the flow of the process is not linear. As validation and evaluation is performed after each of the steps, it might be necessary to go back to previously conducted steps if needed [19]. The beginning of the process involves specifying the real-world problem and breaking it down into a simplified problem and a mathematical optimization model is formulated. The higher the level of simplification is, the solvability of the model increases but its realism is decreased. Finally, an optimization method is applied to solve the optimization model to create a solution for a specific set of input data. The final outcome is the result of the optimization process.

Figure 2: Optimization process flow-chart by Lundgren et al. [19]

Our optimization process starts with the literature study being conducted and data getting collected and processed from different sources. By identifying relevant inputs for the optimization model, such as ambulance stations, acute mental health cases, and defining maximum response times and coverage areas, we form an optimization model. The optimization model is run through an optimization method, which returns an output which, if accepted, is the result of the optimization process. Previous steps are redone when a later step is considered as invalid. For example, when data is insufficient or the method is faulty, a new, updated, optimization model needs to be created.

9 2.4 Evaluation method

In an optimization process, the different steps of the process must be evaluated. The purpose of the evaluation is that the given results should help solve the problem at hand. We evaluate our optimization model using a scenario study. The motivation for using a real-world scenario study as an evaluation method is the fact that real-world data and a real scenario was available. In the scenario study, the optimization model is applied in order to find optimal stations for one, two, and three psychiatric emergency units in the region of Skåne.

2.5 Literature study

The literature study conducted consists of two parts, which are gathering information on mental health in general, the current mental health situation in Sweden, psychiatric emergency units, and previous work on optimal placement of regular and special ambulances. To find articles and studies that are connected to our subject, we used several search engines, the main ones being Google Scholar and the MAU Libsearch, which is a search tool that has a powerful system for filtering searches. As seen in appendix B, which also shows our search terms and results, Google Scholar ended up giving us more material where MAU Libsearch often gave few or no results. In our literature study, we also included multiple articles that were personally recommended to us.

10 3. Data collection and processing

In this study, several types of data are used. Part of the data was received from Region Skåne and the rest was collected from other sources. The first type of data collected was geographical data, which includes latitude and longitude coordinates over ambulance stations, seen in Appendix A, as well as geographically distributed population on a square kilometer level. Each square includes a set of two-dimensional coordinates that defines the area it covers on the map, as well as its number of inhabitants. The population data was the most geographically detailed data available. From Region Skåne we received data about the number of psychiatric cases in Region Skåne, which includes the distribution of cases in municipalities over a year. The region of Skåne is divided into four ambulance districts. The responsibility of ambulance management is distributed between PreMedic AB, Samariten Ambulans AB and Region Skåne [20]. We assigned positional coordinates for each ambulance station. We received data on acute mental health cases by ESS codes from Region Skåne. That data includes the number of cases and shows the case distribution by time of day, municipality and priority codes. The data we used is from 2020 and can be found in Appendix A. Statistiska centralbyrån (SCB) offers downloadable data on their website [21], which is where the population data was retrieved from. Since the scenario in this study only concerns placement of PAP-units in the region of Skåne, the rest of the country’s data has been filtered away. The population data is from the year 2020. Each square kilometer stores information about what region it is located in, what municipality it belongs to, as well as its geographical location and population. Municipality distribution was also retrieved from SCB [22]. The municipalities store information regarding their municipality and geographical location. Municipalities not located in the region of Skåne have been filtered away here as well.

Processing A summarization was made from all the population square boxes within a municipality in order to retrieve its entire population. Using the data on the number of cases per municipality, we were able to calculate the percentage of cases for each municipality, using Equation (1). The total number of cases was calculated by summarizing the number of cases in all municipalities of Skåne. The case distribution is shown in Figure 3.

11 Case percentage municipality = Cases per municipality / Total number of cases. (1)

Figure 3: Calculated percentage of total number of cases for each municipality in the region

In the same manner as we calculated the case percentages for each of the municipalities, the population percentage for each of the considered kilometers covering the considered region was calculated using Equation (2). It should be noted that each of the km2 squares represents a percentage of the municipality’s population. Summarizing all squares inside a municipality returns the municipality´s population.

