key: cord-0690699-2xvm5t0d
authors: Zeb, Anwar; Kumar, Pushpendra; Erturk, Vedat Suat; Sitthiwirattham, Thanin
title: A new study on two different vaccinated fractional-order COVID-19 models via numerical algorithms
date: 2022-02-17
journal: J King Saud Univ Sci
DOI: 10.1016/j.jksus.2022.101914
sha: dc09cba17b356f9f3ace444ab631ed85e2e29716
doc_id: 690699
cord_uid: 2xvm5t0d

The main purpose of this paper is to provide new vaccinated models of COVID-19 in the sense of Caputo-Fabrizio and new generalized Caputo-type fractional derivatives. The formulation of the given models is presented including an exhaustive study of the model dynamics such as positivity, boundedness of the solutions and local stability analysis. Furthermore, the unique solution existence for the proposed fractional order models is discussed via fixed point theory. Numerical solutions are also derived by using two-steps Adams-Bashforth algorithm for Caputo-Fabrizio operator, and modified Predictor-Corrector method for generalised Caputo fractional derivative. Our analysis allow to show that the given fractional-order models exemplify the dynamics of COVID-19 much better than the classical ones. Also, the analysis on the convergence and stability for the proposed methods are performed. By this study, we see that how the vaccine availability plays an important role in the control of COVID-19 infection.

Throughout this pandemic known as COVID-19, we have experimented a great expansion of cases throughout the world. This situation converts into solid actions that affect the population: social isolation, use of masks, etc. Mathematical models play a key role for describing infectious diseases such as COVID-19 expansion. The development and investigation of this type of models provide us tools for describing and characterizing its transmission, and thus, we are able to propose successful techniques to foresee, prevent, and control infections, also to ensure that population is well-being. Till present time, numerous mathematical models see [1, 2, 3, 4, 5] have been considered and analyzed to ponder the spreading of infections. COVID-19, has affected nearly 90% countries across the globe with the infection rate rising rapidly at almost 5% per day. However, the COVID-19 infection behavior is different from nation-to-nation, and is depended on numerous factors. In South Africa, with no exception, almost half a million positive cases have been reported already and is currently one of the five most affected countries globally. To date, various mathematical models have been applied to predict infection rates based on only time-series modes [6, 7] . Very few studies attempted to include other related factors to enhancing the modeling process such as the influence of climatic factors for the disease rapid spread. In the last year, numerical models for COVID-19 plague have been taken into consideration by many scientists with respect to Theorem 1. [23] Let M be a compact metric space and C(M, R) denotes the space of continuous functions when endowed with the supremum norm metric. A set E ⊂ C(M) is compact if and only if E is bounded, closed and equicontinous. Definition 2. [29] The modified Caputo fractional derivative operator, D κ,σ d + , of order κ > 0 is given by:

where σ > 0, d ≥ 0, and n − 1 < κ ≤ n.

Theorem 2. [29] Let n − 1 < κ ≤ n, σ > 0, a ≥ 0 and g ∈ C n [a, b]. Then, for a < t ≤ b,

x=a .

(2)

In order to formulate our COVID-19 model with influence of quarantine class and vaccination, we split the whole population into four different compartments. The first of them is the class of susceptible to disease which is represented as S t , second one is infective or infectious I t , third one is quarantined Q t (in which the infectious peoples are putting for isolation), and last one is the recovered class R t with temporary immunity. The flow of the population is described in the following system of differential equations:

where b describes the enroll rate of the population that directly joins the susceptible class S t , β stands for the contact rate mainly incidence rate at which susceptible class joins infectious class I t , d denotes the out going rate of each class in the form of natural death or migration rate from each class, γ is the recovered rate of infected class to join recovered class R t and ρ is the recovered rate of quarantine class people. Moreover, σ 1 and σ 2 are the disease related deaths rates for infected class and quarantined class, δ shows the relapse rate at which the recovered class R t moves to susceptible class and q represents the vaccine rate, that is, the proportion of the susceptible class that becomes vaccinated with 0 ≤ q ≤ 1. To simulate the past history or hereditary characteristics in the given model (3), we utilized the Caputo-Fabrizio (CF) fractional derivatives instead of classical derivatives. In this matter, we propose the following model of fractional order type

where 0 < κ < 1 and CF c D κ t presents the fractional derivative in the Caputo-Fabrizio sense. For generating more diversity in the fractional-order simulations, we propose another fractional order model in the sense of generalized version of Caputo-type fractional derivative as follows:

where 0 < κ < 1, σ > 0, and C D κ,σ t prsents the fractional derivative in the generalized (or modified) Caputo sense.

