A conformable fractional finite difference method for modified mathematical modeling of SAR-CoV-2 (COVID-19) disease

Syeda Alishwa Zanib; Tamour Zubair; Sehrish Ramzan; Muhammad Bilal Riaz; Muhammad Imran Asjad; Taseer Muhammad

doi:10.1371/journal.pone.0307707

Abstract

In this research, the ongoing COVID-19 disease by considering the vaccination strategies into mathematical models is discussed. A modified and comprehensive mathematical model that captures the complex relationships between various population compartments, including susceptible (S_α), exposed (E_α), infected (U_α), quarantined (Q_α), vaccinated (V_α), and recovered (R_α) individuals. Using conformable derivatives, a system of equations that precisely captures the complex interconnections inside the COVID-19 transmission. The basic reproduction number (R₀), which is an essential indicator of disease transmission, is the subject of investigation calculating using the next-generation matrix approach. We also compute the R₀ sensitivity indices, which offer important information about the relative influence of various factors on the overall dynamics. Local stability and global stability of R₀ have been proved at a disease-free equilibrium point. By designing the finite difference approach of the conformable fractional derivative using the Taylor series. The present methodology provides us highly accurate convergence of the obtained solution. Present research fills research addresses the understanding gap between conceptual frameworks and real-world implementations, demonstrating the vaccination therapy’s significant possibilities in the struggle against the COVID-19 pandemic.

Citation: Zanib SA, Zubair T, Ramzan S, Riaz MB, Asjad MI, Muhammad T (2024) A conformable fractional finite difference method for modified mathematical modeling of SAR-CoV-2 (COVID-19) disease. PLoS ONE 19(10): e0307707. https://doi.org/10.1371/journal.pone.0307707

Editor: Renier Mendoza, University of the Philippines Diliman, PHILIPPINES

Received: August 8, 2023; Accepted: July 10, 2024; Published: October 28, 2024

Copyright: © 2024 Zanib et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: Data sharing is not applicable to this article as no datasets were generated during the current study.

Funding: The authors received no specific funding for this work.

Competing interests: NO authors have competing interests.

1 Introduction

In the modern world, epidemics like Ebola, HIV, HBV, H₁N₁, and malaria are receiving more attention over time, and it is difficult to stop diseases from spreading among the populace. On the other hand, the globe continues to battle already-existing infectious diseases, while on the other sight, shifting global circumstances promote the birth of various viral kinds. The coronavirus shown in Fig 1, which first surfaced in early 2020 and is still not completely under control, is the newest and most potent of these viruses in recent years. While the first instances were found in Wuhan, China, on December 31, 2019, [1–3], the disease’s biological cause has not yet been fully identified. Lung disease has a high mortality rate and may be found all over the world because the World Health Organization (WHO) has declared it a pandemic. If left untreated, it can also lead to the spread of viruses that cause diseases like severe acute respiratory syndrome. The three coronavirus subgroups are alpha, beta, and gamma. SARS-CoV is a member of a fourth new class of viruses known as delta coronaviruses. In the middle of the 1960s, human coronaviruses were first discovered [3].

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Fig 1. COVID-19 transmission, symptoms and vaccination.

