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Edited by: Yasser Aboelkassem, University of Michigan-Flint, United States

Reviewed by: Rashid Jan, University of Swabi, Pakistan

Ercan Çelik, Kyrgyz Turkish Manas University, Kyrgyzstan

Haci Mehmet Baskonus, Harran University, Türkiye

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

This research introduces a sophisticated mathematical model for understanding the transmission dynamics of COVID-19, incorporating both integer and fractional derivatives. The model undergoes a rigorous analysis, examining equilibrium points, the reproduction number, and feasibility. The application of fixed point theory establishes the existence of a unique solution, demonstrating stability in the model. To derive approximate solutions, the generalized Adams-Bashforth-Moulton method is employed, further enhancing the study's analytical depth. Through a numerical simulation based on Thailand's data, the research delves into the intricacies of COVID-19 transmission, encompassing thorough data analysis and parameter estimation. The study advocates for a holistic approach, recommending a combined strategy of precautionary measures and home remedies, showcasing their substantial impact on pandemic mitigation. This comprehensive investigation significantly contributes to the broader understanding and effective management of the COVID-19 crisis, providing valuable insights for shaping public health strategies and guiding individual actions.

The novel coronavirus, later identified as SARS-CoV-2, was first reported in Wuhan, China, in December 2019. Investigations suggested that the virus may have originated in bats and was potentially transmitted to humans through an intermediate host, possibly from a wild animal traded at a seafood market in Wuhan. Zoonotic diseases, where pathogens jump from animals to humans, are not uncommon, and past infectious disease outbreaks have similarly had zoonotic origins. Examples include the H1N1 influenza virus, Ebola virus, and Middle East Respiratory Syndrome (MERS) coronavirus. These viruses undergo genetic mutations or reassortment, enabling them to adapt and infect humans. Studying the origins of such diseases is crucial for prevention and control, and ongoing international research aims to comprehend the circumstances leading to the initial transmission of SARS-CoV-2 to humans Mathematical models play an important role in understanding the transmission dynamics of diseases, providing policymakers with a valuable tool for assessing and evaluating potential health risks.

Mathematical modeling in epidemiology plays a crucial role in understanding and predicting the spread of infectious diseases within populations. By employing mathematical equations and statistical techniques, epidemiologists can simulate the dynamics of disease transmission, assess the impact of interventions, and formulate informed public health strategies. These models often consider factors such as the rate of infection, recovery, and contact between individuals to simulate the progression of an outbreak. Through mathematical modeling, researchers can explore different scenarios, evaluate the Impact of various control measures, and ultimately contribute to the development of evidence-based policies targeted at mitigating the influence of infectious diseases on society. Mathematical models in epidemiology serve as powerful tools for decision-makers in their efforts to prevent, manage, and regulate the transmission of disease in society. Indeed, mathematical modeling has been extensively applied to study various infectious diseases, including malaria, chickenpox, and COVID-19. Each disease presents unique challenges, and mathematical models help researchers and public health officials understand the dynamics of transmission, evaluate the impact of interventions, and make informed decisions. For malaria, models might focus on factors like mosquito breeding habitats, vector behavior, and the impact of insecticide-treated bed nets. In the case of chickenpox, models may consider the age structure of the population and the waning immunity over time. The COVID-19 pandemic has witnessed a surge in mathematical modeling efforts to predict the course of the disease, evaluate the influence of non-pharmaceutical interventions, and guide vaccination strategies. These models provide valuable insights that aid policymakers in designing strategies to control and prevent the spread of contagious diseases. In 2023, the study examined the effectiveness of precautionary measures in managing chickenpox in Phuket by utilizing mathematical modeling and conducting bifurcation analysis to evaluate potential outcomes [

After exposure to the virus, symptoms typically manifest in individuals within a range of two to 14 days. The insidious nature of the coronavirus reveals a concerning aspect as individuals harboring the virus may become contagious up to 48 h before symptoms materialize. This contagious phase extends for a variable span of 10 to 20 days, influenced by the intricacies of one's immune response and the severity of the illness. The initial signs of the virus present a diverse array of symptoms, encompassing the familiar culprits such as cough, fever, and shortness of breath. However, the subtlety lies in the nuanced manifestations including, muscle aches, sore throat, loss of taste or smell, and a range of other discomforts. This temporal intricacy and symptomatic diversity underscore the challenges in identifying and containing the spread of the virus effectively. COVID-19 can manifest as mild illness in some individuals, while others may remain asymptomatic. In severe cases, the disease can lead to respiratory distress. Lingering impact resonates within, leaving lasting impressions on the lungs, heart muscles, and intricacies of the nervous system. Renal failure or, in extreme instances, death.

