Application of Mathematical Modeling in Prediction of COVID-19 Transmission Dynamics

The simplest compartmental model is SIR (‘Susceptible-Infectious-Removed’) where S, I and R are the susceptible, infectious (can infect others) and removed (recovered or dead) [76] populations, respectively. A flowchart of this model is presented in Fig. 3. There are many models that have been used to investigate the transmission dynamics of viruses, and those models are in some way derivatives of the basic SIR model. For the present pandemic circumstances, many researchers have developed SIR-based mathematical models. The model is formulated by a set of nonlinear ordinary differential equations (ODEs) and then solved numerically. The simplest form of nonlinear ODEs can be expressed as in Eq. 1.(Parameters: \(\alpha \)—removal rate, \(\beta \)—infection rate).

In this study [25], an age-structured SIR mathematical model considering social connection matrices based on surveys and Bayesian imputation is presented to inspect the momentum of the SARS-CoV-2 pandemic in India. This study accentuated the importance of both social contact and age structures in appraising the country-specific impacts of the widely used social distancing strategy for controlling and mitigating the virus. In [24], the authors proposed a simplified SIR mathematical model to predict the peak of the disease infection and suggested that the healthcare system could significantly shorten the outbreak period and reduce one-half of the transmission. In another study, [41], an SIR model is used to predict disease (COVID-19) trends and how quarantine decreases infection.

A compartmental model SEIR (‘Susceptible-Exposed-Infectious-Removed’) with S, E, I and R represents the Susceptible, Exposed (infected, but not infectious), Infectious (can infect others) and Removed (recovered or dead) populations. A flowchart of this model is presented in Fig. 4.This model significantly simplifies the mathematical modeling of different infectious diseases such as the novel SARS-CoV-2. Thus, the model is formulated by a set of nonlinear ordinary differential equations (ODEs) and therefore solved numerically. The simplest form of a set of ODEs for the SEIR-based model is as follows:(Parameters and Notations: \(\beta _{F}\)—Behavior change transmission rate, \(\beta \)—Transmission rate between two groups, \(\tau \)—Scaling factor, \(\delta \)—Transmission reduction component, \(\kappa \)—Progression rate, \(\alpha \)—Confirmation rate, \(\gamma \)—removal rate, C—Confirmed and isolated, N—Population)

In the current pandemic, many researchers are adopting the SEIR model to determine the transmission dynamics of COVID-19. Accordingly using the SEIR model, in [32], the authors described transmission dynamics by quantifying the school closure potential effect on the disease and mainly investigated child-to-child infection transmission. In their other work [33], they predicted the pattern of local transmission dynamics based on changes in individuals’ behavior in Korea, and they found a per-capita infection transmission rate that was 8.9 times higher in the local area (Daegu/Gyeongbuk) than nationwide (average). Likewise, the authors of [7, 27‐31, 35, 77‐79] proposed and formulated the model with a set of ODEs considering different transmission pathways and symptoms. They identified transmission dynamics of the virus and suggested different control measures. Earlier, in [80], the authors formulated a SEIRS model for avian influenza that includes bird human interaction by investigating the essential transmission dynamics of the disease based on equilibrium analysis.

The M-SDI (multiple-information susceptible-discussing-immune) dynamic model proposed in [35] was used to understand the types of significant information propagation on social media based on the amount of public discussion and the frequent behavior change (searches/comments) of users. A sample flowchart of this model is presented in Fig. 5. These authors estimated reproduction ratio decreases from 1.7769 to approximately 0.97, which means that the public discussion peak has passed, but it will still progress for a period of time. The model is illustrated mathematically as follows:To explore some of the characteristics of the qualitative nature of the prediction in the M-SDI model, authors have used the LS method for estimating parameters and primary susceptible population. Accordingly they estimated parameters in their model with data of at least 3–4 days’ to predict public discussion trends at different phases.

SUQC (‘susceptible-unquarantined infected-quarantine infected-confirmed infected’) is a compartmental model where S, U, Q and C represent the susceptible (S is similar as in the existing infectious virus transmission models SIR and SEIR), un-quarantined infected (infected and un-quarantined cases different from E in the existing SEIR model), quarantine infected (quarantine infected cases) and confirmed infected population. A flowchart of this model is presented in Fig. 6. Essentially, in [34], the authors developed the SUQC model to describe the transmission dynamics of the novel SARS-CoV-2 and especially the interference effects of the control measures by analyzing the disease outbreak. The method is formulated with a set of ODEs as follows:(Parameters: \(\alpha \)—Infection rate, \(\gamma \)—Quarantine rate, and \(\beta \)—Total confirmation rate.)

