Coronavirus Optimization Algorithm: A Bioinspired Metaheuristic Based on the COVID-19 Propagation Model

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The user has requested enhancement of the downloaded file. Coronavirus Optimization Algorithm: A bioinspired1 metaheuristic based on the COVID-19 propagation model2 F. Mart´ınez-´ Alvarez1∗ , G. Asencio-Cort´es1, J. F. Torres1,3 D. Guti´errez-Avil´es1, L. Melgar-Garc´ıa1, R. P´erez-Chac´on1,4 C. Rubio-Escudero2, J. C. Riquelme2, A. Troncoso1 5 1Data Science & Big Data Lab, Pablo de Olavide University, ES-41013 Seville, Spain6 2Department of Computer Science, University of Seville, ES-41012 Seville, Spain7 Abstract8 This work proposes a novel bioinspired metaheuristic, simulating how the coronavirus spreads and9 infects healthy people. From a primary infected individual (patient zero), the coronavirus rapidly10 infects new victims, creating large populations of infected people who will either die or spread infec-11 tion. Relevant terms such as reinfection probability, super-spreading rate, social distancing measures12 or traveling rate are introduced into the model in order to simulate the coronavirus activity as accu-13 rately as possible. The infected population initially grows exponentially over time, but taking into14 consideration social isolation measures, the mortality rate and number of recoveries, the infected15 population gradually decreases. The Coronavirus Optimization Algorithm has two major advantages16 when compared to other similar strategies. Firstly, the input parameters are already set according to17 the disease statistics, preventing researchers from initializing them with arbitrary values. Secondly,18 the approach has the ability to end after several iterations, without setting this value either. Further-19 more, a parallel multi-virus version is proposed, where several coronavirus strains evolve over time20 and explore wider search space areas in less iterations. Finally, the metaheuristic has been combined21 with deep learning models, in order to find optimal hyperparameters during the training phase. As22 application case, the problem of electricity load time series forecasting has been addressed, showing23 quite remarkable performance.24 Keywords: Metaheuristics, soft computing, deep learning, big data, coronavirus.25 ∗Corresponding author, [email protected] 1 1 Introduction26 The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is a new respiratory virus, causing27 coronavirus disease 2019 (COVID-19), firstly discovered in humans in December 2019, that has spread28 across the globe, having reportedly infected more than 4 million people so far [1]. Much remains un-29 known about the virus, including how many people who may have very mild, asymptomatic or simply30 undocumented infections and whether they can transmit the virus or not [2]. The precise dimensions of31 the outbreak are hard to evaluate [3].32 Bioinspired models typically mimic behaviors from the nature and are known for their successful33 application in hybrid approaches to find parameters in machine learning model optimization [4]. Viruses34 can infect people and these people can either die, infect other people or simply recover after the disease.35 Vaccines and the immune defense system typically fight the disease and help to mitigate their effects36 while an individual is still infected. This behavior is typically modeled by an SIR model, consisting of37 three types of individual: Sfor the number of susceptible, Ifor the number of infectious, and Rfor the38 number of recovered [5].39 Metaheuristics must deal with huge search spaces, even infinite for the continuous cases, and must find40 suboptimal solutions in reasonable execution times [6]. The rapid propagation of the coronavirus along41 with its ability to cause infection in most of the countries in the world impressively fast, has inspired42 the novel metaheuristic proposed in this work, named Coronavirus Optimization Algorithm (CVOA). A43 parallel version is also proposed in order to spread different coronavirus strains and achieve better results44 in less iterations.45 The main CVOA advantages regarding other similar approaches can be summarized as follows:46 1. Coronavirus statistics are not currently known with precision by the scientific community and some47 aspects are still controversial, like the reinfection rate [7]. In this sense, the infection rate, the48 mortality rate, the spreading rate or the reinfection probability cannot be accurately estimated so49 far, due to several issues like the lack of tests for asymptomatic people. However, the current state of50 the pandemic suggests certain values, as reported by the World Health Organization [8]. Therefore,51 CVOA is parametrized with the actual reported values for rates and probabilities, preventing the52 user from performing an additional study on the most suitable setup configuration.53 2. CVOA can