Population percentage in km2 square = Population in square/ Municipality population. (2)

12 Using the results from the previous calculations, we calculate a percentage that represents the probability for a case to appear in a km2 square. The calculation was made using Equation (3), and Figure 4 shows the results using a visual representation of the case probability density in the region of Skåne.

Case probability square = Case percentage municipality * population percentage square. (3)

Figure 4: Visual representation of case probability for each km2 square covering Region Skåne. Solid red squares have the highest populations. White squares areas with no data.

13 4. Optimal Psychiatric Emergency Unit Placement Model

In this section, we present our model for finding the optimal placement of a set of psychiatric emergency units, focusing on helping as many individuals as possible. First of all, it should be mentioned that we split the geographical region under consideration into a set of subregions, denoted R. We let A denote the set of all possible ambulance locations, and sj , for subregion j∈ R, denotes what to maximize coverage for (for example, population or risk probability for each of the subregions). It is assumed that population or risk probability data is available for each j. We let P denote the number of psychiatric emergency units we want to place in the region.

For each combination of ambulance location i ∈ A and subregion j ∈ R, we let dij be a boolean parameter with the value 1 or 0 depending on if subregion j∈ R is covered by ambulance location i ∈ A or not, that is:

For each region j∈ R we define yj , a help variable, in order to prevent counting the number of covered patients in the objective function multiple times in case multiple ambulance stations cover the same subregion. It has the value 1 if j is covered by at least one psychiatric emergency unit and 0 if it is not covered by any psychiatric emergency unit, that is:

There are two sets of decision variables used by the optimization model. The first one is xi (i∈ A) which is assigned the boolean value 1 if a psychiatric unit is placed at location i ∈ A, and 0 if it is not:

14 The objective function of the optimization model is:

(1)

The objective states that we want to find a set of locations that maximize the sum of the s_j value for the covered regions.

The model has three constraints that must be fulfilled. The first one is that the sum of xi , that is, the number of allocated units, must be equal to P, since P defines the number of ambulances to be placed

(2)

The following constraint set forces each of the yj variables to have the value 1 if the corresponding subregion j is covered by at least one ambulance:

(3)

The following constraint set forces each of the yj variables to have the value 0 if the corresponding subregion j is not covered by any ambulance:

(4)

M should be set to a high value, at least larger than P.

15 5. Scenario study

We applied the optimization model in a scenario where the goal was to identify the optimal locations of one, two or three psychiatric emergency units in the Region of Skåne. This allowed us to validate the results generated by the optimization model, hence allowing us to validate the functionality of the model itself.

5.1 Scenario description

Skåne has a population of about 1 393 004, with the largest percentage distributed on the western coast of the region, which includes Malmö, the third largest city in Sweden. Figure 5 displays the population distribution in the entire region. There are 10 hospitals and 26 ambulance stations distributed in the region Skåne. The locations of ambulance stations, which are the candidate locations for the psychiatric emergency units in our study, are shown in Figure 6, which also shows the estimated coverage area.

Figure 5: Percentage of population of each municipality in the region

16 Currently, we restrict the possible psychiatric emergency unit placements in the scenario to 26 ambulance stations, as that is usually where ambulances are placed. The scenario is not restricted to the ambulance station in Malmö, where the currently and only operating PAP-unit is located. The case percentages for each municipality is shown in Figure 3. In Section 3, we also calculate the case probability for each of the 1x1 km2 squares covering Region Skåne. Region Skåne’s goal for all types of priority 1 emergency calls is to reach 90% of the region’s population within 20 minutes. To find an average coverage size used by all ambulance stations, the ambulance station in Malmö was used. Since it is the largest city in Skåne, it has the densest traffic, so the distance travelled per minute is lower than the rest of the region. Using the average distance travelled from the Malmö ambulance station within 20 minutes ensures that any location in any ambulance station coverage area can be reached within that time frame. For each main cardinal direction (North, East, South and West) from the station, the linear distance travelled within 20 minutes was collected using Google Maps’ distance measuring feature. We averaged the travel distance of the four directions, which resulted in a 20 km distance from the Malmö ambulance station. By calculating the coverage area, the input parameter dji (i ∈ A, j ∈ R) is defined. See Figure 6, for an illustration of the calculated coverage areas for each of the ambulance stations.

