1 Introduction

As the COVID-19 pandemic continues, new variants of the virus proceed to emerge and unfold round the world. These special mutations affect how the virus spreads and interacts with the immune system, posing new challenges for public health officers attempting to manipulate the pandemic.

Researchers have recognized and labelled numerous COVID-19 (SARS-COV-2) variants, each with special mutations that influence how the virus spreads and interacts with the immune system [1,2,3,4,5,6]. In this regard, a review is conducted in accordance with Fig. 1.

Fig. 1
figure 1

Structure of the review

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The WHO classifies the SARS-COV-2 editions as Variant of Interest (VOI) and Variant of Concern (VOC) [5]:

A variant is considered a Variant of Interest if it has mutations that are believed or confirmed to cause notable alterations, and has a widespread distribution (e.g., identified to cause numerous groups of infected people, or found in many nations).

A variant of interest becomes a Variant of Concern if it is proven to spread more rapidly, cause more severe illness, evade the body’s immune response, alter its clinical manifestations, or reduce the efficacy of recognised interventions including vaccinations, diagnostics, treatments, and public health campaigns. Figure 2 depicts the review structure of the COVID-19 variants.

Fig. 2
figure 2

COVID-19 variants review

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1.1 Most exquisite COVID-19 variations [4, 5]

  1. a.

    The Alpha Variant (B.1.1.7), first recognized in UK, is more transmissible than the authentic strain of SARS-CoV-2, the virus inflicting COVID-19. This variant includes spike protein mutations, enabling it to bind more tightly to human cells. As a result, it has unfolded swiftly round the world. Research indicates that the B.1.1.7 lineage was more lethal than the original virus and increased the likelihood of hospitalization in the afflicted population. Existing vaccinations, however, have proven effective in preventing it.

  2. b.

    The Beta variant (B.1.351), first detected in South Africa, possesses greater transmissibility than the original strain. There are concerns that this variant may showcase decreased susceptibility to some current vaccines due to mutations in the spike protein, which assist it in dodging the immune system.

  3. c.

    The Gamma variant (P.1), first recognized in Brazil, shares comparable traits with the Beta variant. It is extra transmissible than the original strain and may exhibit decreased susceptibility to current vaccines. Significant mutations in the spike protein enable it in evading the immune system. Infections attributed to it are associated to an appreciably greater viral load than other variants.

  4. d.

    The Delta variant (B.1.617.2) emerged in India and became a cause of global concern due to its even higher transmissibility as compared to the Alpha variant. It carries mutations in the spike protein that beautify its potential to bind tightly to human cells, contributing to its speedy spread. This particular variant is considered to be among the most dangerous Covid variants due to its high fatality rate. The CDC recommended "layered prevention strategies" for each of the immunized and unimmunized in response to this Covid variant.

  5. a

    The Omicron variant, additionally acknowledged as B.1.1.529, is a variant of the SARS-CoV-2 virus that was first recognized in November 2021 in South Africa. The variant has become more difficult to combat globally because of the significantly high number of mutations in its spike protein, a part of the virus that is essential to penetrate human cells.

1.1.1 Key factors about the Omicron variant include

  1. i.

    High quantity of mutations: The Omicron variant has a drastically greater wide variety of mutations in the spike protein compared to preceding variants. These mutations have led to worries about probable variations in transmissibility, degree of severity, and vaccine effectiveness.

  2. ii.

    Global spread: The variant has been detected in several different nations worldwide, elevating concerns about its potential for rapid dissemination.

  3. iii.

    Impact on Vaccine Effectiveness: According to some research, the Omicron variant's capacity to be neutralised by antibodies developed from vaccinations or prior infections may be impacted by some of the mutations. However, more investigation is required to comprehend the practical effects on vaccination efficacy.

  4. iv.

    Public Health Response: Many nations have taken precautionary measures to restrict the propagation of the variant, such as travel prohibitions and increased monitoring.

It's vital to notice that data about the Omicron variant is swiftly evolving, and research is being performed to apprehend its transmissibility, severity, and effect on public health measures.

By incorporating the distinct characteristics of each variant into mathematical models, researchers can more accurately forecast the virus's transmission and evaluate the effectiveness of interventions, including vaccination campaigns and social distancing policies. Public health policies and attempts to create novel vaccines and treatments that are effective against all known viral variants can be guided by this knowledge [5, 6]. These models utilize a combination of differential equations and statistical techniques to depict the dynamics of the pandemic and its interactions with diverse populations.

One significant challenge in modelling the dissemination of COVID-19 variants lies in the virus's rapid mutation, giving rise to new variants that may exhibit increased transmissibility or greater resistance to vaccines. Mathematical models based on differential equations can be adapted to account for these changes by incorporating variables that capture the effects of these mutations.