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In its broadest definition, mathematical modelling is an attempt to use mathematics to explain a phenomenon, an event, and the relationships between them without using mathematics, or it is the process of finding mathematical techniques within these phenomena and occurrences [4]. Through the process of modelling, mathematics is an organized way of thinking that produces answers for occurrences and issues that happen in the actual world. We see that the fundamental principles of mathematical notions have their roots in real phenomena and connections to them when mathematics is applied to the world. Modelling approaches and solution techniques for these problems need to be created because many problems, particularly today, are complicated, non-linear, have memory effects, or have stochastic structures. Although mathematical models cannot offer treatment for a specific infectious disease, they can be used to illustrate and examine potential outcomes of the current dynamics [5]. In a short period, numerous investigations on the mathematical model of the COVID-19 pandemic have been published in the literature. In 2020, Zu et al. [6] created and tested COVID-19 contagion models on the Chinese mainland as well as the efficacy of various control measures. Then, using the sensitivity analysis method, they were able to forecast the efficacy of various intervention options while effectively estimating the epidemic trend and COVID-19 transmission risk. The COVID-19 transmission in China’s final phase of the pandemic was discussed in 2020 by Tang et al. [7] analysis of the efficacy of isolation and quarantine. According to their results, effectively controlling the COVID-19 epidemic has needed further improvements to isolation and quarantine procedures as well as higher detection rates in mainland China. Uncertainty studies that highlight the continuous unpredictability of the epidemic show that these measures have been crucial in the nation’s response. It is crucial to continue working on these techniques to resolve the remaining uncertainties. In 2020, Ahmed et al. [8] used numerical methods and logistic models to analyze a mathematical model of COVID-19, they introduced and analyzed a few COVID-19 models that contain crucial queries regarding international health care and provide crucial suggestions. They suggested using Euler’s method, the second order (RK2), and the fourth order (RK4) Runge-Kutta methods to solve the given equations. In 2020, using numerical simulations, Okuonghae and Omame [9] evaluated the effects of control measures on COVID-19 dynamics, focusing on social distance, face mask use, and testing. They also produced important predictions for the total number of reported cases and the various intensities of control measures used. According to numerical simulations of the model, the disease would eventually vanish from the population if at at least 55 per cent of the population agreed with the social distance limit and roughly 55 per cent of the population utilized face masks efficiently in society. In 2021, Srivastav et al. [10] investigated the COVID-19 pandemic’s behavior in India, to evaluate the effects of the face mask, hospitalization of symptomatic patients, and quarantining of asymptomatic people. They discovered that hospitalization of symptomatic patients, isolation of asymptomatic patients, and regular use of face masks in public places were all useful strategies to decrease the impact of COVID-19 in India. In 2021 Yavuz et al. [11] developed a mathematical model to reveal the effects of vaccine treatment, which has been performed recently, on COVID-19 in this study. In their proposed model, as well as the vaccinated individuals, a five-dimensional compartment system including the susceptible, infected, exposed and recovered population was constructed. According to the research by Pearson et al. [12] in 2021 if the COVID-19 vaccine was reasonable and highly effective, that could be cost-effective even in low and middle-income populations. In 2021, Alzahrani et al. [13] developed a model by taking into consideration the environmental contributions of the latent, infected and asymptomatic infected population. The model under consideration is taken in the form of a fractional order ABC derivative. In 2023, Suganya & Parthiban [14] reviewed a mathematical model look at the quantitative analysis and dynamical behaviors of a novel coronavirus, with a focus on the Caputo fractional derivative. In 2023, Jose et al. [15] studied a deterministic mathematical model for Dengue Fever (DF) and Zika virus (ZIKV) co-infection transmission dynamics was formulated and analyzed. In 2023, Jose et al. [16] developed a mathematical model depicting the transmission dynamics of Chickenpox by incorporating a new parameter denoting the rate of precautionary measures. In 2023, Ouncharoen et al. [17] explored a nonlinear SEIR model for COVID-19 transmission dynamics, investigating its stability, reproduction number, and simulations using classical and fractional order methods. Graphical representations accompany the study’s findings. In 2024, Abdulwasaa et al. [18] addressed the intricate link between poverty and corruption by developing a mathematical model. Through linear analysis and Eviews software, indicators are examined, leading to predictions of poverty rates. The model, framed with Caputo fractional derivatives, undergoes nonlinear analysis and numerical simulations, with comparisons to real data for validation. Various mathematical models are developed to observe biological diseases using Ordinary Differential Equations (ODEs) as a framework [19–22] Jumarie defined a few basic derivative formulae for fractional calculus in [23], by proposing Modified R-L fractional derivative [24]. Afterwards, in [25–27], a few conflicts regarding Jumarie formulae were raised. So, to resolve those problems, a new definition of fractional derivatives was defined by Khalil et al., in [28]. We have used the above definition, to study the model of alcohol consumption in Spain, which is very helpful in better understanding the model. In 2013, Mickens et al. [29] examined how conservation laws restrict finite difference discretization for coupled population systems using Mickens’ nonstandard finite difference (NSFD) methodology. They identify various conservation law types and illustrate NSFD discretization through popular population models, highlighting their importance in numerical integration challenges. In 2023, Obiajulu et al. [30] analyzed a novel fractional-order mathematical model Using efficient finite difference methods, controlling the co-circulation of dengue and COVID-19, ensuring solution uniqueness via Banach’s fixed-point theorem and stability analysis around the infection-free equilibrium. Numerical solutions with the NSFD approach converge to disease-present or -free equilibrium, regardless of initial conditions or fractional orders. During the literature study, adding further compartments to the model, such as those that represent vaccinated and quarantined individuals, can produce more accurate findings. This is because both vaccination and quarantine when considered separately, have the potential to affect the disease’s spread and management significantly. The model may better reflect the dynamics of the real-world scenario and give more precise insights into the efficacy of these measures. We will also design the finite difference approach of the conformable fractional derivative using the Taylor series. This numerical method will give us the high convergence solution of the system of equations which is the main objective of this study. Section 2: This section will cover the modified COVID-19 transmission model with quarantine and vaccination class. Section 3: The reproduction number will be found by using the Next-generation Method of the modified model and also checking its sensitivity analysis. Local and global stability at disease-free equilibrium is also discussed in this chapter. We present the existence of a solution and its uniqueness. Section 4: In this section, we approximate the finite difference method of conformable derivative. After discretization, the COVID-19 model’s results and discussions will be discussed. Section 5: This section will cover the conclusion.

