Mathematical Investigation of the Infection Dynamics of COVID-19 Using the Fractional Differential Quadrature Method

Comment

Response

The recent global challenge of the COVID-19 pandemic has deeply impacted both public health and the worldwide economy. The use of mathematical modeling has emerged as an essential approach to grasp and interpret the dynamics of infectious diseases.

In light of this, could you elaborate on how the differential quadrature method (DQM) has been employed to anticipate the behaviors of COVID-19 through a fractional mathematical model?

The differential quadrature method (DQM) is a numerical technique used to solve differential equations by approximating the derivatives with a set of discrete points. This method has been employed in various fields, including engineering and applied mathematics, to solve differential equations with high accuracy.

In the context of anticipating the behaviors of COVID-19 through a fractional mathematical model, the DQM was utilized to solve the fractional differential equations that describe the dynamics of the disease. Fractional calculus is a branch of mathematics that generalizes the concept of derivatives and integrals to non-integer orders, allowing for a more accurate representation of complex phenomena with memory effects.

The following procedures were taken while applying the DQM to a fractional mathematical model of COVID-19:

1. Discretization the fractional derivatives: Convert the fractional derivatives into discrete points using the DQM. This involves approximating the fractional derivatives by a combination of discrete values at specific points in the domain.

2. Approximation the fractional differential equations: The fractional derivatives in the governing equations were replaced with the discrete approximations obtained in the previous step. This results in a system of algebraic equations.

3. Solving the system of algebraic equations: The system of algebraic equations derived from the fractional differential equations was solved by DQM with the aid of Matlab program. The DQM approximated the derivatives by using a weighted combination of function values at discrete points, allowing for an accurate numerical solution.

4. Analyzing the results: Once the system of equations was solved, we analyzed the obtained results to understand the behaviors and dynamics of COVID-19 predicted by the fractional mathematical model. This analysis can provide insights into the spread of the disease, the effectiveness of control measures, and the potential impact on public health and the economy.

5. Finally, by employing the differential quadrature method (DQM) to solve the fractional differential equations governing the dynamics of COVID-19, researchers can gain a better understanding of the behaviors and patterns of the disease. This can aid in decision-making processes, such as implementing appropriate public health interventions and economic policies, to mitigate the impact of the pandemic.

How have uniform and non-uniform polynomial differential quadrature (PDQM) and discrete singular convolution (DSCDQM) been harnessed to elucidate the dynamics of susceptible, infected, exposed, recovered, and deceased individuals within the context of this pandemic?

Uniform and non-uniform polynomial differential quadrature (PDQM) and discrete singular convolution (DSCDQM) techniques have indeed been utilized to analyze the dynamics of susceptible, infected, exposed, recovered, and deceased individuals in the context of the COVID-19 pandemic.

1. In PDQM, a uniform mesh was employed to discretize the fractional differential equations governing the spread of the disease. This uniform discretization allowed for the approximation of derivatives within the equations. By applying PDQM, the resulting system of algebraic equations was solved, providing valuable insights into the dynamics of COVID-19. This method can help determine the distribution of individuals among different compartments (such as susceptible, infected, exposed, recovered, deceased) over time, and predict the future course of the pandemic.

2. On the other hand, non-uniform PDQM considers a non-uniform mesh that may be more suitable for capturing the intricacies of the system. Since the dynamics of infectious diseases like COVID-19 can vary in different regions or populations, non-uniform PDQM allows for a more precise representation of the underlying dynamics. This method can account for variations in population density, demographic factors, and other spatial or temporal characteristics that affect the spread of the disease.

3. Additionally, discrete singular convolution differential quadrature method (DSCDQM) was employed to analyze the dynamics of COVID-19. DSCDQM is a numerical discretization technique that utilizes singular convolution to approximate derivatives in differential equations. It provides a high-order approximation, enabling accurate solutions to the equations governing the pandemic dynamics. By utilizing DSCDQM, we effectively captured the detailed behavior of the disease, including rapid changes and local variations, which are crucial for effective modeling and decision-making during the pandemic.

4. Overall, the utilization of uniform and non-uniform PDQM and DSCDQM techniques allows researchers to gain a comprehensive understanding of the dynamics of COVID-19. These approaches provide insights into the distribution of individuals among different groups and help in predicting the future course of the pandemic, aiding decision-makers in formulating effective strategies to combat the spread of the virus.

Moreover, could you provide insights into the comparison drawn between the methods presented and the modified Euler method?

§  We modified the paragraph in page 9 to provide insights into the methods presented.

§  Here, we can compare here between the methods presented and the modified Euler method as follows:

1. Scope of Application:

   - Uniform and non-uniform PDQM and DSCDQM: These methods are applied to solve fractional differential equations, which can provide a more accurate representation of the dynamics of COVID-19. Fractional differential equations capture memory effects and long-range interactions, which can be relevant in modeling infectious diseases.

