COVID-19 spread algorithm in the international airport network-DetArpds

Evolutionary algorithms

The application of evolutionary algorithms (EAs) has been widespread in the specialized literature on the generation of routes. The study by Ho-Huu et al. (2018) proposed a method to solve route optimization problems with considerable noise using an EA called MOEA/D. The aim of this research was to generate new optimal air routes for noise reduction at Rotterdam Airport. The proposed method achieved high robustness and reliability. In addition, it generated different route replacement strategies, stop condition criteria, and constraint management to enable better decision-making.

Kammoun & Rezg (2018) developed a hybrid system for routing and reprogramming aircraft using a genetic algorithm and timed Petri net. The goal of this study was to minimize the total routing time and the amount of flight rescheduling. The results were encouraging because the system facilitated the management of many flights (including rescheduling) to improve their handling, rendering the processing time more efficient.

Wu et al. (2018) posited a genetic classification algorithm called DA-NSGA-II, to solve the problem of airspace congestion (traffic flow) in China. The objective of this study was to analyze the degree of air traffic congestion, total cost of operations, flight delay times, and all military and civilian flights. The results were optimal, with the DA-NSGA-II model generating improved air routes in a short execution time with a low degree of crossing between aircraft. Alieksieiev (2019), analyzed the problem of routing in airspace by using an EA combined with free routing airspace (FRA) and route availability document methods for better air routing.

Turky et al. (2020) presented a memetic EA that identified the shortest route based on the expected travel time. This method produces a prediction that determines travel times and then optimizes the shortest path by applying the memetic algorithm and deep learning. This method considers factors such as weather conditions, traffic, time of day, and day of the week. The results were satisfactory because the proposed method was highly effective compared to the most advanced procedures.

Cai et al. (2020), analyzed the improvement in air traffic network (ATN) robustness. The study proposed eliminating ATN edges using the Braess paradox, which employs an unmastered genetic classification algorithm (NSGA-II). The results were optimal because they improved robustness by 100% by eliminating fewer than six edges, while the remaining six edges yielded approximately 10% improvement.

Yanzhe & Bingxiang (2020) presented a model for evaluating and predicting the spread of COVID-19 in four countries (China, Italy, Great Britain, and the United States) using the ant colony algorithm and SIR (susceptible, infectious, or recovered) model. The objectives of their study were to analyze the epidemiological situation of COVID-19 and to examine the trend of the spread of the virus. The results of their proposed model were efficient because it reliably predicted the epidemiological trend, helping government decision-making and reducing infection rates and the spread of the pandemic. In addition, the research confirmed that reducing mass travel and closing severely affected areas could effectively slow the speed of the COVID-19 outbreak.

Weerasinghe et al. (2020), proposed a system for optimizing the daily operations of airlines through genetic programming. The objective of this research was to determine optimal flight routes, fleet allocation, gate assignment, and crew assignment. The results were satisfactory because the proposed system obtained reasonable results in a short period. In addition, the system contributed to the optimization of decision-making in Airline Operations Control Centers (AOCCs).

A study carried out by Fijar Awalivian, Suyanto & Sa’Adah (2021) detailed the serious problem of the spread of the COVID-19 virus, which affected flight scheduling and passenger-carrying capacity. In this study, the multi-objective antlion optimization (MALO) method was implemented, which solved flight scheduling and aircraft redirection problems under the current pandemic conditions. The results obtained were optimal because the MALO method can support large programming tasks by quickly converging a large dataset.

The algorithm used real data from the International Civil Aviation Organization (2021) for testing. The results were encouraging, as they demonstrated a good focus on finding a route between the departure and destination points and implemented a better strategy to generate improved short or optimal routes.

Cai, Ang & Alam (2021) presented a method to reduce the risk of collisions on Singapore’s air routes. This method used two EAs: NSGA-II and NSGA-III. The objective of the study was to analyze the risk of collision in each traffic flow. The results were optimal, and the research contributed to air traffic management both strategically and tactically.

Dynamic programming

Azis et al. (2018) presented clear examples of a dynamic programming (DP) algorithm, in which they compared conventional and heuristic methods to solve the problem of the shortest path. The data used for these studies were obtained from the Losari Beach route. The results were optimal since the conventional method provided a not better solution (shorter time) compared to the heuristic method. Moreover, the conventional method provided a less optimal local route.