Suppose that R 4 + = (S, I, Q, R)|S, I, Q, R ≥ 0 . From [24] and utilizing a generalized mean value theorem and a fractional comparison principle, the proof of the following theorem is achieved. We state the analysis for the Caputo-Fabrizio fractional model (4) and it is straightforward to obtain the corresponding analysis for the generalised Caputo one(5).

Theorem 3 (Positivity and boundedness). Let (S 0 , I 0 , Q 0 , R 0 ) be any initial data belonging to R 4 + and S t , I t , Q t , R t , the corresponding solution of model (4) to the given initial data. The set R 4 + is positively invariant. Furthermore, we have lim sup

For all t ≥ 0, with the help of generalized mean value theorem [24] and system (7), we can conclude that S t , I t , Q t , R t ≥ 0. First equation of system (4) implies that

By utilizing the fractional comparison principle, it follows that lim sup

Second equation of the system (4) implies that

which implies that lim sup

In a result, the second estimate of (6) is obtained. While third equation of the system (4) gives us

for enough large value of t. This follows the third estimate of (6). Finally, the fourth equation of system (4), implies that

for enough large value of t and the fourth estimate of (6) holds.

Diseases Free Equilibrium (DFE) point of system (3) is given by

For the reproductive number of model (3) , suppose that y = (I t , S t ) and using next generation matrix approach [3] , we have

where Jacobian of F and V are

At E 0 , we have

Hence, reproductive number for model (3) is

.

The results about the positive endemic equilibrium point are contained in the next theorem. Proof. Endemic equilibrium point (EEP) is obtained from system (3), by putting right hand side of each equation equal to zero, we have

Now, from last equation of system (13), we have

By the values of S * , I * , Q * and R * , it is clear that a unique EEP E * exists, if ψ 0 > 1.

Theorem 5. The model (3) is locally stable at E 0 for ψ 0 < 1 and unstable for ψ 0 > 1.

Proof. The Jacobian of model (3) is

Along E 0 , it implies that

which follows that all the eignvalues are negative if ψ 0 < 1 and eigenvalue λ 2 is positive for ψ 0 > 1. Hence, we conclude that the system (3) is locally stable under the condition ψ 0 < 1 and unstable for ψ 0 > 1.

Theorem 6. The model (3) is globally stable, if ψ 0 > 1 at E 0 .

Proof. First, we define the Lyapunov function V(t), for the system as:

Then differentiating the equation (17) with respect to time, we have

By manipulating along the point E 0 , we get

Remark 1. The simulations of stability of E * is an important mathematical term, but in this paper, we particularly focus on the case ψ 0 < 1 to find effective manners to prevent the epidemic.

Since last few years, a lot of work has been done in the field of existence of solution for different types of fractional differential equations by using techniques from fixed point theory. In order to fulfill this requirement for the proposed model, we use the procedure which has been recently proposed by Verma et al. in [23] . For this purpose, we rewrite our model in a compact form given by:

Now the above system (18) converts to the following fractional Volterra integral form when we apply CF integral operator on it of order 0 < κ < 1,

Now we derive the analysis for S t (t) and it is straightforward to mention that the given analysis will exist in a similar way for the other model equations of (18) .

{κ} be the minimum and maximum weight of the variable non-integer order κ on [0, T ]. Now, we recall the following hypothesis to explore our main observations:

Theorem 7. Assume that hypothesis [X 2 ] holds and there exists C > 0 (constant) such that

Since C = [(1 − κ * )N c + κ * N c T ] < 1, using Banach fixed point theorem, we conclude that the operator O has a unique fixed point. Then, the model (18) has a unique solution.

Theorem 8. Assume that statements [X 1 ] − [X 2 ] hold and 0 < (1 − κ * )N c < 1. Then the system (18) has at least one solution.

Proof. First, we show the operator O 1 is a contraction. Indeed, it is given S t ∈ T where T = {S t ∈ B : S t ≤ w, w > 0} is a closed convex set it follows that

Hence O 1 is a contraction. Now to demonstrate that the second operator O 2 is compact we can see that O 2 is continuous and compact for any S t ∈ T , then O 2 is contraction as G 1 is continuous, then

Hence, the operator O 2 is equicontinuous. As a consequence of Theorem 1, O 2 is compact. Now by referring to the analysis given in section 5 of Verma et al. [23] , we concludes that the given system has at least one solution.

