Simultaneous response to multiple disasters: Integrated planning for pandemics and large-scale earthquakes

Abstract

Since the beginning of COVID-19, individuals who have SARS-CoV-2 infectious have brought a heavy burden on the healthcare system. Unavoidably, along with pandemics, large-scale disasters, which are possibly emerging, may double the current health crisis. For a powerful disaster response plan, the health services should be prepared for the overwhelming number of disaster victims and infected individuals The proposed framework determines the appropriate number and location of temporary healthcare facilities for large-scale disasters while considering the burden of ongoing pandemic diseases. In this study, first, a multi-period, mix-integer mathematical model is developed to find the location and number of disaster emergency units and disaster medical facilities. Second, we develop an epidemic compartmental model to stimulate the negative effects of the disaster on disease spread and a mixed-integer mathematical model to find optimal number and the location of pandemic hospitals and isolation centers. To validate the mathematical models, a case study is conducted for a district of Istanbul, Turkey.

1. Introduction

The United Nations Office for Disaster Risk Reduction (UNDRR) defines a disaster as “a serious disruption of the functioning of a community or society involving widespread human, material, economic or environmental losses and impacts, which exceeds the ability of the affected community or society to cope using its own resources” [1]. Disaster management aims to mitigate the devastating impacts of extreme incidents, such as geologic disasters (earthquake, landslide, and tsunami), climatic disasters (famine, hurricane), biological disasters (epidemic diseases), social disasters (wars, terrorist attacks) and technological disasters (industrial and transportation failures) [2].

The world is presently in the battle against a pandemic disaster, coronavirus disease 2019 (COVID-19). Apart from COVID-19 pandemic, people all around the world are also vulnerable to other disasters. Japan gave public notifications related to dual disasters during the COVID-19 pandemic [3]. Under conditions of COVID-19, some communities had to deal with the threat of floods [4]. For instance, As the COVID-19 outbreak goes on, the toughest flood recorded in 2020 left 285 dead in Kenya [5]. In the literature Researchers also pointed out the earthquakes occurred during COVID-19 [[6], [7], [8]].

While healthcare systems all over the world are under the pressure of COVID-19, existing service capacity is expected to have difficulty in responding another disaster. Therefore, emergency and health systems should be designed for the preparedness of multiple disasters. WHO gave notification on preparation for possible large-scale incidents while managing COVID-19. Turkey, located on one of the most active seismic belts, is found at risk of earthquakes which is possible to occur during the COVID-19 pandemic [9].

Mediterranean-Alpine-Himalayan belt, in which 25% of worldwide earthquakes occurred, meanly causes a devastating earthquake in every five years in Turkey [10]. As of the end of 2019, three deathful earthquakes have been experienced in East Anatolia and Aegean Districts of Turkey. Elazığ earthquake with a magnitude of 6.8, Iran-Khoy earthquake with a magnitude of 5.9 and Aegean earthquake with a magnitude of 6.6 on the Richard scale occurred in January, February and October 2020 [11]. Respectively, these disasters left the thousands of injured people behind and killed 169 victims in total [12]. More fortunate than the time Aegean earthquake occurred, disaster victims were not subjected to an infectious disease in Elazığ and Iran-Khoy earthquake.

Aegean earthquake is recorded as the most fatal seismic event occurred in 2020 [12]. This disaster hit the third most populous city in Turkey, Izmir, where 3000 constructions became no useable, and 13 of them were ruined [13]. United Nations Office for the Coordination of Humanitarian Affairs (OCHA) categorized the disaster as “earthquake” and “epidemic” [14]. Because of the chaotic environment aftermath of the earthquake, preventive measures such as social distancing were disrupted. The number of deaths caused by COVID-19 tripled after the occurrence of earthquake due to the uncontrolled contact, stated by the governor of Izmir [15].

Rapidness and effectiveness in medical response is crucial for the first 72 h aftermath of a disaster. Due to the disruption of non-pharmacological measures, the augmented number of infected individuals is expected to force healthcare system together with disaster victims. Owing to the damage probabilities and occupied capacity of hospitals, existing healthcare facilities may not be sufficient in responding dual disasters. In this point, temporary healthcare facilities can enlarge the existing capacity while providing rapid access. They have critical role in early emergency medical care and act as temporary facilities until damaged hospitals regain their functionality.

In this study, the proposed framework determines the appropriate number and location of temporary healthcare facilities for large-scale disasters while considering the burden of an on-going pandemic diseases.

The contributions of this study can be summarized as follows.

• •In humanitarian logistics, temporary healthcare facility location models and their applications are considerably significant and extensively studied in the literature [[16], [17], [18], [19], [20], [21], [22], [23], [24], [25]]. Each of the above-mentioned research papers studied either a specific natural disaster or focused on infectious disease outbreaks. Only Anparasan and Lejeune [25] connected an earthquake case and an epidemic disease, but they considered the 2010 cholera outbreak in Haiti, which was occurred aftermath of an earthquake. Therefore, their research is related to compounded disasters, which cause one another. However, in our study, we focus on disasters occurred independently, which is the current issue in COVID-19 outbreak. To the authors best knowledge, there is no research study suggested a healthcare network design for a case of large-scale disaster occurred during a pandemic outbreak.
• •In the first stage of this study, our purpose is to increase the effectiveness of emergency and following medical response operations. Accordingly, we develop a multi-period mix-integer mathematical model determining the location of emergency treatment facilities which can be converted to full-fledged healthcare facilities under dimension and set-up time restrictions.
• •For the sake of resource efficiency in emergency medical care, we enable medical team rotation between facilities under distance and resource utilization threshold.
• •To reflect the uncertainty aftermath of a disaster, demand stochasticity and possible road damages are considered.
• •In second stage of this study, we develop a compartmental model to forecast the number of infected individuals with their illness severity degree aftermath of a large-scale earthquake occurred during COVID-19. Within the concept of analyzing potential impact of earthquakes in the outbreak, the closest study is conducted by [26]. In the second stage of this study, different than [26], we integrate a disease spread model and a healthcare network configuration where the burden of both earthquake victims and infected cases are analyzed.
• •Further, a mathematical model is developed to find the optimal number and the location of COVID-19 hospitals and isolation centers. Once the coronavirus 2019 disease waked, isolation hospitals/healthcare facilitates has become an important measure in controlling the spread of the infectious disease, and the affirmative effects of institutional-based isolation has extensively been analyzed in the literature [[27], [28], [29], [30], [31]]. Even though, several studies are published on setting up isolation hospitals and the region selection criteria, none of the study proposed an optimization-based methodology to determine the location of dedicated pandemic hospital and isolation center which are supposed to serve to mildly symptomatic individuals after an earthquake.

The remainder structure of this paper is as follows. In Section 2, we represent the literature on facility location problems in humanitarian logistics and studies focused on infectious disease outbreaks. Methodology is introduced in Section 3. Data collection and analysis are represented in Section 4. The results, discussions, and conclusions of this study with the statements about future research are given in Section 5.

2. Literature review

The structure of this literature review section is composed by two main streams. We analyzed the literature related to the logistical aspects of disaster management, concentrated primarily on the facility location problems. Besides, we briefly reviewed the past studies related to pandemic and epidemic diseases.

The role of humanitarian logistics in disaster management has become more apparent after the Indian Ocean Tsunami in 2004 [32]. A remarkable number of studies have applied Operation Research (OR) methodologies to increase the efficiency of pre- and post-disaster logistics operations [[33], [34], [35]]. reviewed the existing literature extensively and showed how the OR applications are wide in the field of disaster management.

[36] surveyed facility location models in humanitarian logistics in depth and pointed out the importance of proposed problem. This area of research can be classified as shelter-site location problem, warehouse location problem and emergency medical location problem [37]. Various studies considered the problem of infrastructural damages aftermath of a disaster and developed models to select temporary housing site [[37], [38], [39], [40], [41], [42], [43]]. Supplying relief item is another humanitarian logistics problem extensively studied by OR community [[44], [45], [46], [47]]. To reflect awaited chaotic environment right after a disaster, dynamic modelling techniques are also applied by scholars. Within this concept [[48], [49], [50], [51], [52], [53]], suggested multi-period facility location models for relief and blood supply.

In order to increase the capacity and accessibility of healthcare services, temporary healthcare facilities are suggested to be established aftermath of a disaster [54]. categorized temporary emergency healthcare facilities into two main types: medication dispensing sites, in which medical equipment and medicines are supplied, and temporary medical centers, in which medical treatments are performed. However, these two facility types have common modelling approaches, in terms of objective functions and constraints.

In the context of dispensing sites [55], proposed conventional facility location models for the placement of medical service facilities while considering terrorist attack scenarios. An ant colony optimization algorithm was introduced by Ref. [56] to deal with the same problem presented by Ref. [55]. A heuristic approach was developed by Ref. [57] to locate emergency facilities under demand uncertainty in a bio-terror attack [58]. introduced a methodology to select supply storages while considering the transportation of medical relief commodities under uncertain parameters [59]. proposed a mathematical model to locate disaster response centers and compared the performance of deterministic and stochastic models.

1999 Marmara earthquake is an instance where existing healthcare facilities were insufficient in capacity due to the sudden influx of casualties [60,61]. Scholar addressed this problem and suggested to set temporary healthcare facilities aftermath of an earthquake [[16], [17], [18],22]. [16] suggested a dynamic location-distribution model including both the transportation of relief supplies and wounded people. Two-stage heuristic approach was developed to locate temporary emergency facilities serving victims with minor injuries while aiming to reduce patient influx at hospitals. Different than [16,17] disregarded the distribution of relief supplies and focused on casualty transportation and settlement of medical centers. In the proposed two-level location-routing problem, uncertainties related to healing rates and network conditions were considered. A two-stage stochastic optimization methodology was introduced by Ref. [18] to determine the locations of field hospitals under the failure probability of existing hospitals. The impact of triage classification was demonstrated in a two-stage stochastic programming by Ref. [22]. [19] proposed a robust model to place both relief bases and temporary medical facilities. The suggested humanitarian relief network plan considered casualty transportation, relief goods and blood supplies under possible damages to pathways and relief-bases [24]. introduced a bi-objective location-routing problem while minimizing relief time of casualties and total cost. Similar to Ref. [24]; multi-objectivity on location and allocation of temporary medical centers and services was studied by Ref. [21]. Limited resources such as hospital bed, medical personnel and vehicle are regarded, and an earthquake case is performed.

The time interval of response operations varies because of different disaster characteristics. For instance, short-term planning problems are considered in a case of earthquake [16,17] since demand influx is expected in hours or days aftermath of an earthquake. Within the concept of pandemic and epidemic diseases, logistics problems of mass vaccination, such as distribution, can be classified within the same group since it occurs in a short-term [62]. On the other hand, medium-term planning problems are also placed in the humanitarian logistics literature related to pandemic and epidemic outbreaks [[23], [25], [63]].

An epidemic-logistics model was represented by Ref. [20] in order to locate temporary treatment centers with isolation units while minimizing number of infectious. Anparasan and Lejeune [25]considered 2010 Cholera outbreak in Haiti, a secondary disaster aftermath of an earthquake, and developed a complex integer linear programming to place treatment facilities, distribute medical personnel and transport infected patients [23]. considered the same epidemic-logistic problem earlier introduced by Ref. [20] and proposed an integrated framework, including a compartmental model and a logistic plan to locate isolation wards in a case of an influenza outbreak in China.

Table 1 demonstrates the summary of the studies related to temporary healthcare facility location problem.

Table 1.

Literature review.

Study	Parameter type																Other
	Logistic attribute			Period setting		Objective(s)	Deterministic			Deterministic/scenario analysis			Stochastic			Problem type	Disaster type		Patient classification	Transportation type
	L	A	D	MP	SP		F	R	D	F	R	D	F	R	D		Earthquake	Outbreak		Ambulance	Helicopter
[16]	✓	✓	✓	✓		Minimizing unsatisfied demand	✓	✓	✓							MIP	✓		✓	✓	✓
[17]	✓	✓		✓		Minimizing travel time Minimizing waiting time Minimizing cost	✓				✓	✓				MIP	✓			✓
[18]	✓	✓			✓	Minimizing distance		✓				✓	✓			IP	✓
[19]	✓	✓	✓	✓		Minimizing cost								✓		MIP	✓			✓*
[20]	✓	✓				Minimizing new infectious and fatalities	✓	✓	✓							MIP		✓
[21]	✓	✓	✓	✓		1. Maximizing survivals 2. Minimizing cost	✓	✓	✓							MIP	✓		✓	✓	✓
[25]	✓	✓				Maximizing number of patients transported	✓	✓	✓							IP		✓	✓
[22]	✓	✓			✓	Minimizing cost						✓	✓	✓		IP	✓		✓
[24]	✓	✓			✓	1.Minimizing time 2.Minimizing cost	✓	✓							✓	MILP	✓		✓	✓	✓
This study	✓	✓		✓		Minimizing time				✓	✓	✓				MIP	✓	✓	✓	✓

L: Location; A: Allocation; D: Distribution.