stop the solutions exploration after several iterations, with no need to be configured.54 That is, the number of infected people increases over the first iterations, however, after a certain55 number of iterations, the number of infected people starts decreasing, until reaching a void infected56 set of individuals.57 3. The coronavirus high spreading rate is useful for exploring promising regions more thoroughly58 (intensification) while the use of parallel strains ensures that all regions of the search space are59 2 evenly explored (diversification).60 4. Another relevant contribution of this work is the proposal of a new discrete and of dynamic length61 codification, specifically designed for combining Long Short-Term Memory networks (LSTM) with62 CVOA (or any other metaheuristic).63 There is one limitation to the current approach. Since there is no vaccine currently, it has not64 been included in the procedure to reduce the number of candidates to be infected. This fact involves an65 exponential increase of the infected population in the first iterations, and therefore an exponential increase66 of the execution time for such iterations. This, however, is partially solved with the implementation of67 social isolation measures to simulate individuals who cannot be infected during a particular iteration.68 A study case is included in this work which discusses the CVOA performance. CVOA has been used69 to find the optimal values for the hyperparameters of a LSTM architecture [9], which is a widely used70 model for artificial recurrent neural network (RNN), in the field of deep learning [10]. Data from the71 Spanish electricity consumption have been used to validate the accuracy. The results achieved verge on72 0.45%, substantially outperforming other well-established methods such as random forest, gradient-boost73 trees, linear regression or deep learning optimized with other metaheuristics. The code, developed in74 Phyton with a discrete codification, is available in the supplementary material (along with an academic75 version in Java for a binary codification).76 Finally, the need to further study the performance of well-established fitness functions [11] is ac-77 knowledged. However, given the relevance that this pandemic is acquiring throughout the world and the78 remarkable results achieved when combined with deep learning, this work is shared with the hope that79 it inspires future research in this direction.80 The rest of the paper is organized as follows. Section 2 discusses related and recent works. The81 methodology proposed is introduced in Section 3. Section 4 proposes a discrete codification to hybridize82 deep learning models with CVOA and provides some illustrative cases. A sensitivity analysis on how83 populations are created and evolved over time is discussed in Section 5. The results achieved are reported84 and discussed in Section 6. Finally, the conclusions drawn and future work suggestions are included in85 Section 7.86 2 Related works87 There are many bioinspired metaheuristics to solve optimization problems. Although CVOA has been88 conceived to optimize any kind of problems, this section focuses on optimization algorithms applied to89 hybridize deep learning models.90 It is hard to find consensus among the researchers on which method should be applied to which91 problem, and, for this reason, many optimization methods have been proposed during the last decade to92 3 improve deep learning models. Generally, the criterion for selecting a method is its associated performance93 from a wide variety of perspectives. Low computation cost, accuracy or even implementation difficulty94 can be accepted as one of these criteria.95 The Virus Optimization Algorithm was proposed by Liang and Cuevas-Ju´arez in 2016 [12] and later96 improved in [13]. However, as many other metaheuristics, the results of its application are highly depen-97 dent on its initial configuration. Additionally, it simulates generic viruses, without adding individualized98 properties for particular viruses. The results achieved indicate that its usefulness is beyond doubt.99 One of the most extended metaheuristics used to improve deep learning parameters is genetic algo-100 rithms (GA). Hence, a LSTM network optimized with GA can be found in [14]. To evaluate the proposed101 hybrid approach, the daily Korea Stock Price Index data were used, outperforming the benchmark model.102 In 2019, a network traffic prediction model based on LSTM and GA was proposed in [15]. The results103 were compared to pure LSTM and ARIMA, reporting higher accuracy.104 Multi-agents systems have also been applied to optimize deep learning models. The use of Particle105 Swarm Optimization (PSO) can be found in [16]. The authors proposed a model based on kernel principal106 component analysis and back propagation neural network with PSO for midterm power load forecasting.107 The hybridization of deep learning