Mathematical modelling based on differential equations proves to be a potent instrument for forecasting the transmission of COVID-19 variants. Leveraging extensive databases of COVID-19 cases, it becomes feasible to construct models that precisely anticipate the dissemination patterns of diverse virus variants. Differential equations serve as mathematical representations elucidating the evolution of a quantity over time. In the realm of COVID-19 modelling, these equations aptly depict the rate at which the virus propagates within a population, elucidating the influence of factors such as social distancing, vaccination rates, and travel restrictions on the virus's spread.

The classification of virus variants is a crucial aspect of understanding and managing infectious diseases. However, existing variant classification systems face several limitations, especially when dealing with rapidly evolving viruses like SARS-CoV-2. Some critical points to consider are listed below:

  1. 1.

    Rapid Evolution of the Virus:

    The SARS-CoV-2 virus evolves quickly, leading to the emergence of new variants with distinct genetic signatures. This rapid evolution challenges the ability of classification systems to stay up to date with the ever-changing landscape of viral genomes.

  2. 2.

    Variant Identification Challenges:

    Identifying and characterizing new variants depends on comprehensive and timely genomic surveillance. However, not all regions or countries have the capacity for robust surveillance, potentially resulting in underreporting or delayed identification of emerging variants.

  3. 3.

    Diversity of Variants:

    Current classification systems may oversimplify the diversity of variants by grou** them into specific lineages. This simplification might not capture the full spectrum of genomic changes, potentially missing important variations that could have clinical or epidemiological significance.

  4. 4.

    Lack of Standardized Nomenclature:

    The nomenclature for variants can vary across different organizations and countries, leading to confusion and making it challenging to establish a unified global understanding of the virus's evolution. A standardized naming system is crucial for effective communication and collaboration.

  5. 5.

    Limited Functional Information:

    Variant classification often relies on genetic data, but the functional consequences of many genetic changes may not be fully understood. Integrating functional information into classification systems is essential for predicting the potential impact of variants on transmission, severity, and vaccine efficacy.

  6. 6.

    Bias in Sequencing Data:

    The availability of genomic data is not uniform globally, leading to potential biases in variant surveillance. Overrepresentation of certain regions or populations in sequencing efforts may skew our understanding of variant distribution and prevalence.

  7. 7.

    Dynamic Clinical Implications:

    The clinical significance of variants may evolve over time as more data becomes available. Variants initially considered "variants of interest" may later be reclassified as more information is gathered on their transmissibility, severity, and vaccine escape potential.

  8. 8.

    Adaptability to Diagnostics and Vaccines:

    Some variants may affect the accuracy of diagnostic tests or the effectiveness of existing vaccines. Continuous monitoring and adaptation of diagnostic tools and vaccines are essential to address the challenges posed by new variants.

  9. 9.

    Ethical and Social Implications:

    The classification of variants can have significant implications for public health measures, travel restrictions, and vaccine deployment. Ethical considerations surrounding the communication and management of variant information need careful attention to avoid unnecessary panic or stigmatization.

Overall, while variant classification systems play a crucial role in understanding and managing infectious diseases, the evolving nature of viruses presents ongoing challenges. Addressing these limitations requires a collaborative and dynamic approach, incorporating advances in genomics, epidemiology, and clinical research.

1.2 Various mathematical models

Several mathematical models have been proposed to forecast the spread of COVID-19. Some of the commonly employed mathematical models of along with their respective equations are mentioned below [1, 7,8,9,10,11]:

1.2.1 Compartmental models

Compartmental models operate on the premise that the population can be segmented into different compartments based on their disease status. Widely utilized in epidemiology, these models offer a straightforward means of illustrating the dynamics of infectious diseases. The most frequently applied compartmental model for COVID-19 is the SIR (Susceptible-Infected-Recovered) model [12, 13]. The equations for this model (Eq. 1 to Eq. 3) are provided below:

$$dS/dt= -\beta S I$$
(1)
$$dI/dt= \beta S I - \gamma I$$
(2)
$$dR/dt= \gamma I$$
(3)

where: S: the number of susceptible individuals, I: the number of infected individuals, R: the number of recovered individuals, N: the total population size (N = S + I + R), β: the effective contact rate between susceptible and infected individuals, γ: the recovery rate.

1.2.2 SEIR model

The SEIR model stands out as one of the most widely employed models for infectious disease modelling. Following transmission, COVID-19 and its variants manifest an evolution phase or exposed condition in an individual before the onset of noticeable symptoms. The SIR model faces limitations in accurately capturing the spread of the disease in such instances. To address the necessity of incorporating the exposed or latent condition into a mathematical model, the SEIR (Susceptible-Exposed-Infected-Recovered) model was formulated.