17 Figure 6: Calculated coverage areas for ambulance stations in the region of Skåne

5.1.1 Scenario cases

As a part of the scenario study we solve the optimization model for one, two and three psychiatric emergency units with two different considerations of sj , namely case probability and population.. The P parameter is the number of psychiatric units to consider. For our scenario, R represents 1x1 km2 squares, each square represented by j. Apply the optimization twice for each of one, two, and three units, where sj represents either the population or case probability for a square, depending on which one of these we want to maximize.

18 5.1.2 Solution generation

The combinations were generated using a programmed script. A list of the ambulance stations and a number X, which represents the length of the combinations that are to be generated, were passed as arguments. The combination length is the number of psychiatric emergency units to be placed. To ensure the number of combinations created by the script was correct, it was verified by this combination formula (binomial coefficient):

(5) k! is a subset of the set n! and in our particular scenario n = 26 and k = 1, 2 or 3 when finding optimal locations for 1, 2 and respectively 3 psychiatric emergency units in a collection of 26 ambulance stations. The number of combinations generated was proven to be correct. For each combination of ambulance stations, the combined ambulance stations’ coverage areas were combined. To prevent population and probability data from being counted multiple times due to overlapping coverage areas, we merged the coverage areas. It should be noted that the scenario we use is small enough where it’s reasonable to make complete calculations of possible combinations of psychiatric unit placements. Scenarios that are more complicated might require a more sophisticated method to solve this problem.

5.2 Conditions

There are several conditions for applying this scenario to our model, and due to the limited data available, we had to estimate some of the input data needed by the model.

1. Population statistics from 2020 2. Number of cases on a municipality level. We used expected value optimization to estimate cases on a 1x1 km2 level. 3. It is assumed that the PAP-unit is always available to respond to acute emergencies. 4. A lack of data for ambulance response times in Skåne means the coverage area for the ambulance stations were estimated with the help of Google Maps. According to our contact person at Region Skåne, the goal for response times in Skåne is a maximum of 20 minutes.

19 5.3 Results

By using the optimization model and the solution generator, we were able to identify the optimal placement of one, two and three PAP-units in the Skåne region for two different considerations (values on sj ): 1. Probability percentage for acute psychiatric case out of the total in Skåne 2. Population out of the total in Skåne

5.3.1 Placement of one PAP unit

In Figure 7, we show a comparison of the placement of one PAP-unit in Malmö and Lund respectively, considering both case probability and population. The map shows the coverage areas for the stations and which municipalities that are covered. The station in Malmö covers 43% of the expected cases in Skåne and 567.000 (~40,7%) out of the total population in Skåne. Lund’s station covers 42% of the expected cases in Skåne and 589.000 (~42,3%) out of the total population.

Figure 7: Optimal placements of one PAP unit Malmö best probable case percentage coverage, Lund highest population coverage

20 5.3.2 Optimal placement of two PAP units

In Figure 8, it can be seen that the optimal locations of two psychiatric emergency units, considering case probability coverage, are Helsingborg and Malmö. The stations in Helsingborg and Malmö cover 58,61% out of the total expected cases in Skåne.

Figure 8: Optimal case chance coverage for two PAP units Helsingborg & Malmö | Case chance covered: 58,61%

21 In Figure 9, we show the optimal locations for population coverage when placing two psychiatric emergency units in Region Skåne. The optimal locations are Helsingborg and Lund. Together, the units placed in these two stations cover 794.000 persons, roughly 52% of Skåne´s population.