models with PSO was also explored in [17] but, this time, the authors108 applied the methodology with image classification purposes.109 Ants colony optimization (ACO) models have also been used to hybridize deep learning. Thus, Desell110 et al. [18] proposed an evolving deep recurrent neural networks using ACO applied to the challenging111 task of predicting general aviation flight data. The work in [19] introduced a method based on ACO112 to optimize a LSTM recurrent neural networks. Again, the field of application was flight data records113 obtained from an airline containing flights that suffered from excessive vibration.114 Some papers exploring the Cuckoo Search (CS) properties have been published recently as well. In115 [20], CS was used to find suitable heuristics for adjusting the hyper-parameters of another LSTM network.116 The authors claimed an accuracy superior to 96% for all the datasets examined. Nawi et al. [21] proposed117 the use of CS to improve the training of RNN in order to achieve fast convergence and high accuracy.118 Results obtained outperformed those than other metaheuristics.119 The use of the artificial bee colony (ABC) optimization algorithm applied to LSTM can also be found120 in the literature. Hence, and optimized LSTM with ABC to forecast the bitcoin price was introduced in121 [22]. The combination of ABC and RNN was also proposed in [23] for traffic volume forecasting. This122 time the results were compared to standard backpropagation models.123 From the analysis of these works, it can be concluded that there is an increasing interest in using124 metaheuristics in LSTM models. However, not as many works as for artificial neural networks can be125 found in the literature and, none of them, based on a virus propagation model. These two facts, among126 others, justify the application of CVOA to optimize LSTM models.127 4 3 Methodology128 This section introduces the CVOA methodology. Thus, Section 3.1 describes the steps for a single strain.129 Section 3.2 introduces the modifications added to use CVOA as a parallel version. Section 3.3 suggests130 how the input parameters must be set. Section 3.4 includes the CVOA pseudo codes.131 3.1 Steps132 Step 1. Generation of the initial population. The initial population consists of one individual, the133 so-called patient-zero (P Z). As in the coronavirus pandemic, it identifies the first human being infected.134 If no previous local minima have been found, a random initialization for the P Z is suggested.135 Step 2. Disease propagation. Depending on the individual, several cases are evaluated:136 1. Each infected individual has a probability of dying (P DIE), according to the COVID-19 death137 rate. Such individuals cannot spread the disease to new ones.138 2. The individuals who do not die, will cause infection to new individuals (intensification). Two types139 of spreading are considered, according to a given probability (P SUP ERSP READER):140 (a) Ordinary spreaders. Infected individuals will infect new ones according to a regular spreading141 rate (SP READIN G RAT E).142 (b) Super-spreaders. Infected individuals will infect new ones according to a super-spreading rate143 (SUP ERSP READING RAT E).144 3. There is another consideration, since it is needed to ensure diversification. Both ordinary and super-145 spreaders individuals can travel and explore solutions quite dissimilar. Therefore, individuals have146 a probability of traveling (P T RAV EL) to propagate the disease to solutions that may be quite147 different (T RAV ELER RAT E). In case of not being traveler, new solutions will change according148 to an ORDINARY RAT E. Note that one individual can be both super-spreader and traveler.149 Step 3. Updating populations. Three populations are maintained and updated for each generation.150 1. Deaths. If any individual dies, it is added to this population and can never be used again.151 2. Recovered population. After each iteration, infected individuals (after spreading the coronavirus152 according to the previous step) are sent to the recovered population. It is known that there is a153 reinfection probability (P REINF ECT ION). Hence, an individual belonging to this population154 could be reinfected at any iteration provided that it meets the reinfection criterion. Another155 situation must be considered, since individuals can be isolated simulating they are implementing156 the social distancing measures. For the sake of simplicity, it is considered that an isolated individual157 is sent to the recovered population when the isolation probability is met (P ISOLAT ION).158 5 3. New infected population. This population gathers all individuals infected at each iteration, ac-159 cording to the procedure described in the previous steps. It is possible that repeated new infected160 individuals are created at each iteration and, consequently, it is recommended to remove such161 repeated individuals