This model is adapted to the data of daily released confirmed cases to analyze the outbreaks of the diseases in Hubei, Wuhan, and four other first-tier cities in China. Authors have demonstrated authentic predictions of the transmission trends considering multiple characteristics, which include high infectivity, time delay and intervention effects. However, SUQC can quantify ariables and parameters regarding the intervention effects of the outbreaks. According to the simulation results the method further provides guidance in controlling disease spread.

BHRP (Bats-Hosts-Reservoir-People) is a network model for simulating the transmission of the virus, where B, H, R and P represent the bats (probable infection source), hosts (unknown but probably wild animals), reservoir (seafood market) and people (exposed population) developed in [36]. A flowchart of this model is presented in Fig. 7. In this method, the authors ignored the Bats-Host transmission network and presented the BHRP model in a simplified RP (Reservoir-People) model. They divided people into five different compartments: susceptible, exposed, symptomatic infected, asymptomatic infected and removed. The simplified model is illustrated mathematically as follows:(\(y_{p}\)—death rate (people), \(\sigma \)—multiple of transmissibility (\(A_{p}\) to \(I_{p}\)), \(\delta _{p}\)—infection rate (asymptomatic people), \(\beta _{p}\)—transmission rate (\(I_{p}\) to \(S_{p}\)))

They estimated the basic reproduction number according to their numerical illustration by assessing the transmissibility of the virus based on a simplified RP model, and they found values of 2.30 (reservoir-person) and 3.58 (person-person). The results showed that the transmissibility of SARS-CoV-2 is higher than that of the MERS in the Middle East and is similar to that of serious respiratory syndrome. They also found that the transmissibility is smaller than that of MERS in Korea.

SEIHR (Susceptible-Exposed-Symptomatic infectious-Hospitalized-Removed) is a virus transmission deterministic model where S, E, I, H and R represent the susceptible, exposed, symptomatic infectious, hospitalized and removed (recovered or death) populations. A flowchart of this model is presented in Fig. 8. With this model, in [37], the authors estimated the size of the outbreak and the reproduction number and found that, if the rate of transmission decreases, the outbreak of the disease ends early and the virus infection cases also decrease. The mathematical implementation of the model is illustrated with a system of nonlinear ODEs as follows:(\(\beta \)—Transmission rate, \(\sigma \)—Progression rate, \(\alpha \)—Isolation rate, \(\gamma \)—Removal rate).

In the study the authors did not consider natural deaths and births, infections during latency, asymptomatic infections, or re-infected cases. However, according to their simulation results, they suggested different social awareness activities such as wearing masks and social distancing, to reduce the fast transmission of the virus.

An extended compartmental model Theta-SEIHRD (Susceptible-Exposed-Infectious-Hospitalized-Recovered-Dead) where S, E, I, H, R and D represent the susceptible, exposed (incubation period), infectious (undetected but still can infect others), hospitalized (recovered or dead), recovered (previously detected and previously undetected but infectious) and dead populations, was developed in [38]. A flowchart of this model is presented in Fig. 9. The model is illustrated mathematically as follows (presented concisely here for simplicity):(Parameters: \(\gamma \)—Transition rate for different compartment, \(\beta \)—disease contact rate for different compartment \(\theta \)—fraction of infected people).

This model investigates the transmission dynamics of the disease considering the known major characteristics of the disease as the presence of infectious undiscovered as well as detected cases and the distinct sanitary and infectiousness conditions of hospitalized individuals. This model is able to estimate the required number of beds needed in hospitals, and it can calculate the basic reproduction number.

A compartmental model SEIPAHRF (susceptible class-exposed class- symptomatic and infectious class-super spreaders class-infectious but asymptomatic class- hospitalized class-recovery class-fatality class) divided into eight epidemiological compartment where S, E, I, P, H, R and F represent the susceptible individuals, exposed, symptomatic and infectious, super-spreaders, infectious but asymptomatic, hospitalized, recovered and dead populations, was developed in [23]. A flowchart of this model is presented in Fig. 10. The model is illustrated mathematically with a system of nonlinear ODEs as follows:(\(\beta \)—transmission coefficient (infected individuals), \(\beta '\)—transmission coefficient (super-spreaders), m—Relative transmissibility (hospitalized patients), \(\sigma \)—infectious rate (from exposed people), \(\rho _{1}\)—Rate (exposed individual become infected I) \(\rho _{2}\)—Rate (exposed individual become super-spreaders), \(\gamma _{a}\)—Rate (hospitalized), \(\gamma _{b}\)—Recovery rate (without hospitalized), \(\gamma _{c}\)—Recovery rate (hospitalized patients), \(\delta _{b}\)—Death rate (infected class), \(\delta _{p}\)—Death rate (super-spreaders class), \(\delta _{h}\)—Death rate (hospitalized class))