Considering a fixed population size N = S + E + I + R, where an infected person transmits the disease with a mean transmission rate β to a susceptible person, and the rate of transition from the exposed state to the infected state is σ (the duration of the mean exposed period or latent phase is 1/σ). The individual recovers at the end of the infectious time, with a mean recovery rate γ (the mean transmission period in any person is 1/γ), and is classified as a member of the Recovered (R) class. The system of differential Eqs. (4–7) describing the transmission in accordance with the basic SEIR model is provided by [12,13,14]:

$$dS/dt=-\beta S I$$
(4)
$$dE/dt= \beta S I- \sigma E$$
(5)
$$dI/dt= \sigma E-\gamma I$$
(6)
$$dR/dt= \gamma I$$
(7)

An expansion of the SIR model involves incorporating an additional compartment for individuals who have been exposed to the virus but have not yet become infectious, with or without vaccination data. The model Eqs. (8–12) for the spread of COVID-19 variants can be formulated as follows [9, 12,13,14,15]:

$$dS/dt= -\beta S(I+\eta V)N$$
(8)
$$dE/dt= \beta S(I+\eta V)N - \alpha E$$
(9)
$$dI/dt= \alpha E -\gamma I$$
(10)
$$dR/dt= \gamma I$$
(11)
$$dV/dt= \xi V(1 -V/C)- \varphi I V$$
(12)

where: S: the number of susceptible individuals, E: the number of exposed individuals, I: the number of infected individuals, R: the number of recovered individuals, V: the number of vaccinated individuals, N: the total population size (N = S + E + I + R + V), β: the effective contact rate between susceptible and infected individuals, η: the relative transmissibility of the variant compared to the original strain, α: the rate of transition from the exposed compartment to the infected compartment, γ: the recovery rate, ξ: the vaccination rate, C: the maximum vaccine coverage, φ: the vaccine eficacy against the variant.

In the system of equations, Eq. 8 signifies the rate of change of susceptible individuals, proportionate to the number of infected and vaccinated individuals, along with the effective contact rate between them. Equation 9 denotes the rate of change of exposed individuals, proportionate to the number of susceptible individuals and the effective contact rate, subtracted by the rate of transition to the infected compartment. Equation 10 signifies the rate of change of infected individuals, proportionate to the number of exposed individuals and the rate of transition from the exposed compartment, subtracted by the recovery rate. Equation 11 represents the rate of change of recovered individuals, proportionate to the number of infected individuals and the recovery rate. Equation 12 denotes the rate of change of vaccinated individuals, proportionate to the vaccination rate, subtracted by the rate of infection of vaccinated individuals [1].

1.2.3 Agent-based models

Agent-based models (ABMs) are individual-based models which simulate the behaviour of individual agents and their interactions with other agents and the environment [14, 15].

1.2.4 Spatial models

Spatial models include spatial and temporal aspects into the model to comprehend the geographical spread of the disease. These models are helpful for identifying local or regional patterns of disease transmission. The model equations are comparable to the compartmental models or ABMs, but they incorporate additional factors to account for spatial elements [14, 15].

1.2.5 Network models

Network models describe the transmission of disease via a network of contacts between individuals. Each person is represented as a node in a network, and the edges between nodes represent the contacts between individuals [13,14,15]. The spread of disease may be modelled by replicating the dynamics of disease transmission through the network.

The goal of the review is to discover the utility of differential equations based mathematical frameworks for predicting the spread of COVID-19 variants. The paper aims to investigate how these frameworks can be tailored to comprise the special traits and transmission dynamics of distinct COVID-19 variants, offering insights into manageable influence on disease spread.

The pursuit of sustainable health and well-being has emerged as a paramount global concern, especially given the unprecedented challenges posed by pandemic, environmental changes, and lifestyle-related diseases. This review aims to present an overview of the current state across various domains contributing to sustainable health. It encompasses advancements in medical research, public health initiatives, technological innovations, and holistic well-being frameworks.

The research highlights the application of mathematical modelling with techniques particularly tailormade for predicting the spread of COVID-19 variants. It explores how differential equations-based models, such as compartmental models (e.g., SIR, SEIR) and spatial models (e.g., reaction–diffusion, metapopulation), can be tailored to seize the precise traits of different variants.

The research provides insights into the possible differences in transmission dynamics, such as determining the basic reproduction number (R0), incubation duration, and illness progression. These insights aid in understanding how variants may manifest within populations and the manageable challenges they present throughout the disease's management phases.

By offering predictions and insights into the unfolding of COVID-19 variants, the paper helps inform policymakers on the predictable trajectory of the disease, the effectiveness of management measures and public health interventions, such as vaccination campaigns, social distancing measures, or targeted management techniques and the want for variant-specific strategies.