Figure 9: Optimal population coverage for two PAP units Helsingborg & Lund | Population covered: 793.916 (~52%)

22 5.3.3 Optimal placement of three PAP units

The optimal locations of three psychiatric emergency units, when focusing on case probability coverage, is Helsingborg, Kävlinge and Svedala, which is shown in Figure 10. The units placed in these three stations are expected to cover 68,2% of the expected cases in Skåne.

Figure 10: Optimal case chance coverage for three PAP units Helsingborg, Kävlinge & Svedala | Case probability covered: 68,23%

23 In Figure 11, we show the optimal locations of three psychiatric emergency units, which are Helsingborg, Kävlinge and Svedala, when considering population coverage. Together, the emergency units located in these three stations are expected to cover 937.000 persons, roughly 61,4% out of Skåne´s total population.

Figure 11: Optimal population coverage for three PAP units Helsingborg, Kävlinge & Svedala | Population covered: 936.571 (~61,4%)

24 5.4 Analysis

In the previous section, we present the results from the application of our optimization model for the given scenario, that is, placement of psychiatric emergency units in Region Skåne. Below, we analyze the results, as well explaining and comparing the different psychiatric emergency unit placements.

Table 1: Summary of results

Psychiatric Case Optimal stations Population Expected Units covered cases covered

1 Case Malmö 567.000 (~40,7%) 43,3% probability

1 Population Lund 589.000 (~42,3%) 42,0%

2 Case Helsingborg & 772.000 (~55,5%) 58,6% probability Malmö

2 Population Helsingborg & Lund 794.000 (~57,0%) 57,3%

3 Case Helsingborg, 937.000 (~61,4%) 68,2% probability Kävlinge & Svedala

3 Population Helsingborg, 937.000 (~61,4%) 68,2% Kävlinge & Svedala

The results show that our model is in fact working, and that it is able to find the optimal location of psychiatric emergency units regardless of what the sj parameters represent. In our study the sj :s represent either population or case probability. It should be emphasized that the result from the optimization model depends on the quality of the input data. While the model itself can make use of real data, our scenario lacked some data that the model takes as input, thus making the results a combination of real-world data, calculations, and estimations. While we had real case data received from Region Skåne, the case probability for each km2 square had to be estimated. By multiplying each square’s population percentage with its municipality’s case percentage of the total in Skåne we could estimate the probability of cases distributed in Skåne. The population data was collected from the Central Bureau of Statistics in Sweden, which means the population is mostly correct, aside from being data from the previous year. More data would allow for more accurate calculation and estimation, which would create a more accurate result providing a closer representation of the real world.

25 Using population as the metric to maximize the coverage for, the results from the optimization model show that one psychiatric emergency unit should be placed at the ambulance station in Lund to provide coverage for as many people as possible within a radius of 20 kilometers. This is 22.000 more persons than if it is placed in the Malmö station. The coverage area when placed in Lund also covers the largest cities of the 5 municipalities closest to its center, including the largest city: Malmö. Placing two psychiatric units results in an optimal placement of both units in the western part of Skåne. Helsingborg and Malmö become the optimal locations when focusing on covering the highest possible percent of case probability. When using population as the maximization metric, the optimal locations are Lund and Helsingborg. This connects to the results when one unit is placed, with Malmö and Lund switching places. Helsingborg, Kävlinge and Svedala are the optimal locations for three psychiatric emergency units, both in terms of population and probability of cases. The coverage area for these units' stations includes Skåne's three largest municipalities, Malmö, Helsingborg, and Lund. The case percentage distribution between the municipalities in Skåne differs slightly from the population percentage, as seen in Figure 12. Figure 13 shows how the municipalities deviate from the trend where cases increase with the population. As the population grows larger, so does the gap between the cases and population in the municipalities. For our scenario, this shows that the difference between placements for maximizing population coverage and maximizing risk probability coverage shrinks as more psychiatric units are placed and the municipalities with the largest populations are covered.