MP: Multi period; SP: Single period.

F: Facility condition; R: Route condition; D: Demand.

Apart from medical facility location decisions, number of researchers handled food distribution [63], hospital-based resource allocation [62,64,[65], [66]], healthcare delivery [[67], [68], [69]] problems during an infectious outbreak via mathematical modelling or simulation tools. Although influenza is prioritized among other diseases within the existing literature [70,71], one of the recent literature review study published by Ref. [71] demonstrated that studies related covid-19 pandemic have already became visible. The impact of mitigation strategies such as social distancing, quarantine, isolation during COVID-19 pandemic is analyzed by scholars [72] via disease spread models. Further, various papers focused on the impact of COVID-19 on the healthcare systems [68,69] and related impacts in various field [73,74].

In this study, we introduce two integrated facility location-allocation model to determine the number and location of temporary healthcare facilities, which serve in post-disaster early emergency phase and provide assistance to handle the burden of infected patient influx. Suggested healthcare network design proposes a response plan for the large-scale disaster under the burden of on-going pandemic diseases. In the first stage of this study, we develop a multi-objective multi-period mixed integer mathematical model for the early emergency medical care, corresponding to 0–72 h aftermath of a disaster under the uncertain number of casualties and the failure probability of damaged hospitals and roads. Because of the expected increase in the number of infected cases aftermath of an earthquake occurred during COVID-19, healthcare systems should be prepared for a heavier load. The need of following-up care for disaster-related trauma and infected cases is considered in the second stage of this study. In order to reduce the number of infectious and increase the responsiveness of healthcare systems for infected and non-infected cases, we develop an epidemic-logistic model and suggest establishing dedicated healthcare facilities and isolation centers.

In the succeeding section, the methodology of this study is described in detail.

3. Methodology

3.1. Model-I: facility location and allocation model for emergency medical care and advanced trauma life support

In Model-I, three healthcare facilities (HF) are considered: Disaster Emergency Units (DE), Disaster Medical Centers (DMC) and functioning hospitals. The general framework of the healthcare service network design aftermath of a disaster is demonstrated in Fig. 1 .

Fig. 1.

Network design of healthcare services aftermath of a disaster.

DE, which has similar functionality to disaster medical aid centers [75] and casualty treatment stations [76], can be established and starts serving immediately aftermath of a disaster. In these rapid installable facilities, injured victims receive emergency care assistance before being transferred to full-fledged healthcare facilities. DE's capacity is determined by the number of medical teams assigned. For the sake of rapidness and efficiency at emergency services, medical teams can be rotated between facilities, including DEs and the emergency units of functioning hospitals. DEs can be converted to Disaster Medical Facility (DMF). Intensive care units (ICUs) and non-intensive care units (non-ICUs) are the resources of full-fledged healthcare facilities, including functioning hospitals and DMFs. DMFs are full-fledged hospitals and provide advanced healthcare services. There are three different sizes of DMF and, it is assumed that only one type can be established at a candidate location. Because of the transportation and installation of medical equipment, DMF cannot be set-up before a certain time depending on the facility size. Last, resource unit occupied by casualties differs in terms of severity of injury, and it is assumed that the capacity is not fully allocated to disaster victims. Structural and functional problems at existing hospitals can emerge because of earthquakes. In the accordance with Hospital Disaster and Emergency Plan (HAP), published by the Ministry of Health of Turkey, compulsory or precautionary evacuation may be required [77]. Therefore, the functioning healthcare facilities where patients at damaged hospitals will be transferred should be determined in advance. In this study, we consider that damaged hospitals lose their functionality immediately aftermath of a disaster, and patients in these facilities are sent to functioning facilities within an acceptable period. Last, it is assumed that a certain amount of the capacity of functioning hospitals is seized by COVID and non-COVID patients. By following the procedures of Simple Triage and Rapid Treatment (START) system, wounded victims are classified in terms of their injury scores (Schultz et al., 1996 [75]). Walking victims who do not require advanced medical attention are called as slightly wounded casualties. Victims with serious morbidities, who need further treatment at full-fledged healthcare facilities, are referred as heavily and moderately wounded casualties. All casualties are assumed to be transported to the emergency units. Thereafter, moderately, and heavily wounded casualties are transferred to functioning hospitals or DMFs for advanced treatment. It is not allowed to remain unserved heavily and moderately wounded casualties due to the great importance of on time intervention. On the other hand, slightly wounded casualties do not require medical examination in a timely manner. Therefore, delay on demand satisfaction is allowed only for them. To avoid underestimating hidden traumas, backordered demand of slightly wounded victims is satisfied in farther time periods.

3.1.1. Formulation of Model-I

The proposed multi-period, mix integer linear mathematical model selecting the appropriate number and location of DEs and DMFs is formulated as.

Sets and indices:
$N,n$	: Set and index of neighborhoods,$n\in N$
$H,h,h^{\prime },h^{″}$	: Set and index of healthcare facilities (HF), ${h,h}^{\prime },h^{″}\in H$
$H_E$	: Set of functioning hospitals, $H_E\subset H$
$H_M$	: Set of candidate Disaster Emergency Units (DE), $H_M\subset H$
$H_F$	: Set of damaged hospitals, $H_F\subset H$
$C,c$	: Set and index of casualty types, $c_{hw},c_{mw},c_{sw}\in C$
$Z,z$	: Set and index of hospital-based resources
$J,j$	: Set and index of full-fledged field hospital types
$T,t,g$	: Time periods, $t,g\in T$
Parameters
${tnh}_{nh}$	: Travel time (min) from neighborhood $n$ to HF $h$, where $h\in \left(H_E\cup H_M\right)$
${ird}_{nh}$	: Distance increase ratio from neighborhood $n$ to HF $h$, $whereh\in \left(H_E\cup H_M\right)$
${cam}_c$	: Capacity of an ambulance for casualty type $c$
${thh}_{h^{\prime }h}$	: Travel time (min) from HF $h^{\prime }$ to HF $h$, where $h^{\prime },h\in H$
${irh}_{h^{\prime }h}$	: Distance increase ratio from HF $h^{\prime }$ to HF $h$, where $h^{\prime },h\in H$
${pam}_z$	: Capacity of an ambulance for disaster victims who need hospital-bed $z$
${Dca}_{cn}^t$	: Number of casualty type $c$ at neighborhood $n$ at time period $t$
$L$	: A large number
${hmt}_h$	: Number of medical team at emergency unit of HF $h$, where $h\in H_E$
$emt$	: Number of medical team initially assigned to DEs
$r_c$	: Resource unit occupied by casualty type $c$ at emergency units
$\lambda$	: Capacity of one medical team
$\rho$	: Threshold value for the minimum travel time of a medical team between emergency units of HFs, where $H_E\cup H_M\subset H$
$\beta$	: Threshold value for the minimum unused capacity of emergency units
${\tau }_{cz}$	: Percentage of casualty type $c$ who need hospital-bed $z$, where $\sum _{z\in Z}{\tau }_{cz}=1$
$\varnothing$	: Threshold value for the number of rotated medical teams
$\sigma$	: Threshold value for the number of medical teams
$d{ep}_{zh}$	: Number of patients in the need of hospital-bed $z$ at HF $h$, where $h\in H_F$
${at}_z$	: Acceptable time period that patients who need hospital-bed $z$ can wait until they are transported another healthcare facility.
${disr}_z$	: Discharge rate of patients in the need of hospital-bed $z$
${cap}_{zh}$	: Capacity of hospital-bed $z$ at HF $h$, where $h\in \left(H_E\cup H_M\right)$
${caf}_{zj}$	: Capacity of hospital-bed $z$ at field hospital type $j$
${size}_j$	: Size of field hospital type $j$
${area}_h$	: $\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{f}\mathrm{c}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{H}\mathrm{F}h$, $whereh\in H_M$
${to}_j$	: Set-up time of field hospital type $j$
$nmc$	: The number of disaster medical facilities allowed to be opened
$ned$	: The number of emergency units allowed to be opened
Decision variables
$w_{cnh}^t$	: Number of casualty type $c$ sent to emergency unit of HF $h$, where $h\in \left(H_E\cup H_M\right)$, from neighborhood $n$ at time period $t$
${transw}_{z{h}^{\prime }h}^t$	: Number of casualties in the need of hospital-bed $z$ transferred from HF $h^{\prime }$ to HF $h$, where $h^{\prime },h\in \left(H_E\cup H_M\right)$ , at time period $t$
$trans{p}_{z{h}^{\prime }h}^t$	: Number of patients in the need of hospital-bed $z$, transferred from HF $h^{\prime }$, where $h^{\prime }\in H_F$, to HF $h$, where $h\in \left(H_E\cup H_M\right)$, at time period $t$
${rmed}_{h^{\prime }h}^t$	: Number of rotated medical teams to HF $h$ from HF $h^{\prime }$, where $h^{\prime },h\in \left(H_E\cup H_M\right)$, at time period $t$
${amed}_h^t$	: Number of arrived medical teams in HF $h$, where $h\in \left(H_E\cup H_M\right)$, at time period $t$
${dmed}_h^t$	: Number of departed medical teams from HF $h$, where $h\in \left(H_E\cup H_M\right)$, at time period $t$
${med}_h^t$	: Number of medical teams at HF $h$, where $h\in \left(H_E\cup H_M\right)$, at time period $t$
$t{w}_{ch}^t$	: Number of casualty type $c$ need to be transferred to full-fledged HFs (including functioning hospitals and TMCs) from the emergency unit of HF $h$, where $h\in \left(H_E\cup H_M\right)$ , at time period $t$
${ar}_{zh}^t$	: Number of patients in the need of hospital-bed type $z$ and sent to HF $h$, where $h\in \left(H_E\cup H_M\right)$, at time period $t$
${distw}_{zh}^t$	: Number of patients in the need of hospital-bed type $z$ discharged from HF $h$, where $h\in \left(H_E\cup H_M\right)$, at time period $t$
${avcap}_{zh}^t$:	: Available capacity of hospital-bed $z$ at HF $h$, where $h\in \left(H_E\cup H_M\right)$, at time period $t$
${intcap}_{zh}^t$	: Initial capacity of hospital-bed $z$ at HF $h$, where $h\in H_M$, at time period $t$
$v_{jh}^t$	: Number of field hospital type $j$ established at DMC $h$, where $h\in H_M$, at time period $t$
$y_{h{h}^{\prime }}^t$	$:\{\begin{array}{c}1,\mathrm{i}\mathrm{f}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{e}\mathrm{a}\mathrm{m}\mathrm{i}\mathrm{s}\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{e}\mathrm{d}\mathrm{t}\mathrm{o}\mathrm{H}\mathrm{F}h\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m}\mathrm{H}\mathrm{F}{h}^{\prime },\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\\ h,h^{\prime }\in \left(H_E\cup H_M\right),\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{d}t\\ 0,\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}$
$x_h$	: $\{\begin{array}{c}1,\mathrm{i}\mathrm{f}\mathrm{c}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{D}\mathrm{E}h,whereh\in H_M,\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{n}\\ 0,\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}$