In [23], the authors (F Ndairou and colleagues) studied the stability of the free equilibrium of the basic reproduction number and investigated the sensitivity by considering the variation in its parameters. They analyzed and simulated the current outbreak considering some important aspects of virus transmission and gave an acceptable approximation based on the data in Wuhan, China. They also fitted the model with daily confirmed cases. According to their findings, the numerical results reflect the real scenario of the Wuhan outbreak.

To quantify the individual human-to-human variation of COVID-19 transmission and observe the outbreak sizes in affected countries, in [40], the authors proposed an interesting mathematical model applying a branching process model where the number of secondary transmissions was assumed to follow a negative binomial distribution.

By assuming that the secondary transmissions (offspring distributions) of COVID-19 cases were independently and identically distributed in a negative binomial distributions. The authors calculated likelihood following a previous study [81]. They computed the probability mass function for a final cluster size resulting from s initial cases following the formula below:Adjusting the future growing issue for cluster size, the research team introduced a corresponding likelihood functions presented below:To compute the total likelihood, the research group applied a final likelihood cluster size of certain countries with other ongoing outbreak countries following the formula below:For statistical analysis, they used a Markov chain Monte Carlo (MCMC) method with 95% credible intervals (CrIs). To determine the best method, they compared negative-binomial branching process model with a Poisson branching process model. However, they used simulations to investigate potential bias caused by under-reporting ( failure to fully report), one of the major limitations of their study.

To estimate the epidemic size, predict disease transmission trends and forecast the advancement of the disease, the basic SIR model (Fig. 3) was applied to the present COVID-19 pandemic to understand the effect of interventions such as quarantine, social distancing on disease spread and to control and mitigate virus outbreaks. We found three different studies [24, 25, 38] based on the basic SIR mathematical model. In [38], the authors used an SIR model to predict disease (COVID-19) trends and the extent to which quarantine decreased infection. In [24], the authors proposed a simplified SIR mathematical model to predict the peak of disease infection. In [25], an age-structured SIR mathematical model considering social connection matrices based on surveys and Bayesian imputation is presented to determine the momentum of the SARS-CoV-2 pandemic in India. According to the simulation results these studies accentuated the importance of the strategy of social distancing, improvement of the healthcare system, age-structuring and other intervention measures for controlling and mitigating COVID-19 outbreaks. To prevent the rise of disease infection they evaluated different policies in the well-defined contexts of COVID-19. In the presented models some of the parameters and variables were estimated, and some were composed from other published articles. To formulate the models and numerical illustrations they used a set of nonlinear ordinary differential equations (ODEs) (for more details see Overview section). In mathematical simulation reasonable data availability for better estimation and prediction was a challenge. In most cases, they used short and limited epidemiological data, and in some cases, unreasonable released data. As their main objectives were to identify the impacts of lockdowns/ quarantines, infection rates and social distancing through parameter estimation, these models were suggested for disease mitigation policies according to their numerical illustrations.

The SEIR is a simple compartmental model and different from SIR; in the SEIR model (Fig. 4) the exposed compartment and a parameter (the movement from the exposed compartment to the infected compartment) are added to describe the transmission dynamics of exposed individuals. This framework significantly simplifies the mathematical modeling of the present novel SARS-CoV-2 pandemic. Recently, several studies have focused on mathematical modeling and have adopted the SEIR model with a mathematical implementation (for more details see Overview section) to estimate transmission and to predict the trends of COVID-19 outbreaks. In their studies, they considered multiple transmission pathway mechanisms, including environment-to-human and human-to-human routes, in the infection dynamics, emphasizing the role of the environmental source in transmission and growth. These studies applied nonlinear ODEs and MCMC methods to calculate the basic reproduction number. This parameter thus provides valuable data on of the current COVID-19 outbreaks.