from this population before the next iteration starts running.162 Step 4. Stop criterion. One of the most interesting features of the proposed approach lies on its ability to163 end without the need of controlling any parameter. This situation occurs because the recovered and dead164 populations are constantly growing as time goes by, and the new infected population cannot infect new165 individuals. It is expected that the number of infected individuals increases for a certain number of itera-166 tions. However, from a particular iteration on, the size of the new infected population will be smaller than167 that of the current one because recovered and dead populations are too big, and the size of the infected168 population decays over time. Additionally, a preset number of iterations (P ANDEMIC DU RAT ION)169 can be added to the stop criterion. The social distancing measures also contributes to reach the stop170 criterion.171 3.2 Remarks for a parallel CVOA version172 It must be noted that it is very simple to use CVOA in a multi-virus version since it can be implemented173 as a population-based algorithm, when considering the pandemic as a set of intelligent agents each of174 them evolving in parallel. In contrast to trajectory-based metaheuristics, population-based focuses on175 the diversification in the search space.176 For this case, a new variable must be defined, strains, which determines the number of strains that177 will be launched in parallel. Each strain can explore different regions and can be differently configured178 so that each of them intensifies with their own rates.179 Several considerations must be done for this case:180 1. Every strain is run independently, following the steps in the previous section.181 2. A wise strategy must be followed to generate P Zs for each strain. For instance, it is suggested182 the generation of P Zs evenly spaced or, at least, with high Hamming distances. That way, the183 exploration of distinct regions of the search space is facilitated (diversification).184 3. The interaction between the different strains is done by means of dead and recovered populations,185 that must be shared by all the strains. Operations over these populations must be handled as186 concurrent updates [24].187 4. New infected populations, on the contrary, are different for each strain and no concurrent operations188 are required.189 6 5. This version may help to simulate different rates for different strains. That way, if there is any190 initial information about the search space, some strains could be more focused on diversification191 and some others on intensification.192 Depending on the hardware resources and how busy they are, every strain may evolve at different193 speeds. This situation should not pose any problems since it is known that the pandemic evolves at194 different rates and starts at different time stamps depending on region of the world.195 Last, another application can be found for this parallel version. CVOA simulates an SIR model and196 consequently, any other global pandemic can be modeled by using the specific rates. Different pandemics197 could be run in parallel.198 3.3 Suggested parameters setup199 Since CVOA simulates the COVID-19 propagation, most of the rates (propagation, isolation or mortality)200 are already known. This fact prevents the researcher from wasting time in selecting values for such rates201 and turns the CVOA into a metaheuristic quite easy to execute.202 However, it must be noted that the current rates are still changing and it is expected they will vary203 over time, as the pandemic evolves. Maybe these values will not be stable until 2021 or even 2022. The204 suggested values have been retrieved from the World Health Organization [25] and are discussed below:205 1. P DIE. An infected individual can die with a given probability. The case-fatality ratio (CFR)206 [26] varies by location, age of person infected and the presence of underlying health conditions but,207 currently, this rate is set to almost 5% by the scientific community [27]. Therefore, P DIE = 0.05.208 2. P SUP ERSP READER. It is the probability that an individual spreads the disease to a greater209 number of healthy individuals. It is believed that this situation affects to a 10% of the infected210 population [28], therefore, P SUP ERSP READER = 0.1. After this condition is validated, two211 situations can be found:212 (a) ORDINARY RAT E. If the infected individual is not a super-spreader, then the infection213 rate (also known as reproductive number, R0) is 2.5. It is suggested that this rate is controlled214 by a random number in the range [0,5].215 (b) SUP ERSP READER RAT E. If the infected individual turns out to be a super-spreader,216 then up to 15 healthy individuals can be infected. It is suggested that this rate is controlled217 by a random number in the range [6,15].218 3. P REIN F ECT ION. This is a very controversial