3.1.2. Model-I

$$\mathit{M}\mathit{i}\mathit{n}\mathit{i}\mathit{m}\mathit{i}\mathit{z}\mathit{e}{f}_1=\sum _{t\in T}\sum _{c\in C}\sum _{n\in N}\sum _{h\in \left(H_E\cup H_M\right)}\frac{{tnh}_{nh}\left(1+{ird}_{nh}\right)w_{cnh}^t}{{cam}_c}+\sum _{t\in T}\sum _{z\in Z}\sum _{h^{\prime }\in \left(H_E\cup H_M\right)}\sum _{h\in \left(H_E\cup H_M\right)}\frac{{thh}_{h^{\prime }h}\left({1+irh}_{h^{\prime }h}\right)trans{w}_{z{h}^{\prime }h}^t}{{pam}_z}+\sum _{t\in T}\sum _{z\in Z}\sum _{h^{\prime }\in H_F}\sum _{h\in \left(H_E\cup H_M\right)}\frac{{thh}_{h^{\prime }h}\left({1+irh}_{h^{\prime }h}\right){transp}_{z{h}^{\prime }h}^t}{{pam}_z}$$ (Obj1)

$$\sum _{h\in \left(H_E\cup H_M\right)}{w}_{cnh}^t={Dca}_{cn}^t,{c=c}_{hw}andc=c_{mw},\forall n\in N,\forall t\in T$$ (a1)

$$\sum _{t\in T}\sum _{h\in \left(H_E\cup H_M\right)}{w}_{cnh}^t=\sum _{t\in T}{Dca}_{cn}^t,{c=c}_{sw},\forall n\in N$$ (a2)

$$\sum _{g\in T|g\le t}\sum _{h\in \left(H_E\cup H_M\right)}{w}_{cnh}^g\le \sum _{g\in T|g\le t}{Dca}_{cn}^g,{c=c}_{sw},\forall n\in N,\forall t\in T$$ (a3)

$$\sum _{h^{\prime }\in \left(H_E\cup H_M\right)|h^{\prime }\ne h}r{med}_{h^{\prime }h}^t=a{med}_h^t,\forall h\in \left(H_E\cup H_M\right),\forall t\in T$$ (a4)

$$\sum _{h^{\prime }\in \left(H_E\cup H_M\right)|h^{\prime }\ne h}r{med}_{h{h}^{\prime }}^t=d{med}_h^t,\forall h\in \left(H_E\cup H_M\right),\forall t\in T$$ (a5)

$${med}_h^t\le L{x}_h,\forall h\in H_M,\forall t\in T$$ (a6)

$${med}_h^{t+1}={med}_h^t+a{med}_h^{t+1}-d{med}_h^{t+1},\forall h\in \left(H_E\cup H_M\right),\forall t\in T|t\le T-1$$ (a7)

$${med}_h^0\le {hmt}_h,\forall h\in H_E$$ (a8)

$${med}_h^0\le emt{x}_h,\forall h\in H_M$$ (a9)

$$\sum _{c\in C}\sum _{n\in N}{r}_c{w}_{cnh}^t\le {\lambda med}_h^t,\forall h\in \left(H_E\cup H_M\right),\forall t\in T$$ (a10)

$${thh}_{h{h}^{\prime }}{y}_{h{h}^{\prime }}^t\le \rho ,\forall h,h^{\prime }\in \left(H_E\cup H_M\right)|h\ne h^{\prime },\forall t\in T$$ (a11)

$${\lambda med}_h^t-\sum _{c\in C}\sum _{n\in N}{r}_c{w}_{cnh}^t\le \beta y_{h{h}^{\prime }}^t+L\left(1-y_{h{h}^{\prime }}^t\right),\forall h,h^{\prime }\in \left(H_E\cup H_M\right)|h\ne h^{\prime },\forall t\in T$$ (a12)

$$L\left(1-y_{h{h}^{\prime }}^t\right)\ge \sum _{h^{″}\in \left(H_E\cup H_M\right)|h^{\prime }\ne h}{y}_{h^{″}h}^t,\forall h,h^{\prime }\in \left(H_E\cup H_M\right)|h\ne h^{\prime },\forall t\in T$$ (a13)

$$r{med}_{h{h}^{\prime }}^t\le L{y}_{h{h}^{\prime }}^t,\forall h,h^{\prime }\in \left(H_E\cup H_M\right)|h\ne h^{\prime },\forall t\in T$$ (a14)

$$\sum _{h^{\prime }\in \left(H_E\cup H_M\right)|h^{\prime }\ne h}\sum _{t\in T}r{med}_{h^{\prime }h}^t\le {\varnothing med}_h^0,\forall h\in \left(H_E\cup H_M\right)|h\ne h^{\prime },\forall t\in T$$ (a15)

$${med}_h^t\ge {\sigma med}_h^0,\forall h\in \left(H_E\cup H_M\right),\forall t\in T$$ (a16)

$$\sum _{n\in N}{w}_{cnh}^t={tw}_{ch}^{t+1},{c=c}_{hw}andc=c_{mw},\forall h\in \left(H_E\cup H_M\right),\forall t\in T|t\le T-1$$ (a17)

$${tw}_{ch}^0=0,{c=c}_{hw}andc=c_{mw},\forall h\in \left(H_E\cup H_M\right)$$ (a18)

$$\sum _{h\in H_E\cup H_M}{transw}_{z{h}^{\prime }h}^t=\sum _{c_{hw}{,c}_{mw}\in C}{\tau }_{cz}t{w}_{c{h}^{\prime }}^t,\forall z\in Z,\forall h^{\prime }\in H_E\cup H_M,\forall t\in T$$ (a19)

$$\sum _{t\in T|t={at}_z}\sum _{h\in H_E\cup H_M}trans{p}_{z{h}^{\prime }h}^t={dep}_{z{h}^{\prime }},\forall z\in Z,\forall h^{\prime }\in H_F$$ (a20)

$$\sum _{h^{\prime }\in H_E\cup H_M}{transw}_{z{h}^{\prime }h}^t+\sum _{h^{\prime }\in H_F}trans{p}_{z{h}^{\prime }h}^t={ar}_{zh}^t,\forall z\in Z,\forall h\in H_E\cup H_M,\forall t\in T$$ (a21)

$$\sum _{g\in T|g\le t}\sum _{h^{\prime }\in H_E\cup H_M}{disr}_z{\left(1-{disr}_z\right)}^{t-g}trans{w}_{{zh}^{\prime }h}^g={distw}_{zh}^t,\forall z\in Z,\forall h\in H_E\cup H_M$$ (a22)

$${avcap}_{zh}^0={cap}_{zh},\forall z\in Z,\forall h\in H_E$$ (a23)

$${avcap}_{zh}^{t+1}={avcap}_{zh}^t-{ar}_{zh}^{t+1}+{distw}_{zh}^{t+1},\forall z\in Z,\forall h\in H_E,\forall t\in T|t\le T-1$$ (a24)

$$v_{jh}^t=0,\forall j\in J,\forall h\in H_M,\forall t\in T|t\le {to}_j$$ (a25)

$${avcap}_{zh}^{t+1}\le L\left(\sum _{j\in J}\sum _{g\in T|g\le t}{caf}_{zj}{v}_{jh}^g\right),\forall z\in Z,\forall h\in H_M,\forall t\in T|t\le T-1$$ (a26)

$${intcap}_{zh}^t=\sum _{j\in J}{caf}_{zj}{v}_{jh}^t,\forall z\in Z,\forall h\in H_M,\forall t\in T$$ (a27)

$${avcap}_{zh}^{t+1}={intcap}_{zh}^t+{avcap}_{zh}^t-{ar}_{zh}^{t+1}+{distw}_{zh}^{t+1},\forall z\in Z,\forall h\in H_M,\forall t\in T|t\le T-1$$ (a28)

$$\sum _{j\in J}\sum _{t\in T}{size}_j{v}_{jh}^t\le {area}_h{x}_h,\forall h\in H_M$$ (a29)

$$\sum _{j\in J}\sum _{t\in T}{v}_{jh}^t\le 1,\forall h\in H_M$$ (a30)

$$\sum _{h\in H_M}{x}_h\le ned$$ (a31)

$$\sum _{j\in J}\sum _{t\in T}\sum _{h\in H_M}{v}_{jh}^t\le nmc$$ (a32)

$$w_{cnh}^t\ge 0,\forall c\in C,\forall n\in N,\forall h\in \left(H_M\cup H_E\right),\forall t\in T$$ (a33)

$${transw}_{z{h}^{\prime }h}^t\ge 0,\forall z\in Z,\forall h,h^{\prime }\in \left(H_M\cup H_E\right),\forall t\in T$$ (a34)

$$trans{p}_{z{h}^{\prime }h}^t\ge 0,\forall z\in Z,\forall h^{\prime }\in H_F,\forall h\in \left(H_M\cup H_E\right),\forall t\in T$$ (a35)

$${rmed}_{h{h}^{\prime }}^t\ge 0,\forall h,h^{\prime }\in \left(H_E\cup H_M\right),h\ne h^{\prime },\forall t\in T$$ (a36)

$${amed}_h^t\ge 0,\forall h\in \left(H_E\cup H_M\right),\forall t\in T$$ (a37)

$${dmed}_h^t\ge 0,\forall h\in \left(H_E\cup H_M\right),\forall t\in T$$ (a38)

$${med}_h^t\ge 0,\forall h\in \left(H_E\cup H_M\right),\forall t\in T$$ (a39)

$$t{w}_{ch}^t\ge 0,c_{hw},c_{mw}\in C,\forall h\in \left(H_M\cup H_E\right),\forall t\in T$$ (a40)

$${ar}_{zh}^t\ge 0,\forall z\in Z,\forall h\in \left(H_M\cup H_E\right),\forall t\in T$$ (a41)

$${distw}_{zh}^t\ge 0,\forall z\in Z,\forall h\in \left(H_M\cup H_E\right),\forall t\in T$$ (a42)

$${avcap}_{zh}^t\ge 0,\forall z\in Z,\forall h\in \left(H_M\cup H_E\right),\forall t\in T$$ (a43)

$${intcap}_{zh}^t\ge 0,\forall z\in Z,\forall h\in \left(H_M\cup H_E\right),\forall t\in T$$ (a44)

$$v_{jh}^t\ge 0,integer,\forall j\in J,\forall h\in \left(H_E\cup H_M\right),\forall t\in T$$ (a45)

$$y_{h{h}^{\prime }}^t\in \left\{\mathrm{0,1}\right\},\forall h,h^{\prime }\in \left(H_E\cup H_M\right),h\ne h^{\prime },\forall t\in T$$ (a46)

$$x_h\in \left\{\mathrm{0,1}\right\},\forall h\in H_M$$ (a47)

The model formulated above aims to minimize the total travel time of disaster victims. The transportation of casualties to emergency units and full-fledged HF and the transportation of patients at damaged hospital are considered. All casualties are carried by ambulances. Similar to Ref. [22]; we assume that an ambulance is able to carry one heavily, three moderately and six slightly wounded casualties at once. Likewise, one patient in the need of ICU and three patients in the need of non-ICUs can be transferred by an ambulance as well. Due to the damages that may occur on the roads, an increase in the transportation time is expected. The uncertain duration of possible damages on road is reflected by the distance increase rate presented by Ref. [22]. Constraints in (a1) guarantee that heavily and moderately wounded victims are sent to emergency units on time. Different than heavy and moderate casualties, the proposed model allows delay in the transportation of minimal casualties. Constraints in (a2) ensure that all slightly wounded victims are transported within the planning time horizon. Constraints in (a3) prevent HFs from serving slightly wounded victims before demand occur. For the sake of resource efficiency, the proposed model allows medical team rotation between emergency units, so the available capacity is updated at each period. Constraints in (a4) and (a5) calculate the number of arrived and departed medical teams. The assignment of any medical team to non-opened DE is prevented by constraints in (a6). Constraints in (a7) update the available number of medical team at each emergency unit. Constraints in (a8) and (a9) assign the initial number of medical team at functioning hospitals and DEs respectively. Constraints in (a10) ensure that the available capacity of the emergency units of HFs is not exceeded. Constraints in (a11) assure that medical teams cannot be rotated between two HFs if the threshold value for the distance is exceeded. Constraints in (a12) state that the unutilized capacity of the emergency units where a new medical team assigned should be less than the corresponding threshold value. Otherwise, new medical teams are not allowed to be assigned. Constraints in (a13) enable emergency units receiving medical team if and only if any team is not sent to another facility from the same emergency unit at the same time. Constraints in (a14) do not allow to rotate medical team between facilities if the conditions are not satisfied. Constraints in (a15) prevent the number of rotated medical teams from exceeding the initial capacity more than the specified threshold value. Constraints in (a16) prevent the number of medical teams from remaining under the predetermined threshold value. We assume that heavy and moderate casualties receive emergency medical care assistance and need to be transferred to full-fledged HF within the next period. Constraints in (a17) calculate the number of casualties who need to be transferred to a full-fledged HF. Constraints in (a18) prevent calculation of ICU and non-ICU support at period t0. Constraints in (a19) and (a20) ensure that the casualties in the need of ICU and non-ICU support at emergency units are sent to a full-fledged HF. Constraints in (a20) assure that patients at damaged hospitals are transferred to functioning or temporary HFs within an acceptable time. Respectively, constraints in (a21) and (a22) calculate the number of arrived and discharged casualties and patients sent from damaged hospitals. Respectively, Constraints in (a23) and (a24) assign and update the initial ICU and non-ICU capacity of functioning hospitals. DMF cannot be installed immediately due to the transportation and complexity of the equipment to be installed. Constraints in (a25) prevent opening any DMF before the time for installation. Constraints in (a26) prevent assigning the capacity before the period DMF are established and exceed the maximum capacity after establishment. These facilities admit patients one period after installation. Constraints in (a27) calculate the initial capacity of DMF. Constraints in (a28) update the available capacity of ICU and non-ICU beds of DMF. While constraints in (a29) prevent the DMF from exceeding the area where it will be established, it also ensures that DMF are set up at the points where DEs are opened. Constraints in (a30) allow opening of one type of DMF at candidate locations. Respectively, constraints in (a31) and (a32) ensure that in total $nmc$ DE and $ned$ DMF serve for disaster victims. (a33-a44) are non-negativity, and (a45-a47) are integrality constraints.