2 Model formulation

The provided set of equations represents a mathematical model for the dynamics of a population concerning the spread of a disease, possibly COVID-19. This model is compartmental, specifically an S_αE_αQ_αU_αV_αR_α model, where individuals are categorized into different compartments based on their health shown in Fig 2.

Susceptible (S_α) are individuals in this compartment who are not infected.
Exposed (E_α) are individuals in this compartment who have a disease-causing pathogen in their bodies but are not showing any overt clinical symptoms.
Infected (U_α) are individuals who become infectious and can spread the disease to others.
Quarantine (Q_α) those who are infected but do not have any viral symptoms.
Vaccinated (V_α) those who are vaccinated.
Recovered (R_α) whose are recovered.

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Fig 2. COVID-19 model.

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(1) The Eq (1) describes the rate of change of susceptible individuals. It includes factors such as the natural birth rate (β), the transmission from exposed to susceptible individuals (a₁E_α), those who have not been exposed to the disease (a₂), and the natural death rate (d). (2) The Eq (2) represents the rate of change of exposed individuals. It considers the transmission from susceptible to exposed individuals (a₁E_αS_α), the impact of vaccination (σV_αE_α), and factors such as the progression to active infection, quarantine, natural death, and death due to the disease (b₁U_α + b₂Q_α + d + c₁). (3) The Eq (3) describes the rate of change of individuals in the quarantine compartment. It includes terms representing the movement of exposed individuals to the quarantine compartment (b₂E_αQ_α) and the factors influencing the exit from the quarantine compartment, such as the recovery rate (η), natural death rate (d), and death due to the disease in the quarantine compartment (c₂). (4) The Eq (4) represents the rate of change of individuals in the infectious compartment. It considers the transmission from exposed to infectious individuals (b₁E_αU_α) and the factors influencing the transition out of the infectious compartment, including the recovery rate (w), natural death rate (d), and death due to the disease in the infectious compartment (c₃). (5) The Eq (5) describes the rate of change of vaccinated individuals. It includes terms representing the movement of individuals who have not been exposed to the disease to the vaccinated compartment (a₂S_α) and factors influencing the exit from the vaccinated compartment, such as the impact of exposure to the disease (σE_α) and natural death rate (d). (6) The Eq (6) represents the rate of change of individuals in the recovered compartment. It includes terms representing the movement of individuals from the quarantine compartment to the recovered compartment (ηQ_α), individuals from the infectious compartment to the recovered compartment (wU_α), and factors influencing the exit from the recovered compartment, such as natural death rate (d) discussed in Table 1.

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Table 1. Description of physical parameters.

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We will utilize the Khalil conformable derivative, as defined in [28], to explore the memory effects within the model. (7) If C is differentiable the, (8) To establish the following conformable model of COVID-19 as described in Eqs (1)–(6): (9) with initial conditions, (10)

3 Model analysis

In this section, we will comprehensively discuss the differential analysis of the system, including the invariant region, positivity of solution, disease-free equilibrium point, basic reproduction number, sensitivity analysis, local and global stability at the disease-free equilibrium point, and the existence and uniqueness of the system.