issue, since the scientific community does not219 agree on whether a recovered individual can be retested positive or not. As claimed by the WHO,220 no study has evaluated whether the presence of antibodies to COVID-19 confers immunity to221 7 subsequent infection by this virus in humans [29]. Some tests performed in South Korea suggest222 a rate of 2% according to the Korea Centers for Disease Control and Prevention [30]. Therefore,223 P REIN F ECT ION = 0.02, but this value will be reevaluated, for sure, in the near future.224 4. P ISOLAT ION. This value is uncertain because countries are taking different measures for social225 isolation. This parameter helps to reduce the exponential growth of the infected population after226 each iteration. In other words, this parameter helps to reduce R0and it is crucial to ensure the227 pandemic ends. Therefore, a high value must be assigned to this probability. It is suggested that228 P ISOLAT ION ≥0.7, since this value ensures R0<1 (please refer to Figure 5 to see discussion).229 5. P T RAV EL. This probability simulates how an infected individual can travel to any place in the230 world and can infect healthy individuals. It is known that almost a 10% of the population travel231 during a week (simulated time for every iteration) [31], so P T RAV EL = 0.1.232 6. SOCIAL DIST ANCING. It is the number of iterations without social distancing measures. Since233 the populations grow exponentially at the beginning of the pandemic, this value must be carefully234 selected and must be set according to the size of the problem. Empirical values that suit for any235 codification vary from 7 to 12, so it is suggested that 7 ≤SOCIAL DIST ANCING ≤12.236 7. P ANDEM IC DURAT ION. This parameter simulates the duration of the pandemic, that is, the237 number of iterations. Currently, this data is unknown so this number can be adjusted to the size238 of the problem. It is suggested that P ANDEMIC DURAT ION = 30.239 8. strains. This parameter should be adjusted according to the size of the problem and the hard-240 ware availability, and it is difficult to suggest a value suitable for all situations. But a tentative241 initial value could be five, in an attempt to simulate one different strain per continent. Therefore,242 strains = 5. Another important decision that must be made is how to initialize every P Z associated243 with the strains. When just one strain is considered, P Z is suggested to be randomly initialized.244 However, with strains > 1 the user should search for orthogonal P Zs and to uniformly distribute245 them in the search space. This strategy should help to cover bigger search spaces in less iterations246 and to explore individuals with maximal distances.247 3.4 Pseudo codes248 This section provides the pseudo code of the most relevant functions for the CVOA, along with some249 comments to better understand them.250 3.4.1 Function CV OA251 This is the main function and its pseudo code can be found in Algorithm 1. Four lists must be maintained:252 dead, recovered, infected (the current set of infected individuals) and new infected individuals (the set253 8 5 CVOA sensitivity analysis372 This section discusses several aspects about the sensitiveness of CVOA to different configurations. Hence,373 Section 5.1 evaluates the evolution of the populations for a different number of strains. Section 5.2 assesses374 the performance when other well-known viruses are modeled. Finally, Section 5.3 provides information375 about R0and how it varies when social distancing measures change.376 5.1 Sensitivity to the number of strains377 This section provides an overview on how populations evolve over time and how the search space is378 explored, when a different number of strains is used.379 A binary codification has been used, with 20 bits, to conduct this experimentation. A simple fitness380 function has been evaluated, f(x) = (x−15)2, because the goal of this section is to evaluate the growth of381 the populations, and not to find challenging optimum values. This function reaches the minimum value382 at x= 15, that is, f(15) = 0.383 According to Section 3.3, the following configuration has been used: P DIE = 0.05, P ISOLAT ION =384 0.8, P SUP ERSP READER = 0.1, P REIN F ECT ION = 0.02, SOCIAL DIST ANCING = 8,385 P T RAV EL = 0.1 and P ANDEM IC DURAT ION = 30.386 Every experiment has been launched 50 times and, on average, the optimum value was found during387 the iteration number 13, 6 and 3, for 1, 4 and 8 strains, respectively.388 Figure 2 illustrates the evolution of the new infected population over time, for 1, 4 and 8 strains. The389 number of new infected people increases exponentially during the first SOCIAL DIST ANCING = 8390 iterations because R0>0 but, from iteration 9 on, an acute decrease is reported because R0becomes less391 than 0. This fact is controlled by P ISOLAT ION = 0.8 (a deeper study on R0and P ISOLAT ION392 can be