The detailed study of disease momentum is a prevailing theme for many mathematicians and biologists. We can observe numerous works, such as those by Haq et al. [

The fractional-order derivative is a broader interpretation, extending beyond the conventional scope of the integer-order derivative. The Caputo fractional derivate has been chosen not only because of its capability to integrate the memory effect on the model but also the completeness of the analytical tools to provide the dynamical behaviors of the model which does not exist on the other fractional derivative such as the Caputo-Fabrizio [

From a mathematical standpoint, the diffusion of the disease is expected to reduce when the effective reproduction number remains bounded by 1. Additionally, a research on controlling the Spread mechanisms of a virus has been proposed by Edward et al. [

The study of COVID-19 transmission in Thailand is pivotal in predicting the spread of the disease, offering invaluable insights for estimating healthcare resource requirements such as hospital beds, ventilators, and medical staff. By comprehensively analyzing transmission rates, incubation periods, and the effectiveness of public health interventions, this research illuminates strategies to curtail further spread and maintain control over the outbreak. Through identifying patterns of virus transmission, including peak infection periods and potential hot spots, the study provides essential guidance for mitigation efforts. Moreover, it enhances our understanding of the multifaceted factors influencing viral spread, empowering us to make informed decisions. Utilizing metrics like the reproduction number, we can forecast future transmission rates with greater accuracy. Notably, recognizing the impact of various parameters in the model is crucial, as it can significantly alter outcomes, underscoring the importance of precision in pandemic modeling and response strategies.

This paper is divided into several sections. Second section provides fundamental definitions and concepts of calculus in fractions. In Section 3, we delve into the fractional model of COVID-19 transmission. The discussion of the uniqueness and existence of the solution is presented in Section 4. Finally, Section 5 outlines the numerical method employed for solving the model, accompanied by the presentation of numerical results.

The main highlights:

The study introduces a unique mathematical model for COVID-19 transmission that goes beyond traditional models by incorporating both integer and fractional derivatives. This innovation allows for a more accurate representation of the complex dynamics involved in the spread of the virus.

The research conducts a thorough analysis of the mathematical model, specifically examining equilibrium points and the reproduction number. This provides a deeper understanding of the stability and characteristics of the system, offering insights into critical aspects of COVID-19 transmission dynamics.

The study employs fixed point theory to rigorously establish the occurrence of a unique solution to the proposed mathematical model. This contributes to the mathematical foundation of the model and enhances confidence in its predictive capabilities, demonstrating the robustness of the approach.

The research utilizes the generalized Adams-Bashforth-Moulton method to obtain estimated solutions for the mathematical model. This computational technique allows for practical implementation and analysis, facilitating the exploration of the model's behavior and outcomes.

In this model, individuals can be categorized into six groups: susceptible individuals (S), exposed individuals (E), quarantined individuals(Q), symptomatic infected individuals (I), asymptomatic infected individuals (A), and removed individuals (R), including cured and dead individuals. In the SEQIAR epidemiological model, the quarantine compartment plays a crucial role in representing individuals who have been exposed to the infectious agent but are temporarily isolated from the general population. The Quarantine compartment accounts for those who have closely interacted with individuals carrying the infection and are placed under quarantine to avoid the potential spread of the disease. This compartment acknowledges the incubation period during which individuals may not yet exhibit symptoms but can transmit the infection. The inclusion of a quarantine compartment enhances the SEIAR model's realism by capturing the impact of public health measures, such as isolation and quarantine, in controlling the transmission of infectious illness. Through the incorporation of the quarantine compartment, the SEQIAR model becomes a valuable tool for emulating and understanding the momentum of epidemics and assessing the effectiveness of interventions in mitigating disease transmission. The entire population is represented by N, where

with,

_{0}, _{0}, _{0}, _{0}, _{0} are the initial conditions of the system.

Referencing the detailed provided in

Detailed description of parameters of the model (

Λ = |
N is the total number of individuals and n is the birth rate |

μ | The death rate of individuals |

α | The transmission rate of S to E |

γ | The transmission rate of E to Q |

σ | The proportion of transmission rate of individuals from E to A |

θ | Transmission rate of individuals from E to A |

π | The proportion of transmission rate of individuals from Q to A |

ϕ | Transmission rate of individuals from Q to I |

μ_{1} |
The death rate of individuals in Q |

μ_{2} |
The death rate of individuals in A |

μ_{3} |
The death rate of individuals in I |

χ_{1} |
Recovery rate of individuals from A to R |

χ_{2} |
Recovery rate of individuals from I to R |

ψ | Transmission rate of individuals from Q to R |

ψ_{1} |
Proportion of individuals who take home remedies |

Transfer coefficient from A to I |

In this section, we've examined the invariant region, the positivity of the solution, the presence of equilibria, disease-free and disease-endemic equilibrium points, the basic reproduction number and also conducted stability analysis.