3.1 Invariant region

To find the invariant region of system of equations (9) with non-negative initial conditions (10) solution is bounded, taking total population In the absence of disease, take the derivative of concerning ρ.

We obtain (11) after solving (11) and ρ → ∞, then, (12) which is the feasible solution set of a system of equations are bounded.

3.2 Positivity of solution

Theorem 3.1. If are positive in the feasible set Ω, then the solution set, of system of equations is positive ∀ρ ≥ 0.

Proof. Taking the first equation from the system of equations, (13) after simplification, (14) similar to another system of equations. Therefore, we can say the solution set of all systems of equations is positive for ρ ≥ 0.

3.3 Disease-free equilibrium point (DFEP)

For the case, the population has no infectious individuals of COVID-19, Then disease-free equilibrium point is, (15)

3.4 Basic reproduction number

The next-generation matrix method is used to calculate the basic reproduction number R₀ [31]. To determine R₀, we first derive the transmission matrix from the system of equations (9) at the disease-free equilibrium point. (16) Next, we derive the transition matrix from the system of equations (9) at the disease-free equilibrium point. (17) We then compute the product , which reflects the overall transmission potential considering both new infections and transitions between compartments: (18) Finally, the basic reproduction number R₀ is derived from the dominant eigenvalue of : (19)

This expression for R₀ provides insight into how various parameters affect the transmission dynamics of the infection. This shows that an increase in the transmission rate β or the progression rates a₁ and σ will raise R₀, indicating a higher potential for the disease to spread. Conversely, higher recovery or transition rates d and c₁ can reduce R₀, highlighting the importance of timely interventions and effective disease management strategies, shown in Fig 3.

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Fig 3. Combined figure showing the behavior of R₀ in two different plots.

(A) Reproduction number R₀ between a₁ and a₂, (B) Reproduction number R₀ between σ and a₂, (C) Reproduction number R₀ between β and c₁, (D) Reproduction number R₀ between β and σ.

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3.5 Sensitivity analysis

The factors contributing to this disease spread and persistence in the community are examined using sensitivity analysis. Our focus is on the variables that cause a greater variance in the basic reproduction number.

Sensitivity indices of R₀.

The sensitivity index used to compute the corresponding variance in the state variable caused by the changing of a parameter. These indices have been calculated using the definition from [32]. The following definition of the sensitivity index is presented as partial derivatives: (20) The sensitivity indices of Eq (19) are given as follows, (21) (22) (23) (24) (25) (26)

The explanation above demonstrates that the basic reproduction number R₀ is most sensitive to variations. If β rises, R₀ will rise in proportionally the same way and β if falls, R₀ will fall in proportionally the same way. The link between c₁, and d is inversely proportional with R₀, therefore, an increase in either of these will result in a drop in R₀. It makes sense to concentrate on lowering these since, a₁, a₂, and σ are more susceptible to changes than R₀ redshown in Fig 4. In other words, this sensitivity analysis shows us that preventing problems is preferable to fixing them. In addition to playing an important role in controlling the transmission of a virus, it is vital to remember that vaccinations also significantly contribute to reducing the impact of the disease.

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Fig 4. Sensitivity analysis.

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3.6 Local stability of DFEP

Theorem 3.2. The disease-free equilibrium point is locally stable if R₀ < 1 and unstable if R₀ > 1 [33].

Proof. The Jacobian matrix of system of equations (9) at disease-free equilibrium point, (27) The characteristic equations are, (28) where, (29) consist of solutions to the characteristic equation. We observe that χ_1,2,3,4,5 have negative numbers, and if R₀ < 1, then it shows that χ₆ is negative, indicating that DFEP is asymptotically stable locally. The proof is now completed.

3.7 Global stability

The concept proposed by Castillo-Chavez et al., [34] may be used to state the following results, (30) where, (31) When the system is at the DFEP, the infected and uninfected populations are represented by the values and , respectively. The condition for global stability at the DFEP in the epidemiological model is given by: (32) (33)

Theorem 3.3. The system of equations (9) are globally asymptotically stable if R₀ < 1 at DFE point.