   - Modified Euler method: The modified Euler method is typically used for solving ordinary differential equations (ODEs) and may not directly handle fractional derivatives. However, it can still be used for modeling certain aspects of COVID-19 dynamics that do not explicitly involve fractional calculus.

2. Accuracy and Convergence:

   - Uniform and non-uniform PDQM and DSCDQM: PDQM and DSCDQM can offer high accuracy and convergence when solving fractional differential equations. By using appropriate grid points and weighting functions, these methods can provide accurate approximations of the solution.

   - Modified Euler method: The modified Euler method is a lower-order numerical integration method for ODEs. It has moderate accuracy and convergence, but it may not be as accurate as higher-order methods. Therefore, it may not capture complex dynamics accurately, especially in the case of COVID-19 modeling, which can involve intricate interactions and behaviors.

3. Computational Efficiency:

   - Uniform and non-uniform PDQM and DSCDQM: The computational efficiency of PDQM and DSCDQM depends on the specific implementation and the complexity of the problem. These methods may require significant computational resources, particularly when dealing with a large number of grid points or complex equations.

   - Modified Euler method: The modified Euler method is computationally efficient and relatively simple to implement. It requires fewer computational resources compared to more advanced numerical methods. This efficiency can be advantageous when dealing with large-scale COVID-19 models or scenarios that require real-time simulations.

How do the execution times of the code differ, and what implications does this have for the efficiency of DQM?

§  The execution times of the code using differential quadrature methods (DQM) can vary depending on several factors, including grid point density, weighting functions, and computational resources (hardware and software resources).

§  If the DQM code has long execution times, it may limit the ability to perform real-time or time-dependent simulations. In scenarios where timely results are required, long execution times can be a hindrance.

§  As a result, DSCDQM is the best approach for allocating time and effort toward obtaining results.

Additionally, could you delve into the key findings that emerged from this investigation?

Thank you for your kind comment. Section 4 was modified to explain the key findings that emerged from this investigation

How does the fractional order factor into the outcomes, and how does the approach of DQM shed light on the evolution of susceptible, exposed, deceased, asymptomatic, recovered, and infected individuals? Particularly, how does the evolution of infected individuals unfold in the initial stages of the investigation, given the rapid spread of the disease?

This suggestion is very constructive, and for this reason, we modified Section 4.3 to shows the different effects of fractional order on the evolution of susceptible, exposed, deceased, asymptomatic, recovered, and infected individuals.

page 12-13

Further, From the Figs. 3-8, one can observe that the considered model extremely depends on the order and offers more degree of flexibility. As we increase the values of the η, we see that the solution tends to integers order solution. The growing and decaying rate of various classes of model is different at different fractional order. Therefore fractional calculus can be helpful in understanding the transmission dynamics of COVID-19. Here we, remark that at smaller fractional order the decay process is faster while the growth rate is slow. Increasing the fractional order the process of decay may become slow while, the grow rate goes on raising. Further, the fractional order has great impact on the transmission dynamics of the proposed model. Also, it helps in better understanding of physical behaviour of spreading of infection in a community. Moreover, the adopted numerical methods can be used as a fruitful technique to achieve computational results for such type nonlinear problems. The concerned growth or decay process of various compartments is faster slightly at lower fractional order as compared to greater value of η. These graphical representations provide valuable insights into the dynamics of various population groups during the course of infection. The analysis demonstrates the impact of fractional order on the susceptible, exposed, infected, recovered, deceased, and asymptomatic populations. These findings contribute to a deeper understanding of the disease's progression and can aid in formulating effective strategies for disease management and control.

Lastly, could you discuss the observed relationship between the number of recovered individuals and the prevalence of infected individuals, and how this insight informs our understanding of the pandemic's trajectory?

Thank you very much for your suggestion. Indeed, it is a good idea to show this relationship in our paper.  

page 13-14

The relationship between the number of recovered individuals and the prevalence of infected individuals provides valuable insights into the trajectory of a pandemic, such as COVID-19. Understanding this relationship helps in assessing the progression of the disease, estimating the effectiveness of control measures, and predicting future trends. Here are some key observations:

From the Figs. 4-6, the number of recovered individuals has a direct impact on the prevalence of infected individuals. As more individuals recover, the pool of susceptible individuals decreases, leading to a decline in the transmission of the virus. This helps to slow down the spread of the disease and reduce the overall prevalence of infected individuals over time. The relationship between recovered and infected individuals is closely linked to the concept of herd immunity. When a significant portion of the population becomes immune either through infection or vaccination, it creates a protective barrier that limits the transmission of the disease. This reduces the prevalence of infected individuals and helps to control the pandemic. By closely monitoring the relationship between the number of recovered and infected individuals, public health officials and policymakers can gain insights into the trajectory of the pandemic. They can assess the effectiveness of interventions, make informed decisions regarding control measures, and predict future trends in disease transmission. This understanding helps in managing healthcare resources, planning vaccination campaigns, and implementing targeted measures to mitigate the impact of the pandemic.

