A Fuzzy Rule-Based Compartmental Modeling for SARS-CoV-2

Diﬀerent epidemiological compartmental models have been presented to predict the transmission dynamics of the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) which is the most burning issue all over the world right now. In this study, we have proposed a new fuzzy rule-based Susceptible-Exposed-Infected-Recovered-Death (SEIRD) compartmental model to delineate the intervention and transmission heterogeneity in China, New Zealand, United States and Bangladesh for SARS-CoV-2 viral infection. We have introduced a new dynamic fuzzy transmission possibility variable in the compartmental model. Through our model, we have presented the correspondence of the intervention measures in relaxing the transmission possibility. We estimated that the peak in the US might arrive during the last half of August and for Bangladesh, it might occur during the ﬁrst half of August, 2020 if current intervention measures are not violated. We have modeled a prediction scenario for Bangladesh if current intervention measures are violated due to Eid-ul-Azha. We further investigated what might happen if Bangladesh government reopens everything from September, 2020. We suggested various eﬀective epidemic control policies for the authority of Bangladesh to ﬁght against the virus. We concluded analyzing the current scenario of Bangladesh suggesting that extensive tests must be carried out collecting more samples of the asymptomatic individuals along with the symptomatic cases and also proper isolation and quarantine measures should be maintained strictly to contain the epidemic sooner.

Zakaria Shams Siam formulated the methodology, designed the Maple code for solving the fuzzy rule-based compartmental model, conducted literature studies and wrote the manuscript. Rubyat Tasnuva Hasan plotted all the figures and tables, conducted the predictive analyses for all the countries. Zakaria Shams Siam and Rubyat Tasnuva Hasan formatted the manuscript. Hossain Ahamed and Samiya Kabir Youme performed the fuzzy rule-based analyses in MATLAB, conducted literature studies and provided compartmental model fitting information with the real data. Soumik Sarker Anik provided necessary information for result discussion and reviewed the fuzzy results. Soumik Sarker Anik and Sumaia Islam Alita performed necessary data collection and conducted literature studies for model simulation and prediction. This study was supervised by Rashedur M. Rahman. All the authors reviewed the manuscript and are accountable for collecting, analyzing, and interpreting the data for this study.

Introduction
As of December 2019, a drastic upsurge of respiratory disorder due to the novel severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) that is also known as COVID-19, originated in Wuhan, China [1]. To be believed having a zoonotic origin, this virus has similar qualities as 2 severe acute respiratory syndrome (SARS)type coronavirus chains or sequences being secluded in bats in the course of 2015 to 2017 [2]. This suggests that the rackets' coronavirus and the 2019 novel coronavirus allocate a modern common antecedent, hence the 2019 novel coronavirus (2019-nCoV) can be regarded as a SARS-like pathogen and consequently titled as SARS-CoV-2 [3]. Fever, dry cough, breathing difficulty, fatigue, bilateral lung infiltration in most acute cases etc. are some typical symptoms, whereas nausea, diarrhea and vomiting are some non-respiratory manifestations of SARS-CoV-2 infection, and the respiratory symptoms are similar to those caused by the severe acute respiratory syndrome coronavirus (SARS-CoV) and the Middle East respiratory syndrome coronavirus (MERS-CoV) infections [4]. Currently, this COVID-19 outbreak is increasing swiftly over most of the countries of the whole world and since the new infected cases were increasing at an alarming rate, this disease was proclaimed as a pandemic on March 11, 2020 by the World Health Organization (WHO) [5]. According to the last update till July 28, 2020, total 16,643,498 people have been reported as coronavirus positive cases, 656,545 died because of this disease but only 10,231,837 got recovered and so far, 213 countries and territories are infected [6]. So, these numbers are clearly showing that this single disease is the number one concern for today's whole world right now. So, it has now become the necessity of the current time that we understand and predict this infection transmission process realistically and also how to stop this pandemic as soon as possible to save the world. The study of infection dynamics of COVID-19 has become really complicated due to the several factors involved with it and hence disease control is getting more challenging. These several factors include: first, we are still uncertain regarding the genesis of the epidemic. Second, transmission can take place from the asymptomatic people or even inanimate carriers [1]. Third, vaccines are still not available. Fourth, the symptoms are very much alike as that of traditional flu in human body. As a result, for controlling this disease at present, we are heavily dependent upon the different intervention measures, such as isolation of the symptomatic positive cases, quarantine of the susceptible and asymptomatic cases, improvement of the medical facilities, maintaining national lockdown, social distancing, largescale sample-testing etc. to relax the transmission rate gradually. However, it is notable here that the transmission of the epidemic and the intervention measures taken by the country/community against the transmission are conversely interrelated in a nonlinear way. The number of active cases along with death cases due to the virus turn down substantially as containment measures (also known as, the treatment function, T(I)) are taken seriously to hinder the proliferation of the infection [1,[7][8][9]. So, it is precise that a transmission heterogeneity will occur provided that diversity in the intervention measures takes place. A good number of modeling studies have been conducted using different classical compartmental epidemiological