3.2. Model-II: projecting COVID-19 spread after an earthquake

To predict the COVID-19 spread after an earthquake, a compartmental model for COVID-19 transmission is constructed. Two integrated cases are generated considering the conditions before and after a seismic event. First, we take the model introduced by Ref. [78] into the account as a basis. Then, it is modified to SEI ¹ IS ¹ I ² I ³ TR-D compartmental model, where isolated and non-isolated individuals with mild disease and transferred individuals with critical disease are added as new classes. The connection of different compartments is demonstrated in Fig. 2 . Accordingly, individuals are classified in one of the following stages: susceptible (S), exposed (E), infected individuals with mild disease who are not isolated (I ¹), infected individuals with mild disease who are isolated (IS ¹), infected individuals with severe disease (I ²), infected individuals with critical disease (I ³), infected individuals with critical disease who are transferred back to non-ICUs from ICUs (TI ³), recovered (R) and dead (D). The population size is presented as N = S + E + I ¹ + IS ¹ + I ² + I ³ + TI ³ + R + D.

Fig. 2.

SEI¹IS¹I²I³TR-D disease spread model.

Individuals who are not infected but at the risk of infection are called Susceptible (S). Individuals move from stage (S) to exposed stage (E), in which they are asymptomatic without being infected. The progression rate from stage (E) to infected stage (I ¹ and IS ¹), where the individual is symptomatic with mild disease, occurs at a rate of α, and individuals are isolated at a rate of θ. Isolated (IS ¹) and non-isolated (I ¹) infected individuals with mild disease progress to severe infection (I ²) at a rate of $p^1$ or recover at a rate of ${\gamma }^1$. Severely infected patients (I ²) recover at a rate of ${\gamma }^2$ or progress to a critical stage (I ³) at a rate of $p^2$. Individuals with critical infection (I ³) recover from the critical condition and return to the normal service. The progression rate of the individuals at stage (I ³) to the transferred individuals (TI ³) is ${\gamma }^2$. Individuals with critical infection (I ³) die at a rate of μ. Individuals who are recovered from the disease are included in class (R), and at this stage it is considered that the individuals have become immune and will not be sick again. Infected individuals (I ¹ , IS ¹ , I ² , I ³ , TI ³) transmit infection at different rates. The transmission rates of infected individuals are defined as ${\beta }_1,{\beta }_{\mathrm{s}1},{\beta }_2,{\beta }_3,{\beta }_{\mathrm{t}3}$. To observe the effect of social distancing measures on the spread of the disease before and after the earthquake, p value is defined that affects β contact rate. p takes a value between 0 and 1. A value of 1 indicates that there is no practice to prevent social contact. The gradual decrease of this value means the gradual increase in the precautions.

3.3. Model-III: facility location and allocation model for COVID-19 hospitals and temporary isolation centers

Considering the logistic network schema represented in Fig. 3 , we propose a framework to determine the number and location of designated COVID-19 hospitals (CH) and temporary isolation centers (IC) in a case of infectious disease outbreak after an earthquake.

Fig. 3.

General schema of the suggested logistic network configuration.

Assumptions for the proposed mathematical model are given as follows:

According to the health conditions of infected individuals, patients are classified. Respectively, individuals with mild, severe, and critical diseases are represented in three stages: I ¹, I ² and I ³. Assumptions based on the infected individuals are made by following Hill [78]. Infected individuals cannot reach second class (I ²) and third class (I ³) unless they pass through first class (I ¹). Similarly, infected individuals cannot enter third class (I ³) before reaching second class (I ²). Due to the life-threatening health conditions of infected individuals in third class (I ³), only patients with critical disease can be recorded as dead. Infected individuals are treated at dedicated healthcare facilities: COVID-19 hospitals and isolation centers. COVID-19 hospitals are considered as full-fledged hospitals where only infected individuals with severe (I ²) and critical conditions (I ³) are treated. Different than COVID-19 hospitals, isolation centers only monitor and serve to infected individuals in class (I ¹), who need basic medical treatment. In this point, isolation centers have similarity to Fangcang hospitals, which were established as field hospitals to isolate individuals with minor symptoms in Wuhan, China [79].

The medical examination and intervention of individuals with mild disease (I ¹) are initially made at COVID-19 hospitals, and then they are transferred to isolation centers. Infected individuals whose symptoms are deteriorated into severe conditions (I ²) are transferred back to COVID-19 hospitals and treated at non-ICUs. Infected individuals whose symptoms are deteriorated into critical conditions (I ³) are transferred to ICUs. Based on the medical consultant opinions, we assumed that if patients in class (I ³) are recovered from critical conditions, they are transferred back to non-ICUs to complete their treatment.

The goal of the multi-period MILP location model is to find the optimal location decisions of COVID-19 hospitals and isolation centers. Also, allocation decisions of infected patients from demand points to designated facilities are decided. The notifications and formulation of the mathematical model are given as follows:

Formulation of Model-III.

Additional sets and indices:
$I,i$	Set and index of candidate isolation centers (IC), $i\in I$
Additional parameters
${tdh}_{nh}$	Travel time (min) from neighborhood $n$ to HF $h$, where $h\in \left(H_E\cup H_M\right)$
${thi}_{hi}$	Travel time (min) from HF $h$, where $h\in \left(H_E\cup H_M\right)$, to isolation center $i$
${tih}_{ih}$	Travel time (min) from isolation center $i$ to HF $h$, where $h\in \left(H_E\cup H_M\right)$
$I_{nk}^1$	Number of non-reported individuals with mild disease at neighborhood $n$ at time period $k$
${IS}_{nk}^1$	: Number of reported individuals with mild disease at neighborhood $n$ at time period $k$
$R_h^1$	: ICU capacity at candidate COVID-19 hospital $h$
$R_h^2$	: non-ICU capacity at candidate COVID-19 hospital $h$
$R_i^3$	: Bed capacity at candidate isolation center $i$
${\psi }_1$	: Threshold value for the number of CH
${\psi }_2$	: Threshold value for the number of IC
Additional decision variables
${PI}_{nhk}^1$	: Number of non-isolated individuals with severe disease at neighborhood $n$ sent to HF $h$ at time period $k$
${PIS}_{nhk}^1$	: Number of new isolated individuals with mild disease at neighborhood $n$ sent to HF $h$ at time period $k$
${TI}_{hik}^1$	Number of new individuals with mild disease transferred to isolation center $i$ from HF $h$ at time period $k$
${RI}_{ik}^1$	: Number of new recovered individuals with mild disease at isolation center $i$ at time period $k$
${PI}_{ik}^2$	: Number of new individuals with severe disease at isolation center $i$ at time period $k$
${TI}_{ihk}^2$	: Number of new individuals with severe disease transferred to HF $h$ from isolation center $i$ at time period $k$
${RI}_{hk}^2$	: Number of new recovered individuals with severe disease at HF $h$ at time period $k$
$I_{hk}^2$	: Number of individuals with severe disease at HF $h$ at time period $k$
${PI}_{hk}^3$	: Number of new individuals with critical disease at HF $h$ at time period $k$
${RI}_{hk}^3$	: Number of recovered individuals with critical disease who transferred to non-ICU's from ICUs of HF $h$ at time period $k$
${TIC}_{hk}^3$	: Number of individuals with severe disease transferred to non-ICU's from ICUs of HF $h$ at time period $k$
${dh}_h$	: $\{\begin{array}{c}1,\mathrm{i}\mathrm{f}\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{h}\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}h\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{a}\mathrm{s}\mathrm{C}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}19\mathrm{h}\mathrm{o}\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{t}\mathrm{a}\mathrm{l}\\ 0,\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}$
${ic}_i$	: $\{\begin{array}{c}1,\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}i\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\\ 0,\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}$
${TI}_{hk}^3$	: Number of new individuals with severe disease transferred to non-ICU's from ICUs of HF $h$ at time period $k$
$I_{hk}^3$	: Number of individuals with critical disease at HF $h$ at time period $k$
${DI}_{hk}^3$	: Number of new individuals with critical disease died

3.3.1. Model-III

$$Minimize{f}_2=\sum _{h\in H_E\cup H_M}\sum _{n\in N}\sum _{k\in K}{{tdh}_{nh}(PI}_{nhk}^1+{PIS}_{nhk}^1)+\sum _{h\in H_E\cup H_M}\sum _{i\in I}\sum _{k\in K}{thi}_{hi}{TI}_{hik}^1+\sum _{h\in H_E\cup H_M}\sum _{i\in I}\sum _{k\in K}{tih}_{ih}{TI}_{ihk}^2$$ (Obj2)

$$\sum _{h\in H_E\cup H_M}{PI}_{nh\left(k+1\right)}^1={p^{1}I}_{nk}^1,\forall n\in N,\forall k\in K|k\le K-1$$ (b1)

$$\sum _{h\in H_E\cup H_M}{PIS}_{nhk}^1={IS}_{nk}^1,\forall n\in N,\forall k\in K$$ (b2)

$$\sum _{i\in I}{TI}_{hik}^1=\sum _{n\in N}{PIS}_{nhk}^1,\forall h\in \left(H_E\cup H_M\right),\forall k\in K$$ (b3)

$${RI}_{i\left(k+1\right)}^1=\sum _{h\in H_E\cup H_M}{\gamma }^1{TI}_{hik}^1,\forall i\in I,\forall k\in K|k\le K-1$$ (b4)

$${PI}_{i\left(k+1\right)}^2=\sum _{h\in H_E\cup H_M}{p}^1{TI}_{hik}^1,\forall i\in I,\forall k\in K|k\le K-1$$ (b5)

$$\sum _{h\in H_E\cup H_M}{TI}_{ihk}^2={PI}_{ik}^2,\forall i\in I,\forall k\in K$$ (b6)

$${RI}_{h\left(k+1\right)}^2=\sum _{i\in I}{\gamma }^2{I}_{hk}^2,\forall h\in \left(H_E\cup H_M\right),\forall k\in K|k\le K-1$$ (b7)

$${PI}_{h\left(k+1\right)}^3=p^2{I}_{hk}^2,\forall h\in \left(H_E\cup H_M\right),\forall k\in K|k\le K-1$$ (b8)

$$I_{h\left(k+1\right)}^2=\sum _{i\in I}{TI}_{ih\left(k+1\right)}^2+\sum _{n\in N}{PI}_{nh\left(k+1\right)}^1+I_{hk}^2-\left({RI}_{h\left(k+1\right)}^2+{PI}_{h\left(k+1\right)}^3\right),\forall h\in \left(H_E\cup H_M\right),\forall k\in K|k\le K-1$$ (b9)