Prakash, Piplani & Desai (2018) proposed an algorithm to manage aircraft on a single runway through DP. The aim of the study was to maximize aircraft entry and exit performance. The results demonstrated that the algorithm was easy to implement, maximized the operational performance of aircraft control, and provided optimal solutions in real time.

Brown, Hirabayshi & Wickramasinghe (2019) presented a model for air route planning that considers wind factors for medium-and long-haul flights. The objective of this study was to generate, draw, and optimize air trajectories through dynamic programming on the free route of the North Pacific. The results were effective because the proposed method provided several alternate routes for the aircraft. Moreover, this method facilitated the calculation of optimal trajectories (shorter routes) with shorter flight times and minimum fuel consumption (based on the wind variable).

Tian et al. (2019) provided details of a model for minimizing the ecological operating costs (GDOC) of cruise aircraft. The objective of this study was to analyze fuel consumption, flight time, and distance at the cruise stage using DP. The results were encouraging since the method generated a sustainable optimal flight plan for the environment by reducing altitude and speed. In addition, the model minimized the operating costs of civil aviation. Yu & Wang (2019) produced another interesting article in the same area of DP. They proposed a model to improve the efficiency of allocating connecting flights in China through dynamic scheduling. The main objective of the study was to collect data such as flight times and optimal flight routes. The results indicated that the proposed model maximized the economic benefits of the airlines by optimizing flight times.

Chen & Solak (2020) conducted another recent study in the same context for developing a flight management model for the Metropolitan Airport of Wayne County in Detroit (United States). This model employed DP to determine the optimal number of aircraft that could be directed to the runway tail and to identify optimal output-measurement policies. The results indicated that the model improved the efficiency of air traffic management and outbound operations, helping to minimize the overall costs for airlines.

Ntakolia, Caceres & Coletsos (2020), presented a model for air traffic management that used DP. The aim of this study was to improve aircraft traffic safety and increase the efficiency of the air network. The results were acceptable, as they generated optimal flight paths based on the variables of flight delays, flight durations, ground waiting delays, flight cancellations, flight speed deviations, and flight level alterations.

Recently, Yi, Tong & LiuXi (2020) detailed a spatial digital grid model for monitoring short-term flight conflicts in real time. The objective of this study was to compare the grid coordinates of each node, determine the grid coordinates of the nodes in possible collisions, and classify the type of collision using DP. The results were optimal because a reliable prediction of possible collisions was generated. In addition, the model suggested possible optimal solutions for conflict resolution through the strategic selection of maneuvers according to the performance index.

Ahmed, Bousson & Coelho (2021) proposed a modified dynamic programming (MDP) approach to flight path optimization. The objective of this research was to generate optimal air routes with minimum fuel consumption, and the results obtained were successful. By using MDP, they generated optimal routes that improved flight efficiency by reducing fuel consumption by 10.1% and yielded a 10.99% reduction in aircraft gas emissions.

In the latest study conducted in this field by de Campos, Vieira & Munari (2021), a method for the routing of aircraft with crew assignment was presented using DP. The objective of this study was to generate routes with lower operating costs. The results were acceptable, with the model generating optimal routes that reduced operating costs by 23%.

Dijkstra algorithm

Dijkstra’s algorithm (DJK) uses graphs to determine the shortest path based on the weight of each edge. An example application of this algorithm was presented by Adacher & Flamini (2018). In this study, the direct costs of robust routes were analyzed with respect to the shorter routes to reduce congestion and air delays by applying Dijkstra’s algorithm. Moreover, a robust route algorithm was developed to calculate robust air routes based on a real traffic network database obtained from Flightradar (2021) study. The study obtained a valid alternative route for each network point, which allowed aircraft to reach their destination within a certain time limit, with optimal results. One of the most relevant works in the application of Dijkstra’s algorithm was by Gao et al. (2018), in which a method was proposed for solving the problem of the shortest path in a network of domestic airlines in China. This study was based on Dijkstra’s algorithm, Fenwick’s tree algorithm, and the analytic hierarchy process (AHP), with the aim of quantitatively evaluating the accessibility of airports. The proposed method used data provided by the Chinese airline network from July 1, 2016, to July 7, 2016. The study concluded that the proposed method could become a guide for airlines and civil aviation administration for further development and management.