Using the following theorem we can demonstrate the nonnegativity,

Theorem 3.1. Solutions of the all dynamic attribute (S(t),E(t),Q(t),A(t),I(t),R(t)) with initial condition satisfy

where,

Now, we get the following expression from integrating the above equation

Now we can show the positive invariance for all variables

by integrating the above equation, we get

Hence the proof.

We examined the model (

Theorem 3.2. The set of feasible solution of the system of

Upon integration of the aforementioned equation, the result is obtained as, N(t) is a constant.

Accordingly, with the constant magnitude of the population, we obtain that all feasible solutions of each of the system of

In examining the transmission of an infection, the disease-free equilibrium (DFE) represents a population state crucial for understanding when the disease is not widespread. The determination of the disease-free equilibrium involves setting E, Q, A, I, and R to zero in the system of

The solution to the algebraic equations allows us to identify the system's equilibrium points. The disease-free equilibrium point is reached when the absence of any ailment prevails, given by

And also, if _{0} > 1, then the system (

where,

The stability of ^{0} relies on the basic reproductive number _{0}, which represents the average number of secondary cases generated by a COVID-19 infected individual throughout their contagious period when introduced to a population of susceptible individuals without any interventions. We examine the equilibrium's stability using the next-generation operator. Referring to the notations in Van den Driessche and Watmough [

and

The Jacobian matrix for F and V at _{0} obtained as

and

^{−1} is known as the next generation matrix for the system, then the basic reproduction number is defined as

Theorem 3.3. The disease-free equilibrium, _{0} < 1 and unstable if _{0} > 1.

_{0} as in Yaagoub et al. [

Then,

Then, after calculation, we will have

Since the arithmetic mean either exceeds or equals the geometric mean, so

Moreover, if _{0} < 1, then we will have _{1} < 1 and _{2} < 1, therefore we will have ^{0} is globally asymptotically stable.

Within this section, we usher in a moderating influence into the system, initiating a transformation where the conventional time derivative yields to the nuanced impact of the Caputo fractional derivative. This shift, however, induces a dimensional incongruity between the right and left facets of the equation. To harmonize this dissonance, we introduce a pivotal auxiliary parameter, aptly labeled as ξ and characterized by dimensions in seconds. This strategic introduction of ξ serves as a calibrated lever, skillfully adjusting the fractional operator. The overarching goal is to orchestrate a balanced equation where both sides coalesce seamlessly, sharing a harmonized dimensionality.

In light of the explained methodology, the fractional model for coronavirus transmission, valid for

with initial conditions of the system of

The equilibrium points of the fractional order system are determined by solving the given equations.

we obtain equilibrium points same as in model (_{0} is known as the basic reproduction number which is obtained using the next-generation method. So, to find _{0}, first we have to consider the system as follows,

Then the basic reproduction number of the Fractional model is the same as the model (

In this section, we are going to show that the system has a unique solution, we write the equation as follows,

By basic definitions we can write these as,

Now we will show that the kernels _{m}, 1 ≤

Theorem 4.1. The kernel _{1} fullfil the Lipchitz condition and contraction if the inequality holds 0 ≤ μ + α(δ_{3} + _{4}) < 1

let, _{1} = (μ + α(δ_{3} + _{4})), where, ||_{3}, ||_{4}, is function with boundary, so ||_{1}(_{1}(_{1})|| ≤ _{1}||(_{1}(_{1} the Lipchitz condition is satisfied and if 0 ≤ (μ + α(δ_{3} + _{4})) < 1, then _{1} is a contraction.