$${TI}_{h\left(k+1\right)}^3={\gamma }^3{I}_{hk}^3,\forall h\in \left(H_E\cup H_M\right),\forall k\in K|k\le K-1$$ (b10)

$${DI}_{h\left(k+1\right)}^3=\mu I_{hk}^3,\forall h\in \left(H_E\cup H_M\right),\forall k\in K|k\le K-1$$ (b11)

$$I_{h\left(k+1\right)}^3={PI}_{h\left(k+1\right)}^3+I_{hk}^3-\left({DI}_{h\left(k+1\right)}^3+{TI}_{h\left(k+1\right)}^3\right),\forall h\in \left(H_E\cup H_M\right),\forall k\in K|k\le K-1$$ (b12)

$${RI}_{h\left(k+1\right)}^3={{\gamma }^{2}TCI}_{hk}^3,\forall h\in \left(H_E\cup H_M\right),\forall k\in K|k\le K-1$$ (b13)

$${TCI}_{h\left(k+1\right)}^3={TI}_{h\left(k+1\right)}^3+{TT}_{hk}^3-{RI}_{h\left(k+1\right)}^3,\forall h\in \left(H_E\cup H_M\right),\forall k\in K|k\le K-1$$ (b14)

$$I_{hk}^3\le R_h^1{dh}_h,\forall h\in \left(H_E\cup H_M\right),\forall k\in K$$ (b15)

$$I_{hk}^2+{TCI}_{hk}^3\le R_h^2{dh}_h,\forall h\in \left(H_E\cup H_M\right),\forall k\in K$$ (b16)

$$\sum _{h\in H_E\cup H_M}{TI}_{hik}^1\le R_i^3{ic}_i,\forall i\in I,\forall k\in K$$ (b17)

$$\sum _{n\in N}{PIS}_{nhk}^1\le L{dh}_h,\forall h\in \left(H_E\cup H_M\right),\forall k\in K$$ (b18)

$$\sum _{h\in H_E\cup H_M}{dh}_h\le {\psi }_1\text{,}$$ (b19)

$$\sum _{i\in I}{ic}_i\le {\psi }_2,$$ (b20)

$${PI}_{nhk}^1\ge 0,\forall n\in N,\forall h\in \left(H_E\cup H_M\right),\forall k\in K$$ (b21)

$${PIS}_{nhk}^1\ge 0,\forall n\in N,\forall h\in \left(H_E\cup H_M\right),\forall k\in K$$ (b22)

$${TI}_{hik}^1\ge 0,\forall h\in \left(H_E\cup H_M\right),\forall i\in I,\forall k\in K$$ (b23)

$${RI}_{ik}^1\ge 0,\forall i\in I,\forall k\in K$$ (b24)

$${PI}_{ik}^2\ge 0,\forall i\in I,\forall k\in K$$ (b25)

$${TI}_{ihk}^2\ge 0,\forall i\in I,\forall h\in \left(H_E\cup H_M\right),\forall k\in K$$ (b26)

$${RI}_{hk}^2\ge 0,\forall h\in \left(H_E\cup H_M\right),\forall k\in K$$ (b27)

$$I_{hk}^2\ge 0,\forall h\in \left(H_E\cup H_M\right),\forall k\in K$$ (b28)

$${PI}_{hk}^3\ge 0,\forall h\in \left(H_E\cup H_M\right),\forall k\in K$$ (b29)

$${TI}_{hk}^3\ge 0,\forall h\in \left(H_E\cup H_M\right),\forall k\in K$$ (b30)

$$I_{hk}^3\ge 0,\forall h\in \left(H_E\cup H_M\right),\forall k\in K$$ (b31)

$${DI}_{hk}^3\ge 0,\forall h\in \left(H_E\cup H_M\right),\forall k\in K$$ (b32)

$${RI}_{hk}^3\ge 0,\forall h\in \left(H_E\cup H_M\right),\forall k\in K$$ (b33)

$${TIC}_{hk}^3\ge 0,\forall h\in \left(H_E\cup H_M\right),\forall k\in K$$ (b34)

$${dh}_h\in \left\{\mathrm{0,1}\right\},\forall h\in \left(H_E\cup H_M\right)$$ (b35)

$${ic}_i\in \left\{\mathrm{0,1}\right\},\forall i\in I$$ (b36)

The objective function (obj 2) of Model-III minimizes the summation of expected travelled distance of infected individuals. Late reported infected individuals with mild disease are assumed to be admitted to COVID-19 hospitals after their health conditions are deteriorated to severe conditions. Constraints in (b1) ensure that late reported individuals with severe disease are sent to COVID-19 hospitals from each neighborhood. Reported individuals with mild disease is assumed to be admitted to COVID-19 hospitals and directly transferred to isolation centers. Accordingly, constraints in (b2) guarantee that reported individuals with mild disease are sent to COVID-19 hospitals from each neighborhood. Constraints in (b3) indicates that the number of individuals with mild disease at COVID-19 hospitals should be transferred to isolation centers. Herein, individuals with mild disease may be recovered, or their health conditions are deteriorated to severe conditions. Respectively, constraints in (b4) calculate the number of recovered individuals with mild disease, and constraints in (b5) calculate the number of individuals whose symptoms are deteriorated into severe conditions at isolation centers. Constraints in (b6) calculate the number of individuals transferred to COVID-19 hospitals from isolation centers. Constraints in (b7) calculate the number of recovered individuals who are transferred to COVID-19 hospitals from isolation centers. Constraints in (b8) calculate the number of individuals whose symptoms are deteriorated into critical conditions. Constraints in (b9) update the current number of individuals with severe disease at each COVID-19 hospital. Constraints in (b10) calculate the number of transferred individuals who are sent to non-ICUs from ICUs. Constraints in (b11) indicate the number of individuals died from COVID-19. Constraints in (b12) calculate the current number of individuals with critical disease. We assumed that if individuals in class ($I_3$) are recovered from critical conditions, they are transferred back to non-ICUs to complete their treatment. Constraints in (b13) calculates the number of recovered individuals transferred from ICUs. Constraints in (b14) update the current number of individuals transferred from ICUs. Constraints in (b15), (b16) and (b17) assure that the number of individuals does not exceed the capacity of selected facilities. Herein, we consider that the pre-determined percentage of existing capacity of candidate COVID-19 hospitals are allocated to infected individuals from the outside of the considered part of the district. Constraints in (b18) prevent the assignment of any individuals with mild disease to non-COVID 19 hospitals. Constraints in (b19) and (b20) assure that the number of opened facilities does not exist the predetermined threshold value. Constraints in (b21-b36) are non-negativity and binary assignments.

4. Data collection and analysis

A case study for a district Bakirkoy, Istanbul, in which a large-scale earthquake is expected in near future, is performed for computational studies. The earthquake hazard and risk assessment of Bakirkoy, İstanbul is recently updated and reported by the collaboration of Istanbul Metropolitan Municipality (IMM) and Boğaziçi University Kandilli Observatory and Earthquake Research Institute (BU-KOERI). In the associated booklet [80], the potential damages and losses aftermath of an earthquake is projected, and night-time and day-time scenarios are generated. Night-time scenario is the most pessimistic one with a magnitude of 7.5Mw, and it is assumed that the registered population in the entire region is in the household. Bakirkoy has 15 neighborhoods, and demand points are considered as neighborhood administration units. In the night-time scenario, 581 heavily wounded casualties, 4601 slightly wounded casualties and 2701 hospitalizations are expected. We assume that hospitalized disaster victims correspond to heavily and moderately wounded casualties. Model-I is solved for nine consecutive time periods, representing 72 h aftermath the earthquake, where rapid changes in demand and service capacity occur. As it is indicated above, we obtain the total number of injured disaster victims from (IMM and BU-KOERI, 2020). By considering the fatal seismic events occurred in Turkey in last decades, we assume that 55%, 80% and 95% of casualties are expected to reach healthcare services in the first 24, 48 and 72 h respectively [[11], [60], [61],[81], [82]]. In this study, the number of injured people who will need medical support at each period is generated by Monte Carlo simulation considering the parameters indicated by Refs. [[11], [60], [61],[81], [82]]. The results are given in Appendix Table A1. After the initial treatment at emergency units, heavy and moderate casualties are transferred to the full-fledged HFs. Considering the need of intensive care of casualties in 2011 Van Earthquake, we assume that the ICUs should be reserved for 13% of heavily wounded casualties [82]. Further, the discharge rate of moderate casualties from the full-fledged HFs is arranged as 0.2. There are eight functioning hospitals in Bakirkoy. Due to the risk of tsunami after an earthquake, we disregard two hospitals located at the point where horizontal flood distance to Marmara coastline reaches approximately 360 m, according to the Bakirkoy Tsunami Risk Analysis and Action Plan booklet [83]. The ICU and non-ICU capacity of hospitals are gathered from the publishment of the MoH of Turkey [84] and gazette news [[85], [86]]. The number of medical team at emergency units is identified according to the minimum standard rule of emergency services, given by the MoH of Turkey [87]. Appendix Table A2 demonstrates the emergency unit and hospital-bed capacity of functioning hospitals at Bakirkoy. Note that 72% and 55% of ICU and non-ICU beds at functioning hospitals occupied by COVID-19 and non-COVID-19 patients according to the COVID-19 statistics given by the MoH of Turkey [88]. Emergency assembly points are considered as candidate locations for DEs, which can be converted to DMFs. 15 candidate locations are selected according to their surface area, distance to coastline and distance to emergency main arterial roads [[89], [90]]. Capacious DMFs have the disadvantage of set-up time. For these reasons, three type of DMF with 30-beds, 50-beds and 150-beds are considered. By doing so, we can benefit from areas located at critical hubs yet not suitable for broad DMF. The area and capacity of these temporary facilities are gathered from U-project website [91]. We assume that 30-beds, 50-beds and 150-beds can be installed in one, two and three time period respectively. The property of field hospitals is shown in Appendix Table A3. Last, [92] expresses that a medical team can examine nine patients with minor injuries, or three patients with serious injuries simultaneously. Accordingly, we assume that a medical team has nine-unit capacity. Slightly wounded casualties seize one-unit and heavily casualties seize three-unit. Four medical teams are assumed to be assigned to DEs initially. The coordinates of functioning and candidate HFs and distances between relevant nodes are gathered from google maps [93]. We set the vehicle velocity to 50 km/h and reflect the expected increase in travel time caused by possible road damages with distance increase rate, uniformly distributed between 3% and 38% as given by Ref. [22]. We set the threshold value for the minimum travel time of a medical team as five km and threshold value for the minimum unused capacity of emergency units as 10 units. The threshold coefficients for the number of rotated medical teams and the available medical teams are arranged as three and 0.5. To reflect the transmission of COVID-19 in Bakirkoy, Istanbul, we collect the statistical data from local resources and relevant existing literature. The parameters and literature are given in Appendix Table A5, and the values used before and after the earthquake presented in Appendix Table A6. Sinan Erdem Stadium, CNR Expo Fair Center, Turkish Airlines Training Center and Florya Metin Oktay Stadium are selectes as the candidate isolation centers.

5. Results, discussions and conclusions

5.1. Analysis of Model-I

The problem is solved via MIP using CPLEX methodology in GAMS.

5.1.1. Effects of expansion in medical team

We examine the effect of different numbers of medical teams, DE and DMF, and the results are shown in Fig. 4 . The unsatisfied demand of casualties (a) and patients (b) decreases as the number of medical teams at emergency units and DE and DMF increases. Decreasing unsatisfied demand results in a significant decline in the total travel time of disaster victims (c). This is mainly because of the reduction in total unsatisfied demand penalty cost. On the other hand, when the number of satisfied demand increase, the total travel time of treated casualties and victims increase (d). The reason lying behind this situation is the fact that since the treatment capacity is enlarged, more victims are sent to medical facilities instead of the dummy hospital.

Fig. 4.

Analysis of different numbers of medical teams, DE and DMF.

