key: cord-0803654-lbvxugtj
authors: Panwar, Virender Singh; Sheik Uduman, P.S.; Gómez-Aguilar, J.F.
title: Mathematical Modeling of Coronavirus Disease COVID-19 Dynamics using CF and ABC Non-Singular Fractional Derivatives
date: 2021-02-04
journal: Chaos Solitons Fractals
DOI: 10.1016/j.chaos.2021.110757
sha: 6c5f982fdd517bc6b66bc32af9653447b58d73bf
doc_id: 803654
cord_uid: lbvxugtj

In this article, Coronavirus Disease COVID-19 transmission dynamics were studied to examine the utility of the SEIR compartmental model, using two non-singular kernel fractional derivative operators. This method was used to evaluate the complete memory effects within the model. The Caputo–Fabrizio (CF) and Atangana–Baleanu models were used predicatively, to demonstrate the possible long–term trajectories of COVID-19. Thus, the expression of the basic reproduction number using the next generating matrix was derived. We also investigated the local stability of the equilibrium points. Additionally, we examined the existence and uniqueness of the solution for both extensions of these models. Comparisons of these two epidemic modeling approaches (i.e. CF and ABC fractional derivative) illustrated that, for non-integer [Formula: see text] value. The ABC approach had a significant effect on the dynamics of the epidemic and provided new perspective for its utilization as a tool to advance research in disease transmission dynamics for critical COVID-19 cases. Concurrently, the CF approach demonstrated promise for use in mild cases. Furthermore, the integer [Formula: see text] value results of both approaches were identical.

mitigating public health risks [12] [13] [14] [15] [16] [17] [18] . In epidemiology, deterministic mathematical models have an important function in investigating the dynamics of infectious diseases. Very recently, Srivastava et al. [19, 20] investigated fractional calculus and fractal-fractional calculus applications in modeling the dynamics of the Ebola virus with three different kernels using numerical simulations; also we can see the fractional models of Diabetes in [21] . Whereas, a comparative study of the fractional clock chemical model's numerical results (with a finding of an exact result) was carried out in [22] . Recently published articles have been formulated on the transmission dynamics of COVID-19. However, current research is mainly restricted to classical integer-order models, logistic models and delay or stochastic differential equation models [23] [24] [25] [26] [27] [28] [29] [30] . Recently, the AtanganaBaleanu fractional derivative was applied in order to create a mathematical model for the dynamics of the novel corona virus in [31] and pointed out the need for evaluating the comparison with other derivative operators.

Often, in epidemiological analysis, diseases caused by viruses grow exponentially with a fixed reproduction rate. Similarly, the growth curve for the number of COVID-19 diagnoses outside China are exponential, as demonstrated by [32, 33] . The following research [34] supports the likelihood that due to various prevention measures, growth decreased exponentially. Until present, research has not yet explored comparisons of the CF and ABC fractional derivative operators against the transmission dynamics of the COVID-19 SEIR model, which has non-singular kernels, the memory effects, crossover properties and other important properties when compared to the integer order derivative. The findings above make the CF fractional derivative and ABC fractional derivative promising options for use as a mathematical tool in modeling the transmission dynamics of COVID- 19. In the following sections, the COVID-19 SEIR standard incidence compartmental ODE model is extended to the fractional differential equation model. This is accomplished by using two different non-singular kernel fractional derivatives, both CF and ABC fractional derivative operators. This approach is used to study the trajectory of transmission of infection. The expression for the basic reproduction number (using the next generating matrix method) is derived and local stability of the equilibrium points are analyzed. The existence and uniqueness of both COVID-19 fractional models are proved via the fixed-point theorem. We also analyze the changing trends of the COVID-19 pandemic with graphs for numerical simulations of both CF and ABC approach models. Considering the timeline for disease onset to clinical recovery and severity of illness, we discuss the suitability of using both operators, accompanied by relative similarities, as well as their differences. Lastly, we conclude the article by analyzing the importance of disease transmission models and discuss their future implications for disease control policy development at both national and International levels to interrupt or minimize transmission chains in humans.

In this section, we present the basic definitions of the new fractional derivatives with exponential and Mittag-Leffler kernels.