Proof. To prove condition (32), the model (9) can be set by, the disease-free equilibrium point is given, (34) and the system, (35) (36) By solving Eq (36), the equation has a unique equilibrium point, (37) hence, for the condition (32) X⁰ is globally asymptotically stable is satisfied. Now, to verify the second condition (33). (38) and, (39) (40) (41) (42) this shows that, (43) This result proves the conditions (32) and (33), indicating that the system is globally asymptotically stable when R₀ < l at the DFEP. This completes the proof of Theorem 3.3.

3.8 Existence and uniqueness of solution

In differential calculus, the existence and uniqueness of solutions are crucial, as emphasized in numerous studies in [35–37]. In this section, we demonstrate the existence of solutions for the non-linear system of equations in the COVID-19 model (9) using fixed point theory with results proved in [35].

For non-linear system, (44) Now, let’s start the procedure (45) Now, we define the kernels (46)

Theorem 3.4. If the following inequality in proven, then the kernels Λ₁, Λ₂, Λ₃, Λ₄, Λ₅ and Λ₆ satisfy the Lipschitz assumptions and contractions. (47) where ∥S_α ∥ ≤ k₁, ∥ E_α ∥ ≤ k₂, ∥ Q_α ∥ ≤ k₃, ∥ U_α ∥ ≤ k₄, ∥ V_α ∥ ≤ k₅, ∥ R_α ∥ ≤ k₆, p₁ = a₁k₂ + a₂ + d, p₂ = a₁k₁ + σk₄ + b₂k₃ + b₁k₄ + b₂k₃, p₃ = b₂k₂, p₄ = b₁k₂, p₅ = σk₂, p₆ = d.

Proof. Consider S_α1 and S_α2 are two functions for the kernel Λ₁, then (48) k₁ = ∥S∥ is bounded function of p₁ then, (49) when E₁ and E₂ are two bounded functions for the kernel λ₂, then similarly, (50) when Q₁ and Q₂ are two bounded functions for the kernel λ₃, then similarly, (51) when U₁ and U₂ are two bounded functions for the kernel Λ₄, then similarly, (52) when V₁ and V₂ are two bounded function for the kernel Λ₅, then similarly, (53) when R₁ and R₂ are two bounded functions for the kernel Λ₆, then similarly, (54) therefore the Λ₁, Λ₂, Λ₃, Λ₄, Λ₅, Λ₆ satisfy the Lipschitz conditions.

If 0 ≤ p₁, p₂, p₃p₄, p₅, p₆ < 1, then p₁, p₂, p₃, p₄, p₅ and p₆ also contraction for Λ₁, Λ₂, Λ₃, Λ₄, Λ₅, Λ₆ respectively. This is the proof of this theorem.

Now consider the kernels Λ₁, Λ₂, Λ₃, Λ₄, Λ₅, Λ₆ and rewrite the system of equations, (55) Now proceed with the recursive formula, which is as follows, (56) where, (57) It can also be written in sequential term differences which are as follows, (58) this system of equations implies that, (59) Now, we take both sides of the system of equations, then kernels satisfy the Lipschitz condition. Now triangle inequality applies to a system of equations, then we have, (60) we have, (61) The following theorem may be derived from these findings.

Theorem 3.5. The modified COVID-19 model offers a solution under the condition that can be formed τ_max property, (62) Proof. Consider the function S_α(ρ), E_α(ρ), Q_α(ρ), U_α(ρ), V_α(ρ) and R_α(ρ) are the bounded and having the kernels Λ₁, Λ₂, Λ₃, Λ₄, Λ₅, Λ₆ satisfied the Lipschitz condition. We apply the recursive method to a system of equations, (63) so this is the solution of the COVID-19 model, we suppose that, (64) It is shown that the term in Eq (64) hold, ∥Z1_n(ρ)∥ → 0, ∥Z2_n(ρ)∥ → 0, ∥Z3_n(ρ)∥ → 0, ∥Z4_n(ρ)∥ → 0, ∥Z5_n(ρ)∥ → 0, ∥Z6_n(ρ)∥ → 0, so we have, (65) Similarly for others, (66) (67) (68) (69) and, (70) apply recursive relation, then we obtain (71) taking at τ_max point, we get (72) as r → ∞ apply both sides, then using the result of theorem 3.4, then we get,

Theorem 3.6. if (73) then modified COVID-19 model has a unique system of solutions.