The insights gained from these models can help us understand variant behaviour better, guide future study, and improve awareness of the COVID-19 pandemic worldwide.

Overall, the study aims to highlight the contribution of mathematical modelling for understanding how COVID-19 variants emerge, hence providing valuable insights and guiding public health strategies for monitoring and regulating these variants.

2 Deterministic and statistical based prediction models

Deterministic models such as the SIR and SEIR models are based on differential equations and lay emphasis on mathematical driven equations for the outcome, but statistical models are data driven models whose accuracy depends upon the quality of data taken. Figure 3 depicts the flow of review and the mathematical modelling-based prediction methodologies.

Fig. 3
figure 3

Review on methods of prediction

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A compartmental model for the pandemic with time-delayed categorization was implemented to prevent the excessively rapid spread of infected individuals. Two-time delays are included into the model and its parameters to represent the disease's incubation period and the quarantine period for those who encounter infected persons to study the threshold dynamics [16]. Based on the Holt-Winters approach, a statistical model was created to estimate the Covid-19 operational forecasts. This model works effectively for prediction forecasts at various levels of administration in India. In case of Covid-19, Holt-Winters models occupy the live trends and the stages according to time series [17].

A novel susceptible-exposed-infected-hospitalized-removed (SEIHR) model that takes into consideration human migrations and asymptomatic infectivity was developed. This model demonstrated superiority over traditional SIR and SEIR models, especially in complex and nonlinear environments. It offered both validation and future forecasts for COVID-19 transmission [18].

LS (least squares) support parametric optimisation was used to investigate and propose the SpID model for COVID-19 fatality estimation of COVID-19 deaths in Turkey. The model predicts that the number of infected and lost people would decrease in 300 days, while the number of suspicious deceased will reach its minimum in around 1000 days [19]. A mathematical model that uses auto and linear regression techniques for future forecasts was planned to predict infected, recovered, and fatal cases [20].

Equations drawn from reduced and general logistic models were used to propose an epidemiological model that predicted how the epidemic would spread throughout Spain, Italy, and the United Kingdom. Taking recovered people into consideration improved the accuracy of this model [21]. For the prediction of COVID-19 Susceptible-Infected-Recovered (SIR) version and FB-Prophet version for time collection evaluation are implemented. SIR modelling is extra intuitive and explainable, however calls for lots of trial and mistakes and assumptions. The FB-Prophet prediction procedure is easy, and accuracy is likewise higher as compared to SIR modelling. How such non-stop and unprecedented elements lead us to layout complicated approach, as it`s time to apply data-pushed, mathematically verified approaches with the cap-potential to composition of parameters dynamically and involuntarily over the years are discussed [22]. To decide a way to examine local hazard associated with COVID-19, a two-segment modelling method is evolved thinking about demographic and monetary criteria [23]. Riccati equation with adaptively predicted parameters is proposed for learning model on facts-guided detection and concatenation of infected population. In this, distribution of the epidemic time-path into five “Riccati modules” indicating principal infected population is shown. The acquired parameter estimates imply slow infectivity rate, even though the contemporary signal is predicted to be the largest [24].

The Table 1 provides a diverse array of methodologies and techniques employed in predicting and understanding COVID-19 dynamics. While each study contributes to the field, there are some common strengths and challenges.

Table 1 Summary of literature review based on the technique used, considered parameters and the outcome
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Strengths:

  • Many studies leverage advanced models (e.g., SIR, SEIRD, SEIHR) to capture the complexity of virus transmission.

  • The incorporation of various parameters, including clinical, demographic, and environmental factors, enhances the models' comprehensiveness.

  • Some models, such as the innovative Riccati equation approach, demonstrate adaptability and uniqueness in tracking the epidemic trajectory.

Weaknesses:

  • Several studies acknowledge limitations, such as underreporting, model constraints, or data biases.

  • The lack of standardized nomenclature and varied data availability globally may impact the generalizability of some models.

  • Limited functional information on genetic changes and variations in some classification models may restrict a comprehensive understanding.

Author contributions:

  • Models like the Holt-Winters and ARIMA demonstrate applicability at different administrative levels and emphasize the need for monitoring and forecasting.

  • The integration of time delays, consideration of human migrations, and the emphasis on adaptability enhance the sophistication of certain models.

  • Deep learning and hybrid algorithms contribute to identifying trends and forecasting methods.

  • Prediction mainly depends upon geographic and demographic conditions, so its area specific and change according to place. Even learning model is proposed with time and non-linearity of the spread of infection, so current data plays vital role for prediction and can vary according to situation to situation.