5.1.2. Effects of resource sharing between emergency units

For the sake of rapidness and efficiency, medical teams can be rotated between emergency units by expanding the threshold value of travel time ($\rho$). In this section, the impacts of maximum time that medical teams are allowed while traveling between medical units are analyzed. Fig. 5 demonstrates the objective function values sensitivity considering the number of medical teams and the variation in the maximum travel time allowed. Herein, (a) demonstrates the improvement rate in the travel time of disaster victims, including unsatisfied demands, and (b) only shows improvement rate in the travel time of the treated casualties. According to the results, sharing resources between emergency units can only decrease the travel time of treated disaster victims; on the other hand, as it is given in Appendix Table A4, it does not impact the number of unsatisfied demands of casualties and patients. This is the reason why a drastic change in (a) is not observed. ıt is also perceived that allowing the medical teams to travel between emergency units is significant if the number of the medical teams at is more limited. For instance, in Fig. 5 (b), while the improvement rate of objective function value is varied between %18-%20 in the case where 5 medical teams are assigned, this rate is decreased to %10-%15 in the case where 11 medical teams are assigned.

Fig. 5.

Analysis of different numbers of medical team and threshold value for travel time.

5.1.3. Effects of medical facilities

Throughout this section, Model-I is solved for 36 different cases, generated by varying the number of DE and DMF. We set the number of the medical teams and the threshold value for travel time to seven and five respectively. To interpret results comprehensively, the objective function values are provided in two different ways: unsatisfied demand is included (UDI) and unsatisfied demand is not included (UDNI) in Table 2 . The first column in Table 2 shows the number of established DMF, and the second row demonstrates the number of established DE. Predictably, when the number of both facilities increases, the total travel time of casualties and patients declines.

Table 2.

Analysis of different numbers of DE and DMF on the objective function values.

Number of DE
		1		3		5		7
		UDI	UDNI	UDI	UDNI	UDI	UDNI	UDI	UDNI
Number of DMF	0	1,293,353	2553	991,297	2897	695,844	3581	440,303	4094
	1	1,293,207	2407	991,036	2636	689,177	3177	402,709	3999
	3	–	–	990,772	2372	688,790	2790	387,210	3610
	5	–	–	–	–	688,647	2647	386,932	3332
	7	–	–	–	–	–	–	386,808	3208
	9	–	–	–	–	–	–	–	–
	11	–	–	–	–	–	–	–	–
	13	–	–	–	–	–	–	–	–
	14	–	–	–	–	–	–	–	–
		9		11		13		14
		UDI	UDNI	UDI	UDNI	UDI	UDNI	UDI	UDNI
Number of DMF	0	297,502	4365	287,510	4453	278,534	4793	278,533	4791
	1	222,517	4379	212,499	4440	203,515	4773	203,511	4769
	3	143,153	4535	133,092	4555	124,024	4803	124,021	4799
	5	117,292	4292	79,521	4589	70,400	4784	70,396	4780
	7	116,978	3978	64,276	4277	18,345	4925	14,813	4950
	9	116,850	3851	63,997	3997	14,071	4471	4620	4620
	11	–	–	63,890	3890	13,864	4264	4371	4371
	13	–	–	–	–	13,770	4170	4263	4263
	14	–	–	–	–	–	–	4247	4247

Results are rounded to the nearest integer.

UDI: Unsatisfied demand is included.; UDNI: Unsatisfied demand is not included.

However, in contrast to a consistent decrease in UDI, UDNI increases as the number of DE increases. The reason lying behind this situation is the fact that when the number of satisfied demand boosts, as shown in Table 3 , with enlarging capacity, the total time travel of treated disaster victims increases. In Table 3, slight increases in the number of unsatisfied patients and casualties are observed in the despite of the growing capacity of medical centers. This is due to the trade-offs between patients and slight, severe, critical casualties.

Table 3.

Unsatisfied demand of casualties and patients.

	Number of DE
		1		3		5		7		9		11		13		14
		Casualty	Patient	Casualty	Patient	Casualty	Patient	Casualty	Patient	Casualty	Patient	Casualty	Patient	Casualty	Patient	Casualty	Patient
Number of DMF	0	4407	1212	3448	1009	2557	904	758	758	682	733	623	708	582	684	581	684
	1	4381	1185	3429	984	2445	761	571	571	519	545	477	520	421	497	105	711
	3	–	–	3416	977	2404	720	476	476	347	347	290	321	260	298	260	298
	5	–	–	–	–	2456	721	407	407	289	276	187	187	153	164	153	164
	7	–	–	–	–	–	–	417	417	279	266	150	150	34	34	25	25
	9	–	–	–	–	–	–	–	–	273	260	150	150	24	24	0	0
	11	–	–	–	–	–	–	–	–	–	–	150	150	24	24	0	0
	13	–	–	–	–	–	–	–	–	–	–	–	–	24	24	0	0
	14	–	–	–	–	–	–	–	–	–	–	–	–	–	–	0	0

Results are rounded to the nearest integer.

Moreover, the improvement in the objective function values is more apparent in cases where the number of DE increase since casualties who are in the need of first aid are considerably more than patients who are in the need of advanced treatment.

5.2. Analysis of SEI¹IS¹I²I³TR-D disease spread model on post-disaster period

To analyze the spread of the disease, 18 scenarios are generated on SEI ¹ IS ¹ I ² I ³ TR-D model by varying the pandemic day that disaster occur (DD), the degree of non-pharmacological measures (p) taken before (MBE) and after (MAE) the earthquake.

In post-disaster period, individuals who lost their lives due to the earthquake are excluded from the number of individuals in the society, and the results are analyzed for 120 days.

5.2.1. Effect of the disaster day

The effect of disaster day occurred in COVID-19 is evaluated based on the maximum number of infected individuals, day maximum number of infected individuals, the maximum need of non-ICU and ICU and number of deaths. Scenarios are generated based on the assumption that the earthquake is occurred on 30th, 60th and 90th day of COVID-19 pandemic, and are compared in three groups (Table 4 ). In the first group, it is assumed that no infected individual is isolated (θ: 0), and Scenario 1, Scenario 7 and Scenario 10, where p (MBE) and p (MAE): are taken as 0.8 and 1 are evaluated. In the second group, it is assumed that no infected individual is isolated (θ: 0), and Scenario 3, Scenario 8 and Scenario 11, where p (MBE) and p (MAE): are taken as 0.8 and 0.8 are evaluated. In the third group, it is assumed that 30% of infected individual is isolated (θ: 0.3), and Scenario 3, Scenario 8 and Scenario 11, where p (MBE) and p (MAE): are taken as 0.8 and 0.8 are evaluated. The results are shown on Table 5 .

Table 4.

Scenarios generated model in the post-disaster period.

Scenario Number	Disaster Day (DD)	p-value before the earthquake (p - MBE)	p-value after the earthquake (p - MAE)
S1	30	0.8	1
S2	30	0.8	0.9
S3	30	0.8	0.8
S4	30	0.8	0.7
S5	30	0.8	0.6
S6	30	0.8	0.5
S7	60	0.8	1
S8	60	0.8	0.8
S9	60	0.8	0.6
S10	90	0.8	1
S11	90	0.8	0.8
S12	90	0.8	0.6
S13	30	0.7	1
S14	30	0.6	1
S15	30	0.5	1
S16	30	0.7	0.8
S17	30	0.6	0.8
S18	30	0.5	0.8

Table 5.

Evaluation of the disaster day.

Group	Parameters	Scenario Number	Disaster day (DD)	Maximum number of infected individuals	Day when maximum number of infected individuals recorded	Non-ICU needs	ICU needs	Number of deaths
1	θ:0 p (MBE):0.8 p (MAE):1	S1	30	76,174	46	8300	2114	3353
		S7	60	76,158	34	8286	2113	3353
		S10	90	74,932	22	8159	2085	3353
2	θ:0 p (MBE):0.8 p (MAE):0.8	S3	30	67,663	54	7498	1958	3341
		S8	60	67,503	39	7498	1957	3341
		S11	90	66,546	25	7385	1385	3341
3	θ:0.3 p (MBE):0.8 p (MAE): 0.8	S3	30	55,646	65	6314	1698	3295
		S8	60	55,693	47	6311	1697	3292
		S11	90	55,305	29	6266	1687	3294

Results are rounded to the nearest integer.

In the first group, the postponement of the earthquake from the 30th day to 60th day and to 90th day reduces the maximum number of infected and number of patients in need of ICU and non-ICU. Also, it backdates the day that the maximum number of infected numbers recorded. No effect on the number of death due to COVID-19 is obtained in grouped scenarios. In the second group, similar outputs are obtained. Herein, due to increased p value, the number of infected patients and the number of individuals who lost their lives are less than Group 1. In Group 3, unlike Group 1 and Group 2, it is assumed that 30% of the infected patients are isolated. Approximately, 15% and %1 decrease are obtained in the maximum number of infected individuals and number of deaths. Also, there are negligible fluctuations recorded in the maximum number of infected and the number of individuals who died.

5.2.2. The effect of MBE on post-disaster period

In this part, the effect of measures to reduce social contact before the disaster on the epidemic spread model after the disaster are analyzed.

In Group 1, it is assumed that no action is taken after the earthquake, and MBE value is gradually reduced. It is recorded that the precautions taken before the earthquake did not cause a significant change in the maximum number of infected individuals, a decrease of 0.1 in MBE cause an average of three days delay on the day the maximum number of infected individuals is recorded. In Group 2, it is assumed that 30% of infected individuals are isolated. As in Group 1, the gradual decrease in MBE does not significantly change the maximum number of infected individuals and causes an average of 3 days delay in the day when the maximum number of infected is recorded. On the other hand, the results show that when 30% isolation after the earthquake decreases the maximum number of infected by 11,000 on average, and the day the maximum value is recorded is delayed by an average of 10 days. Last, in Group 3, it is assumed that MBE is 0.8 and 30% of the infected individuals are isolated. While the effect of MBE is similar on outcomes received in Group 1 and Group 2, the maximum number of infected and the lowest lag in the day received obtained due to the precautions taken. The results given in Table 6 are visualized in Fig. 6 .

Table 6.

Evaluation of the MBE.

Group	Parameters	Scenario Number	p (MBE)	Maximum number of infected individuals	Day when maximum number of infected individuals recorded
1	θ:0 DD:30 p (MAE):1	S1	0.8	76.174	46
		S13	0.7	76.211	49
		S14	0.6	76.183	50
		S15	0.5	76.061	53
2	θ:0.3 DD:30 p (MAE):1	S1	0.8	65.700	55
		S13	0.7	65.679	58
		S14	0.6	65.699	60
		S15	0.5	65.668	63
3	θ:0.3 DD:30 p (MAE):0.8	S3	0.8	55.646	65
		S16	0.7	55.672	69
		S17	0.6	55.649	71
		S18	0.5	55.710	75

The results are rounded to the nearest integer.

Fig. 6.

Evaluation of MBE on the number of infected individuals.

5.2.3. The effect of MAE and isolation measures on post-disaster period

In Table 7 , the effect of the measures taken after an earthquake (MAE) that occurred on the 30th day of the pandemic are demonstrated. Herein, it is assumed that MBE is 0.8. From Scenario 1 to Scenario 6, the degree of post-earthquake measures (MBE) is gradually reduced. In addition, in each scenario, the isolation percentage of infected individuals (θ) is increased by 0.1 from 0 to 0.5, and six cases are examined.

Table 7.

Results of S1, S2, S3, S4, S5 and S6.