Xiao, Cai & Abbass (2018) presented a hybrid algorithm of indirect and direct coding to solve the problem of air traffic flow in the Chinese network. This algorithm controlled the weights in each arc to select optimal flight paths based on the current air traffic. It was based on a heuristic technique using Dijkstra’s algorithm to generate different types of routes. The results obtained from the proposed method demonstrated that it surpassed the direct coding method in terms of efficiency and effectiveness.

An interesting article in the same area was presented by Dabachine, Bouikhalene & Balouki (2019), in which a model for aircraft routing at Swiftair Casablanca Airport was proposed using Dijkstra’s algorithm. The aim of this study was to reduce flight delays and overall travel costs. The model provided different optimal paths in a short computational time, facilitated the planning of trajectories, and supported real-time decision-making.

Petrov (2019) presented a method for routing aircraft at low altitudes using Dijkstra’s algorithm, the distance, and the magnitude of height change. The results were acceptable, and the proposed method generated hidden routing through a mountain range and a route that avoided radar detection zones.

Dudi et al. (2020) presented a robust methodology called Airspace Map, which was aimed at planning routes with high aircraft safety (crew members and passengers) in unfamiliar environments. Their methodology was based on Dijkstra’s algorithm and Voronoi’s diagram. The results demonstrated that the Airspace Map helps avoid redundant obstacles in the planning of air trajectories in a short time and with adequate precision. Similarly, Jazzar et al. (2020) developed a methodology for generating shorter routes with low cost using Dijkstra’s algorithm. This methodology used real data from the route between Beirut and New York City. The results were encouraging, as the system generated different optimal routes. This methodology can help decision-making, cost saving, and travel time reduction.

Zhao et al. (2020) proposed a method for preventing air traffic collisions using Dijkstra’s algorithm. The objective of this study was to create a collision risk map. The results indicated that the model was efficient and safe, generating routes with minimum cost and a low probability of collision.

Finally, Maristany de las Casas, Sedeño-Noda & Borndörfer (2021), developed a new algorithm called the multi-objective Dijkstra algorithm (MDA), which solved the problem of the multi-objective shortest path. Through the simplification of tags, it parallelized some subpaths, facilitating the calculation of a complete minimum set of efficient paths for a given instance. The data used for this research were provided by Airway Networks. The results of the proposed MDA algorithm provided a response time that was improved by a factor of between 2 and 9 compared to Martins’ algorithm. Weiszer, Burke & Chen (2020), proposed an aerial routing algorithm called NAMOA, which determined the shortest route in ground operations based on multi-target trajectories. The study applied heuristic functions based on cost preferences, running time, and fuel consumption. The algorithm was developed in Python with a set of real data from Doha International (DOH), Hong Kong International (HKG), and Beijing Capital International (PEK) airports. The results were encouraging, with the algorithm finding a set of optimal solutions in a single run in a short time.

Neural networks

Lin, Wei Zhang & Liu (2019), presented a model called ConvLSTM, which analyzed the spatial traffic characteristics and predicted air traffic flow with real data. This model was based on an end-to-end deep-learning approach using a CNN and a recurrent neural network. The results indicated that the model predicted the distribution of traffic flow at different flight levels accurately and stably.

In another work on neural networks by Liu et al. (2019), a model was proposed based on a deep neural network (R-3DCNN) for air traffic management. The results were optimal, and the proposed model incorporated temporal and spatial dependencies of air traffic flow, providing solid and accurate predictions pertaining to the distribution of flight levels.

Wang et al. (2019) proposed a system for the detection and resolution of conflicts in air traffic in China based on reinforcement deep learning (DRL). The results were acceptable since the system generated an optimized trajectory within 200 ms while avoiding conflicts and changes in the angles of the course of the aircraft.

Han et al. (2019) developed a method for predicting aerial trajectories using a gated recurrent unit (GRU) neural network. The objective of this study was to accurately generate air routes that guarantee safety and efficiency in air traffic operations. The results indicated that the method could generate reliable real-time predictions with high accuracy.

Wang et al. (2020), analyzed the spread and identification of highly suspected cases of COVID-19 based on the Internet of Things (loT) and graph theory with a reinforced learning approach. The results were encouraging and could help decision-making to effectively reduce the rate of epidemiological reproduction of the infection. In addition, the use of these techniques could help in the early identification of COVID-19 cases. On similar lines, Ma & Tian (2020) developed a trajectory prediction method based on the aerodynamics of a ship. This method used deep learning combined with a convolutional neural network (CNN) and long-term memory (LSTM). The results demonstrated that the method achieved a prediction error of between 21.62% and 52.45% compared to other models. In addition, the model could provide important information for decision-making within air traffic management.