models. A compartmental model comprises of different groups of individuals of the population characterized by the disease. The Susceptible-Infected-Recovered (or, Removed) or, SIR model can be regarded as one of the most straightforward models initially used by Kermack et al [10,11] to conduct a quantitative prediction on the dynamics of an epidemic. Different mathematical and statistical models that had been used in epidemiology [12][13][14] are also being exploited to estimate as well as understand the transference dynamics of SARS-CoV-2 [15]. To project more realistic scenarios and prediction, more compartments should be appended, for instance, Giordano et al formulated a new model (Susceptible-Infected-Diagnosed-Ailing-Recognized-Threatened-Healed-Extinct, concurrently SIDARTHE model) to predict the dynamics and also project the intervention measures taken by Italy against COVID-19 [16]. Yang et al conducted compartmental method incorporating multiple transmission pathways of COVID-19 considering the environmental effect [1]. Jiang et al conducted statistical analysis on COVID-19 and used the Susceptible-Exposed-Infected-Diagnosed and Quarantined-Suspected-Recovered or SEIQDR model [17]. Li et al conducted prediction-based mathematical modeling of COVID-19 using the SEIQDR model [18]. In all these compartmental models, the most sensitive and crucial parameter is the transmission rate, β that is contingent on another crucial variable, the effective reproduction number, R t [7,8]. R t indicates the original mean number of the newly infected people caused by each infectious person at time, t [7,[19][20][21]. At t = 0 (at the initial stage of the epidemic when, unknowingly no intervention measures are taken), R t =R 0 , that is known as the basic reproduction number [22]. If the value of the reproduction number goes upper than 1, the epidemic grows exponentially provided that no intervention measures are taken and if lesser than 1, then epidemic decays exponentially provided that intervention measures are strictly maintained [7,22]. If the number becomes 1, then the disease stays in the system. A number of studies on SARS-CoV-2 analyzed based on R t . For instance, Wang et al went for real data fitting of Wuhan, China using phase-based R t values [7]. Chatterjee et al also made use of R t and applied the Susceptible-Infected-Recovered-Death (SIRD) model on different Indian states along with different countries for prediction. The contemporary progress in the realm of Artificial Intelligence (AI) has driven us to exploit different data mining algorithms on the real datasets to extract novel findings. A number of studies on SARS-CoV-2 have applied these types of algorithms, such as the adaptive neuro-fuzzy inference system (ANFIS), multilayered perceptron-imperialist competitive algorithm (MLP-ICA), Genetic Algorithm (GA) etc. with the traditional compartmental and statistical approaches to achieve better accuracy in prediction conducted by Pinter et al [23], Al-qaness et al [24], Alsayed et al [25] and others. All the classical compartmental models use system of ordinary differential equations (ODE) and totally ignore the different uncertainties or impreciseness involved in the dynamic system of the infection process. The theory of fuzzy logic [26], a sub-domain of AI [27], can be applied in this context with the classical model to deal with different uncertainties and extract new knowledge from the real data of infection and death, since there are many real life applications of fuzzy logic [28] including disease epidemiological modeling [29,30]. Different infectious disease-based epidemiological studies have been conducted where fuzzy technique was applied to the classical Susceptible-Infected (SI) or SIR model to deal with different outspreads [9,31,32]. Many existing literatures using classical compartmental models on SARS-CoV-2 have worked with constant β but this is not the case in reality. Other studies used phase-based R t values to estimate phase-based β values [7,8,16], but in reality, β is a continuous time-dynamic function which is contingent on the interventionmeasure variables and many other known and unknown factors. In this study, we have constructed this parameter β as a dynamic fuzzy linguistic variable to incorporate the transmission and intervention heterogeneity in the infection dynamics of SARS-CoV-2 and also fitted our fuzzy rule-based Susceptible-Exposed-Infected-Recovered-Death (SEIRD) model with the datasets of real active cases and death cases of China, New Zealand, United States (US) and Bangladesh [6,33]. As a developing country, Bangladesh is very much at risk due to several reasons in this epidemic at present since the number of sample-tests is decreasing over time. We have delineated different probable scenarios of the execution of intervention measures in Bangladesh. To sum up, we are incorporating a fuzzy rule-based model to the classical SEIRD model to deal with the intervention and transmission heterogeneity in the infection process. This type of fuzzy rule-based work incorporated to a compartmental model, along with the real data fitting and comparison among different countries in the context of SARS-CoV-2 transmission and intervention heterogeneity taken on a quantitative (scale-based) measure, has not been implemented in the literatures. This model has allowed us to analyze a scale-based quantitative and comparative study among different countries in the context of the sufficiency in maintaining the preventive measures. This study will be important to make the better decision at a specific time while taking different intervention measures in opposition to SARS-CoV-2. We have organized the arrangement of this study as follows. We discuss the related works more elaborately in section 2. We formulate our model in section 3. We present our results, discuss and analyze the results in detail along with our observation on the efficacious epidemic control policies for Bangladesh to deal with the pandemic and the limitations with possible future works in section 4. In section 5, we conclude our study summarizing everything.