2.1 Deterministic models

Several model mechanisms feature the use of differential equations to model the pandemic. One of the most widely used models-the SIR model is based on a system of ordinary differential equations, which may represent the spread of the infection and possible predictions in terms of supporting data to ascertain the real situation [39].

2.1.1 Model based on multi objective optimization

Researchers in the estimation have proposed to assess the model boundaries, define a multi-objective problem, and perform proper design mining of the acquired information through reliable data. First, the model bounds serve as the improvement factors, and the accessible information is treated like a multi-objective streamlining problem. Streamlining is used to evaluate these limits. An innovative idea that expands the parameters of compartmental epidemiological models is presented [40].

2.1.2 Hidden nodes model

All numerical models manage the number of recognized cases. Nonetheless, practically speaking quite possibly the individual might be infected and unidentified. Toward this path, a model considering dynamic and secret groups is proposed by researchers in [41] through expecting that one round off infected patient interacts with others.

2.1.3 Intervention and control information model

The evolution of the population from one segment to the next can influence the contact level of individuals infected with a disease. To address this, the movement of individuals is integrated into spreading models in several studies. Notably, [42] explores the progression of infected individuals within the spreading model. Governmental preventive measures, such as lockdowns and law-enforced quarantine, are recognized in [43] as interventions that can reduce both the spreading rate and mortality rate. Additionally, compartmental modelling is applied to formulate differential equations for analysing the transmission of the pandemic in Morocco. To reduce the complexity an agent-based method with artificial intelligence is used and it allows customising for different scenario. These two different methods suggest successful models. Differential equations-based models [44] are reviewed for spread of Covid -19 and nonlinear ordinary differential equations (ODEs) are analysed. To address the minimization problem, approaches are introduced that rely on a blend of global techniques, encompassing approaches such as covering methods, nature-like algorithms, and multi-level gradient methods, along with local techniques like gradient methods and the Nelder-Mead method.

Epidemic models of susceptible, infectious, and recovered types [45], formulated with ordinary differential equations, are used to evaluate the effectiveness of diverse control strategies involving vaccination and antiviral treatment. These models prove highly valuable for comprehending the impacts of various factors on the transmission and control of infectious diseases. Despite of lockdown in India numbers of infected cases are rapidly increasing [46,47,48,49]. New models are projected to predict the spread and consequence of preventive measures and followed methodology is based on differential equations. Model mainly considers critical epidemiological conversion and communication parameters of COVID-19 [50]. Some other prediction models presented by authors are shown in Table 2.

Table 2 Proposed predictive models
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The study explores the interaction and importance of fractional-order derivatives within the Susceptible-Infected-Quarantine (SIQ) model [60] concerning the coronavirus, considering the impact of lockdown measures. Various scenarios utilizing adjusted fractional order values are presented to address the fractional-order SIQ differential equation model. Modelling [61] involves the use of fractional differential equations to analyse bandpass filter outcomes related to the peaks of a windowed signal. The study links differ-integrator parameters, determining long-range dependence on estimated instantaneous frequencies, and compares it to a benchmark non-iterative signal reconstruction method (SPSI).

2.2 Statistical models

The impact of natural factors that decide the transmission of COVID-19 is studied and accounted in statistical model [62]. The model broke down the four most impacted regions in China and five regions in Italy and considered ecological elements such as relative humidity, temperature and breeze pace that might influence the spread of the infection. The Tolerant Information Based Algorithm falls under the category of information-driven models, computing the future mortality rate using currently available data. It primarily predicts the death rate through a differential equation (DE) [63]. Another study has reported that the mortality rate is influenced by environmental factors and temperatures. In [64], a forecasting attempt is made on the epidemic cycle of COVID-19. Subsequently, a prediction model is developed using Gaussian distribution theory by considering the epidemic features at various stages of COVID-19. The larger part factors that impact the spread of the infection incorporates in model are fundamental propagation number, infection hatching period and everyday contamination number.

From above mathematical models it can be observed that imaginary parameter, roots and eigen values are recognised in mathematical models approaches. These approaches can be utilized for extensive analysis and to structure the linear and non-linear models. To attain featured parametric output in single flow both batch optimization and iterative approaches together are utilised together in parametric modelling. Overlooking of the random variables can be possible in database model. This ignorance is attained through eliminating the stochasticity in the data which has deterministic impact from batch approach of optimisation. These models can elaborate the important information about strong and weak phase of the current real time systems [65]. Many casualties’ analysis model was proposed based on parametric approach for Covid-19, recently shown in Table 3.