Similarly, Shi, Pan & Xu (2020) suggested a system based on ArcGIS 10.0 for the management of aerial trajectories through the LSTM network. The aim of this study was to provide ancillary decisions (or options) for air system operators. The results demonstrated that the system could predict flight paths with high accuracy and efficiency, supporting decision-making in the short term. Saâdaoui, Saadaoui & Rabbouch (2020) proposed a model for predicting traffic conditions in the US air network by decomposing time series and artificial neural networks (feedforward). The results were optimal since the model could generate accurate forecasts within acceptable time.

Lu et al. (2021) proposed the use of neural networks (NNs) to examine the effects of various transport networks on the heterogeneous spread of COVID-19 in China. They proposed a model called transport proximity deep neural network weighted regression (PDNNWR), which combined spatial heterogeneity for the spread of the pandemic, the relevance of transport proximity in human movement, and the accuracy of a deep neural network. The PDNNWR method achieved a more accurate prediction than the geographically weighted regression method. The results indicated that the spread of the virus through the air transport network was particularly high, even without any direct flight connections to the epicenter of the pandemic.

Finally, Choi & Kim (2021), proposed a model for air traffic management using artificial neural networks. The objective of this study was to predict the arrival and departure capacity of Atlanta’s Hartsfield–Jackson International Airport (ATL), using the multilayer perceptron (MLP), recurrent neural networks (RNN), and LSTM models. The results indicated that the artificial neural network approach is effective in predicting aircraft arrivals and departures.

Traveling salesman problem

As an example of the traveling salesman problem (TSP), Wang, Yu & Liu (2018) developed an algorithm using the probability density distribution and the Bayesian formula based on the TSP. The aim of the investigation was to generate a rescue plan for air accidents. According to the results, the algorithm could generate a search plan by identifying the area and generating an optimal search path. In addition, the algorithm was 4% faster than the current strategies.

Duque et al. (2018) proposed an algorithm for the management of optimal routes that also used the TSP in addition to the ant colony optimization (ACO) strategy. The objective of this study was to determine the best flights for visiting certain cities. The results were acceptable since the algorithm generated optimal routes in a short time while also considering the cost of the trips.

Similarly, Muren et al. (2019) developed an air logistics model using incremental heuristic and mixed optimization algorithms. The results indicated that the proposed model performed better than exact and intelligent algorithms. In addition, the model generated optimal aircraft routing with approximately 99.5% accuracy within a short computational time (according to the number of nodes).

Marques, Russo & Roma (2019) developed a model using the TSP, which employed different heuristic and metaheuristic optimization algorithms. The goal of the model was to find the best schedule, route, and set of flights for any unrestricted flight request. The results were acceptable, and the model generated different routes with and prices that were up to 35% cheaper, allowing users to save time and money.

Alrasheed et al. (2019), proposed a model for the optimization of air transport services using the TSP and a local search algorithm. The main objective of the study was to find an optimal route that passed only once through each defined air terminal. The results facilitated the generation of low-cost air trajectories between airports in a short time.

In addition, Ahmad, Muklason & Nurkasanah (2020) recently developed an algorithm for the optimization of air routes using the TSP and a search algorithm called the hybrid tabu. The aim of this study was to find travel routes with the lowest possible costs and then compare them with the great deluge algorithm. Their algorithm was tested with several datasets (both small and medium). The results were encouraging because different optimal routes were generated with a 48.54% confidence level compared to the great deluge algorithm.

Pylyavskyy, Kheiri & Ahmed (2020), proposed a method for the optimization of flight connections based on the TSP through the reinforcement learning algorithm. The main objective of this study was to determine economical air routes between two airports. The results indicated that the proposed method could generate ideal routes while minimizing total travel costs. Tomanová & Holý (2020) proposed a heuristic method using an ant colony algorithm based on a modified TSP. The objective of the research was to determine optimal routes supported by the Hamiltonian cycle with minimal cost. The results were acceptable, and the method generated effective trajectories according to the parameters of cost, distance, and city per day in a short time.