26 Figure 12: Cases and population percentage comparison

Figure 12 and 13 show that there is a linear correlation between the number of cases and population. The correlation coefficient r is ~0.98 which means there is a strong correlation between the two.

Figure 13: Diagram showing deviation from the case increase with population trend

27 6. Discussion

The results from our scenario study indicate that our optimization model is appropriate in order to identify the optimal location of psychiatric emergency units in a geographical region. Dividing a region into subregions which contain data used as a metric to maximize, and defining coverage areas for psychiatric unit locations, our model generates a combination of ambulance stations where psychiatric emergency units should be allocated, with the intent to be as efficient as possible. The identified optimal placements of psychiatric units in Skåne were not entirely unexpected. As seen in the results for our scenario and in Figure 12, the areas that have the highest populations generally also have the largest number of cases, which means that optimizing the placement to maximize the population covered would cover almost as many probable cases as covered by placement optimized for maximizing the percentage of probable cases covered. Our analysis showed a strong correlation between the number of cases and population. If this correlation is not a coincidence, due to the data from that year possibly being an outlier, this could mean that using population as a maximization goal might be as efficient as using the case probability. As an example, due to the very small difference between the coverages for risk and population, the current placement of the current PAP-unit in Malmö might be the optimal one in both considerations even if Lund covers 22.000 more people based on the results. It is also possible that the results would have looked significantly different if we had estimated the coverage areas specifically for each station. This leads to the optimal placements considering population- and case probability maximization being on the western part of Skåne which means sparsely populated municipalities, already down in the priority list, will not benefit from the services provided by the psychiatric emergency units. From an efficiency perspective, it is obvious that resources should be allocated where they will make the biggest difference and that is how the optimization model works. This creates an equality problem where rural areas are left out even more as the urbanization of our society continues. Especially when a psychiatric unit has its hands full with multiple cases in a city and cannot make it on time to individuals outside the city. In those cases, a regular ambulance will be dispatched. There is also an obvious economic factor to be considered, as special care ambulance units cost money and placing such a unit in a rural area might not be profitable in the long run. Even if individuals’ lives will be saved, it can be irrational, from an economic perspective, if the return of investment is too small. When placed where it is needed, the use

28 of a psychiatric emergency unit leads to indirect economic savings, as Carlborg et al. [6] established based on their studies of a psychiatric unit operating in the region of Stockholm. Our study was conducted as part of the PAP-project by Region Skåne and as stated in Section 5.1, the region’s goal is to reach 90% of the population within 20 minutes, so the main focus of our study was on solving the equity problem that arises by maximizing efficiency without taking into consideration the economical or equality standpoint. We see areas where the quality of the results might decrease. For instance, the estimated coverage areas are based on regular driving times estimated by the Google Maps API and the coverage area we estimated for the Malmö station was applied to every station in Skåne. The shape of the coverage area makes large parts cover inaccessible areas. This could have been improved by exploring additional possible psychiatric unit locations, such as hospitals, fire stations, for example. Due to the lack of response time in the data from SOS Alarm and several time constraints, we had to make estimations, which were obvious limitations. If we had more detailed data about the hourly cases during each season or month for each year, we could also have applied the model to scenarios with multiple factors. Furthermore, the ambulance data used was from 2020 which might not represent the number of acute mental health cases in the considered region in the long-term. It would have been interesting to get hands on data showing the progression of mental health cases to create a more accurate prognosis of the risk probability in Skåne. The collaboration with the PAP project means that we could get a better sense about the problem. We had one introductory meeting, where we were able to gather information to understand the purpose of the project to shape our own study’s goals and objectives. We also had communication with the project owner where we asked for details about the data we received and got general information about the PAP-project to ensure that the project was consistently moving forward in the right direction and to prevent any ambiguity that we encountered.