Related Works
In this section, we will discuss the studies more elaborately that have worked on the intervention measures and transmission heterogeneity using compartmental modeling for SARS-CoV-2. Wang et al [7] predicted the positive cases in Wuhan using phase-based R t . However, they have divided the timeline for Wuhan into 4 phases. The time period from December 01, 2019 to January 23, 2020 was considered to be the first phase where R 0 was taken as 3.1. January 24, 2020 to February 02, 2020 was considered to be the second phase having R 0 value as 2.6. The next (third) phase from February 03, 2020 to February 15, 2020 had R 0 as 1.9. They assumed the fourth phase from February 15, 2020 onwards having R 0 value as 0.9 or 0.5, depicting two probable scenarios. According to their estimation, the infection peak will occur in Wuhan either on February 23, 2020 (R 0 in the fourth phase = 0.9 and 58,077-84,520 will be infected) or on February 19, 2020 (R 0 in the fourth phase = 0.5 and 55,869-81,393 will be infected). However, one of the limitations of this study is -using phase-based constant R t values might induce bias in the prediction since R t is a dynamic variable. Giordano et al [16] formulated the SIDARTHE model on Italy to introduce the reflection of the containment measures. They have changed the parameters of their model over time to show the heterogeneity. They have independently varied R 0 values over time. They took R 0 = 2.38 at t = 1 (t means the number of days that have been passed since the initial time, t 0 taken in their model simulation), R 0 = 1.66 at t = 4, R 0 = 1.80 at t = 12, R 0 = 1.60 at t = 22, R 0 = 0.99 at t = 28 and R 0 = 0.85 at t = 38 and thus depicted the success of the containment measures taken by the government of Italy. Clearly, their approach is similar to that of Wang et al [7]. Chatterjee et al [8] studied the progression of SARS-CoV-2 in Indian context exploiting the SIRD model incorporating the impact of different containment measures. To implement the influence of lockdown in their model, they assumed that β is constant before lockdown. After the lockdown is imposed at t = τ , time dependent β(t), which is the infection rate, decreases over time. They assumed that it decreases exponentially according to these functions: [8,15,34], where ζ is the infection/interaction parameter, T is the time delay (in days) before the lockdown-impact is detectable. According to their model prediction, the active peak infection in India will occur at the climax of June or the onset of July, 2020. However, their assumption regarding the concept of a constant β before the imposition of lockdown is not that much realistic since, β can vary due to many reasons, even before the declaration of the national lockdown in reality as intervention measures can be maintained or violated in the local areas of a country to different degrees then. Wang et al [1] conducted the Susceptible-Exposed-Infected-Recovered-Viral Concentration (SEIRV) model to project multiple transmission pathways for SARS-CoV-2. They have stated that with the imposition of different intervention measures, the growth of infection will fall down over time. They have used the function β I (I) = β I0 1+cI , where the positive constant β I0 is the maximum transmission rate before taking the intervention measures, c is a coefficient that adjusts the transmission rate and β I (I) is the transmission rate at a specific time point after taking the intervention measures when I number of infected cases exist. All of the studies described above used modified classical compartmental models to demonstrate the transmission and intervention heterogeneity. Since no additional intelligent system was used, no further new knowledge was extracted. Though a few studies analyzed the risk and preventive factors of COVID-19 using fuzzy rulebased model [35], no real prediction was drawn by incorporating any mathematical modeling. In this study, we made an attempt to incorporate a fuzzy rule-based intelligent system to the classical SEIRD model to illustrate the transmission and intervention heterogeneity. As a result, we utilized the scale-based quantitative comparison of intervention measures taken among different countries and in different scenarios as our new knowledge. In the subsequent section, we will formulate our model.

Fuzzy Rule-Based SEIRD Compartmental Model Formulation
In our proposed model, there are 5 compartments (5 different groups of individuals) in total in the whole population, N. They are: susceptible (S), exposed (E), infectious (I), recovered (R) and dead (D). In this section, we will first introduce all these necessary components and parameters we have used in our model as well as result analysis. t = The number of days passed since the initial time, t 0 (t = 0) considered in the model. For modeling the outspread of SARS-CoV-2 in different countries, t 0 for each country will be different due to different onset time of the disease in that particular country [6]. S(t) = The number of the susceptible individuals in the population at time, t. These individuals are not infected yet but vulnerable to the infection. S 0 = S(0) = The initial number (at t = 0) of the susceptible individuals. The determination of S 0 might be quite strenuous. The whole population of a region might be susceptible to the disease provided that the vaccine or antibody is unavailable at the moment. Although most of the epidemiological studies conventionally assumed that S 0 ≈ N , different demographic characteristics (societal, economic or geophysical factors) of a place can manipulate the value of S 0 significantly. Therefore, we have estimated maximum and minimum S 0 using 2 approaches. First, we got the lower bound of S 0 by using the formula of S 0.min . S 0.min = N ( N umber of total cases at the elevated plain of the inf ection growth curve N ). This formula denotes the lower bound of S 0 since many asymptomatic and mild symptomatic cases might have been unreported for those countries [8]. Second, we got the upper bound of S 0 by using the formula of S 0.max . S 0.max = N ( N umber of total positive cases identif ied N umber of total tests conducted ). This formula denotes the upper bound of S 0 since a country can have many regions isolated where the population might not be susceptible. So, S 0 for a country may potentially vary in the range [S 0.min , S 0.max ] due to the demographic and other factors of that country. E(t) = The number of the people that are infected but lack of the manifestation of infection at time, t. They can be regarded as 'inactively infected'. This compartment is necessary for COVID-19 disease dynamics because clinical evidence shows that during the first 2-14 days (or 2-27 days) after being infected by the virus, also known as the incubation period, infected individuals may not develop any kind of symptoms but are still able to transmit the disease to other people even being unaware of their infection [36]. E 0 = E(0) = The initial number of the exposed individuals. I(t) = The number of active cases at time, t. All these people are both infected and infectious. They possess the symptoms of contagion. They can disseminate the disease to the susceptible individuals. I 0 = I(0) = The initial number of the infectious individuals (active cases). R(t) = The cumulative total number of the recovered individuals who got cured from the disease at time, t. We have assumed in this model that they will not become susceptible to the disease again. R(0) = The initial number of the recovered individuals. D(t) = The cumulative total number of the deceased (due to the viral infection) individuals at time, t. D(0) = The initial number of the deceased (due to the viral infection) individuals. T E = The average incubation period. An individual takes T E amount of time on an average to go from the E state to the I state [37]. We find the average exposure rate, σ by applying the formula σ = 1 T E [1]. T I = The infectiousness period. On an average, an individual takes T I amount of time in the I state (before going to the R or D state) [37]. We find the average recovery rate, γ by applying the formula γ = 1 T I [1] α = The proportion of I that moves to D. So, 1 − α indicates the proportion of I that moves to R [37]. β = This parameter is known as the transmission rate of infection in the literatures. However, in our proposed fuzzy model here, to delineate the transmission and intervention heterogeneity, we take this parameter as a dynamic fuzzy linguistic variable, β(I s , Q, M, t) which depends on 4 other independent variables and it indicates the possibility of transmission of infection to take place due to the contacts between the susceptible individuals and the positive infectious cases at time, t when 3 other independent variables (I s , Q and M that will be discussed shortly) will hold in the surrounding environment. This implies that, unlike the other classical models, this β can contain some more possible and also some less possible values because of the independent variables involved with it. Since t is already introduced before, the 3 other independent variables (all are fuzzy linguistic variables) are discussed below. I s = The score (on a Likert Scale from 0 to 10) of the sufficiency in the isolation measures of the symptomatic positive cases taken by a country. This score quantifies the efficacy of the taken isolation measures of the infected cases up to that time (from the beginning) to contain the epidemic. Q = This score (on a Likert Scale from 0 to 10) evaluates the efficacy of the quarantine measures (taken by a particular region or country) of the susceptible and asymptomatic cases up to that time (from the beginning) to contain the epidemic. M = This score (on a Likert Scale from 0 to 10) weighs the efficacy of the medical facilities of a country or region up to that time (from the beginning) to contain the epidemic.