Guevara & Penas (2020) developed the intelligent infectious diseases algorithm (IIDA), which could locate the sources of contagion and determine the survival rate of the COVID-19 virus using kernel density (KDE), ant colony optimization algorithms, and the TSP. The main goal of this study was to reduce the spread of the virus and decrease the number of infections. The algorithm was tested with data from obtained from New York State. The results were acceptable since the IIDA algorithm managed to determine areas with higher rates of infection and mortality and generate optimal routes of medical care within a reasonable processing time.

Finally, Bulbul & Kasimbeyli (2021) developed a hybrid method using Gasimov’s algorithm and the ant colony algorithm based on the TSP. The aim of the research was to generate optimal routing for aircraft maintenance. This method analyzed the traffic of air routes and maintenance effectively, generating optimal routes that did not violate any defined routes.

Graph theory

Bae et al. (2018) proposed an algorithm to solve routing and flight scheduling problems using graph theory and the first-come-first-served algorithm. The results were optimal, with the algorithm solving iteration problems in routing by generating flights with multiple trajectories and aircraft speed profiles.

Along similar lines, Bhapkar, Mahalle & Dhotre (2020), analyzed the impact of the COVID-19 pandemic on the world by using the graph theory approach. The main objective of the study was to understand and graph the spread of COVID-19 infections. The results indicated different situations of contagion through geolocation. In addition, with a 1% growth rate of infections and an infection rate of 5, the infections doubled in just 15 days.

Small et al. (2020) proposed a model for analyzing the spread of COVID-19 in the cities of Australia and Perth. This model used graph theory to generate a network topology and the SIR model to determine the dynamic transmission of diseases. The results provided a useful prediction for the generation of control strategies and effective policy formulation to reduce the rate of infection. In addition, it was determined that strict compliance with isolation could decrease the probability of spreading COVID-19 by 80%.

Wijayanto & Ridho (2020) developed a model for managing future flight paths. Their model used graph theory for link prediction by mapping existing airports as vertices and flight paths as links. The method achieved an average prediction accuracy of 90%.

Neretin, Budkov & Ivanov (2020), analyzed the problem of implementing navigation routes in civil aviation using the A-star method of graph theory. The results were satisfactory since the proposed method provided four different optimization criteria considering wind conditions, no-fly zones, and aircraft performance.

Serafino (2020) developed a methodology using graph theory for determining the best trajectory in terms of fuel consumption, climatic conditions, and reduction of gas emissions. The results provided a trajectory with high precision for multi-objective routes in a short time.

Ekpenyong et al. (2020), presented a model called Spatio-GraphNet for tracing the people infected by COVID-19 using graph theory. The results were optimal, as the Spatio-GraphNet model generated real-time spatial information on sources of contagion and the predictive behavior of the spread of COVID-19. In addition, the use of graph theory facilitated tracking infected people effectively and promptly. Ivashchuk et al. (2021) proposed a method for analyzing Ukraine’s air traffic management using graph theory. The aim of the study was to examine the configuration of the flight path network. The results were acceptable since the method generated optimal flight paths that would help improve air navigation.

Gaeta, Loia & Orciuoli (2021), presented a method called 3WD for analyzing the spread of COVID-19. The 3WD method used graph theory and proximal three-way decision-making, and the data were provided by the Italian Department of Civil Protection. The results were encouraging, helping decision-making through contagion thresholds and critical infrastructure.

Finally, Isufaj, Koca & Piera (2021) conducted another study using graph theory for modeling air traffic, with the main objective of simplifying air traffic management. The results were encouraging, as method generated indicators that could measure several structural properties of air traffic, including edge density (graph size), strength (severity of interdependencies), clustering coefficient (proximity of each aircraft), and degree of proximity. In addition, the model provided a detailed view of how aircraft interactions can change over time.

Markov chain

Tolić, Kleineberg & Antulov-Fantulin (2018), analyzed the time taken for the spread of pandemics using three different empirical networks. A framework for the simulation of Poisson’s SIR model was developed to find the shortest route by applying Markov chains that could identify epidemic spread times using Poisson’s mathematical model. The proposed framework could determine the estimated time of spread and detect possible sources of contagion.

Similarly, Ayhan, Costas & Samet (2018) proposed a novel system for air conflict management in long-range aircraft. This system used real data from 16 airports in Europe and the United States. The objective of the system was to analyze and detect air trajectories that violated protected areas using the Viterbi algorithm and Markov chains. The results proved efficient, and the system generated different alternative routes that did not violate the protected airspace.