Table 3 Authors proposed work
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These models consider infectious rate, cure rate, and mortality rate but these features were optimised. So, these models are not considered as multidimensional models. Even their approaches are simply statistical and may be not succeeded to analyse virus dynamics that can change uncertainly or internally. Many other models were proposed for vaccination [74]. Data from AstraZeneca, Bharat Biotech, and several other councils indicate that SIR models have been proposed to address issues encountered by national administrations [75, 76]. Apart from population different individuals have different immune systems that may cause severe allergies [77] when they get vaccinated. Mostly approaches used by models are fully data-driven and effortlessly flexible for local and global procedure or convention. These models provide interim COVID-19 prediction with current practical accounts or records of infections.

When conducting a critical evaluation of mathematical models, it is important to consider various aspects, including their strengths, weaknesses, assumptions, and real-world applicability.

Mathematical models to predict disease dynamics should be easy to interpret and adaptable to different scenarios and datasets. It is crucial to evaluate the model’s efficiency in terms of computational resources required and how well it predicts outcomes and matches real world data. The sensitivity of the model is to changes in its parameters should be tested for better accuracy. It is vital to examine how sensitive the model is to data quality and changes in input data and whether it generalizes well to new data. The balance between model complexity and simplicity needs to be evaluated based on the specific needs of the application. The trade-off between bias and variance in the model’s performance must be considered.

Models may also present several weaknesses such as inability to capture noise in the data (overfitting) or lack complexity to capture patterns (underfitting). Some models may fail to identify situations where the underlying assumptions of the model may be violated and the impact on its performance. In addition, many models assume a linear relationship between variables and fail to discuss the implications if this assumption does not hold. Any ethical implications of using the model in real-world scenarios should be discussed.

2.3 Deterministic and statistical-based prevention models

All models reviewed and examined below have shared investigation of spreading of pandemic and the precautionary measure with vaccination drive challenges and can be termed as preventions models. With this consideration, transmission dynamics of COVID-19 has been modelled mathematically [78, 79] [80].

A brief review of some work considered in emergency for last few decades is shown in Table 4

Table 4 Review on working pandemic or in emergency situation
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Requirement of the time is to build models based on differential equations with profound learning in computational science and medication. Some finds have been done with medication to prevent infection [86, 87]. Number of models are reviewed and analysed with their need and outcomes for prediction and prevention.

2.4 Precision medicine and personalized healthcare

The emergence of precision medicine has revolutionized healthcare by tailoring medical interventions to individual characteristics, such as genetic makeup, lifestyle, and environmental factors. Personalized healthcare, enabled by advances in genomics and data analytics, enhances treatment efficacy and minimizes adverse effects, thus contributing to a more sustainable and patient-centric healthcare model [88].

2.5 Eco-friendly healthcare practices and green hospitals=

Sustainable health extends beyond individual well-being to encompass environmentally conscious healthcare practices. The concept of green hospitals involves integrating eco-friendly technologies, renewable energy sources, and waste reduction strategies into healthcare facilities. Such initiatives contribute not only to environmental sustainability but also to the overall health of communities [89].

2.6 Digital health technologies and telemedicine

The rapid evolution of digital health technologies has paved the way for innovative solutions in healthcare delivery. Telemedicine has gained prominence, providing remote access to medical services, reducing healthcare disparities, and minimizing the environmental impact associated with unnecessary travel [90].

2.7 Mind–body interventions and holistic well-being

Holistic approaches to well-being emphasize the interconnectedness of physical and mental health. Mind–body interventions, including mindfulness, yoga, and meditation, have demonstrated positive effects on stress reduction, mental health promotion, and overall quality of life. Integrating these practices into healthcare can contribute to sustainable well-being [91].

2.8 Community health promotion and social determinants

Addressing social determinants of health is essential for sustainable well-being. Community-based health promotion initiatives, focusing on education, socioeconomic factors, and access to healthcare, play a vital role in fostering resilience and reducing health inequities [92].

2.9 Global health security and pandemic preparedness

The events of recent global pandemics have underscored the importance of robust health security systems and effective pandemic preparedness. Investments in research, surveillance, and international collaboration are critical components of a sustainable approach to global health challenges [93].

2.10 Data-driven public health strategies

Data-driven decision-making is pivotal in modern public health. Utilizing big data analytics and artificial intelligence, public health professionals can identify trends, predict outbreaks, and formulate evidence-based strategies for disease prevention and control, thereby contributing to sustainable health outcomes [94].

Reliable and accurate data is essential for building models that can effectively predict and respond to the dynamics of the pandemic. Various challenges, biases, and limitations associated with the data used in COVID-19 prediction models are detailed below:

1. Data Collection Challenges:

  • Testing Disparities: Variability in testing rates and practices can lead to underreporting or misrepresentation of the true number of cases.