Maximum number of infected individuals
Percentage of isolated individuals (θ)
		0	0.1	0.2	0.3	0.4	0.5
p - MAE	1	76,174	73,150	69,630	65,700	60,906	55,348
	0.9	72,279	69,029	65,346	61,065	56,089	50,282
	0.8	67,663	64,191	60,237	55,646	50,458	44,417
	0.7	61,899	58,194	54,022	49,228	43,793	37,585
	0.6	54,791	50,887	46,434	41,474	35,896	29,629
	0.5	45,880	41,708	37,112	32,040	26,489	20,446
Number of infected individuals
Percentage of isolated individuals (θ)
		0	0.1	0.2	0.3	0.4	0.5
p - MAE	1	224,652	224,482	224,167	223,592	222,551	220,670
	0.9	224,410	224,084	223,523	222,568	220,940	218,158
	0.8	223,879	223,271	222,296	220,730	218,205	214,096
	0.7	222,732	221,629	219,957	217,411	213,503	207,415
	0.6	220,273	218,298	215,454	211,320	205,237	196,122
	0.5	214,938	214,415	206,555	199,770	190,148	176,213
Number of deaths
Percentage of isolated individuals (θ)
		0	0.1	0.2	0.3	0.4	0.5
p - MAE	1	3353	3351	3346	3337	3322	3294
	0.9	3350	3345	3336	3322	3298	3256
	0.8	3342	3333	3318	3295	3257	3196
	0.7	3325	3308	3283	3245	3187	3096
	0.6	3288	3258	3216	3154	3063	2927
	0.5	3208	3156	3083	2982	2838	2630

We evaluate the scenarios S1, S2, S3, S4, S5 and S6, where MAE is gradually tightened and assumed that no individual is isolated after the earthquake. Herein, the maximum number of infected individuals is decreased by 5.1%, 11.2%, 18.7%, 28.1% and 39.8%, and the number of individuals who lost their lives decreased by 0.1%, 0.3%, 0.8%, 1.9%, and 4.3% respectively. When the 30% of infected individuals are isolated, respectively, the maximum number of infected individuals and lost is decreased by 51.2% and 10.6% in S6 compared to S1. Fig. 7 demonstrates the impact of different isolation degrees (θ) on S1 and S6. In (S1), the number of death and maximum number of infected individuals decrease by 0.1% and 4% when the number of infected individuals isolated are increased from 0% to 10%. In the same scenario, when the isolation degree is increased to 30%, the number of death and maximum number of infected individuals decrease by 0.5% and 13.8% respectively. In S6, when the isolation rate is increased from 0 to 10%, the number death and maximum number of infected individuals approximately decrease by 2% and 9%. When the rate of isolated individuals is increased to 30%, the maximum number of infected individuals decreases by 30% and the number of deaths decreases by 7%. According to the results obtained, when the measures become serious, more effective results are obtained from isolation.

Fig. 7.

Evaluation of isolation degree on S1 and S6.

In Appendix, Table A7, Table A8, demonstrate results of S7, S8, S9, where DD is 60th and S10, S11, S12, where DD is 90th.

5.3. Analysis of Model-III

It is assumed that along with the existing hospitals, h8, h9, h14, h15, h18 are opened as temporary hospitals. The locations of existing hospitals, temporary treatment facilities and candidate isolation hospitals are shown in Fig. 8 .

Fig. 8.

Existing hospitals, temporary health facilities and candidate isolation centers.

Model-III is solved in GAMS using the CPLEX optimization program. The obtained results are optimal. The model is run under the parameters of S6 of SEI ¹ IS ¹ I ² I ³ TR-D disease spread model. Appendix Table A9, Table A10, Table A11, demonstrate the results of opened CH and IC considering different isolation degree. In the results, isolation centers are allowed to be open under the constraints (19) and (20). Sinan Erdem Sports Hall is selected under single facility constraint. In cases where two or more facilities are allowed to be opened, Florya Metin Oktay Stadium is chosen as the isolation center together with Sinan Erdem Sports Hall.

Fig. 9 demonstrates the objective function value of Model-III considering S6 conditions and varying value of isolation degree (θ). In cases where 10% and 20% of infected individuals are isolated, opening one or two COVID-19 hospital are not sufficient to serve all infected individuals. While 30% of infected individuals are isolated, only one CH is expected to meet all infected patients’ demand. Herein it is obvious that increasing isolation degree has a drastic impact on the objective function value due to decreasing demand of infected individuals. While increasing IC one from two improves the objective function value 8% in average, establishing more than two IC does not affect the objective function value.

Fig. 9.

Objective function value of Model-III under S6 parameters varying isolation degree (θ).

6. Discussions

During the latest pandemic, the overwhelming number of infected individuals have been in the need of medical care. For this reason, hospital-based resources were majorly allocated to patients with COVID-19. Dedicating the hospital resources to infected individuals might be useful to cope with the current infectious disease. On the other hand, in a sudden large-scale disaster, healthcare service capacity should be ready to accept disaster victims. Moreover, after a devastating earthquake, the healthcare system is expected to be forced by the increased number of infected individuals due to increase in social contact in society. Since the beginning of the pandemic, many countries have experienced natural disasters. For instance, at the beginning of the COVID-19 pandemic, an earthquake with a magnitude of 5.5 on the Richter scale with its aftershocks happened in the capital of Croatian [8]. An Earthquake with 2575 deaths was recorded in Haiti in 2021. In the same year, floods occurred in India and China [94]. Along with the rest of the world, in the middle of the pandemic, Turkey experienced a deathful disaster, an earthquake with a magnitude of 6,6. Those disasters that the world experienced show the need for a robust response plan for multiple disasters. As it is mentioned previously, disasters during a pandemic are likely to require having a plan considering the increasing number of infected individuals and its impact on healthcare services. The long duration of COVID-19 pandemic demonstrates that it is necessary to have a strong response plan for other disasters with the on-going pandemic. Since the huge unknowns that the world has face at the beginning of the pandemic, a response plan for multiple disasters could not be designed. Nowadays, healthcare experts obtained experience in the management of pandemic. Therefore, the disaster management plan during pandemics can be strengthen.

7. Conclusions

In this study, we point out the importance of preparedness for multiple disasters and introduce an integrated framework for post-disaster response operations in a pandemic. In the first phase of this study, a methodology is developed to determine the optimal number and location of temporary emergency service units and full-fledged treatment facilities to provide post-disaster health care services to the victims with severe, moderate, and mild injuries. In the presented multi-period mixed integer mathematical model, the expected damage to roads and existing hospitals are considered. The rotation of health teams, capacity constraints, and occupancy rate of the patient beds due to COVID-19 are considered. In the second phase of the study, the differential equations based SEIR epidemic model is presented to predict the increase in the number of infected patients after an earthquake. Then, a multi-period mix-integer programming model is developed to select COVID-19 hospitals and isolation centers reserved for infected individuals. The impact of uncertainties brought by disasters have been examined with various sensitivity analyzes and different scenarios.

According to the results of Model-I, the total travel time of disaster victims reduces as the number of medical facilities increase. Moreover, allowing the medical professionals travel between healthcare facilities outcomes a positive result in terms of the number of treated patients under limited resources. Scenario analysis in Model-II demonstrates that the disaster day impacts the maximum number of infected individuals. Moreover, after the earthquake, the measures taken to protect social contact is very significant. The impact of isolation and the measures taken in society is also observed in the results of Model-III, where the number and location of COVID-19 hospitals and isolation centers are selected.

Decision-makers can benefit from the developed framework for single and multiple disasters. Model-I can be used to optimize the number and location of temporary healthcare facilities for only disaster victims. Then, Model-II and Model-III can be integrated with the results of Model-I, and this framework able the decision-makers to deal with victims in the need of medical attention due to earthquakes and pandemics. Apart from an earthquake disaster, Model-II and Model-III can also support decision-makers to control the spread of pandemic disease and select dedicated healthcare facilities for pandemic patients. The proposed model is generic; therefore, it can be applied to other disasters and regions. As a case study, the developed framework is applied to a district of the metropolitan city of Turkey. Since the number of citizens and the expected demand aftermath of an earthquake are very high, this case study provides a satisfaction result for the applicability of the developed methodology.

We encountered some limitations in this research, and it can be extended in several directions. In the first mathematical model, significant parameters: demand and possible damages to roads are considered as stochastic parameters, however, the initial capacity of healthcare facilities is taken as a deterministic parameter. To reflect the uncertainty after an earthquake, the initial capacity of healthcare facilities can be considered as non-deterministic. To analyze the spread of disease, a differential based SEIR model is developed, and the impact of earthquake on social interaction is reflected according to published news. Therefore, it might not reflect the practiced environment for different policies and human behaviors. The impact of social distract after an earthquake can be investigated in detail according to the behavior of society by using agent-based simulation approach. For the future studies, model-I and model-III can be converted to stochastic optimization model to obtain the decisions under uncertain parameters.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix.

Table A.1.

Distribution of the expected number of casualties over the planning time horizon

Neighborhood	Time period
	24-h			48-h			72-h
	t1	t2	t3	t4	t5	t6	t7	t8	t9
Ataköy 1. Kısım	21	18	4	1	3	17	6	3	4
Ataköy 2-5-6. Kısım	162	95	17	1	28	77	18	8	38
Ataköy 3-4-11. Kısım	88	68	15	0	20	54	10	3	20
Ataköy 7-8-9-10. Kısım	137	90	17	7	25	89	20	20	33
Basınköy	84	76	18	5	18	62	19	11	18
Cevizlik	57	39	10	3	14	38	8	5	19
Kartaltepe	358	237	49	6	85	214	44	15	97
Osmaniye	174	121	25	12	31	129	23	22	53
Sakızağacı	92	54	11	0	16	54	12	8	21
Şenlikköy	275	183	52	10	62	144	38	11	63
Yeşilköy	84	63	8	7	17	51	10	4	28
Yeşilyurt	355	207	42	7	70	222	40	26	88
Yenimahalle	65	43	14	2	21	50	12	1	24
Zeytinlik	60	41	10	1	16	27	9	3	21
Zuhuratbaba	221	129	34	10	36	143	23	14	56
Total	2233	1464	326	72	462	1371	292	154	583

Table A.2.

Capacity of functioning hospitals at Bakırköy

Hospital	Index	Emergency service			Hospital-bed			Resource
		Doctor	Nurse	Medical team	Non-ICU bed	ICU bed	Total
Acibadem Bakırköy hospital	h1	2	2	2	75	22	97	(MoH of Turkey, 2018 [87]; Sozcu,2020)
Ethica İncirli hospital	h2	2	2	2	42	18	60	(MoH of Turkey, 2018 [87]; Sozcu,2020)
Özel Çamlık hospital	h3	2	2	2	27	7	34	(MoH of Turkey, 2018 [87]; Sozcu,2020)
Özel Yaşar hospital	h4	2	2	2	11	2	13	(MoH of Turkey, 2018 [87]; Sozcu,2020)
İstanbul Bakırköy Dr. Sadi Konuk Training and Research hospital	h5	10	16	10	612	79	691	(MoH of Turkey, 2018 [87]; SB, 2017)
Prof. Dr. Murat Dilmener Emergency hospital	h6	15	24	15	576	432	1008	(MoH of Turkey, 2018 [87]; SB, 2020)

The number of medical team at h5 and h6 are taken as proportioned to the number of hospital beds due to the different hospital classification.

Table A.3.

Properties of field hospitals

Field hospital	Index	Area	ICU	non-ICU	Set-up time (period)
30-bed	j1	2385.5	4	30	1
50-bed	j2	2385.5	4	50	2
150-bed	j3	6181.2	4	150	3

Table A.4.

Unsatisfied demand of casualties and patients

Threshold value for the travel time of medical teams	Number of medical teams
	5 teams					7 teams					9 teams					11 teams
	Casualty				Patient	Casualty				Patient	Casualty				Patient	Casualty				Patient
	c1	c2	c3	Total		c1	c2	c3	Total		c1	c2	c3	Total		c1	c2	c3	Total
0 min	43	473	1144	1660	516	0	276	13	289	276	0	180	0	180	180	0	162	0	162	164
5 min	45	468	1144	1657	516	0	276	13	289	276	0	171	0	171	180	0	153	0	153	164
10 min	33	501	1144	1678	535	0	276	13	289	276	0	173	0	173	180	0	153	0	153	164
15 min	33	401	1144	1578	468	1	274	13	288	275	0	172	0	172	180	0	154	0	154	164

Table A.5.

Variables of Compartmental Model

Parameter	Description	Value	Literature
IP	Average incubation period	5 days	[95]
DMI	Average mild infections duration	6 days	Viral shedding: [96], Time from symptoms to hospitalization [97,98]:
MR	Average rate of mild infectious	81%	[99]
SR	Average rate of severe infectious	14%	[99]
CR	Average rate of critical infectious	5%	[99]
FR	Fatality rate	2.35%	WHO (2020b)
DHo	Average severe infections hospitalization duration	4 days	[68]
TICUD	Average duration of ICU admission until death or recovery	8 days	[68]

Table A.6.