For simplicity in our model, we have assumed that the above 3 fuzzy variables can cover all possible intervention measures that could be taken by a country or community to slow down the pace of the growth of COVID-19. The determination of these variables is very much imprecise. For example, I s cannot be just determined by the number of isolation beds in the hospitals of a specific country since, many self-isolation can occur as well that cannot be recorded precisely. Likewise, Q can also be both reported (number of home-quarantine and institutional quarantine) and unreported (every unreported susceptible person can be considered as quarantined to some degree of fuzzy membership on a scale from 0 to 1). Moreover, M might incorporate the number and quality of the sample tests conducted, number of hospitals in that country, medical laboratory facilities, diagnosis procedures and accuracy, overall treatment system in the hospitals, intensive care facilities, number of beds in the hospitals, the approachability of medical staffs, the availability of necessary medical instruments and everything else related to the medical system of that country. For these reasons, we have considered these variables as linguistic (Low -[0, 0, 4], Medium - [3,5,7] and High -[6, 10, 10]) variables and also counted them on a scale from 0 to 10. By 'Low' here, we mean the intervention measures taken up to that time from the beginning are still ineffective to contain the epidemic. Likewise, by 'Medium' and 'High' here, we mean the intervention measures taken up to that time are moderately effective and highly effective respectively. We considered these as fuzzy variables since we wish to incorporate these variables to a fuzzy expert system to extract new knowledge [29]. We have taken R t as our fuzzy output variable and we took it as a linguistic variable because Imai et al [22] also assumed different levels of transmission (low, moderate and high transmission level) in their study. The scales of different levels of R t will 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 be different for distinct countries because of the demographic factors and different epidemic trends of different countries. For China, we have taken the scale of R t as: low -[0, 0, 0.02], medium -[0.01, 1.2, 4] and high - [3,15,15]. For New Zealand and the US, we have taken the scale of R t as: low -[0, 0, 0.01], medium -[0.005, 1.3, 4] and high - [3,12,12]. For Bangladesh, we have taken the scale of R t as: low -[0, 0, 0.01], medium -[0.005, 1.7, 3] and high - [2,10,10]. By low, medium and high R t , we mean low risk, moderate risk and high risk of transmission respectively. We have then used the fuzzy Mamdani model [29,38] for building fuzzy linguistic rules between fuzzy inputs and output since Mamdani model is more applicable in projecting the epidemic system because of its expertise to aggregate the expert knowledge and experience into the model despite the unavailability of the functional information. After obtaining the fuzzy output, R t at each t point from the Mamdani model, we calculate our dynamic variable, β(I s , Q, M, t) by using the formula β(I s , Q, M, t) = R t γ [7,8], where R t is a defuzzified output. Then we have used this dynamic β(I s , Q, M, t) parameter inside the compartmental SEIRD model. The following system (1) is our model or the system of ODE.  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  We finally solved the SEIRD model (1) for S(t), E(t), I(t), R(t) and D(t) curves and then fit the predicted I(t) and D(t) curves with the real active cases and death cases of different countries respectively. The flowchart of the complete methodology of this study is presented in Fig. 3.  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 In the subsequent section, we will present all the results derived from our model.

Results and Discussion
Here we have described our results step by step. Concurrently, we have discussed the crucial observations drawn from our results.

Determination of Transmission Possibility from Fuzzy Intervention Variables and the Efficacy of the Fuzzy Model
As discussed in section 3, we have exploited the fuzzy Mamdani model to delineate both transmission and intervention heterogeneity. The fuzzy Mamdani model has 3 fuzzy inputs (I s , Q and M) and 1 fuzzy output (R t ). The combination of these 4 fuzzy subsets resulted in total 81 rules from where we carefully analyzed and selected 34 rules relevant in our model and showed these rules in table 1. We ran the model with max-min inference and for defuzzification, we applied the centroid or center of area technique.
Using this fuzzy rule-based system, we have extracted different values of R t (transmission variable) based on different values of I s , Q and M (intervention variables) over time of a particular country depending on how much efficacy the taken intervention measures hold. There might be more than one combination of the values of 3 fuzzy input variables to produce the same fuzzy output value but we took the most relevant combination of the 3 input values to produce a specific R t at a particular t which is more suitable for fitting with the real data eventually and also consistent with the overall real situation of each country in this pandemic validated by verified information and the built fuzzy rules. From the rules, it is perceptible that if the intervention measures are taken fruitfully so far, or in other words, up to a certain t point, if the taken fuzzy input values are optimal to mitigate the transmission possibility to some degree, then R t will be kept minimum and thus transmission possibility, β(I s , Q, M, t) will be lower at that time and the possibility of the growth of more active cases and death cases will gradually fall down at the end of that time point. For building the fuzzy rules, we relied on the epidemiological and clinical information regarding SARS-CoV-2 and the expert knowledge we got from verified internet sources. For example, our selected first rule says, if isolation measures of the symptomatic cases (I s ), quarantine measures of the susceptible and asymptomatic cases (Q) and medical facilities (M) provided by a country turn out to be ineffective (represented by the linguistic term 'Low' in our fuzzy rules) up to a certain point of time, then the effective reproduction number (R t ) at that time point will be 'High' (high risk of transmission), meaning high number of secondary infections will occur due to a primary infectious case at that time period. Thus, transmission possibility, β(I s , Q, M, t) will also get high then. As a result, more active and death cases will grow over time. This model is efficacious because we can establish and take out new information regarding the correspondence between the taken intervention measures and the transmission   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 65  (1), we can extract model I(t) and D(t) curves and compare them with the real active cases and death cases data of that country respectively to both predict and determine the prediction accuracy of the model. Now if the fitting of the model I(t) and D(t) curves with the real data of a country gets pretty good (meaning, the model provides pretty similar information as the real information regarding the number of active cases and death cases of a country; in this study, for verification, we have fitted our model with China and New Zealand in the subsequent subsection since 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 the epidemic is almost over in these 2 countries so that we can verify our model as well as the selected fuzzy intervention values), then in the inverse way, we can say that the fuzzy output values (R t ) were correctly estimated by the fuzzy Mamdani model to predict the scenario as yet. This infers that the fuzzy input values we set in the first place were correct to project the present, past and future epidemic scenario. With the help of this complete fuzzy rule-based model, we have predicted the future possible scenario in the subsequent subsections using the SEIRD model in (1) provided that the ongoing intervention measures are continued to be held or violated by the community or country.