, then the Caputo-Fabrizio (CF) fractional derivative is defined as [35] :

where F(τ ) is a normalization function satisfying

Definition 2. Let 0 < τ < 1, then the integral of the fractional order τ for a function Υ(t) is defined as [35, 36] : 

where E τ represent the Mittag-Leffler function and B(τ ) is a normalization function satisfying

[37] Let 0 < τ < 1, then the integral of the Atangana-Baleanu derivative in the Caputo sense, fractional order τ for a function Υ(t) is defined as:

Considering the transmission method is human to human, we chose a model for the transmission dynamics of COVID-19 based on the SEIR compartmental model (susceptible exposed infected recovery model) with standard incidence [38] . Assuming all individuals are susceptible, and that infected individuals can spread the disease, this model was represented via the nonlinear system below.

The equations above created four compartments, for a total number of individuals labelled at time t. As such, these were categorized as (S(t)) susceptible individuals whom were not considered infected but as having the capacity to become infected, (E(t)) exposed individuals are those who have been infected but are not infectious, (I(t)) infectious individuals are those who are symptomatic and clinically tested. Lastly, (R(t)) removed individuals are those who have recovered from the disease. The total population was denoted by N such that N = S(t) + E(t) + I(t) + R(t) was considered constant with an assumption of birth rate represented as n, and death rate as m. All parameters were positive constants described as (Π = nN ) rate of recruitment of individuals, (β) transmission rate of the disease upon contact with symptomatic infected individuals. Moreover, the parameter (ψ −1 ) represented the latency period of 3 to 5 days and (γ −1 ) represented the infectious period within the range of 2 to 14 days [39] .

The ODE model for COVID-19 was extended to the fractional order model using the CF derivative below:

Similarly, the ODE model for COVID-19 was extended to the fractional order model with Atangana-Baleanu in the Caputo sense (ABC) fractional derivative provided below:

The initial conditions involved throughout the analysis were

Reproduction number is a crucial figure within the mathematical analysis of any disease model:-it aids in determining if an epidemic will likely occur. The reproduction number R 0 of the model represents the anticipated sum of infectious cases generated by one infectious individual within a population of susceptible individuals. The value of R 0 for the aforementioned models, utilized the next-generation matrix method given by Van Den Driessche and Watmough [40] . Findings indicated the relevant Jacobian matrices F and V were associated with the rate of appearance of new infections and with net rate out of the corresponding compartments, respectively were given by

Thus, we find

solving for eigen values of B we find,

and λ 2 = 0.

The reproduction number R 0 is given by the dominant eigenvalue. precisely,

If R 0 < 1 then the disease would likely self-terminate. However, if R 0 > 1 then the disease would likely prevail and become a pandemic if containment procedures are not initiated. We found two biologically meaningful equilibria. One was disease-free equilibrium T 0 and an endemic equilibrium (EE)T 1 .

The disease-free equilibrium (DFE) T 0 of the system (7), (8) was found by taking zero value for the derivatives side, considering there are no exposed individuals.

Thus, by substitution in the aforementioned systems we found,

The endemic equilibrium points T 1 were derived by considering a population of infected individuals and all equations of models (7), (8) are equal to zero. Effectively denoted interms of R 0 as

In this subsection we worked on the stability analysis of the COVID-19 model (7, 8) .

The DFE T 0 of the COVID-19 model (7, 8) was locally asymptotically stable if and only if R 0 < 1.

Proof. For the proof, we obtained at DFE T 0 , the Jacobian matrix below,

The characteristic equation for the Jacobian matrix mentioned above is the form:

For R 0 < 1 the P (λ) equation has all positive coefficients and by the criteria of Routh-Hurwitz for the second order polynomial a i > 0 for i = 0, 1, 2. The DFE T 0 of the COVID-19 model (7, 8) is locally asymptotically stable for R 0 < 1.

Theorem 2. The EE T 1 of the COVID-19 model (7, 8) is locally asymptotically stable for R 0 > 1 and unstable for R 0 < 1.

Proof. For the proof, we obtain at EE T 1 , the Jacobian matrix below,

The characteristic equation for the Jacobian matrix mentioned above is the form:

where,

It is clearly shown that for R 0 > 1 the above equation has all positive coefficients by the criteria of Routh-Hurwitz for the third order polynomial a 1 a 2 > a 3 for i = 0, 1, 2, 3. The EE T 1 of the COVID-19 model (7, 8) is locally asymptotically stable for R 0 > 1 and unstable for R 0 < 1.

In this article, we have taken a nonlinear fractional order model into consideration. The following section focuses on investigation of both the existence and uniqueness of the solution concerning the aforementioned model (7) by applying the fixed-point theory.

The CF model is represented by Eq. (7) in applying the fractional integral [36] , our findings include the following;

The notations [36] , when organized within the context of the problem yield the following results;

We interpret for clarity;

Theorem 3. If the inequality given below holds

then, the kernel C 1 justifies the Lipschitz condition and contraction.