Proof. Suppose different system of solution such as , then it may write, (74) Apply norm on both sides (74) and results of kernels which fulfil the Lipschitz condition. We can write it as, (75) then, (76) consequently, (77) This shows that the model has a unique solution. which is the complete proof of the theorem.

4 Numerical simulations

In this section, we explore the numerical results for the model we developed, considering various vaccination strategies. Furthermore, the essential assumption of the FDM is based on Taylor’s theorem proved from [38], which asserts the following: (78) For the fractional case, we have: (79) Where shows the k^th order derivative. For conformable fractional derivatives, we have (80)

4.1 FDM approximations for conformable fractional derivatives

Using the fractional order approximations for the forward finite difference method as discussed in [39, 40] and based on the results mentioned in Eq (80), we derive the following: (81)

Theorem 4.1. The relationship between classical and fractional order derivatives using the finite difference method can be expressed by the generalized recurrence relation presented in Eq (81), (82) Proof: (83) Using fractional order Taylor series, (84) From Eq (83), we get (85) From Eq (84), we get (86) Comparing Eqs (85) and (86), we get After rearranging, we get (87) or also can be written as, (88) Applying this method to the system of equations (9), we obtain: (89) Now, to establish the stability, consistency, and convergence of the system, the analysis will be based on the novel scheme.

4.2 Stability analysis

According to [41], in order to establish the stability of first equation from system (89), we initially simplify the analysis by neglecting certain factors, (90) After simplification, we have Where and R₂ = a₁E_α(n)+ a₂+ d. Then, Equation (4.2) yields: This shows that first equation of system (89) is stable. Similarly, we can prove other equations of system (89).

4.3 Consistency

As the grid interval and time step size approaches zero, the truncation error vanishes. Consistency, as discussed in [41], assesses the accuracy with which the finite difference method approximates the ordinary differential equation (ODE). We consider first equation from system (9) to be: (91) Using the Taylor series, we have After Substitution in Eq (91), we have (92) Therefore, the truncation error for the Eq (91) is as follows: The truncation error τ → 0 when Δρ → 0. Similarly, we can prove other equations of system (89).

4.4 Convergence

Using the Lax-Richtmyer Equivalence theorem mentioned in [42], Then governing equations of the mathematical model are convergent.

4.5 Graphical behavior

Now, we have plot the system of equations (89) using the parameters values mentioned in Table 2.

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Table 2. Values of parameters.

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This figure illustrates the plot of the variable S_α(ρ), which likely represents the number of susceptible individuals in society over time ρ (days), for different values of the parameter ϕ = 0.4, 0.6, 0.8, and 1. Fig 5 shows the dynamic change in the number of susceptible individuals over a period of 40 days. The results indicate that the number of susceptible individuals is increasing, suggesting a decreasing chance of infection. As ϕ increases, this decrease in the infection rate becomes more pronounced. For lower values of ϕ, the decrease is slower, indicating that the influence of fractional derivatives significantly affects the transition rate from susceptible to exposed or other compartments.

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Fig 5. Susceptible individuals.

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The variable E_α(ρ) seems to be plotted in this figure for various values of the parameter ϕ=0.4,0.6,0.8,1, which indicate the number of exposed people in society throughout a period ρ(days). Fig 6 shows the dynamic change in Exposed Individuals in 40 days. The number of exposed individuals initially increases, peaks, and then decreases. Higher ϕ values lead to a faster rise and fall in the exposed population. This suggests that with higher ϕ, exposed individuals either progress to other compartments (like quarantined or infected) more quickly or recover faster.

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Fig 6. Exposed individuals.

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For various values of the parameter ϕ=0.4,0.6,0.8,1, this figure appears to be an illustration of the variable Q_α(ρ), which may indicate the number of Quarantined individuals in society during a period ρ(days). Fig 7 shows the dynamic change in Quarantined Individuals in 40 days. The quarantined population increases initially and then decreases. Higher Φ values result in a quicker increase in the quarantined population. This indicates that the fractional derivative parameter accelerates the rate at which exposed individuals are moved to quarantine.

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Fig 7. Quarantined individuals.

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Fig 8 shows the dynamic change in Infected Individuals in 40 days. For various values of the parameter ϕ=0.4,0.6,0.8,1, this figure appears to be an illustration of the variable U_α(ρ), which may indicate the number of Infected individuals in society during a period ρ (days). After an initial rise, the infected population begins to decrease due to vaccination. Increased ϕ values lead the infected population to grow and stabilize more quickly than the quarantined population. This demonstrates how the onset of infection speeds up by fractional derivatives.