Initial Conditions and Parameters Estimation: Solving the SEIRD Model
In this study, we have conducted our analyses with 4 different countries -China, New Zealand (NZ), the United States (US) and Bangladesh. For predictive analyses, we used the SEIRD model in (1). However, this type of model is highly sensitive to the estimated parameters and chosen initial conditions. So, all of these parameters should be estimated as correctly as possible to project the transmission scenario realistically. However, making a perfect estimation is not possible due to the simplicity of the compartmental modeling, unknown information regarding the virus and also because the epidemic is still ongoing. In the model, we took t 0 at January 02 for China, February 28 for NZ, February 15 for the US and March 26 for Bangladesh based on the onset time of the epidemic in these countries. As discussed in section 3, using the concept of S 0.max and S 0.min [8], we have estimated S 0 = 555000 for China, S 0 = 600000 for NZ, S 0 = 25000000 for the US and S 0 = 1550000 for Bangladesh for the best fit with the corresponding real data. S 0 might seem relatively less for China compared to NZ but we claim that this is true because the whole people of China were not susceptible to get infected. According to the daily report of NHC, only Hubei Province (the capital city of this province is Wuhan which was the primary source of transmission) had over 80% of new active cases of the whole China and also, after January 23, 2020, public transportations from Wuhan to outside of Wuhan were strictly stopped [7]. As a result, these intervention measures kept the S 0 of the whole China much lesser compared to the huge population of China. We considered I 0 = 41 for China [7,39], I 0 = 1 for NZ [6], I 0 = 12 for the US [6] and I 0 = 33 for Bangladesh [6] based on the real active cases at t 0 of each country. We have estimated E 0 = 20I 0 for China and NZ and E 0 = 30I 0 for the US and Bangladesh based on the estimation of the literatures [7]. We have considered the initial recovered, R(0) = 0 and the initial death, D(0) = 0 in the model [7,40]. However, we took D(0) = 5 for Bangladesh based on the real data [6]. We have set T E = 5 days (σ = 1 5 days −1 ) for China and US, T E = 4.5 days (σ = 1 4.5 days −1 ) for NZ, T E = 6 days (σ = 1 6 days −1 ) for Bangladesh, T I = 14 days (γ = 1 14 days −1 ) for China and US, T I = 12.5 days (γ = 1 12.5 days −1 ) for NZ, T I = 19 days (γ = 1 19 days −1 ) for Bangladesh, α = 0.0365 for China, α = 0.0102 for NZ, α = 0.022 for the US and α = 0.01 for Bangladesh. All these chosen parameter values are concurrent with the estimation of those in the published literatures [1,7,8,[40][41][42] and the real infection datasets of these countries as yet [6]. Using these parameter values and initial conditions, along with the values of β(I s , Q, M, t) (described in subsection 4.1), we have solved system (1) for determining the predicted S(t), E(t), I(t), R(t), D(t) curves and also the total cases curve over time for China, NZ, US and Bangladesh. The predicted I(t) and D(t) curves (considering that the current intervention measures will be maintained) for these 4 countries are delineated in Fig. 4 with a fitting with their corresponding real active cases and death cases respectively.  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62 63 64

Verification of the Complete Model: Fitting with the Real Datasets
In this subsection, we have verified our fuzzy incorporation technique to the SEIRD model by showing the incorporation of the real active cases and death cases of China and NZ to our predicted model curves since these 2 countries have almost contained this epidemic already [6] (although to sustain this situation, they have to be very careful to continue taking measures against any future possible transmission of the virus). So, applying our model on the real infection datasets of these 2 countries will be a good way of verifying our fuzzy incorporation technique as well as the validity of our selected fuzzy rules and input values. We have selected the fuzzy input values based on the information of the intervention measures taken in these countries. For simplification, we have considered the first two values after the decimal points of every fuzzy input and output variables in the Mamdani model. In China, their intervention measures heterogeneity resulted in creating four phases described by Wang et al [7]. Concurrent with their information, in the first phase Consequently, this led to producing maximum β = 0.79 on February 01, 2020 due to the unawareness of people in January. From January 23 onwards, China (mainly Wuhan and Hubei Province) started taking strict measures [7] gradually. Wang et al [7] and Chatterjee et al [8] described that the taken strict measures require some amount of latent time for being effective at mitigating the transmission rate. This implies that the efficacy of the intervention measures will be visible after some days if the measures are stringent enough to contain the epidemic. So, this verifies the accuracy of our estimation of maximum β to be held on February 01, 2020 for China that was produced from the fuzzy Mamdani model by our fuzzy input values set at t = 30, since the measures taken by China was stringent enough [43,44] to fight and win the infection eventually. From January 23 till February 02 [7], isolation and quarantine measures were continuously being established in China. So, they were shifting towards taking moderately effective control measures from ineffective control measures. From February 03 onwards [7], the medical facilities provided by China to contain the disease got very much effective. Accordingly, we have set the values of the intervention variables from the moderately effective scale towards the highly effective scale over time. For instance, to project the reality, at t = 40 (February 11, 2020), we set I s = 6.43, Q = 6.97 and M = 7.38. For these values, β turned out to be 0.086 approximately from the Mamdani model which   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 was even less than the β at t 0 due to the upsurge in taking effective intervention measures. At t = 60 (March 02, 2020), we set I s = 6.44, Q = 7 and M = 7.63 (that means the taken measures up to that time got more effective) taking the latency time required for the effectiveness of the containment measures into our consideration. As a result, at t = 60, β turned out to be very less (β = 0.12), meaning the transmission risk got lesser over time due to the strict intervention measures. We have set the values of the fuzzy intervention variables in the similar process for NZ, US and Bangladesh as well (according to the information on the efficacy of the distinct intervention measures they took individually as a