Similarly, the Lipschitz condition for _{m}, 1 ≤

where, ||_{1}, ||_{2}, ||_{3}, ||_{4}, ||_{5}, ||_{6} and _{2} = (θ + γ + μ), _{5} = (ϕ + μ + μ_{1} + ψ + ψ_{1}), _{4} = (χ_{1} + μ + μ_{2}), _{3} = (μ + μ_{3} + χ_{2}), _{6} = μ are bounded functions, if 0 ≤ _{m} < 1, _{m}, 2 ≤

with initial conditions of the system

Proceeding, we evaluate the norm of the first equation within the described system of equations, then

From Lipchitz conditions, we have

similarly, we obtain

thus we can write,

Now, we are going to show that the uniqueness of the solution

Theorem 4.2. A system of solutions given by the fractional COVID-19 model exist if there exist _{1} such that

Thus, we can say the system has a

we assucume that,

so,

Now we repeat the method and we obtained,

at _{1}, we get

Now take the limit on the recent equation and let n tends to ∞, we obtain

similarly we can show that ||_{in}(

Hence the proof.

Now, we going to show the _{1}(_{1}(_{1}(_{1}(_{1}(_{1}(

now we take the norm of the equation

By using lipschitz condition

Then

Theorem 4.3. the solution of the model (

Then we can say, ||_{1}(_{1}(

In this context, we employ the numerical technique known as the Generalized Adams-Bashforth-Moulton (ABM) method [

Now, the equation mentioned above can be expressed as an equivalent Volterra integral.

When tackling integration challenges, the Adams-Bashforth-Moulton method stands out as a reliable numerical technique. Let _{n} = ^{+}, so we can express the system through its representation.

where,

in which,

and

In this section, initially, we conducting a data exploration by plotting a bar chart of Daily Cumulative Cases with Slope Lines (

Daily cumulative COVID cases with slope lines.

COVID cases distribution illustrated with violin plot.

The integrated violin and box plot depicted in

Subsequently, we conduct a numerical modeling simulation utilizing authentic data to model the transference momentum of COVID-19 in Thailand. The approach involves the estimation of certain parameters, drawing from existing literature, while the remaining parameters are meticulously adjusted to enhance model accuracy. We fitted the COVID-19 Model (^{1}

In this study, we conducted numerical simulations to validate the outcomes of our proposed model, employing the Caputo fractional derivative and solvers implemented in the MATLAB programming language. The purpose of these simulations was to complement and validate the analytical results obtained. To provide a visual illustration, we chose baseline values of parameter (refer to

Parameter values of model.

Λ | 0.009718 | [ |

μ | 0.008159 | [ |

α | 1.0791 | Fitted |

γ | 1/7 | [ |

σ | 0.8 | Fitted |

θ | 1/5.2 | [ |

π | 0.5 | Fitted |

ϕ | 1/15 | Fitted |

μ_{1} |
0.015 | [ |

μ_{2} |
0.015 | [ |

μ_{3} |
0.015 | [ |

χ_{1} |
0.13978 | [ |

χ_{2} |
0.13978 | [ |

ψ | 0.65 | Fitted |

ψ_{1} |
1/7 | [ |

0.001 | Fitted |

For data fitting, we gathered daily authenticated cases of COVID-19 in Thailand from May 19, 2022, to June 22, 2022, sourced from the World Health Organization (WHO), as outlined in

The daily confirmed cases of COVID-19 in Thailand from May 19, 2022, to June 22, 2022.

19-May | 6,305 | 1-Jun | 4,563 | 14-Jun | 1,833 |

20-May | 6,463 | 2-Jun | 2,560 | 15-Jun | 2,263 |

21-May | 5,377 | 3-Jun | 2976 | 16-Jun | 2,153 |

22-May | 4,739 | 4-Jun | 3,001 | 17-Jun | 1,967 |

23-May | 4,099 | 5-Jun | 3,236 | 18-Jun | 2,272 |

24-May | 4,144 | 6-Jun | 2,162 | 19-Jun | 1,892 |

25-May | 5,013 | 7-Jun | 2,224 | 20-Jun | 1,784 |

26-May | 4,813 | 8-Jun | 2,688 | 21-Jun | 1,714 |

27-May | 4,837 | 9-Jun | 3,165 | 22-Jun | 2,387 |

28-May | 4,488 | 10-Jun | 2,836 | ||

29-May | 3,649 | 11-Jun | 2,501 | ||

30-May | 3,854 | 12-Jun | 2,475 | ||

31-May | 3,955 | 13-Jun | 1,801 |

Daily cumulative cases of COVID-19 from May 19, 2022 to June 22, 2022.