Now we write the solution of the proposed system in CF sense applying two-step Adams-Bashforth algorithm. Our time interval is [a, T ] with the step width h = T −a N , where N is the sample size. Let S t j be the numerical approximation of S t (t) at t = t j , where t j = 0+jh and j = 0, 1, ..., N . Writing the equations of S t (t) at the uniform grid points (t 0 , t 1 , t 2 , ..., t j−1 , t j , t j+1 ), we get the estimations at distinct grid point values. For doing it, first we consider the equivalent Volterra CF integral equation for S t (t) which is,

So the estimations at t j are

and at t j+1

Subtracting equation (29) from (28), we get

Now, by applying linear interpolation to G 1 (t, S t (t)) and employing trapezoid rule on the integral part, we obtain

where ∆t = t j − t j−1 . Hence, we have established the numerical approximation for S t (t) as

As a consequence, the solution of the proposed CF model (18) states as follows:

Theorem 9. The proposed numerical scheme (32) is unconditionally stable if (particularly for first model equation)

Proof. Given S t (t) the solution of (27), we have that:

Making j → ∞, we get

Clearly, the second part of the above inequality goes to zero when j → ∞. Now, if ||G 1 (t j+1 , S t (t j+1 )) − G 1 (t j , S t (t j ))|| → 0 as j → ∞, we conclude that the given scheme is stable.

Theorem 10 (Convergence). Let the solution of CF D κ t S t (t) be S t (t). Then there exist Γ, such that O k κ ≤ Γ. Proof. Starting from equation (30) and performing linear interpolation, we have

Simplifying further, we arrive at the numerical solution with the truncation term

where the truncation term is written as

Then taking the norm, we have

Hence, the solution has a convergence result.

In this section, we derive the all necessary plots by using the above given scheme. We use the initial populations S t (0) = 100, I t (0) = 10, Q t (0) = 5, R t (0) = 0, and parameter values d = 0.001, β = 0.003, δ = 0.003, γ = 0.002, b = 10, η = 0.05, ρ = 0.003, σ 1 = 0.003, σ 2 = 0.002, q = 1(this value is just an assumption) which are taken from the literature of COVID-19 cases in China [14, 17] . In the collection of Figure 1 , the subfigures 1a, 1b, 1c, 1d are devoted to show the variations in S t , I t , Q t , and R t against the time variable t. Here, the variations in the dynamics of the model can be clearly explored at different derivative order values. We can observe that when the fractional order values changes then the differences between phases of the plot lines increases. Figure 2 reflects the relations between the given classes. Subfigure 2a plots the variations of S t versus I t , subfigure 2b plots the corresponding ones for S t versus Q t and 2c plots the variations of S t versus R t . Finally, subfigure 2d plots R t against I t . The fractional order values which have been considered are κ = 0.75, 0.85, 0.95, 1. Now we intend to explore the role of vaccine on the given model classes. For this purpose, we change the value of the vaccination fraction q to simulate the model structure. Here, in the family of Figure 3 , the subfigures 3a, 3b, 3c, 3d demonstrate the variations in S t , I t , Q t , and R t against the time variable t at the vaccination fraction q = 0, where all other values are same as used above. Similarly, Figure 4 shows the corresponding ones when the vaccination fraction q = 0.5. By the comparison of these figures, we can easily observe that when the value of vaccination fraction q increases then the population of infectious humans decreases. So, vaccine availability is one of the most important control measure to reduce the infection of COVID-19. 6. Solution of the generalised Caputo fractional model (5)

In this concern, to prove the existence of unique solution of the proposed modified Caputo type fractional order model, we again write the given model into compact form as

Now we just adopt the first equation of the above system to derive the necessary results.

The equivalent Volterra integral equation of the proposed IVP is Theorem 12.

[17] (Uniqueness). Consider S t (0) ∈ R, K > 0 and T * > 0. Also, let 0 < κ ≤ 1 and m = ⌈κ⌉. For the set G as given in Theorem 9 and assume G 1 : G → R be continuous. Assume that G 1 agrees to the Lipschitz condition with respect to the second variable, i.e.

for some constant V > 0 which does not dependent to t, S t 1 , and S t 2 . Then, a unique solution S t ∈ C[0, T ] exists for the IVP (40a) and (40b).