Proof. Assume S and S 1 are two functions, with that we have

Applying the triangular inequality on Eq. (28) gives

Let

Hence, the Lipschitz condition is obtained for kernel C 1 and 0 ≤ βa N + m < 1 provides C 1 , which satisfies contraction.

Similarly, for the other kernels the Lipschitz condition and contraction can be performed and written as shown;

Taking the aforementioned kernels Eq. (26) becomes

We focus on the following recursive formulae, given as

Along with

the initial conditions. The subsequent expressions for the difference of successive terms are expressed as:

It is obvious that

Applying the norm on Eq.(35), yields;

Applying the triangular inequality, Eq.(37) reduces to

Since, the kernel C 1 satisfies the Lipschitz condition, we have

Therefore, we conclude;

Similarly, we gain the following results

Considering the above results, we present the following theorem:

Theorem 4. The COVID-19 fractional model (7) has a solutions if we can find a t 0 , such that:

Proof. Considering Eqs. (40) and Eqs. (41), along with the fact that functions S(t), E(t), I(t) and R(t) are bounded and the kernels justify the Lipschitz condition. We have the following relation employing the recursive method:

Thus, we demonstrate solutions exist and also satisfy continuity, for the COVID-19 model (7) . For the sake of clarity, the above functions are the solution of Eq.(7). We suppose

Therefore, we get

After repeating same process, then at t 0 we obtained

As n approaches infinity, taking limit on Eq.(45) we get ||G n (t)|| → 0. Similarly, we find ||H n (t) → 0||, ||J n (t) → 0|| and ||K n (t) → 0|| .

For clarity on uniqueness for the solutions of the model (7), assuming S 1 (t), E 1 (t), I 1 (T ) and R 1 (t) are a distinct set of solutions pertaining to Eq.(7), then

Considering the fact that kernel satisfies the Lipschitz condition and taking norm on Eq.(46), prompts to the inequality given below:

Theorem 5. If the following inequality holds

then a unique solution for the COVID-19 model (7) exists.

Proof. With the condition that (47) holds, taking

So, we obtain

Then,

In the same manner, we gain

Which verifies the proof for uniqueness of the solutions for COVID-19 model (7).

In this section we will prove the existence and uniqueness of solution for model (6) with ABC fractional derivative operator represented by Eq. (8) .

Implementing the fractional integral to both sides of Eq.(8), the model can be written as follows:

The notations [37] , when organized within the context of the problem yield the following results;

We interpret for clarity;

Theorem 6. If the inequality given below holds

then, the kernelC 1 justifies the Lipschitz condition and contraction.

Proof. Assume S and S 1 are two functions, with that we have

Applying the triangular inequality on Eq. (55) gives

where ||S(t)|| ≤s, ||R(t)|| ≤r , ||I(t)|| ≤ā and ||E(t)|| ≤b are bounded functions, we get

Hence, the Lipschitz condition is obtained for kernelC 1 and 0 ≤ βā N + m < 1 pro-videsC 1 , which satisfies contraction.

Similarly, the other kernelsC 2 ,C 3 andC 3 satisfy the Lipschitz condition and contraction.

Taking the aforementioned kernels Eq.(53) becomes

We focus on the following recursive formulae, given as

Along with initial conditions:

The subsequent expressions for the difference of successive terms are expressed as:

It is obvious that

R n (t) = n i=1η i (t).

(62)

Applying the norm on Eq.(61), yields;

Applying the triangular inequality, Eq.(63) reduces to

Since, the kernelC 1 satisfies the Lipschitz condition, we have

Therefore, we conclude;

Similarly, we gain the subsequent results

Considering the above results, we present the following theorem:

The COVID-19 fractional model (8) has a solutions if we can find a t 0 , such that:

Proof. Considering the Eqs. (66) and (67), along with the fact that functions S(t), E(t), I(t) and R(t) are bounded and the kernels justify the Lipschitz condition. We have the following relation employing the recursive method:

We demonstrate solutions exist and also satisfy continuity, for the COVID-19 model (8) .

For the sake of clarity, the above functions are the solution of Eq.(8). We suppose

Therefore, we get

After repeating same process, then at t 0 we obtained

As n approaches infinity, taking limit on Eq.(71) we get ||Ḡ n (t)|| → 0. Similarly, we find ||H n (t) → 0||, ||J n (t) → 0|| and ||K n (t) → 0|| .