  • Testing Accuracy: The accuracy of diagnostic tests can affect the reliability of the data. False positives or negatives can lead to distorted model predictions.

2. Temporal and Spatial Variability:

  • Reporting Delays: Delays in reporting cases can impact the timeliness and accuracy of the data used for modelling.

  • Spatial Disparities: Differences in reporting and testing practices across regions can introduce biases in spatial models.

3. Asymptomatic and Mild Cases:

  • Underreporting of Asymptomatic Cases: Asymptomatic cases may go undetected, leading to an underestimation of the true prevalence and affecting the model's predictive power.

  • Mild Case Misclassification: Cases with mild symptoms may not be recognized or reported, contributing to an incomplete understanding of the disease spread.

4. Demographic and Socioeconomic Bias:

  • Testing Accessibility: Availability and accessibility of testing facilities may be influenced by socioeconomic factors, leading to biased representation in the data.

  • Demographic Disparities: Differences in healthcare-seeking behaviour and testing rates among different demographic groups can introduce biases.

5. Public Health Intervention Impact:

  • Effect of Mitigation Measures: Interventions such as lockdowns, social distancing, and vaccination campaigns can impact the dynamics of the pandemic. Models need to account for these changes in the data.

6. Incomplete and Inconsistent Data:

  • Data Reporting Inconsistencies: Inconsistencies in reporting formats, definitions, and data collection methods can pose challenges in aggregating and analysing data consistently.

  • Missing Data: Incomplete or missing data, whether intentional or due to reporting delays, can hinder the model's accuracy and reliability.

7. Data Source Heterogeneity:

  • Integration of Multiple Data Sources: Combining data from diverse sources, such as clinical reports, laboratory results, and surveillance systems, may introduce heterogeneity that needs careful handling.

8. Ethical Considerations:

  • Privacy Concerns: Balancing the need for detailed individual-level data with privacy concerns are crucial and may impact the granularity of the data used in models.

9. Model Validation Challenges:

  • Changing Dynamics: The evolving nature of the pandemic makes it challenging to validate models accurately, especially when the underlying dynamics of the virus may change over time.

10. Communication and Transparency:

  • Communication Challenges: Communicating uncertainties and limitations associated with the data is crucial to maintaining public trust in the predictions and recommendations derived from the models.

Addressing these challenges requires ongoing efforts to improve data quality, enhance testing and reporting practices, and incorporate robust validation techniques into COVID-19 prediction models. It is also essential for researchers and policymakers to transparently communicate the limitations and biases associated with the data used in these models to ensure informed decision-making.

This review provides a comprehensive overview in various domains contributing to sustainable health and well-being in the context of the pandemic. The integration of precision medicine, eco-friendly healthcare practices, digital health technologies, holistic well-being approaches, community health promotion, global health security, and data-driven public health strategies collectively shapes a roadmap towards a more sustainable and resilient healthcare future.

2.11 Observations and discussions

  • COVID-19 and Its Variants: Efforts have been made to provide a comprehensive introduction to COVID-19 and its variants, highlighting the origins of the virus and the emergence of extraordinary variants, each with its special genetic makeup and possible implications for transmission and vaccine efficacy. These aspects set the stage for understanding the need for mathematical modeling in monitoring and predicting the spread of these variants.

  • Factors Influencing Spread: The paper accurately emphasizes the factors influencing the unfolding of COVID-19 and its variants, such as close contact, indoor settings, asymptomatic and pre-symptomatic spread, superspreader events, and travel. Understanding these elements is critical for modeling the dynamics of the disease accurately.

  • Prevention and Control Measures: The discussion of prevention and control measures, including vaccination, masking, physical distancing, hand hygiene, ventilation, testing, contact tracing, and travel restrictions, is well-informed and aligns with current public health guidelines. These measures form the foundation for interventions considered in the mathematical models.

  • Analytical and Numerical Models: The paper effectively distinguishes between analytical and numerical models for COVID-19 prediction. Analytical models, like compartment models and hidden nodes models, rely on mathematical equations to depict the disease's spread, while numerical models, such as statistical and data-driven models, rely on real-world data for predictions.

  • Models Reviewed: The overview summarizes several mathematical models used for COVID-19 prediction. These models range from basic compartmental models like SIR and SEIR to data-driven techniques like the Holt-Winters model, machine learning, and deep learning models. The paper discusses their suitability for different situations and their strengths and limitations.

  • Parametric Modeling: Parametric modeling is highlighted as a simple yet powerful method for analyzing the spread of COVID-19. These models consider infectious rates, treatment rates, and mortality rates but may lack multidimensionality. It's essential to note that parametric models often simplify complex real-world scenarios.