The M-SDI (Fig. 5) was developed to understand the types of significant information propagation on social media based on the amount of public discussion and considering the frequent behavior change (searches/comments) of users. In this work, they focused on the characteristic of users who choose to re-enter related information after they finish a discussion about certain topics. Based on analysis of a model simulation of the multiple-information generation mechanism regarding COVID-19, this model can distinctly predict the evolution of public opinion. They estimated that the reproduction ratio decreases from 1.7769 to approximately 0.97, which means that the public discussion peak was passed, but it will still progress for a period of time. To explore the qualitative nature of prediction in the M-SDI model (Figure 5), the authors used the LS method for estimating parameters and the primary susceptible population. Accordingly, they also estimated parameters in their model with the data of at least 3 to 4 days to predict public discussion trends at earlier phases. The M-SDI model can predict multiple-information development trend during a large-scale community health emergency. The authors used limited data for the estimation of parameters from the most popular Chinese Sina-microblog.

The SUQC model (Fig. 6) was developed to characterize the transmission dynamics of the novel SARS-CoV-2 and especially parameterize the effects of interventions measures. SUQC is not the same as SEIR, as infected people are classified into confirmed, quarantined and un-quarantined in SUQC. Moreover, in this case, un-quarantined people are able to infect susceptible people only, but in SEIR, infected individuals are infectious. Additionally, the quarantine rate in SUQC is used especially to model the influence of quarantine and other control measures. According to the author’s explanation, the model is more effective than other available epidemic models for analyzing the dynamics of the disease. This model was fitted with daily released confirmed cases data to analyze the outbreaks patterns of the diseases in Hubei, Wuhan, and four other first-tier cities in China. This model employs deterministic ODEs and it was needed to interpret certain uncertainties in this case. The authors demonstrated an authentic prediction of transmission trends considering multiple characteristics including high infectivity, time delay and interventions effects. However, SUQC can quantify variables and parameters regarding the intervention effects of the outbreaks. According to the simulation results, they found a reproduction of number greater than 1 before Jan 30, 2020, in Hubei, Wuhan, and other first-tier cities in China except Beijing, and after Jan 30, 2020, they found a reproduction of number less than 1, in all regions, indicating the effectiveness of the control measures. Subsequently the method further provides guidance controlling disease spread.

The Reservoir-People (RP) transmission model (Fig. 7) was developed in [36], considering the routes from reservoir (market) to person and from person to person for severe acute respiratory syndrome SARS-CoV-2. Published data of Wuhan City, China, were used to fit the model. The simulation results found basic reproduction numbers for SARS-CoV-2 of 3.58 from person to person and of 2.30 from reservoir to person. This model might predict interesting transmission chains applying limited data involving a number of important parameters. However, these predictions might not reflect the actual situation of the early stage virus transmission because some parameters were not taken from the accurate database or were applied through assumption.

Overall, the objective of this study was to provide an effective mathematical model to estimate virus transmission as accurately as possible with limited data using more parameters.

The SEIHR (Fig. 8) is a deterministic ODE model developed in [37] and was used to estimate the size of the outbreak and the reproduction number. They also evaluated the effects of different preventive measures. They used the daily data of confirmed cases in Daegu and North Gyeongsang (NGP), the main outbreak regions in Korea. According to the mathematical illustrations they found that if the rate of transmission decreases, the outbreak of the disease ends early; virus infection cases also decrease. They also suggested different social awareness activities such as wearing masks, social distancing and other intervention measures to reduce the fast transmission of the virus. They did not consider natural deaths and births, infections during latency and asymptomatic infections, or re-infected cases.

In [38], the authors proposed an extended compartmental model Theta-SEIHRD (Fig. 9) to investigate transmission dynamics of COVID-19 considering the known major characteristics of the virus. They studied the specific case of China including Hong-Kong, Macao, Taiwan and Mainland China and considered reported data that fit the parameters of the model that can be also useful to estimate the spread of COVID-19 in other countries. The presented model was adapted to the disease and is capable of estimating the progression of undetected and detected cases, deaths and the required number of beds needed in hospitals where the health problem is severe due to the outbreaks in territories, considering several different scenarios. In addition they calculated the basic reproduction number of COVID-19, which was 4.2250, but it fell to less than 1 after Feb 1, 2020, due to control measures. They also included in their model a new approach taking into account ‘Theta’, the fraction of total detected cases (infected), to study the importance of this fraction on the influence of COVID-19.