Numerical simulation of model (

The importance of home remedies in influencing the spread rate of COVID-19 is crucial, particularly during times of quarantine. By employing a combination of precautionary measures and home remedies, individuals can observe a noticeable shift toward recovery. This section aims to demonstrate the impact of a specific parameter, denoted as ψ_{1}. Through the implementation of various levels of ψ_{1}, we can analyze and understand how these home-based interventions contribute to the overall trajectory of recovery. Home remedies, ranging from herbal infusions and nutritional supplements to respiratory exercises, not only bolster the immune system but also play a significant role in managing symptoms and reducing the severity of the illness. As we explore the effectiveness of different levels of ψ_{1} in this context, it becomes evident that personalized and targeted home-based interventions can be instrumental in mitigating the transmission of the virus and expediting the recovery procedure for individuals affected by COVID-19.

In _{1}, and we observe corresponding variations in all compartments. Specifically, we consider values for ψ_{1} as 0.1, 0.2, 0.3, and 0.4. The graph illustrates that an increment in the ψ_{1} parameter leads to a decline in the populations of compartments A, I, Q, and E, while the number of recovered individuals increases. Detailed numerical data supporting these observations can be found in _{1} parameter in influencing the dynamics of the model and its impact on the various compartments.

Numerical simulation of model (_{1}.

Results of fractional model with various of ψ_{1}.

ψ_{1} |
||||
---|---|---|---|---|

6, 285.03 | 5, 944.44 | 5, 683.46 | 5, 477.44 | |

1, 860.46 | 1, 749.55 | 1, 665.07 | 1, 598.71 | |

98, 424.3 | 100, 149 | 102, 286 | 105, 001 |

Numerical simulation of model (

Numerical simulation of model (

In

Numerical simulation of model (

This study has investigated the intricate momentum of COVID-19 spreading by formulating a mathematical model that incorporates the integer fractional-order Caputo derivative. Our exploration encompassed crucial aspects such as identifying the region of feasibility, pinpointing equilibrium points, and calculating the fundamental reproduction number _{0}. The application of fixed-point theory robustly established the existent of a unique solution to the model. Utilizing the Adams-Bashforth scheme, we derived approximate solutions for the system, allowing for a practical understanding of its behavior. Our simulations, based on real-world COVID-19 reports from Thailand, spanning from May 19, 2022, to June 22, 2022, provided valuable insights into equilibrium points and the reproduction number. Our findings confirm that the disease is anticipated to dwindle within the populace when the basic reproduction number is bounded by one, while persistence prevails when _{0} exceeds one. Moreover, our data analysis has deepened our understanding of disease transmission dynamics, offering a nuanced perspective on how COVID-19 behaves in the specific context of Thailand. Notably, our study sheds light on the impact of home remedies and precautionary measures on diminishing the disease and enhancing recovery rates, from _{1} increases from 0.1 to 0.4, causing individuals in compartments like A(t) and I(t) to decrease. This indicates how home remedies influence the rate of recovery even before individuals enter the asymptotically or symptomatically infected classes. In a similar vein, each graphical representation depicting the impact of a parameter provides a detailed insight into how that specific parameter influences the number of individuals involved in a given scenario. These representations serve as visual tools, offering a clear understanding of the dynamics at play within the system under consideration. By observing these graphs, we can discern the trends and patterns associated with the parameter's effect on the population dynamics. Furthermore, these graphical depictions offer valuable information on how we can effectively manage and control the spread of a certain phenomenon. By analyzing the trends illustrated in the graphs, we can identify strategies and interventions that prove effective in mitigating the spread of the phenomenon in question. Whether it involves adjusting certain parameters, implementing targeted interventions, or adopting preventive measures, these graphical representations serve as essential guides in devising strategies to curb the spread and maintain control over the situation. The comprehensive nature of our investigation not only contributes to the theoretical understanding of COVID-19 dynamics but also provides practical implications for managing and mitigating the impact of the pandemic. As we continue to navigate the complexities of the ongoing health crisis, these insights offer valuable guidance for public health strategies and individual actions in the pursuit of controlling and eventually overcoming the challenges posed by COVID-19.

Publicly available datasets were analyzed in this study. This data can be found here:

SE: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Resources, Software, Validation, Visualization, Writing – original draft, Writing – review & editing. SJ: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Project administration, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing. HP: Data curation, Formal analysis, Investigation, Methodology, Resources, Software, Validation, Visualization, Writing – original draft, Writing – review & editing. AJ: Data curation, Formal analysis, Funding acquisition, Investigation, Project administration, Software, Supervision, Validation, Visualization, Writing – review & editing. BO: Data curation, Investigation, Methodology, Resources, Software, Validation, Visualization, Writing – review & editing. ZY: Formal analysis, Investigation, Methodology, Resources, Validation, Writing – review & editing.

The author(s) declare that no financial support was received for the research, authorship, and/or publication of this article.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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