Modify introduction 

“Probing Families of Optical Soliton Solutions in Fractional Perturbed Radhakrishnan–Kundu–Lakshmanan Model with Improved Versions of Extended Direct Algebraic Method”

“Investigating Families of Soliton Solutions for the Complex Structured Coupled Fractional Biswas–Arshed Model in Birefringent Fibers Using a Novel Analytical Technique” 

“Investigation of Fractional Nonlinear Regularized Long-Wave Models via Novel Techniques”

Thanks for these nice and important references. These references are added to the introduction.

check grammatical mistakes

Thanks. The whole manuscript has been revised and checked.

Comment

Response

Explain how to obtain those parameter values in Table 2 and 3.

Thank you for your kind comment. We added Table 1 for a description of the parameters involved in the model. (Page 7-8)

The authors used the model formulation (equations 1-6) from paper [15]. They then analyze from the existing model formulation. There are many studies on this area, but there is no comparison to any existing results.

§  This suggestion is very constructive. There are many studies in this area, however, some have more than six equations and others have less than six, depending on the parameters being studied.

§  Because each study used different data from numerous countries, applying the proposed methods to some of them would be difficult in one paper.

§  So, we will introduce different methods for solving other models in this area.

§  We added some references related to our paper.

Authors mentioned about the COVID-19 but did not include any data for their study. It would be good if they use some COVID-19 data and analyze it and use them for the model parameter values. This can enhance a merit for consideration for publication. Without it, there is no new contribution for publication.

Thank you very much for your suggestion. Indeed, we included Table 1 to include any data utilized in the study in the model. (Page 7)

It is difficult to figure out what are the new material compared to some existing works due to English writing.

The whole paper has been Linguistically revised. The motivation of this manuscript is to use more reliable numerical technique (Fractional differential quadrature method). This method is compared to other relevant techniques to validate the results. Moreover, the stability of the method is emphasized. The execution time and the errors are better than relevant techniques. From the technical point of view, the results are very useful to investigated the dynamics behavior of Covid-19 which is modeled by fractional derivatives. The results are evaluated at different fractional orders from 0 to 1.

Comment

Response

1. The original contributions need to be much better presented in the last paragraph of section \INTRODUCTION". All improvements, if they are, and new results must be described in this paragraph. The advantages of the work are not discussed in the text.

Thanks for your suggestion. The motivation paragraph is added to the introduction.

2. The Abstract should be modified by adding advantages of the proposed method. Please rewrite as new one.

Thanks for your constructive comment. The abstract has been updated.

3. Check the manuscript carefully for typos and grammatical errors.

Thanks for your great comment. The whole manuscript has been grammatically checked.

4. What is the novelty of the work and where does it go beyond previous efforts in the literature?

Thanks for your comment. A paragraph about novelty has been added at the end of the introduction

5. The references list is not at all updated with latest developments and publications.

Thanks for your nice suggestion. More recent references have been added.

6. In general, the typeset equations should be regarded as parts of a sentence and treated accordingly with the appropriate grammatical convention and punctuation. More editing for writing is needed. At the end of all equations must be put “COMMA” or “POINT” according to the typing rules.

Thanks again. We went over the whole manuscript and updated all typos errors according to your comment.

7. All acronyms should be checked.

Thanks. The acronyms were checked.

8. In section “Numerical Results”, please interpret the obtained results in detail. It is important what you conclude to them.

Thank you for your kind comment. Section 4 was modified to explain the key findings that emerged from this investigation

9. At the beginning of the numerical results section, authors should present the configuration of the personal computer used to perform the simulation results.

Thank you for your kind comment, the configuration of the computer used to perform the simulation results is HP Probook 450 G8 Laptop - 11th Intel Core i5-1135G7, 8GB RAM, 512GB PCIe NVMe SSD, 15.6" FHD (1920 x 1080), Intel Iris X Graphics. These configurations were added in page 6

10. Be sure that all the parameters descriptions are mentioned in section “Numerical results ".

Thank you for your kind comment. We added Table 1 for a description of the parameters involved in the model. (Page 7)

11. How did you discuss the local stability of the method?

This suggestion is very constructive, and for this reason, we added Section 4.2 to explain the stability of the method.

Comment

Response

Comment 1: The writing organization is good, but the writing style (more explanation of the results in text) needs to be polished

Thanks for your comment. The whole manuscript has been revised.

Comment 2: What is the robustness of the proposed method

1-   To validate the current method, DQM, the results of uniform PDQM, non-uniform PDQM and DSCDQM-DLK are compare to Modified Euler Method [11]. The comparison is depicted in Fig. 1(a, b).