29 6.1 Validity threats

In our scenario we estimated the response time for the ambulance station in Malmö and applied the same response time for each station in Skåne, which creates a margin of error. If better estimations are done with more data and specific response times for each station, a more accurate result would be provided from the optimization model. Due to the not so geographically detailed data over the cases in the Region of Skåne, estimations were done to calculate the case probability for each km2 in the region. The data that the case probability estimates are based on is from 2020 and might not represent the long-term. We recognize that this could decrease the quality of the results from the optimization model. However, the main contribution of our work is the optimization model, and it seems to perform well even if the data is not ideal. Better data might improve the results, but since the results are reasonably believable, we cannot be certain it would make a difference.

30 7. Conclusions and future research

In order to answer our research question, we created an optimization model that we evaluated using a scenario study. In the scenario study, we used a mix of real and estimated data, and the results indicate that the optimization model performs well. The model generates placements of psychiatric emergency units, with the intent to maximize their efficiency, that is, to cover as many people in need of emergency care as possible. We can therefore conclude that we have created an appropriate model that can be used to identify the optimal location of psychiatric emergency units in a geographical region. Hence, we were able answer our research question:

“What is an appropriate model for identifying the optimal location of psychiatric emergency units in a geographical region?”

Lastly, we acknowledge that there is further research that could be done as future work, and below we outline some research ideas that could be studied to either improve upon our work or solve problems that were not in our scope of research. Since our focus was on efficiency, sparsely populated regions will show smaller improvements. Further research could be done focusing on an equality perspective to improve the coverage in rural areas. It should be mentioned that our optimization model is static, and an interesting idea would be to use tools like simulations and machine learning with additional data, such as response times per municipality, road types, traffic flow and dynamic operating hours, to create a dynamic model where placed units can be allocated and reallocated based on demand. Response times in general would be interesting to study as well for creating more accurate coverage areas. Since our optimization model has only been evaluated against aggregated data in several scenarios, it would be relevant to evaluate the results in practise. A future study could evaluate the optimal placements generated by our optimization model and explore its real-world performance, analyzing their advantages and disadvantages.

31 References

[1] “The WHO special initiative for mental health (‎2019-2023): universal health coverage for mental health,”‎World Health Organization. [Online]‎. Available: https://apps.who.int/iris/handle/10665/310981 [2] “Suicidtankar,” suicidprevention.se. [Online]. Available: https://www.folkhalsomyndigheten.se/suicidprevention/statistik-om-suicid/suicidtankar/. [3] “Suicide,” World Health Organization. [Online]. Available: https://www.who.int/news-room/fact-sheets/detail/suicide. [4] L. Todorova, A. Johansson, and B. Ivarsson, “Perceptions of ambulance nurses on their knowledge and competence when assessing psychiatric mental illness,” Nursing Open, vol. 8, no. 2, pp. 946–956, 2020. [5] “Pilotprojekt - Prehospital Akut Psykiatri (PAP),” Region Skåne. [Online]. Available: https://www.skane.se/Public/Protokoll/Beredningen%20f%C3%B6r%20prim%C3%A4rv%C 3%A5rd,%20psykiatri%20och%20tandv%C3%A5rd/2018-03-28/F%C3%B6rs%C3%B6ksve rksamhet%20psykiatriambulans/Psykiatriambulans%20(2).pdf. [6] A. Carlborg, R. Sibbel, and M. Helgesson, “Health Economic Evaluation of the Psychiatric Emergency Response Team (PAM) in Stockholm County,” 2020. [7] V. Lindström, L. Sturesson, and A. Carlborg, “Patients' experiences of the caring encounter with the psychiatric emergency response team in the emergency medical service—A qualitative interview study,” Health Expectations, vol. 23, no. 2, pp. 442–449, 2020. [8] S. A. Mahdiraji, O. Dahllöf, F. Hofwimmer, J. Holmgren, R.-C. Mihailescu, and J. Petersson, “Mobile stroke units for acute stroke care in the south of sweden,” Cogent Engineering, vol. 8, no. 1, p. 1874084, 2021. [9] O. Dahllöf, F. Hofwimmer, J. Holmgren, and J. Petersson, “Optimal placement of Mobile Stroke Units considering the perspectives of equality and efficiency,” Procedia Computer Science, vol. 141, pp. 311–318, 2018. [10] E. Erkut, A. Ingolfsson, and G. Erdoğan, “Ambulance location for maximum survival,” Naval Research Logistics (NRL), vol. 55, no. 1, pp. 42–58, 2007. [11] V. Bélanger, A. Ruiz, and P. Soriano, “Recent optimization models and trends in location, relocation, and dispatching of emergency medical vehicles,” European Journal of Operational Research, vol. 272, no. 1, pp. 1–23, 2019.