found in Section 5.3). It must be noted that iteration 0 (PZ infection) counts as a regular iteration.393 0 5 10 15 20 25 Iterations 0 20000 40000 60000 80000 100000 120000 New infected 1 strain 4 strains 8 strains Figure 2: Number of new infected individuals for a 20-bit binary codification execution, with 1, 4 and 8 strains. Figures 3 and 4 show the accumulated number of recovered people and accumulated deaths, respec-394 15 tively. Note that deaths and recovered individuals cannot be infected again (except for the individuals395 in the recovered list that can be reinfected with a given probability, P REINF ECT ION). These two396 curves are a direct consequence of the number of new infected people so, once the number of new infec-397 tions decreases or even disappears, these values remain almost constant. Also, it can be observed that398 P ISOLAT ION = 0.8 after SOCIAL DIST AN CIN G = 8 iterations help to flatten the curves. A399 directly proportional relationship is reported between the number of strains and the number of explored400 individuals at the end of the pandemic.401 0 5 10 15 20 25 Iterations 0 50000 100000 150000 200000 250000 300000 350000 400000 450000 Recovered 1 strain 4 strains 8 strains Figure 3: Total number of recovered people for a 20-bit binary codification execution, with 1, 4 and 8 strains. 0 5 10 15 20 25 Iterations 0 5000 10000 15000 20000 25000 Deaths 1 strain 4 strains 8 strains Figure 4: Total number of deaths for a 20-bit binary codification execution, with 1, 4 and 8 strains. Four main conclusions can be drawn from the analysis of these figures:402 1. The number of new infected individuals, accumulated recovered and deaths is directly proportional403 to the number of strains.404 2. The higher the number of strains, the lower number of iterations that are required to reach the405 optimal value.406 3. The number of individuals evaluated increases at each iteration on an almost linear basis, as the407 16 number of strains increases. In case no random numbers were generated, the relationship would be408 directly proportional, that is, four strains would evaluate four times the number of individuals than409 one strain would do.410 4. To reach the optimum values, the search space explored is smaller as the number of strains increases.411 This is due to the generation of PZ evenly spaced, which makes easier to explore wider areas.412 5.2 Sensitivity to the parameters413 Several well-known viruses with deep impact in human beings’ health are modeled in this section, in414 order to assess the CVOA robustness to different input parameters values.415 Middle East Respiratory Syndrome (MERS), Severe Acute Respiratory Syndrome (SARS), influenza416 (seasonal strains) and ebola have been selected, with the parametrization shown in Table 1. It is worth417 mentioning that the modeling of each virus requires much research and a approximate parametrization418 has been used, according to the references in the rightmost column.419 Table 1: Parametrization for other viruses. Disease R0Fatality rate Vaccine Super-spreaders References SARS 1.4-2.5 11% No Yes [35, 36] MERS 0.3-0.8 34.4% No Yes [28, 35, 37] Influenza 0.9-2.1 0.1% Yes No [38] Ebola 1.5-1.9 50% Yes No [39, 40] All experiments have been conducted with four strains and 30 iterations. The viruses with vac-420 cines have been simulated by using P ISOLAT ION = 0.95 after 5 iterations, since this feature is not421 implemented in CVOA.422 Table 2 summarizes the percentage of search space explored and the best fitness found, on average.423 Codifications of 10, 20, 30, 40 and 50 bits have been used, with associated search spaces of length 1024,424 1.05E+6, 1.07E+09, 1.10E+12 and 1.13E+15, respectively. Several findings are revealed:425 1. CVOA finds the optimal values even for the longest codification (50 bits) and it is done by exploring426 a similar search space size as the other configurations do.427 2. SARS is the second best parametrization, reaching remarkable fitness even for 50 bits. But it428 required the evaluation of a greater number of individuals and, therefore, the execution time was429 greater as well.430 3. MERS obtained the poorest results in terms of fitness but it explored a smaller space search. This431 situation may be explained due to the low associated reproductive number (R0<1).432 4. Influenza has obtained slightly worse results in terms of fitness than CVOA but with less solutions433 explored. This configuration may be useful to obtain satisfactory results in a reduced execution434 time.435 17 5. The high death fatality rate of ebola prevents from exploring most of the search space. This fact436 makes difficult to visit optimal values. However, results for 40 bits are satisfactory in terms of437 fitness. For 50 bits, its use is discouraged considering the poor fitness value reached.438 Table 2: CVOA performance when different configurations. 