country) as what we did with China. In Fig. 4(a) -Fig. 4(d), we have fitted our I(t) and D(t) curves with the real active cases and death cases of China and NZ respectively. We have fitted our model with the real data of China from January 02, 2020 to July 22, 2020 and the real data of NZ from February 28, 2020 to July 22, 2020. In Fig. 4(a) - Fig. 4(d), the coefficients of correlation of the model I(t) curves of China and NZ with their corresponding real data of active cases are both around 0.99 and the coefficients of correlation of the model D(t) curves with their corresponding real data of death cases are both around 0.97. Therefore, it is evident that the fitting has been very good with the real datasets (both for the active and death cases of both countries). For China, the peak in active cases took place at t = 47 (February 18, 2020) with 58016 active cases, whereas, our predicted I(t) curve for China has its peak at t = 44 (February 15, 2020) with 57790 active cases. The D(t) curve predicts that the total number of deaths up to July 22, 2020 will be 4687 whereas, real data says this number is 4634 [6]. For NZ, the peak in the number of active cases occurred at t = 38 (April 06, 2020) with 929 active cases, whereas, our predicted I(t) curve has its peak at t = 40 (April 08, 2020) with 931 active cases. The D(t) curve predicts that the total number of deaths up to July 22, 2020 will be 23 whereas, real data says this number is 22 [6]. So, the error at prediction is very much negligible for both countries. From Fig. 4(e) - Fig. 4(h), it is evident that the real data so far has been fitted quite good with our model curves of US and Bangladesh as well. This implies that our fuzzy input values (at all t points) and built fuzzy rules are consistent with containing the information regarding the epidemic and also projecting the epidemic trend achieving a pretty good accuracy. Furthermore, this fuzzy technique allows us to generate dynamic β values over t which is more aligned with the reality, unlike the phase-based β values being used in some good literatures [7] which is not that much realistic since the intervention measures taken by a country always vary continuously from time to time and we just simulated this realistic fact in this fuzzy rule-based compartmental model. All these facts and observations prove the efficacy and strength of our fuzzy model in delineating the intervention and transmission heterogeneity more realistically.

Predictions for the US and Bangladesh (If Current Intervention Measures are Maintained)
In Fig. 4(e) and Fig. 4(f), we have delineated the predicted I(t) and D(t) curves for the US fitted with their real data. And, in Fig. 4(g) and Fig. 4(h), we did the same for Bangladesh. We fitted our model with the real data of the US from February 15, 2020 to July 22, 2020 and the real data of Bangladesh from March 26, 2020 to July 31, 2020. Fig. 4(e) -Fig. 4(h) predicted for US and Bangladesh assuming the fact that the current intervention measures will be maintained in future. As of July 22, 2020, US is the most affected country [6] due to this virus and this country is approaching towards the second wave of infection. According to the model, for the US, the first peak was supposed to occur at t = 96 (on May 21, 2020) with 1225505 active cases, whereas, the first peak actually occurred at t = 105 (on May 30, 2020) with 1169161 active cases. So, the prediction is pretty close to the real data. According to our model, the 2nd peak (biggest peak) might be held at around t = 180 (around August 13, 2020) with 2267376 active cases if US continues with their current containment measures and total 250993 people might die due to the virus by then. This peak numbers have been estimated using the best fit of S 0 with the real data as yet. However, peak size of both active and death cases can increase or decrease if S 0 increases or decreases respectively (delineated in Fig. 4(e) -Fig. 4(h)). So, the number of peak active cases in the US might reach up to total 4199789 and total death cases might reach up to 320237 (using S 0.max ) at around August 27 according to the prediction curves (Fig. 4(e) -Fig. 4(f)). The peak time might shift from around August 04 (with 202022 death cases in total) using S 0.min to August 27 (with 320237 death cases in total) using S 0.max . So, the upsurge in the size of S 0 is shifting the infection peak time more rightward and the peak value more upward. This observation is consistent with the study by Chatterjee et al [8]. Similar prediction results have also been drawn in a study by Villaverde et al [34]. According to their estimation, above 230000 deaths in total might occur in the US by September. Another study by Marwan Al-Raeei [45] estimated that the active cases might reach 1717800 by the middle of August and 1941600 at the onset of September. Also, the number of total deaths might reach 208940 by the middle of August and 244420 at the onset of September. So, these numbers (both peak time and peak value) are pretty similar as our derived results for the US. We have set the fuzzy intervention variables as relevant as possible for the US compared to those of China and NZ to generate these predicted curves for the US that matched with their real data and predicted their future infection. Chatterjee et al [8] discussed that the growth of R t implies that the intervention measures taken by the US were not as stringent as that of the other countries to be effective at containing the epidemic. We set the fuzzy input values consistent with their information and observation regarding the efficacy of the intervention measures taken by the US. For Bangladesh, our model predicted that if current intervention measures are maintained, the peak might occur at around t = 136 (around August 09, 2020) with around 111573 active cases and by then, total 3407 people might die due to the virus. Using S 0.min and S 0.max , we observed from Fig. 4(g) -Fig. 4(h), peak active cases might reach maximum 130348 by August 11 and total death cases might range from 3193 at August 09 to 3780 at August 11 in the peak time in Bangladesh. So, the situation in Bangladesh is not that satisfying rather intimidating from future prediction. A study conducted by Nabi [46] predicted using the data of Bangladesh from January till May 10 that the peak in daily infectious cases might occur on June 11 having around 2209 new cases. However, Bangladesh has already seen 4019 new cases (in 1 day) on July 02 [6]. Another study by Hoque [47] estimated that the peak active cases might occur during the first half of May. However, that was not the reality for Bangladesh. Rather, active cases are still growing [6]. Hence, our prediction for Bangladesh   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 should be more accurate.