Now we construct the numerical solution of the proposed Caputo fractional model using a modified form of the PC algorithm as mentioned in [29] with some appropriate changes. Here we start with Volterra integral equation (41), which provides

Here, first we recall that a unique solution of the proposed model exists under suitable conditions on the function G 1 on some interval [0, T ]. We divide the interval [0, T ] into N non-uniform subintervals {[t k , t k+1 ], k = 0, 1, ..., N − 1} taking the mesh points

here h = T σ N . We now analyse the approximations S t k , k = 0, 1, ..., N, to solve numerically the proposed IVP. First of all, assuming the approximation S t j ≈ S t (t j )(j = 1, 2, ..., k), we estimate S t k+1 ≈ S t (t k+1 ) by means of the integral equation

Substituing z = ξ σ , we get

Here, to simulate the integrals from the right-side of equation (47), we apply the trapezoidal quadrature rule for the weight function (t σ k+1 − z) κ−1 . We shift the function G 1 (z 1/σ , S t (z 1/σ )) by its piecewise linear interpolants with choosing nodes at the t σ j (j = 0, 1, ..., k + 1), and then we get

Substituing the above approximation into equation (47), we get the corrector formula for S t (t k+1 ), k = 0, 1, ..., N − 1,

where

At the end we aim to change the quantity S t (t k+1 ) from the right-side of equation (49) with the predictor term S P t (t k+1 ) that can be calculated by applying the one-step Adams-Bashforth rule to the integral Eqn. (46). We then susbtitute G 1 (z 1/σ , S t (z 1/σ )) by G 1 (t j , S t (t j )) at each integral in equation (47), obtaining

Therefore, our P-C scheme, for approximating S t k+1 ≈ S t (t k+1 ), is given by

where S t j ≈ S t (t j ), j = 0, 1, ..., k, and the predicted value S P t k+1 ≈ S P t (t k+1 ) can be simulated as shown in equation (51) with the quantities a j,k+1 given in (50). We can repeat this procedure to approximate all equations of system (39) . So, the numerical solution formulae for the adopted model (39) can be written as:

where

(54)

6.2.1. Stability analysis Theorem 13. If G 1 (t, S t ) satisfies a Lipschitz condition on the second variable and S t j (j = 1, ..., k + 1) are the solutions of the above approximations (53) and (54). Then, the proposed scheme (53) and (54) are conditionally stable.

Proof. LetS t 0 ,S t j (j = 0, ..., r+1) andS P t r+1 (r = 0, ..., N −1) be perturbations of S t 0 , S t j and S P t r+1 , respectively. Then, the proposed approximation equations are received by analysing Eqs. (53) and (54)

Using the Lipschitz condition, we simulate

|S t 0 |}. Also, as used in [17] , we derive

where γ 0 = max{ζ 0 + θ −κ h κ m 1 a r+1,r+1 Γ(κ + 2) η 0 }. C κ,2 is a +ve constant only depends on κ (Lemma 1 used in [17] ) and h is supposed to be small enough. Lemma 2 as mentioned in [17] gives |S t r+1 | ≤ Cγ 0 . which finishes the proof.

In this section, we check the correctness of our numerical algorithm by simulating number of graphs at different fractional order values κ. Here, we have considered the same initial populations S t (0) = 100, I t (0) = 10, Q t (0) = 5, R t (0) = 0, and parameter values d = 0.001, β = 0.003, δ = 0.003, γ = 0.002, b = 10, η = 0.05, ρ = 0.003, σ 1 = 0.003, σ 2 = 0.002, q = 1 as in the CF sense simulations. In the subfigures 5a, 5b, 5c, 5d, we show the variations in S t , I t , Q t , and R t against the time variable t. Here the variations in the dynamics of the model can be clearly explored at the various derivative order values. We observe that when the fractional order value changes then the differences between phases of the plot lines increase. Also, figure 6 shows the relations between the given classes at various values of κ. More concretely, in subfigure 6a we plot the variations S t versus I t , and in 6b we graph the variations S t versus Q t . Meanwhile, in subfigure 6c we plot the variations S t versus R t , and in 6d we plot the variations R t versus I t . The fractional order values which we used here are κ = 0.75, 0.85, 0.95, 1 as in the case of CF. Now, to simulate the role of vaccine on the proposed modified Caputo model classes, we change the value of the vaccination fraction q. Here, in the family of Figure 7 , the subfigures 7a, 7b, 7c, 7d demonstrate the variations in S t , I t , Q t , and R t against the time variable t at the vaccination fraction q = 0, where all other values are same as used above. Similarly, Figure 8 demonstrates the changes in the model classes when the vaccination rate q = 0.5. By the comparison of Figure 7 and 8, we can easily observe that when the value of vaccination fraction q increases then the population of infectious humans decreases. This clearly means that high vaccine rate gives much safety and become the only way to control the COVID-19. Now, as many countries like India, USA, UK, Spain, and Brazil have a good rate of vaccination which is a strong answer against COVID-19 infection. Vaccine availability alongwith quarantine and other optimal control facilities makes these countries much stronger to fight against this virus. From the given graphical observations, we can observe that the both kernel properties (exponential decay kernel in CF sense and singular kernel in modified Caputo sense) work well to study the given COVID-19 epidemic dynamics. All graphs are performed by using Mathematica software. The variations in the separate classes for both derivatives which are given in Figures 1 and 5 are probably same but the dynamics of the given classes slightly change. This fact can be observed comparing the group of Figures 2 and 6 . It is clearly observed that vaccination fraction q plays a very important role in the given dynamics and increment in the vaccine rate can decrease the Covid-19 infection.