10 bits 20 bits 30 bits 40 bits 50 bits Disease Explored Fitness Explored Fitness Explored Fitness Explored Fitness Explored Fitness SARS 57.32% 0 0.54% 0 6E-03% 1 1E-05% 4 3E-08% 252 MERS 20.34% 0 0.04% 16 1E-02% 36 1E-05% 112 2E-09% 3210 Influenza 13.23% 0 0.02% 0 8E-04% 2 1E-06% 14 1E-08% 310 Ebola 62.93% 0 0.44% 0 7E-02% 4 2E-05% 15 1E-09% 810 COVID-19 15.63% 0 0.21% 0 1.4E-03% 0 1.6E-05 0 2.0E-08 0 It can be concluded that variations in the input parameters values lead to results varying both in439 fitness and execution time. This feature is very useful for the CVOA parallel version, since strains with440 different rates and probabilities can be simultaneously launched. That is, strains aiming at diversifying441 can be combined with strains aiming at intensifying.442 5.3 Sensitivity to the social distancing measures443 In this section an analysis on how P ISOLAT ION modifies ROis conducted. The purpose is to discover444 when R0<1, situation in which the pandemic prevalence declines. A study with a 10-bit to 50-bit445 codification has been done as well as using different number of strains (1, 4 and 8).446 Figure 5 illustrates how R0varies for a 40-bit codification, with probabilities of isolation ranging from447 0 to 1, and with 1, 4 and 8 strains. Quite similar behaviors have been achieved for all codifications.448 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 P_ISOLATION 0.0 0.5 1.0 1.5 2.0 2.5 3.0 R 0 1 thread 4 threads 8 threads Figure 5: R0sensitivity to P ISOLAT ION, in a 40-bit codification. From the analysis of this figure, several conclusions are drawn:449 1. R0is linear and inversely proportional to P ISOLAT ION .450 18 2. The same negative slope is shown, with variations no higher than 10E-2 on average for all codifica-451 tions and number of strains.452 3. R0is less than 1 with P ISOLAT ION values close to 0.65 (and higher). This fact involves a453 decline of the infectious disease.454 6 Results455 This section reports the results achieved by hybridizing a deep learning model with CVOA. Section456 6.1 describes the study case selected to prove the effectiveness of the proposed algorithm. Section 6.2457 describes the dataset used. Section 6.3 discusses the results achieved and includes some comparative458 methods.459 6.1 Study case: electricity demand time series forecasting460 The forecasting of future values fascinates the human being. To be able to understand how certain461 variables evolve over time has many benefits in many fields.462 Electricity demand forecasting is not an exception, since there is a real need for planning the amount463 to be generated or, in some countries, to be bought.464 The use of machine learning to forecast such time series has been intensive during the last years [41].465 But, with the development of deep learning models, and, in particular of LSTM, much research is being466 conducted in this application field [42].467 6.2 Dataset description468 The time series considered in this study is related to the electricity consumption in Spain from January469 2007 to June 2016, the same as used in [43]. It is a time series composed of 9 years and 6 months with a470 10-minute sampling frequency, resulting in 497832 measures.471 As in the original paper, the prediction horizon is 24, that is, this is a multi-step strategy with h= 24.472 The size of samples used for the prediction of these 24 values is 168. Furthermore, the dataset was split473 into 70% for the training set and 30% for the test set, and in addition, a 30% of the training set has also474 been selected for the validation set, in order to find the optimal parameters. The training set covers the475 period from January 1, 2007 at 00:00 to August 20, 2013 at 02:40. Therefore, the test set comprises the476 period from August 20, 2013 at 02:50 to June 21, 2016 at 23:40.477 6.3 Performance analysis478 This section reports the results obtained by hybridizing LSTM with CVOA, by means of the codification479 proposed in Section 4, to forecast the Spanish electricity dataset described in Section 6.2.480 19 Linear regression (LR), decision tree (DT), gradient-boosted trees (GBT) and random forest (RF)481 models have been used with a parametrization setups according to those studied in [44, 45]. A deep neural482 network optimized with a grid search (DNN-GS) according to [43] has also been applied. Another deep483 neural network, but optimized with random search (DNN-RS) and smoothed with a low-pass filter (DNN-484 RS-LP) [46], has also been applied. Furthermore, CVOA has been combined with DNN (DNN-CVOA),485 using the same codification as in LSTM.486 These results along with those of LSTM, and combinations with GS, RS, RS-LP and CVOA are sum-487 marized in Table 3, expressed in terms of the mean absolute percentage error (MAPE). It can be observed488 that LSTM-CVOA outperforms all evaluated methods which have showed particularly remarkable perfor-489 mance for this real-world dataset. Additionally, DNN-CVOA