The Correspondence of the Intervention Measures on the Transference of Infection in China, NZ, US and BD: A Comparative Study among the 4 Countries
In this subsection, we have explored the effects of the taken intervention measures of the 4 countries on their infection trend using our model. National and regional lockdown are major intervention measures to contain the infection when the whole country or different infected regions of the country try to follow isolation, quarantine and other intervention measures very strictly. In table 2, we have showed the relationship between the intervention measures and the transmission speed after the imposition of lockdown in these 4 countries and thus conducted a comparative study among the countries. A lockdown was imposed in Wuhan from January 23 to April 07, 2020 [43]. Their stringent containment measures have been discussed in subsection 4.3. NZ imposed very early national lockdown unlike most other countries to eliminate the infection [48]. Following a 4-stage intervention system step by step, NZ eventually imposed full national lock down on March 25, 2020 having only 102 recorded positive cases and no single death case. US and most other currently infected European countries along with many developing countries in other continents did not take as early measures as them. For being such early and stringent in their lockdown, they had to trade off their economy to some more extent than other   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 countries but as a result of their efficacious intervention measures timely, they have restored their nation's health and almost contained the epidemic already. They showed extra-ordinary performance in contact-tracing activities and conducting large-scale tests. NZ so far conducted 94054 tests per 1 million of their population whereas China tested 62814 per 1 million of their population [6]. Both these countries implemented effective epidemic control measures in all the ways. The peak size of NZ is comparatively very smaller compared to other countries. The WHO has complimented NZ for their time-efficient measures setting as an ideal sample for other countries to follow [48]. US did not implement early lockdown as NZ and they tried to follow the optimal path between health (full lockdown) and economy (lifting or relaxed lockdown). California first imposed the greatest lockdown in the US on March 19 [49,50]. A study by Columbia University concluded that 54000 deaths could have been avoided in the US if the infected states had imposed their lockdown around March 1 [49]. It is hence obvious that both stricter and earlier intervention measures play dramatic role in containing the epidemic. However, following around 60 days lockdown, all the states of the US decided to impose partial lockdown and reopen shops, restaurants with new instruction policy but all the states are not going along with the new guidance [51]. Different states did not want to allow full lockdown and tried to reopen [51] and their taken measures were not that effective as China and NZ [8]. Consequently, as of July 31, 2020, US has reported the highest number of total positive cases (4705889) and death cases (156775) [6]. So, the effect of their intervention measures is quite obvious and visible in their infection scale. Bangladesh, being a developing country and suffering from many other intricate complicacies to fight against the virus (to be discussed in subsections 4.6 and 4.7), imposed lockdown on March 26 [52] but like the US, they reopened public transports and offices from May 31 for economic stability [53] since the socioeconomic context of Bangladesh does not actually support too-lengthened lockdown. However, the overall containment measures taken in Bangladesh during the lockdown and after reopening were very flexible compared to China and NZ and the experts believe that the lockdown was not that effective to contain the epidemic [54]. This situation further deteriorated after reopening shops and markets prior to Eid-ul-Fitr [54]. For analyzing in table 2, we have considered the lockdown onset time of Wuhan and California as the lockdown onset time of China and US respectively. We have included the values of the fuzzy intervention and transmission variables in this table at different time points after the imposition of lockdowns in 4 countries to observe the role of lockdown in containing transmission. From the table, using their intervention measures, China reduced their transmission possibility by approximately 96.67% within their 21st -40th days of lockdown. Then, another 20 days of being in the lockdown, they decreased β again by around 76.67%. And then, β became very less and reached a plateau. So, they closed their lockdown after 76 days. Also, the values of I s , Q and M increased as the time progressed after imposing their lockdown. For NZ, we have explored that the values of β were very less compared to the other 3 countries throughout the period after imposing lockdown. However, their β increased to some extent as days progressed which indicates that even after mitigating the number of positive cases, taking necessary measures should be maintained. However, their intervention variables were always constrained to the range of highly effective measures after lockdown, as can be seen from table 2. US minimized their transmission possibility by ap -1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 proximately 42.86% and 33.33% within their 21st -40th days and 41st -60th days of lockdown respectively. However, from the table, it is evident that after 60 days of being in the lockdown, when they reopened shops and other things, their transmission possibility increased by approximately 12.5% after 20 days and 33.33% after 40 days upon lifting their lockdown. Concurrently, their intervention values got decreased as well after relaxing lockdown. So, according to our results, they should have remained under full lockdown some days more to further mitigate β like China. For Bangladesh, the transmission possibility was higher than the other countries at most of the time points after imposing lockdown, meaning that their lockdown was not effectively stringent. Moreover, all their intervention variables including medical facilities were mostly ineffective even after lockdown. According to our results in table 2, during the 21st -40th days after lockdown in Bangladesh, β increased by 13.33% approximately. Then it decreased by 23.53%, 15.38% and 36.36% sequentially after every 20 days interval. So, the decreasing rate of β in Bangladesh is lesser than China and even US (when US was under full lockdown). So, these results imply that stricter intervention measures should be taken in Bangladesh to contain this epidemic like China and NZ, otherwise, situation might further worsen.