Variables of SEI¹IS¹I²I³TR-D disease spread model

Variable	Description	Parameter	Value
			Before Earthquake	After Earthquake*
Α	Progression rate from stage (E) to stages (I1) and (IS1).	$\frac{1}{IP}$	0.2	0.2
${\mathit{\beta }}_{\mathbf{1}}$	Transmission rate of non-isolated individuals with mild disease.	[78]	0.5	1.02
${\mathit{\beta }}_{\mathit{s}\mathbf{1}}$	Transmission rate of isolated individuals with mild disease (after the eartquake)	Assumed	0.1	0.2
${\mathit{\beta }}_{\mathbf{2}}$	Transmission rate of individuals with severe disease.	[78]	0.1	0.2
${\mathit{\beta }}_{\mathbf{3}}$	Transmission rate of individuals with critical disease.	[78]	0.1	0.2
${\mathit{\beta }}_{\mathbf{t}\mathbf{3}}$	Transmission rate of transferred individuals with critical disease.	Assumed	0.1	0.2
${\mathit{\gamma }}^{\mathbf{1}}$	Recovery rate due mild infection	$\left(\frac{1}{DMI}\right)MR$	0.135	0.135
${\mathit{\gamma }}^{\mathbf{2}}$	Recovery rate of infected individuals with severe disease	$\left(\frac{1}{DHo}\right)-p^2$	0.206	0.206
${\mathit{\gamma }}^{\mathbf{3}}$	Recovery rate of infected individuals with critical disease	$\left(\frac{1}{TICUD}\right)-u$	0.075	0.075
${\mathit{p}}^{\mathbf{1}}$	Progression rate at which infected individuals in class $I_1$ progress to class $I_2$	$\left(\frac{1}{DMI}\right)-{\gamma }^1$	0.032	0.032
${\mathit{p}}^{\mathbf{2}}$	Rate at which infected individuals in class $I_2$ progress to class $I_3$	$\left(\frac{1}{DHo}\right)\left(\frac{CR}{SR+CR}\right)$	0.044	0.044
Μ	Death rate	$\left(\frac{1}{TICUD}\right)\left(\frac{FR}{CR}\right)$	0.058	0.058

*Parameters taken for after earthquake are based on estimation.

Table A.7.

Results of S7, S8 and S9

Maximum number of infected individuals
Percentage of isolated individuals (θ)
		0	0.1	0.2	0.3	0.4	0.5
p - MAE	1	76,158	73,089	69,604	65,664	60,924	55,308
	0.8	67,503	64,059	60,210	55,693	50,481	44,436
	0.6	54,793	50,845	46,419	41,487	35,918	29,678
Number of infected individuals
Percentage of isolated individuals (θ)
		0	0.1	0.2	0.3	0.4	0.5
p - MAE	1	224,650	224,479	224,162	223,586	222,541	220,650
	0.8	223,873	223,263	222,282	220,705	218,145	213,896
	0.6	220,238	218,229	215,295	210,889	203,899	191,544
Number of deaths
Percentage of isolated individuals (θ)
		0	0.1	0.2	0.3	0.4	0.5
p - MAE	1	3353	3351	3346	3337	3321	3291
	0.8	3341	3332	3317	3292	3251	3180
	0.6	3284	3252	3203	3125	2992	2742

Table A.8.

Results of S10, S11 and S12

Maximum number of infected individuals
Percentage of isolated individuals (θ)
		0	0.1	0.2	0.3	0.4	0.5
p - MAE	1	74,932	72,073	68,799	65,031	60,590	55,405
	0.8	66,546	63,270	59,523	55,305	50,421	44,822
	0.6	54,064	50,315	46,192	41,570	36,400	30,681
Number of infected individuals
Percentage of isolated individuals (θ)
		0	0.1	0.2	0.3	0.4	0.5
p - MAE	1	224,614	224,436	224,112	223,531	222,497	220,659
	0.8	223,803	223,185	222,204	220,650	218,175	214,201
	0.6	220,123	218,156	215,349	211,305	205,399	196,587
Number of deaths
Percentage of isolated individuals (θ)
		0	0.1	0.2	0.3	0.4	0.5
p - MAE	1	3353	3350	3345	3337	3321	3293
	0.8	3341	3331	3317	3294	3256	3196
	0.6	3285	3256	3214	3152	3062	2926

Table A.9.

Opened CH and IC in S6 (θ:0.1)

		Number of isolation centers allowed to be opened
		1		2		3		4
		CH	IC	CH	IC	CH	IC	CH	IC
Number of COVID-19 Hospitals allowed to be opened	3	h5, h6, h8	i1	h5, h6, h8	i1, i4	h5, h6, h8	i1, i4	h5, h6, h8	i1, i4
	4	h5, h6, h8, h9	i1	h5, h6, h8,h9	i1, i4	h5, h6, h8, h9	i1, i4	h5, h6, h8, h9	i1, i4
	5	h1, h5, h6, h8, h9	i1	h1, h5, h6, h8, h15	i1, i4	h1, h5, h6, h8, h15	i1, i4	h1, h5, h6, h8, h15	i1, i4
	6	h1, h3, h5, h6, h8, h9	i1	h1, h3, h5, h6, h8, h15	i1, i4	h1, h3, h5, h6, h8, h15	i1, i4	h1, h3, h5, h6, h8, h15	i1, i4
	7	h1, h2, h3, h5, h6, h8, h9	I1	h1, h2, h3, h5, h6, h8, h15	i1, i4	h1, h2, h3, h5, h6, h8, h15	i1, i4	h1, h2, h3, h5, h6, h8, h15	i1, i4
	8	h1, h2, h3, h5, h6, h8, h9, h15	i1	h1, h2, h3, h5, h6, h8, h9, h15	i1, i4	h1, h2, h3, h5, h6, h8, h9, h15	i1, i4	h1, h2, h3, h5, h6, h8, h9, h15	i1, i4
	9	h1, h2, h3, h5, h6, h8, h9, h15, h18	i1	h1, h2, h3, h5, h6, h8, h9, h15, h18	i1, i4	h1, h2, h3, h5, h6, h8, h9, h15, h18	i1, i4	h1, h2, h3, h5, h6, h8, h9, h15, h18	i1, i4
	10	h1, h2, h3, h5, h6, h8, h9, h10, h15, h18, h14	i1	h1, h2, h3, h5, h6, h8, h9, h10, h15, h18, h14	i1, i4	h1, h2, h3, h5, h6, h8, h9, h10, h15, h18, h14	i1, i4	h1, h2, h3, h5, h6, h8, h9, h10, h15, h18, h14	i1, i4
	11	h1, h2, h3, h4, h5, h6, h8, h9, h10, h15, h18, h14	i1	h1, h2, h3, h4, h5, h6, h8, h9, h10, h15, h18, h14	i1, i4	h1, h2, h3, h4, h5, h6, h8, h9, h10, h15, h18, h14	i1, i4	h1, h2, h3, h4, h5, h6, h8, h9, h10, h15, h18, h14	i1, i4

Table A.10.

Opened CH and IC in S6 (θ:0.3)

		Number of isolation centers allowed to be opened
		1		2		3		4
		CH	IC	CH	IC	CH	IC	CH	IC
Number of COVID-19 Hospitals allowed to be opened	1	h5	i1	h5	i1,i4	h5	i1,i4	h5	i1,i4
	2	h5, h8	i1	h5, h8	i1,i4	h5, h8	i1,i4	h5, h8	i1,i4
	3	h5, h8, h9	i1	h5, h6, h8	i1,i4	h5, h6, h8	i1,i4	h5, h6, h8	i1,i4
	4	h1, h5, h8, h9	i1	h5, h6, h8, h9	i1,i4	h5, h6, h8, h9	i1,i4	h5, h6, h8, h9	i1,i4
	5	h1, h5, h6, h8, h9	i1	h1, h5, h6, h8, h9	i1,i4	h1, h5, h6, h8, h9	i1,i4	h1, h5, h6, h8, h9	i1,i4
	6	h1, h5, h6,h8, h9,h18	i1	h1, h5, h6,h8, h15,h18	i1,i4	h1, h5, h6,h8, h15,h18	i1,i4	h1, h5, h6,h8, h15,h18	i1,i4
	7	h1, h5, h6, h8, h9, h15, h18	I1	h1, h3, h5, h6, h8, h15, h18	i1,i4	h1, h3, h5, h6, h8, h15, h18	i1,i4	h1, h3, h5, h6, h8, h15, h18	i1,i4
	8	h1, h3, h4, h5, h6, h8, h9, h18,	i1	h1, h3, h4, h5, h6, h8, h15, h18,	i1,i4	h1, h3, h4, h5, h6, h8, h15, h18,	i1,i4	h1, h3, h4, h5, h6, h8, h15, h18,	i1,i4
	9	h1, h3, h4, h5, h6, h8, h9, h15, h18	i1	h1, h3, h4, h5, h6, h8, h9, h15, h18	i1,i4	h1, h3, h4, h5, h6, h8, h9, h15, h18	i1,i4	h1, h3, h4, h5, h6, h8, h9, h15, h18	i1,i4
	10	h1, h3,h4, h5, h6, h8, h9, h15, h18, h14	i1	h1, h3,h4, h5, h6, h8, h9, h15, h18, h14	i1,i4	h1, h3,h4, h5, h6, h8, h9, h15, h18, h14	i1,i4	h1, h3,h4, h5, h6, h8, h9, h15, h18, h14	i1,i4
	11	h1, h2, h3, h4, h5, h6, h8, h9, h15, h18, h14	i1	h1, h2, h3, h4, h5, h6, h8, h9, h15, h18, h14	i1,i4	h1, h2, h3, h4, h5, h6, h8, h9, h15, h18, h14	i1,i4	h1, h2, h3, h4, h5, h6, h8, h9, h15, h18, h14	i1,i4

Table A.11.

Opened CH and IC in S6 (θ:0.5)

		Number of isolation centers allowed to be opened
		1		2		3		4
		CH	IC	CH	IC	CH	IC	CH	IC
Number of COVID-19 Hospitals allowed to be opened	1	h5	i1	h5	i1,i4	h5	i1,i4	h5	i1,i4
	2	h1, h8	i1	h6, h8	i1,i4	h6, h8	i1,i4	h6, h8	i1,i4
	3	h5, h8, h9	i1	h6, h8, h9	i1,i4	h6, h8, h9	i1,i4	h6, h8, h9	i1,i4
	4	h1, h5, h8, h9	i1	h5, h6, h8, h9	i1,i4	h5, h6, h8, h9	i1,i4	h5, h6, h8, h9	i1,i4
	5	h1, h5, h8, h9, h18	i1	h1, h3, h6, h8, h9	i1,i4	h1, h3, h6, h8, h9	i1,i4	h1, h3, h6, h8, h9	i1,i4
	6	h1, h5, h8, h9,h18	i1	h1, h5, h6, h8, h15, h18	i1,i4	h1, h5, h6, h8, h15, h18	i1,i4	h1, h5, h6, h8, h15, h18	i1,i4
	7	h1, h5, h6, h8, h9, h18	I1	h1, h4, h5, h6, h8, h9, h18	i1,i4	h1, h4, h5, h6, h8, h9, h18	i1,i4	h1, h4, h5, h6, h8, h9, h18	i1,i4
	8	h1, h3, h4, h5, h6, h8, h9, h18,	i1	h1, h3, h4, h5, h6, h8, h9, h18,	i1,i4	h1, h3, h4, h5, h6, h8, h9, h18,	i1,i4	h1, h3, h4, h5, h6, h8, h9, h18,	i1,i4
	9	h1, h3, h4, h5, h6, h8, h9, h15, h18	i1	h1, h3, h4, h5, h6, h8, h9, h15, h18	i1,i4	h1, h3, h4, h5, h6, h8, h9, h15, h18	i1,i4	h1, h3, h4, h5, h6, h8, h9, h15, h18	i1,i4
	10	h1, h3,h4, h5, h6, h8, h9, h15, h18, h14	i1	h1, h3,h4, h5, h6, h8, h9, h15, h18, h14	i1,i4	h1, h3,h4, h5, h6, h8, h9, h15, h18, h14	i1,i4	h1, h3,h4, h5, h6, h8, h9, h15, h18, h14	i1,i4
	11	h1, h2, h3, h4, h5, h6, h8, h9, h15, h18, h14	i1	h1, h2, h3, h4, h5, h6, h8, h9, h15, h18, h14	i1,i4	h1, h2, h3, h4, h5, h6, h8, h9, h15, h18, h14	i1,i4	h1, h2, h3, h4, h5, h6, h8, h9, h15, h18, h14	i1,i4

Data availability

The data that has been used is confidential.