The SEIPAHRF model (Fig. 10), developed in [23], gives a feasible approximation based on data in Wuhan, China, by studying some important aspects of COVID-19 transmission. They divided their model into eight important epidemiological compartments and illustrated the model mathematically with nonlinear ODEs. They investigated the sensitivity by considering the variations in its parameters. They also fitted the model with real data of daily confirmed cases and found the actual scenarios of the Wuhan outbreak. Interestingly they considered many parameters to quantify transmissibility and computed the basic reproduction number, a measure of virus spread based on Jacobian matrices. Limited data accessibility was one of the major limitations of this study.

To estimate the level of over-dispersion in COVID-19 transmission and characterize the sustained transmission chains of human-to-human transmission, this model was introduced. Although only one research publication [40] was found, the mathematical implementation (see Overview section for more details) and statistical analysis of Offspring distribution looks powerful and should have the computational ability needed for a better estimation of moderate uncertainty levels with limited data resources. This was proven by a widely applicable Bayesian information criterion (WBIC). This model produced more accurate estimations than the Poisson branching process model. However, this model can only provide information about the lower boundary of the basic reproduction number RO(95% CrIs: R0 1.4–12) because of the marginal negative binomial distribution. Overall, the majority of model consistency and certainty relies on available homogeneous real data resources and imputed parameters by avoiding stochastic simulation.

Most of these studies did not consider natural deaths and births although demographic factors (birth, death, emigration and immigration) need to be added to the compartmental models for more realistic outcomes. Moreover, for the M-SDI model, the authors used public opinion data, which is different from other models. The Theta-SEIHRD model used ‘Theta’ the fraction of total detected cases (infected) to assess the significance of this fraction on the influence of COVID-19. In the SUQC model, the authors parameterized the interference effects of control measures by characterizing multiple aspects to predict transmission trends. In addition, the SEIHR model demonstrated that wearing masks, maintaining social distances and other intervention measures can reduce the fast transmission of the virus by calculating the effects of different preventive measures. Furthermore, it was found that the offspring distribution model performed well with uniform and homogeneous data. The SEIPAHRF model was fitted by considering many parameters to quantify transmissibility. Subsequently, more data are needed for a better understanding of the transmission dynamics in the context of COVID-19 in the BHRP model.

Accordingly, using more parameters and adding more stages make a model more complex and computationally challenging. Using original values and data collected from the field work certainly moves a model toward an accurate prediction of the near future. More complex models and methods in the future might handle these upcoming challenges.

Very recently, a few other interesting mathematical models have been proposed such as fractional-order models [82, 83], which are used for simulation dynamics [83], qualitative analysis [53] and numerical analysis [82]. A new Caputo-Fabrizio fractional order was also proposed for the SEIASqEqHR model for COVID-19 transmission with a genetic-algorithm-based control strategy [49]. Similar studies were done by a Nigerian research group [82] and a Saudi Arabian group [83]. Computational and theoretical modeling of the transmission dynamics of COVID-19 was also proposed under the Mittag-Leffler power law considering a fractional-order epidemic model [50]. In another study, a fractional order epidemic model with numerical simulation was proposed for global stability analysis using the Lyapunov candidate function [48]. Another study introduced a Haar wavelet collocation approach for a solution to the fractional-order COVID-19 model using the Caputo derivative [52]. A numeric simulation of fractional order was performed on the basis of the Wuhan COVID-19 case study [54]. However, fractional-order model has also been applied using the real cases reported in Saudi Arabia [55], USA [56], Pakistan [58] and Nigeria [57].

A simulation-based study proposed the dispersal effect [84] to understand the dynamics of the disease by boundedness and non-negativity of solutions. Another simulation-based SQIR-type model under the nonlinear saturated incidence rate was also proposed [39]. Another simulation-based study was used for the age-structured modeling of COVID-19 in the USA, UAE and Algeria [51]. Another study provided guidelines on how to run the analysis for COVID-19 with optimal control, stability and simulations [85]. Another simulation-based study presented a global stability analysis [86].

COVID-19 data are increasing day-by-day and the data storage system is becoming rich. This increases the possibility of finding the most accurate model in the near future. In addition, these data encourage researchers to consider even more complex models by mining the available information. However, it is extremely challenging to handle such a high amount of data and to select the right model for accurate future prediction regarding COVID-19 transmission. More computational power (e.g., supercomputers) will be required for analysis.

How good a model is depends on the model parameters used. Since a large dataset invites researchers to implement more complex models, there is always an important issue regarding parameter estimation. Complex models often use simulated data for unknown parameter estimation and become less accurate. In other words, it is always expected to use real data for parameter estimation to predict the real virus transmission rate. Less complex models should be chosen in the case of limited data.