In this study, two new non-classical COVID-19 epidemic models have been proposed. As a novelty, we include vaccine rate. First we have proposed a classical order model and then we have justified the fractional-order models by analysing the positivity and boundedness of solutions. The disease-free and endemic equilibrium points are calculated along with basic reproductive number. We have satisfied the existence of unique solution for both variable order Caputo-Fabrizio and generalised Caputo-type fractional models. We used two different fractional numerical algorithms along with their stability analysis to solve the proposed models. A deep and long discussion on graphical simulations is given making use of Mathematica software. The current study provides a description of the propagation of COVID-19 disease and supporting analysis proves the correctness of our results. In future, the current model can be validated by using real data from different countries. Also, some other fractional derivatives can be used to solve the current dynamical model.

SBDiEM: A new mathematical model of infectious disease dynamics

Mathematical Immunology of Virus Infections

Mathematical Epidemiology

Mathematical epidemiology: Past, present, and future

Mathematical Modeling and Control of Infectious Diseases

Novel fractional order SIDARTHE mathematical model of COVID-19 pandemic

Mathematical Model for Coronavirus Disease 2019 (COVID-19) Containing Isolation Class

Crowding effects on the dynamics of COVID-19 mathematical model

Dynamics of COVID-19 mathematical model with stochastic perturbation

Modeling and forecasting the spread of COVID-19 with stochastic and deterministic approaches: Africa and Europe

Modeling and forecasting of COVID-19 spreading by delayed stochastic differential equations

A mathematical study of a Tuberculosis model with fractional derivatives

A Malaria Model with Caputo-Fabrizio and Atangana-Baleanu Derivatives

A New Study of Unreported Cases of 2019-nCOV Epidemic Outbreaks

A case study of Covid-19 epidemic in India via new generalised Caputo type fractional derivatives

The analysis of a time delay fractional COVID-19 model via Caputo type fractional derivative

Solution of a COVID-19 model via new generalized Caputotype fractional derivatives

Prediction studies of the epidemic peak of coronavirus disease in Brazil via new generalised Caputo type fractional derivatives

Forecasting of COVID-19 pandemic: From integer derivatives to fractional derivatives

Projections and fractional dynamics of COVID-19 with optimal control strategies

Environmental persistence influences infection dynamics for a butterfly pathogen via new generalised Caputo type fractional derivative

A new definition of fractional derivative without singular kernel

Analysis of a novel coronavirus (2019-nCOV) system with variable Caputo-Fabrizio fractional order

Generalized Taylor's formula

The fractional calculus theory and applications of differentiation and integration to arbitrary order

Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications

Applications of fractional calculus in physics

Theory and applications of fractional differential equations

Numerical simulation of initial value problems with generalized Caputo-type fractional derivatives

Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan

A new mathematical modeling of the COVID-19 pandemic including the vaccination campaign

Chaotic dynamics of a fractional order HIV-1 model involving AIDS-related cancer cells

Numerical Solutions and Synchronization of a Variable-Order Fractional Chaotic System

Stability analysis and numerical computation of the fractional predatorprey model with the harvesting rate

The role of prostitution on HIV transmission with memory: A modeling approach

Analysis of an epidemic spreading model with exponential decay law

A robust study on the listeriosis disease by adopting fractal-fractional operators

Dynamics of generalized Caputo type delay fractional differential equations using a modified Predictor-Corrector scheme

A case study of 2019-nCOV cases in Argentina with the real data based on daily cases from March 03

A Study of Mathematical Model for Extended Lognormal Distribution to Obligatory Role of Hypothalamic Neuroestradiol during the Estrogen induced LH surge in Female Ovariectomized Rhesus Monkey

All authors equally contributed to this work.