Lin, Wei Zhang & Liu (2018) proposed a model to generate trajectory plans through machine learning and Markov chains. The objective of the study was to predict flight paths based on the relative movement between positions (i.e., speed, direction, and angular movement). The results demonstrated that the model could generate excellent solutions for predicting trajectories with high accuracy. Similarly, Delahaye, Chaimatanan & Mongeau (2019) developed a methodology for strategically managing the trajectories of 30,000 flights in Europe using a simulated annealing algorithm and Markov Chains (MC) with a Monte Carlo approach. The objective of the study was to plan the trajectories of large-scale aircraft. The results indicated that the methodology generated strategic planning that minimized the number of interactions between trajectories, helping to improve air management in the same period.

Kieckbusch et al. (2019) proposed a model for air traffic flow management that employed Markov chains and reinforcement learning. The aim was to ensure that the aircraft could take off and land safely, with minimal delays. The results were acceptable because the method generated optimal routes that considered different parameters, such as ground delays, decelerations enroute, and detours to less congested airports.

Rodríguez-Sanz et al. (2019) developed a methodology for predicting and evaluating the operational status of the air traffic system at Adolfo Suárez Madrid–Barajas Airport (LEMD) using a Bayesian network and the Markov chain (MC) with a Monte Carlo approach. The results were acceptable, minimizing congestion problems and air delays by 70%. In addition, the method helped decision-making in cases of operational uncertainty.

Sa, Santos & Clarke (2020) presented a methodology for planning airline fleets using Markov chains. The aim was to identify robust fleet plans that met the demands for long-term travel. Their methodology offered superior planning of air fleets with a greater flight frequency to capture a larger part of the market.

He et al. (2020), presented a model for improving the operational safety of the Shanghai airport airspace system using a Markov chain and set pair analysis (SPA). The method proved effective and feasible in accurately describing the characteristics of dynamic change. In addition, the method was simple and easy to operate. Similarly, Ikai, Miyagi & Sakai (2021) presented a method for planning air routes with multiple locations using a Markov chain. The results were adequate, generating optimal logistics planning for air trajectories. In addition, this method could be implemented for both air networks and vehicular traffic.

Guevara & Bonilla (2021) presented an algorithm for preventing the spread of infectious diseases on air routes and in airports from Wuhan in China (WHU) to different countries around the world. The authors applied Dijkstra’s algorithm, fuzzy logic, graphs, and the Markov chain. The proposed algorithm used information from airports, routes provided by flight connections, and COVID-19 spread data from Johns Hopkins University. As a result, the 25.23% airports could be identified as “high dispersion,” meaning that they should adopt more health and mobility controls. With regard to air routes, 30.80% were considered high risk (in terms of spreading the virus), leaving approximately 850,000 passengers infected with the virus in a month.

Fuzzy logic

Fuzzy logic (FL) is widely applied to solve problems involving shortest routes, as exemplified by Savkin & Huang (2017) presented a method for generating optimal trajectories for military aircraft in harsh environments using an algorithm based on fuzzy logic. The results were acceptable since the method generated an optimal route while minimizing trajectory length and threat levels.

Volpe Lovato et al. (2018) presented a hybrid model for detecting and resolving conflicts on air traffic routes that used a genetic algorithm and fuzzy logic. The objective was to calculate the optimal actions in terms of changes in flight levels between the aircraft through a global and dynamic analysis. The results were encouraging, with the model being able to detect and eliminate longitudinal conflicts in advance and support decision-making effectively for tower controllers.

Similarly, Perçin (2018) proposed a method called VIKOR to evaluate and classify the performance and service of airlines in Turkey using fuzzy logic. The objective was to improve the competitive advantage over other countries, and the results could help decision-making for improving air operation management.

In the same area of fuzzy logic, Dudeja (2019) presented a study in which a modified fuzzy algorithm was applied to an empirical database. The main objective was to propose a novel strategy to reduce costs and time consumption using fuzzy rules to overcome the problem of the shortest path. In addition, the modified fuzzy algorithm was compared with different metaheuristic algorithms. The results demonstrated an improvement in coding efficiency, time consumption, and costs in dynamic path generation.

Kaleta & Skorupski (2019) proposed a model and implemented a hierarchical blurred inference system to determine the probability of air accidents. The objective of the study was to evaluate air traffic safety, the probability of air accidents, and the implementation of guidance procedures using controlled flight into the terrain. This was based on fuzzy logic and hierarchical fuzzy inference systems. The model considerably decreased the probability of air accidents and could be used commercially in small aerodromes.