32 [12] P. L. V. D. Berg and J. T. V. Essen, “Comparison of static ambulance location models,” International Journal of Logistics Systems and Management, vol. 32, no. 3/4, p. 292, 2019. [13] M. A. Zaffar, H. K. Rajagopalan, C. Saydam, M. Mayorga, and E. Sharer, “Coverage, survivability or response time: A comparative study of performance statistics used in ambulance location models via simulation–optimization,” Operations Research for Health Care, vol. 11, pp. 1–12, 2016. [14] L. Brotcorne, G. Laporte, and F. Semet, “Ambulance location and relocation models,” European Journal of Operational Research, vol. 147, no. 3, pp. 451–463, 2003. [15] M. Gendreau, G. Laporte, and F. Semet, “Solving an ambulance location model by tabu search,” Location Science, vol. 5, no. 2, pp. 75–88, 1997. [16] V. Schmid and K. F. Doerner, “Ambulance location and relocation problems with time-dependent travel times,” European Journal of Operational Research, vol. 207, no. 3, pp. 1293–1303, 2010. [17] S. T. March and G. F. Smith, “Design and natural science research on information technology,” Decision Support Systems, vol. 15, no. 4, pp. 251–266, 1995. [18] Hevner, March, Park, & Ram. (2004). Design Science in Information Systems Research. MIS Quarterly, 28(1), 75. doi:10.2307/25148625 [19] J. Lundgren, Ronnqvist Mikael, and Varbrand Peter, Optimeringslara. Studentlitteratur, 2008. [20] “Ambulans,” Region Skåne. [Online]. Available: https://www.skane.se/Halsa-och-vard/hitta-vard/Ambulans/. [21] “Statistik på rutor,” Statistiska Centralbyrån. [Online]. Available: https://www.scb.se/vara-tjanster/oppna-data/oppna-geodata/statistik-pa-rutor/. [22] “Digitala gränser,” Statistiska Centralbyrån. [Online]. Available: https://www.scb.se/hitta-statistik/regional-statistik-och-kartor/regionala-indelningar/digitala-g ranser/.

33 Appendix A (data)

Table 1: Case distribution over the municipalities of Skåne Municipality Number of cases Bjuv 86 Bromölla 42 Burlöv 146 Båstad 50 Eslöv 143 Helsingborg 649 Hässleholm 302 Höganäs 64 Hörby 70 Höör 98 Klippan 84 Kristianstad 370 Kävlinge 68 Laholm 1 Landskrona 203 Lomma 44 Lund 478 Malmö 2104 Osby 49 Perstorp 68 Simrishamn 113 Sjöbo 102 Skurup 78 Staffanstorp 86 Svalöv 110 Svedala 66 Sölvesborg 2 Tomelilla 82 Trelleborg 222 Vellinge 118 Ystad 117 Åstorp 82 Ängelholm 206 Örkelljunga 69 Östra Göinge 51 Total 6623