2-   Moreover, the numerical results are tabulated in Tables 2-4 and compared to those of [11] including the execution time of the code.

3-   Figure 1(a, b) and Tables 2-4 show a great agreement with previous work and better execution times.

So, PDQM and DSCDQM have high accuracy and convergence when solving fractional differential equations. By using appropriate grid points and weighting functions, these methods can provide accurate approximations of the solution.

Comment 3: Write about the advantages of the suggested method over other existing methods

The Fractional Differential Quadrature Method (FDQM) is an extension of the traditional Differential Quadrature Method (DQM) that incorporates fractional calculus. FDQM offers several advantages over other existing methods for modeling Covid-19 dynamics. Here are some key advantages of FDQM:

1.   Capturing long-term memory effects:

 Covid-19 dynamics often exhibit long-term memory effects, where the disease spread is influenced by past events and interactions. Fractional calculus provides a mathematical framework to model and capture these memory effects. FDQM enables the incorporation of fractional derivatives into the model equations, allowing for a more accurate representation of the disease dynamics and the influence of historical data.

2.   Non-locality and non-local interactions:

 In the context of infectious diseases like Covid-19, non-local interactions play a significant role. The spread of the disease in one region can affect the dynamics in neighboring regions due to travel, migration, or social connections. FDQM is well-suited for modeling non-local interactions as it can incorporate fractional derivatives that capture the non-local behavior of the system. This capability enables a more realistic representation of the disease propagation across different regions.

3.   Flexibility in modeling complex dynamics:

FDQM provides greater flexibility in modeling complex dynamics associated with Covid-19. Fractional derivatives allow for the modeling of non-linearities, memory effects, and anomalous diffusion processes, which are often observed in the spread of infectious diseases. By incorporating fractional calculus within the DQM framework, FDQM can capture these complexities and provide a more accurate representation of the disease dynamics.

4.   High accuracy and convergence:

 FDQM retains the high accuracy and convergence properties of the traditional DQM. By discretizing the spatial domain into a set of points and approximating the fractional derivatives using a weighted sum of function values at neighboring points, FDQM provides accurate solutions to the fractional differential equations. This accuracy is crucial for reliable predictions and informed decision-making during the Covid-19 pandemic.

5.   Efficient computational implementation:

 FDQM can be efficiently implemented computationally. It reduces the fractional differential equations to a system of algebraic equations, which can be solved using standard numerical techniques. The computational efficiency of FDQM enables the analysis of large-scale Covid-19 models with high spatial and temporal resolutions, allowing for detailed investigations of the disease dynamics and the impact of various interventions.

6.   Integration with existing models and methods:

 FDQM can be easily integrated with existing models and methods used for Covid-19 modeling. It can be combined with compartmental models, network models, or agent-based models to enhance their capabilities in capturing memory effects, non-local interactions, and complex dynamics. This integration allows researchers to leverage the strengths of different modeling approaches and gain a comprehensive understanding of Covid-19 dynamics.

Comment 4: The originality of the paper needs to be stated clearly. It is of importance to have sufficient results to justify the novelty of a high-quality journal paper

Thanks for your constructive comment. The motivation and novelty of the paper has been added at the end of the introduction and the results have been polished.

Comment 5: The authors should check the punctuation especially after the equations.

We appreciate your remarks. The whole manuscript has been revised and checked.

Comment 6: The English style must be revised because there are some typos must be corrected, for example:

Line 24 on page 6 “are compare” should be are “compared”

We appreciate your remarks. The whole manuscript has been revised and checked

Comment 7: How to validate the efficiency of the proposed methods when the underlying problem has no exact solution

That’s right, professor. The underlying problem has no exact solution, so:

1-  Putting the fractional order "" converts the governing equations to traditional ordinary differential equations that have an exact and several numerical solutions and ensure our results.

2-   To validate the current method, DQM, the results of uniform PDQM, non-uniform PDQM and DSCDQM-DLK are compare to Modified Euler Method [8]. The comparison is depicted in Fig. 1(a, b).

3-   Moreover, the numerical results are tabulated in Tables 2-4 and compared to those of [8] including the execution time of the code.

4-   Figure 1(a, b) and Tables 2-4 show a great agreement with previous work and better execution times.

So, PDQM and DSCDQM have high accuracy and convergence when solving fractional differential equations. By using appropriate grid points and weighting functions, these methods can provide accurate approximations of the solution.

Comment 8: (serious question) How to validate the efficiency of the proposed methods when the underlying problem has no exact solution

Comment 9: All references must be written in the Journal style.

Thanks for your comment. The references are written by the editing office of the journal according to the journal template.

Comment 10: The authors have ignored some basic and recent paper in this field.

Thanks for great suggestion. More recent references have been added to the manuscript.