Lovato et al. (2019), proposed a model for controlling conflicts on air traffic routes. The objective was to quantify the levels of longitudinal conflict between the aircraft and provide acceleration based on the level of conflict detected using fuzzy logic. The model could detect and eliminate longitudinal conflicts in advance and function well without compromising safety or violating existing air regulations.

Olive et al. (2020) developed a model for detecting events in aircraft trajectories using fuzzy logic. The objective was to conduct a large-scale analysis of the trajectories in the flight phase, flight plans, airspace, and operating procedures. The model provided highly valuable information for monitoring and situational decision-making.

Heydari, Omrani & Taghizadeh (2020) proposed a model for assessing the efficiency of Iran’s airlines. The goal was to design a network structure for airlines and calculate the efficiency score. The results suggested that the overall efficiency of airline operations is low. In addition, the method provided important information for improving air route operations. Mijovic, Kalic & Kuljanin (2021) developed a fuzzy logic model using metaheuristic algorithms to determine the market share of airlines on long-haul routes in the London airport system. The model significantly increased the efficiency of market share value prediction and could improve the operational performance of airlines.

Finally, Sudakov, Francisco & López De Hierro (2021), developed a method for planning air travel in Russia using fuzzy logic. The aim was to generate a diffuse air traffic matrix (origin–destination) between airports and classify them into potentially promising routes. The method solved the problems of transport route planning, aircraft choice, and the design of new airports. In addition, the model supported decision-making on improving air transport infrastructure.

Table 1 lists the studies analyzed in the literature review. This table highlights the current state of studies in this area, demonstrating extensive research related to route generation. In addition, Table 1 presents the algorithms and techniques employed in these studies with the symbol “ $✓$”. The following criteria are grouped: evolutionary algorithms (EA), dynamic programming (DP), Dijkstra’s algorithm (DJK), neural networks (NN), graph theory (GT), Markov chain (MC), and fuzzy Logic (FL).

The main objective of these works was to find the shortest route through some methodologies and then compare them (or establish rules) to improve the air transport system in a pandemic situation. Many of these studies ended in the analysis phase and focused mainly on improving the air transport system. However, they did not generate routes for the prevention and spread of diseases. The main difference between the studies reported in the literature analysis and the research presented here is that their objective was to determine different optimal routes between airports.

In Table 1, it is possible to identify works that have simultaneously applied one or more algorithms and techniques. The most used methods and algorithms are neural networks (14 studies) and graph theory (14 studies), and the rest of the algorithms have been applied 11 times each. These outcomes indicate that the neural networks and graph theory have been extensively analyzed with optimal research results. For this reason, we decided to apply graph theory (GT) as the fundamental base of our proposed method and Dijkstra’s algorithm (DJK) to offer another approach to the research area.

Airport database IATA

The airport database includes information from all international air terminals worldwide. This dataset was compiled by the International Air Transport Association (IATA) (Aviation & the environment, 2019). The dataset comprises 26,914 records with the following eight variables: IATA code, latitude, longitude, number of routes, country, continent, origin, and destination of the air terminal (Table 2).

The distribution of existing international airports worldwide, classified by continent, where Asia (26.75%), Europe (34.85%), America (21.90%), Oceania (3.90%), and Africa (12.57%). The most significant number of international airport terminals are concentrated in Asia and Europe, which constitute 61.60% of the total number of international airports.

The top 10 countries with the most significant number of international airports are China, which has the highest number of international airports (75), followed by the United States (71), Russia (65), France (44), Italy (32), United Kingdom (32), Germany (31), Japan (31), Mexico (30) and Spain (30).

The top 10 countries with the highest number of international air routes are: The United Kingdom has 1,776 routes, followed by Germany with 1,503 routes, the United States with 1,417 routes, Spain with 1,410 routes, Italy with 1,266 routes, France with 1,231 routes, China with 990 routes, Greece 848 routes and Turkey 670 routes.

Figure 1 shows the geospatial location (latitude and longitude) of the air terminals of all airports in the world, classified by continent. This information allowed us to identify the distance between international air terminals as well as existing connections.

With this information, we could describe the connections between each of the international terminals, as well as their distance and exact position.