outperforms all other DNN configurations490 which confirms the superiority of CVOA with reference to GS, RS, and RS-LP.491 Another relevant consideration that must be taken into account is that the compared methods gen-492 erated 24 independent models, each of them for every value forming h. So, it would expected that493 LSTM-CVOA performance increases if independent models are generated for each of the values in h.494 Table 3: Results in terms of MAPE for CVOA-LSTM compared to other well established methods. Method MAPE (%) LR 7.34 DT 2.88 GBT 2.72 RF 2.20 DNN-GS 1.68 DNN-RS 1.57 DNN-RS-LP 1.36 DNN-CVOA 1.18 LSTM-GS 1.22 LSTM-RS 0.84 LSTM-RS-LP 0.82 LSTM-CVOA 0.47 These results have been achieved with the individual {4,0,8}{9,7,2,7,2,7,10,7}, which decoded495 involves the following architecture parameters:496 1. Learning rate: 10E-04.497 2. Dropout: 0.498 3. Number of layers: 8.499 4. Units per layer: [250,200,75,200,75,200,275,200]500 Finally, Figure 6 depicts the first five predicted days versus their actual values, expressed in watts.501 20 20/08/2013 12:00 21/08/2013 00:00 21/08/2013 12:00 22/08/2013 00:00 22/08/2013 12:00 23/08/2013 00:00 23/08/2013 12:00 24/08/2013 00:00 24/08/2013 12:00 25/08/2013 00:00 Date 22000 24000 26000 28000 30000 32000 34000 36000 Consumption Actual Predictions Figure 6: Actual versus predicted values for the first five days in the test set (in watts). 7 Conclusions and future works502 This work has introduced a novel bioinspired metaheuristic, based on the COVID-19 pandemic behavior.503 On the one hand, CVOA has three major advantages. First, its highly relation to the coronavirus504 spreading model, prevents the users from making any decision about the inputs’ values. Second, it ends505 after a certain number of iterations due to the exchange of individuals between healthy and dead/recovered506 lists.507 Additionally, a novel discrete and dynamic codification has been proposed to hybridize deep learning508 models. On the other hand, it exhibits some limitations. Such is the case for the exponential growth of509 the infected population as time (iterations) goes by.510 Furthermore, a parallel version is proposed so that CVOA is easily transformed into a multi-virus511 metaheuristic, in which different coronavirus strains search for the best solution in a collaborative way.512 This fact allows to model every strain with different initial setups (higher DEAT H RAT E, for instance),513 sharing recovered or dead lists.514 Additional experimentation must be conducted in order to assess its performance on standard F515 functions and find out the search space shapes in which it can be more effective.516 As for future work, some actions might be taken to reduce the size of the infected population after517 several iterations, that grows exponentially. In this sense, a vaccine could be implemented. This case518 would involve adding to the recovered list, at a given V ACCINE RAT E healthy individuals. This rate519 will remain unknown until a vaccine is developed.520 Another suggested research line is using dynamic rates. For instance, the observation of the pre-521 liminary effects of the social isolation measures in countries like China, Italy or Spain, suggests that522 the INF ECT RAT E could be simulated as a Poisson process, but more time and country recoveries is523 required to confirm this trend.524 For the multi-step forecasting problem analyzed, it would be desirable to generate independent models525 for each of the values that form the prediction horizon h.526 21 Finally, further research has to be conducted in order to assess the CVOA performance when applied527 to other fields and combined with other networks.528 Supplementary material529 Along with this paper, an academic version in Java for a binary codification is provided, with a simple530 fitness function in a GitHub repository (https://github.com/DataLabUPO/CVOA_academic). The mas-531 ter branch includes a simple implementation whereas the sets branch provides an optimized version with532 a command line interface. Additionally, the code in Phyton for the deep learning approach is also pro-533 vided, with a more complex codification and the suggested implementation, according to the pseudocode534 provided (https://github.com/DataLabUPO/CVOA_LSTM).535 Acknowledgments536 The authors would like to thank the Spanish Ministry of Economy and Competitiveness for the support537 under project TIN2017-88209-C2.538 References539 [1] T. P. Velavan and C. G. Meyer. The COVID-19 epidemic. Tropical Medicine and International540 Health, 25:278–280, 2020.541 [2] R. Li, S. Pei, B. Chen, Y. Song, T. Zhang, W. Yang, and J. Shaman. Substantial undocumented542 infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV-2). Nature, 368:489–543 493, 2020.544 [3] G. Giordano, F. Blanchini, R. Bruno, P. Colaneri, A. Di Filippo, A. Di Matteo, and M. 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