Different Prediction Scenarios for Bangladesh
In this subsection, we have delineated different possible scenarios for Bangladesh in the near future due to violating the current trend of taken intervention measures.   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  due to selling and purchasing animals, gatherings in cattle market, distributing meats among people etc. They fear that the situation might worsen after 15 days of Eid-ul-Azha than the situation after Eid-ul-Fitr due to the huge activities taken for sacrificing the animals [55,56]. In table 3, we have analyzed what might happen if intervention measures are degraded due to the Eid. From table 3, the  According to our prediction, the peak number of active cases will be in the range from 113255 around August 30 (using S 0.min ) to 175387 around September 03 (using S 0.max ). The best fit of S 0 says the peak will emerge at around August 30 with around 138300 cases. In this peak time, the total deaths could reach from 4231 by August 30 (using S 0.min ) to 5377 by September 03 (using S 0.max ). Best fit of S 0 says a total number of 4626 death cases might occur by August 30 if the current containment measures are violated due to the Eid. So, to avoid this disastrous scenario after Eid, necessary steps should have been taken before Eid to minimize the gatherings. The government should have developed a single system for performing the activities regarding selling, purchasing, slaughtering animals and also distributing meats among people under that system in a few specific regions for reducing the engagement of individuals to a minimal degree [55]. . So, this vast individuals within a very limited region on an average possess pretty good possibility to result in producing more contact rate with the infectious (both asymptomatic and symptomatic) people. Again, Bangladesh is a developing country and hence pausing economic activities for a long time due to lockdown is also detrimental to the economic status of Bangladesh and not feasible as well. So, local authorities are in favor of reopening the economic wheel by coming back to the normal economic life again. But as derived from our results, the likely consequence of this might be so disastrous for Bangladesh. Hence, we need to keep finding an optimal path between the economy and disease prevention strategies, which will be very 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 much challenging for us. However, medical facilities provided by the country are not that much standard because access to proper medical care for every infected individuals has not been possible [52]. Moreover showing symptoms, go for taking tests and then after some days, he/she gets the result as positive, but this only means that even before knowing, maximum infectiousness has already occurred and thus transmission might have already spread unknowingly. This is disastrous because a research study by Yang et al [66] considered that the transmission rate due to the contacts between the asymptomatic exposed and susceptible individuals is five times greater than the contacts between the symptomatic infected and susceptible individuals. So, the exposed individuals are playing a great role in transmitting the infection. According to our fuzzybased results, the peak number of exposed individuals in Bangladesh might be in the range from 33275 (around August 04) to 51280 (around October 17). With this huge asymptomatic people, the situation might be very much disastrous in Bangladesh. Hence, immediate and effective epidemic control policies need to be implemented more strictly against the epidemic. The authorities should abandon the test fees and start taking ubiquitous testing of both asymptomatic and symptomatic individuals. Media should make the mass people more aware concerning the importance of testing asymptomatic samples. Our results delineate that social distancing and other intervention measures should be maintained more strictly and current medical facilities should be ameliorated to contain the epidemic.  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 all these control measures (more elaborately, making an optimal combination of I s , Q and M at all time points from now on for further infection prevention) will save Bangladesh from this pandemic.

Used Tools and Software
For creating the fuzzy membership functions of the fuzzy input and output subsets, building the fuzzy rules and conducting the fuzzy rule-based analyses, we have used MATLAB R2018a software [67] and for solving the SEIRD compartmental model for prediction-based analyses, we have used Maple software tool [68]. All the figures were created and edited in MATLAB R2018a, Maple software tool and Adobe Illustrator 2020. Microsoft Excel 2019 was used to extract the model fitting information for China and NZ.

Limitations and Possible Future Work
In our model, we have assumed no population heterogeneity, meaning that all individuals in the population possess the same immunity and the contact rate is homogeneous. In our future study, we aim to model incorporating population heterogeneity in the infection dynamics in Bangladesh using the theory of Fuzzy logic [69]. In the present study, we have also assumed that transmission will hold from human to human only. We ignored the natural birth rate and death rate of people since the characteristic period for these demographic factors take more time than the period of the epidemic. In our system (1), we have only assumed that infectious (I) individuals can disseminate the disease to the susceptible (S) individuals by incorporating the β(I s , Q, M, t)SI term in (1). However, for SARS-CoV-2, contagion can spread even from the exposed (E) group. To project this information in our modeling, we have a plan to extend this study in future to incorporate the β 2 (I s , Q, M, t)SE term in (1) where, β 2 is the fuzzy transmission possibility due to the contacts between S and E. We expressed the intervention heterogeneity through 3 major intervention measures in the form of fuzzy subsets for simplification. More subtle fuzzy intervention subsets can be added to this model to delineate the intervention heterogeneity more realistically with extracting more new information. We have further assumed that after getting recovered, an individual will no more be susceptible to the disease again (meaning, they will not be infectious again). This assumption is testified by a study in South Korea [70,71]. However, we still do not have any rigid clinical information in favor of this assumption [72,73]. Furthermore, the prediction from the compartmental model is affected to a great extent by the estimation of the used parameters. Since the epidemic is still ongoing, the parameters are dynamically changing each moment. So, in this stage, there will always be some error at predicting the actual values of these parameters. So, the predictions made here might be, to some extent, deviated from the actual scenario in the future. Another big reason behind this is we still do not know every piece of vital information regarding this virus. Besides, being a very simplified and abstract model, the SEIRD model does not include all the real   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 case scenarios (like the subtle clinical information regarding the virus) but adding more compartments to this model can project so. Our system (1) does not include the spatial topological reliance of predictions. A study by Gatto et al [74] included this in their SEIR compartmental modeling study for Italy. In our future work, we wish to incorporate this approach for all the divisions in Bangladesh to make a more reliable and realistic prediction using our fuzzy rule-based compartmental model.