For clarity on uniqueness for the solutions of the model (8), assuming S 1 (t), E 1 (t), I 1 (T ) and R 1 (t) are a distinct set of solutions pertaining to Eq.(8), then

Considering the fact that kernel satisfies the Lipschitz condition and taking norm on Eq.(72), prompts to the inequality:

Theorem 8. If the following inequality holds

then a unique solution for the COVID-19 model (8) exists.

Proof. With the condition that (73) holds, taking

So, we obtain

Then,

In the same manner, we gain

Which verifies the proof for uniqueness of the solutions for COVID-19 model (8).

In this section, we provide the numerical simulations using Matlab for both CF approach and ABC approach model mentioned in Eq. (7) and Eq.(8) respectively. Next, we compare and discuss the following results.

Caputo-Fabrizio sense.

Using Wuhan City as an example, we created the following framework: Π = 0.36, β = 2.5 − 5, m = 0.30, ψ represents the mean latency period of COVID-19 in humans, considering 14 days, ψ = 1/14, γ represents the infectious period in days. (7) and ABC model (8) demonstrates that for τ = 1 with same the initial conditions and parameter values gives identical output. Whereas, for the same non integer values of τ , both models show dissimilar trajectories. Thus, observation finds that smaller τ susceptible populations decrease at a slower rate with the ABC approach compared to the CF approach. It is clearly visible from graph Figs.1(b)-1(c) and Figs.2(b)-2(c) comparisons that the ABC approach provides comparatively more variation in both infected and recovered individuals than the CF approach, which is more likely to suit real data. Another important aspect considers that exposed and infected individuals show a sharp increase for all values of τ due to the high transmissibility of COVID-19 of the disease advocated by [26] . Furthermore, upon deep consideration of the comparisons indicated in graph Fig 1(c) with Fig 1(d) we find a difference of approximately 2 weeks between the peak of infected and recovered individuals. However, notably in Fig 2(c) with Fig 2(d) the difference between the peak points is 3 weeks, calculated at τ = 0.7. The slope of curves for other values of τ share similarities with the findings. This indicates that infected individuals are recovering very quickly, with a delay of approximately 2 weeks, shown by the CF approach, which is valid for mild cases. Conversely, the ABC approach shows approximately a 3-week delay in the transfer of infected individuals to recovered compartments. This is valid for severe or critical disease states, as per the WHO-China Joint Mission on Coronavirus Disease 2019 (COVID-19) report [41] . The effectiveness of both models, when compared, found notable differences under identical parameter values. This is due to the memory properties of the kernel in the definitions of the fractional operators. The Caputo-Fabrizio derivative has an exponential kernel, whereas the ABC approach uses a generalized Mittag-Leffler kernel. The latter shows a partial exponential decay memory, and also power-law memory (see [42, 43] ). It is clear from the above simulation graphs that the model relies upon the fractional order remarkably, for various values of τ it displays a clear difference. The suggested model explores new aspects at the fractional value of τ , which is inappreciable for the model at τ = 1.

Disease-relevant contact increases with an escalation in population size. Thus, Coronavirus disease COVID-19 transmission non-integer order model is considered, using the CF derivative and ABC derivative (with standard incidence) are formulated. Expression for the basic reproduction number along with equilibrium points and their local stability are analyzed. The uniqueness and existence is verified by employing the fixed-point theorem for both CF and ABC models. Numerical simulation graphs for the proposed COVID-19 fractional order models are shown with distinct fractional order values τ ∈ (0, 1] and briefly compared, discussed and investigated. The graphical results demonstrate the CF approach provides better suitability for mild cases (studies suggest approximately 80% of patients have had mild disease). Whereas, the ABC approach provides superior and more flexible results for critical cases. These results show that CF and ABC approach implementation in real life situations are both plausible and doable as per the severity of illness for patient management. For future research work we propose COVID-19 spread for different geographical areas can be achieved by examining the models with relevant parameter values as per data trends of the region. We anticipate this research will provide significance and will thus strengthen the research relevant to COVID-19 transmission dynamics, so that progressive disease control policies are formulated to provide patients with better medical care for all in need. All persons who meet authorship criteria are listed as authors, and all authors certify that they have participated sufficiently in the work to take public responsibility for the content, including participation in the concept, design, analysis, writing, or revision of the manuscript. Furthermore, each author certifies that this material or similar material has not been and will not be submitted to or published in any other publication before its appearance in the Chaos, Solitons & Fractals 

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José Francisco Gómez Aguilar acknowledges the support provided by CONACyT: cátedras CONACyT para jóvenes investigadores 2014 and SNI-CONACyT.

The authors declare no conflict of interest.