COVID-19 dataset

This data repository collects information on the global spread of COVID-19 and was collected by the Center for Systems Science and Engineering at Johns Hopkins University (Center for Systems Science & Engineering (CSSE) at Johns Hopkins University (JHU), 2023). The dataset has 152,736 records collected between February 1, 2020, and January 5, 2022. In addition, it has seven variables: country, date, confirmed cases, deaths, recovered, active cases, and region (Table 3).

The number of COVID-19 infections is categorized by continent, where America is the one with the highest number of positive cases (109,766.438 positive cases), followed by Europe (92,815.435 positive cases), Asia (85,568.026 positive cases), Africa (10,039.737 positive cases), Oceania (85,568.026 positive cases).

The top 10 countries with the highest infection rates are The United States (58,805,186 cases), followed by India (35,109,286), Brazil (22,351,104), United Kingdom (14,015,065), France (10,921,757), Russia (10,601,300), Turkey (9,718,861), Germany (7,360,556), Italy (6,975,465), and Spain (6,922,466).

Figure 2 presents a chronological diagram of COVID-19 infections classified by continent from February 2020 (880,480 confirmed cases) to January 5, 2022 (2,981,694,750 confirmed cases).

Figure 3 presents the timeline of COVID-19 spread in the top 10 countries with the highest number of infections, from January 1, 2020, to January 05, 2022.

Table 4 presents the temporal chronology of the spread of COVID-19 for each country. This information is grouped by date (categories every 15 days) from January 15 to March 15, 2020, in countries where cases of the disease have been identified.

Application of graph theory to air terminals

This section describes the creation of an interconnected network of international air terminals (airports) by applying graph theory (Ekpenyong et al., 2020). The air terminal graph is based on the information obtained from the IATA database (described in section), as follows (Eq. (1)):

(1) $$G=(V_i,V_f,d)$$

Here $V_i(latitud{e}_{Vi},longitud{e}_{Vi})$ is the initial vertex and $V_f(latitud{e}_{Vf},longitud{e}_{Vf})$ is the final vertex. Each of these vertices is defined by the latitude and longitude, which constitute the geographic location of each terminal. The distance d between each of the vertices $V_i$ and $V_f$ is calculated by applying the Euclidian distance of Eq. (2) (Behrens et al., 2018).

(2) $$d=\sqrt{(latitu{d}_{Vf}-latitu{d}_{Vi}{)}^2+(longitu{d}_{Vf}-longitu{d}_{Vi}{)}^2}$$

Graph G has 1,297 vertices (international airports) and 26,914 distances between the vertices (routes between the terminals).

Table 5 presents an example of the application of Eqs. (1) and (2) to create graph G from WHU to eight different airports around the world.

Figure 4 presents an example of graph G with international air terminals, where each line represents a distance $d$ (route) and each terminal represents a vertex V, from WHU to eight different airports in the continents of America, Europe, Asia, and Oceania.

Application of Dijkstra’s algorithm for route generation

In this section, the generation of routes by applying Dijkstra’s algorithm (Bae et al., 2018; Baeza, Ihle & Ortiz, 2017) is described, from the airport of origin ( $V_i$) to the destination airport ( $V_f$). Algorithm 1 (Table 6) describes the pseudo-code for the generation of possible COVID-19 propagation routes on the international airport network, identifying the most crowded terminals and the most common air routes.

These simulations were performed from the terminal of the city with the first reported COVID-19 case in WUH ( $V_{WUH}$) to 10 terminals in five continents, as presented in Table 7.

For the simulation of propagation routes, the Determination of Airport Routes and Prevention of Disease Spread (DetARPDS) algorithm, as defined by Algorithm 2 (Table 8), was developed. This algorithm first generates an optimal path of graph G by applying Dijkstra’s Algorithm 1 (Table 6) from the start vertex $V_{WUH}$ to the destination vertex $V_f$ with the shortest distance. The resulting optimal path contains a list of connections between vertices $V_i$ and $V_f$, which will sequentially eliminate existing intermediate connections to determine possible propagation routes. This procedure is repeated until there is no connection between the start vertex $V_i$ and the destination vertex $V_f$. Subsequently, this is compared with the COVID-19 spread data in Table 4 to determine the routes that comply with the temporal chronology of the contagion. Finally, a set of resulting routes is obtained, wich determines the likeliest routes of the contagion from the origin to each of the selected destinations.