  • Prevention Models: The paper suggests that models for prediction and prevention are crucial for tackling pandemic. Preventive models can help assess the impact of interventions, vaccination strategies, and travel restrictions, contributing to the development of effective control measures.

Drawing parallels with the broader context of sustainable health and well-being, the integration of precision medicine, eco-friendly healthcare practices, digital health technologies, holistic well-being approaches, community health promotion, global health security, and data-driven public health strategies collectively forms a roadmap toward a more sustainable and resilient healthcare future. For instance, precision medicine tailors medical interventions to individual characteristics, contributing to a patient-centric healthcare model. Similarly, eco-friendly healthcare practices, including green hospitals, align with environmentally conscious approaches, ensuring the overall health of communities. Digital health technologies and telemedicine play a vital role in providing innovative and accessible healthcare solutions, minimizing environmental impacts associated with unnecessary travel.

The holistic well-being approach, incorporating mind–body interventions like mindfulness and meditation, contributes not only to stress reduction but also to mental health promotion, aligning with sustainable well-being goals. Community health promotion initiatives, addressing social determinants of health, foster resilience and reduce health inequities, promoting sustainability in well-being. Global health security and pandemic preparedness, as emphasized in recent pandemics, underscore the need for robust health systems and international collaboration, contributing to long-term health sustainability.

Moreover, data-driven public health strategies, pivotal in modern public health, align with the use of mathematical modeling for disease prediction and control. By utilizing big data analytics and artificial intelligence, public health professionals can identify trends, predict outbreaks, and formulate evidence-based strategies for disease prevention and control, thereby contributing to sustainable health outcomes. This synergy between mathematical modelling for COVID-19 and broader sustainable health practices forms a comprehensive framework for addressing current and future health challenges.

In the realm of mathematical modeling for COVID-19 and its variants, several significant research gaps and future aspects merit attention. Firstly, while the paper provides a comprehensive overview of various modeling techniques, it is crucial to underscore the pivotal role of model validation. Future research endeavors must prioritize the development of robust validation processes, using real-world data to enhance the accuracy and reliability of these models, ultimately facilitating more effective decision-making processes. Additionally, the emergence of COVID-19 variants, each with distinct characteristics, underscores the need for variant-specific modelling. Future models should consider the variant-specific factors influencing transmission and vaccine efficacy. By doing so, these models will be better equipped to adapt to the evolving nature of these viral strains.

Moreover, a significant research gap lies in the integration of behavioral elements into mathematical models. Public compliance with preventive measures and vaccine acceptance are pivotal aspects of the pandemic's dynamics and neglecting them in the models can result in a lack of realistic representation. Integrating these behavioral factors will enable a more accurate portrayal of disease spread and the development of higher quality public health strategies. Furthermore, while the paper predominantly focuses on modeling within specific regions, there is a growing need for research on models with universal applicability that can be adapted to various global contexts. The global nature of the pandemic necessitates mathematical models that are not constrained by geographical boundaries.

Lastly, as the pandemic evolves, there is a growing demand for models capable of accurate long-term forecasting. Short-term predictions are undeniably valuable, but long-term models should consider the interaction of evolving variants, vaccination efforts, and population immunity dynamics over extended periods. Addressing these research gaps and incorporating future aspects, such as improved validation processes, variant-specific modeling, behavioral data integration, global applicability, and long-term forecasting, will substantially enhance the utility of mathematical modeling in guiding effective public health strategies for managing COVID-19 and its variants.

3 Conclusion

In summary, the research on mathematical modelling for COVID-19 and its variants demonstrates notable strengths, including its comprehensive coverage of various modelling techniques and the integration of insights from sustainable health practices. The paper effectively introduces COVID-19 and its variants, emphasizing the importance of mathematical modelling in understanding their spread. It aptly highlights factors influencing transmission, prevention and control measures, and the distinction between analytical and numerical models. The overview of various mathematical models for COVID-19 prediction is thorough, encompassing parametric modelling and the significance of prevention models. Moving on to the discussion on variant classification systems, the paper successfully identifies critical limitations in existing systems. It underscores the challenges posed by the rapid evolution of the SARS-COV-2 virus, issues with variant identification, oversimplification of variant diversity, lack of standardized nomenclature, limited functional information, biases in sequencing data, dynamic clinical implications, adaptability to diagnostics and vaccines, and ethical and social considerations. This analysis sheds light on the complex landscape of variant classification, providing valuable insights for future improvements. In conclusion, the paper effectively outlines the importance of mathematical modelling in addressing the challenges posed by COVID-19 and its variants. By addressing the identified research gaps and incorporating insights from sustainable health practices, future research can contribute to more advanced models, fostering accurate predictions and enabling more effective public health responses to the evolving landscape of COVID-19 variants.