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Fig 8. Infected individuals.

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For various values of the parameter ϕ=0.4,0.6,0.8,1, this figure appears to be an illustration of the variable V_α(ρ), which may indicate the number of Vaccinated individuals in society during a period ρ(days). Fig 9 shows the dynamic change in Vaccinated Individuals in 40 days. These results show that the vaccinated population starts at zero and increases over time so we can say that the chance of infection is reduced. Higher ϕ values result in a more rapid increase in the vaccinated population. This suggests that fractional derivatives enhance the vaccination rate or the effectiveness of vaccination over time.

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Fig 9. Vaccinated individuals.

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This figure appears to be a plot of the variable R_α(ρ), possibly representing the number of Recovered individuals in society over sometime ρ(days), which gives more efficient results for different values of the parameter ϕ=0.4,0.6,0.8,1. Fig 10 shows the dynamic change in Recovered Individuals in 40 days. These results show that the rate of Recovered people is continuously increasing so we can say that the chance of infection is reduced, and the situation is going to be under control. Higher Φ values lead to a quicker rise in the recovered population, indicating faster recovery rates when the influence of fractional derivatives is stronger.

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Fig 10. Recovered individuals.

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4.6 Discussion

In our model, conformable fractional derivatives play a pivotal role by incorporating fractional-order dynamics, which blend classical and modern mathematical approaches. The versatility of this derivative lies in its capacity to converge to the solutions of the classical model (1) as the fractional parameter ϕ → 1. This feature highlights the adaptability of the fractional model, ensuring consistency and continuity across different models. Moreover, the high convergence demonstrated by our finite difference technique, as illustrated in Table 3, underscores its efficacy and reliability in computational simulations, especially evidenced by the substantial reduction in error with increasing discretization levels (N). Furthermore, Simulations were conducted using the Runge–Kutta fourth-order method which reveal a direct correlation between vaccination rates and infection rates, as depicted in Fig 11. The inverse relationship observed suggests that as vaccination rates increase, infection rates correspondingly decrease. This correlation underscores the pivotal role of vaccination strategies in controlling and mitigating the spread of infectious diseases. It highlights the essential role that vaccination programs perform in preventing the spread of disease and emphasizes their significance as an essential component of public health campaigns.

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Fig 11. RecBehavior of COVID-19 model at integer order derivative.

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Table 3. Comparison of numerical scheme accuracy across different time levels N for various values of ϕ.

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5 Conclusion

In conclusion, our work presents a thorough examination of how vaccine treatment affects COVID-19 dynamics. We have better understood the interaction between vaccination, disease transmission, and control measures by combining a six-dimensional compartmental model and using conformable derivatives. Our findings emphasize the significance of including vaccination and quarantined strategies in mathematical models since they considerably impact the disease’s overall dynamics and the fundamental reproduction number (R₀). The stability studies of the above equilibrium points have been examined in the following, and it has been demonstrated that the DFE is asymptotically stable when R₀ < 1 and unstable when R₀ > 1. We designed a finite difference approach for the conformable fractional derivative using the Taylor series achieving a highly convergent solution for the system of equations. We have calculated the efficiency of vaccination in preventing the spread of COVID-19 using careful mathematical simulations and sensitivity analysis. The boundedness, positiveness, and positiveness of the solutions have all been established. We have demonstrated the existence and uniqueness of the solutions using the Lipschitz condition. Future work may explore using another fractional derivative on a modified mathematical model of experimental data.

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Figures

Abstract

1 Introduction

2 Model formulation

3 Model analysis

3.1 Invariant region

3.2 Positivity of solution

3.3 Disease-free equilibrium point (DFEP)

3.4 Basic reproduction number

3.5 Sensitivity analysis

Sensitivity indices of R0.

3.6 Local stability of DFEP

3.7 Global stability

3.8 Existence and uniqueness of solution

4 Numerical simulations

4.1 FDM approximations for conformable fractional derivatives

4.2 Stability analysis

4.3 Consistency

4.4 Convergence

4.5 Graphical behavior

4.6 Discussion

5 Conclusion

References

Sensitivity indices of R₀.