Most COVID-19 studies compute transmission of COVID-19 following only the nature of survival dynamics. However, lost dynamics (where, when and how people die) might also be valuable information for future prediction. Just computing the results of COVID-19 death might not help us to understand its true dynamics. Combination of survival and lost dynamics might provide us a true picture of the reality.

Although the COVID-19 data storage/backup system is becoming richer by the day, people worldwide, with or without symptoms, are afraid of being tested for COVID-19 and having to self-quarantine with [38, 87]. The actual number of people who need to quarantine might never be known unless a safe and remote personal testing kit is developed and people testing positive are encouraged people not to hide that information from the proper authorities. For mathematical modeling, a quarantine might be a challenging stage regarding parameter estimation, which might impact in the final transmission rate of COVID-19.

Scientists cannot guarantee that one person cannot catch COVID-19 twice even though increased immunity was observed in a patient after recovery. An online news portal reported that a Chinese group found evidence of it being caught twice in a monkey study. Thus, the possibility of reinfection might not be precluded. The model researchers might not be able to separate the already infected from upcoming analyses. Most recently, several types of vaccines (e.g., mRNA) have been introduced and priority has been given to medical workers and the elderly. However, it has been heard that very few people are getting infected after vaccination. It is predicted that COVID-19 might continue flu like virus and people might need to be vaccinated after a certain period each year.

Human civilization may face enormous challenge near new future mutations, specifically in glycoproteins [88]. Scientists have already spotted thousands of generic material changes in the coronavirus and its mutations worldwide. However, not all of the mutations infect all parts of the world. For country- or region-specific mathematical modeling, it is also important to include the number of mutations of the virus (if possible). This information can be investigated for weather/area/temperature-based virus transmission. A new strain called SARS-CoV-2 VUI 202012/01 or “B.1.1.7.” has caught attention which was first discovered in the southeast UK in September 2020. It might be more contagious than other available strains (approximately 70% faster than the usual one) and is gradually spreading other countries. This strain of virus might infect any age people including children.

Which model fits well depends on the available data and the goal of the analysis. The researchers’ main goal is to implement a specific model or an improved version of the model to investigate virus transmission. However, applying different models (e.g., SIR and SEIR) in the same study might provide the range and center of true expected virus transmissions. In addition, limitations on parameters between different models might be overcome or minimized.

Human civilization has learned about COVID-19 and passed the second attack (often called second wave) almost all around the world [89]. Almost every countries have made wearing a mask mandatory in all indoor public areas. Currently, 3rd wave is running following new variants and people might still be infected and the data will be stored. We are not sure about how many waves will come in future. For mathematical modeling, researchers should consider this issue and add few new stage ( following the number of waves) of the COVID-19 transmission with new parameters based on the data information. This model might help us to predict the timing and strength of upcoming waves.

Next, we identify some of the possible future directions of mathematical modeling.

So far, no guaranteed antiviral treatment has been recommended regarding COVID-19 [90]. The only way to fight against this contagious disease is to emphasize preventive healthcare systems and implement necessary rules and regulations that depend on the spreading nature of virus transmission. Here, a decision support system (DSS) [91] can play a very important role by having priority guidelines (e.g., social distancing/lockdown) administered by national and international authorities in an effective and efficient way. The outcome of mathematical modeling can be plugged into this DSS system. However, the necessary supporting information can be collected from the corresponding authority.

Most AI implementations for COVID-19 analysis rely on a convolutional neural network following X-ray and CT images. However datasets are also available for aggregated case reports, management strategies, the healthcare workforce, the demography and the mobility during the outbreak. Interestingly, both mathematical modeling and AI have shown their ability as reliable tools to fight against the COVID-19 pandemic [92]. These tools altogether might lead to the new insights on this pandemic. Very recently, a deep learning algorithm was introduced for the modeling and forecasting of COVID-19 in the five worst affected states of India [93].

The field of mathematics has been expanding over time with new methods and implementations. Most COVID-19 studies have applied ordinary differential equations (ODEs) by adding suitable parameters. However, partial differential equations (PDEs) can also be useful in similar studies when other parameters do not change with respect to a specific parameter. To solve the equations (e.g., ODEs) with higher numbers of unknown variables (known as parameters in COVID-19 studies), Galois theory [94] can be used. However, to investigate the specific effects of similarities between different areas or countries of COVID-19, Polya’s theorem [95] can be applied. Fuzzy logic can also be used as it is able to handle uncertainties by considering several membership functions based on the transmission dynamics trends data of COVID-19.