34 Table 2: Coordinates for ambulance stations Ambulance station Latitude Longitude Broby 56.2535703 14.0660815 Bromölla 56.0722563 14.464683 Båstad 56.42746714996647 12.843828242446657 Eslöv 55.8394312 13.3259442 Helsingborg 56.0841177 12.7385205 Hässleholm 56.1618624 13.772954 Höganäs 56.21171446249775 12.555455056173871 Höllviken 55.421329068489044 12.955403190775971 Hörby 55.8497983 13.6410687 Kristianstad 56.03248 14.1696393 Landskrona 55.8726473 12.8650553 Lund 55.69326 13.2233697 Kävlinge 55.7668779 12.9931558 Malmö 55.55625416358296 13.029740808220682 Osby 56.3848089 13.9814788 Perstorp 56.1307426 13.3861089 Simrishamn 55.5528437 14.3362695 Sjöbo 55.6263862 13.7012821 Skurup 55.484284993355715 13.503005156511929 Svalöv 55.90908292959666 13.107930864341728 Svedala 55.51591282654796 13.240273513984663 Tomelilla 55.5459288 13.9662952 Trelleborg 55.38129295040362 13.161358880340487 Ystad 55.4446408 13.8363457 Ängelholm 56.242407 12.8579202 Klippan 56.1877 13.056823

35 Table 3: Psychiatry ESS codes and what they refer to (translated from Swedish)

ESS Code Meaning

80 Depressive episode

81 Mania/Hypomania

82 Anxiety

83 Sleep disorders

84 Psychotic symptoms

85 Missbruk/ Beroende

86 Self-harming behaviour

88 Crisis reaction/hard stress

89 Dementia/ Disorientation

90 Non-acute psychiatric problem

98 Danger assessment

99 Suicide-risk assessment

152 Mental illness (minors)

Table 4: SOS call priorities

Priority Name Description

1 Ambulance - Severe life-threatening symptoms that requires examination, care or treatment at pickup-location and during transport - Severe life-threatening transfer between medical facilities

2 Priority/Urgent - Severe but not life-threatening symptoms that need transport examination, care or treatment at pickup-location and during transport - Severe but not life-threatening transfer between medical facilities

3 Transportation - Non-severe transportation from home that requires examination, care or treatment during transport - Non-severe transfer between medical facilities and from a medical facility to home of patient that require examination, care or treatment during transport

36 Appendix B (literature study)

Table 1: Search term, filters, and number of results

Search phrase Filters MAU Libsearch Google Scholar

“ambulance location models” match title only 0 13

“ambulance location models” - 0 58 500

“mobile psychiatric unit” match title only 2 24

“mobile psychiatric unit” - 2 115 000

“mobile psychiatric unit” AND match title only 0 “location”

Table 2: Name of literature and how it was found

Name Found by Search phrase (if any)

Mobile stroke units for acute stroke care in Personal recommendation - the south of sweden.

Health Economic Evaluation of the Personal recommendation - Psychiatric Emergency Response Team (PAM) in Stockholm County.

Patients' experiences of the caring encounter Personal recommendation - with the psychiatric emergency response team in the emergency medical service—A qualitative interview study.

Perceptions of ambulance nurses on their Personal recommendation - knowledge and competence when assessing psychiatric mental illness.

Comparison of static ambulance location MAU Libsearch “ambulance location models. models”

Ambulance location and relocation models. Google Scholar ambulance location models

Coverage, survivability or response time: A MAU Libsearch “ambulance location comparative study of performance statistics models” used in ambulance location models via simulation–optimization.

37 Optimal placement of Mobile Stroke Units Personal recommendation - considering the perspectives of equality and efficiency.

Ambulance location for maximum survival. Google Scholar “ambulance location”

Design and natural science research on Personal recommendation - information technology.

Design Science in Information Systems Personal recommendation - Research.

Optimeringslara Personal recommendation -

Mobile stroke units for acute stroke care in Personal recommendation - the south of sweden

Recent optimization models and trends in Snowballed - location, relocation, and dispatching of emergency medical vehicles

Solving an ambulance location model by tabu Google Scholar “ambulance location” search

Ambulance location and relocation problems Google Scholar “ambulance location” with time-dependent travel times

38