key: cord-0333356-yxba8rdu
authors: ALLA HAMOU, A.; AZROUL, E.; Hammouch, Z.; Lamrani alaoui, A.
title: On dynamics of fractional incommensurate model of Covid-19 with nonlinear saturated incidence rate
date: 2021-07-23
journal: nan
DOI: 10.1101/2021.07.18.21260711
sha: 335a70a5aec751f42c16d7be02cf6cd6a21bba3b
doc_id: 333356
cord_uid: yxba8rdu

In December 2019, a new virus belonging to the coronavirus strain has been discovered in Wuhan, China, this virus has attracted world-wide attention and it spread rapidly in the world, reaching nearly 216 countries in the world in November 2020. In this chapter, we study the fractional incommensurate SIQR (susceptible, infections, quarantined and removed) COVID-19 model with nonlinear saturated incidence rate using Atangana-Baleanu fractional derivatives. The existence and uniqueness of the solutions for the fractional model is proved using fixed point theorem, the model are shown to have two equilibrium point (disease free and an endemic equilibrium). Some numerical simulations using Euler method are also carried out to support our theoretical results. We estimated the value of the fractional orders and the parameters of the proposed model using the least squares method. Further, the sensitivity analysis of the parameter is performed as a result, our incommensurate model gives a good approximation to real data of COVID-19.

At the end of December 2019, an unidentified virus was found in Hubei province, China. The responsible virus was later confirmed as a new coronavirus [1] . The World Health Organization (WHO) named the virus as the 2019 novel coronavirus (2019-nCoV) or COVID -19 In [34] the authors suggested changing the fractional SIAR model for HIV/AIDS with treatment compartment from classical commensurate model to incommensurate Caputo-Fabrizio fractional model by giving each equation a different order from the other equations. In this chapter we will study the general SQIR epidemiological model for COVID-19 with saturated incidence rate and using ABC-fractional operator. We will compare the results obtained for the fractional model with the classical model by predicting the model using Least Squares method and real data of Mexico from April 12 to October 28, 2020. Then the novelty of this paper is to use a different fractional order for each equation of the SQIR epidemic system with AB-fractional operators, this is fractional incommensurate SQIR model with time fractional derivative and saturated incidence rate.

The paper is organized as follows. In section 2, we present the Background of Atangana-Baleanu derivative and integral operators. The classical and fractional models are formulated in Section 3. The fixed point theorem is applied to prove existence and uniqueness results in section 4. The disease-free equilibrium and their local stability are given in section 5. In section 6 the Euler numerical method is used to determine the numerical solutions of the fractional proposed model. The model parameters are estimated according to real data of Mexico and the influence of fractional order has been shown in section 7 . Finally, we gave some remarks and conclusions in section 8.

We first give some preliminary on the definitions of fractional derivatives will be used in this paper. [35] . Definition 1. Let Ψ ∈ H 1 (η, η), η > η, α ∈ (0, 1) then the Atangana-Baleanu-Caputo (ABC for short) derivative is given as

where B is a function fulfills the condition B(1) = B(0) = 1. and E α is the generalized Mittag-Leffler function given by

where Γ is Gamma function.

The corresponding integral is given by the flowing definition. Remark 1. When α → 0 in the above definition we find the initial function. Moreover, we find the Riemann integral if the fractional order α turns to 1.

We consider in this part the model of COVID-19. The population is divided into four compartments, the first is the susceptible class S(t), Infected class I(t), Quarantined class Q(t) and Recovered class R(t).

The transfer diagram for this model is described by Figure 1 and the classical version of this model formulated by the flowing system of four nonlinear ODEs :

with the initial conditions S(0) = S 0 0, Q(0) = Q 0 0, I(0) = I 0 0 and R(0) = R 0 0. 

In this section, we change the classical derivative in (1) by ABC-fractional derivatives mentioned in Section 2 .

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The copyright holder for this preprint this version posted July 23, 2021. ;  Now remplacing the classical derivative in (1) by Atangana-Baleanu-Caputo derivative we obtain by moment closure, the following system of ABC-fractional ODEs

with the initial conditions S(0) = S 0 0, Q(0) = Q 0 0, I(0) = I 0 0, R(0) = R 0 0 and ABC D α i t , i = 1, 2, 3, 4, is the ABC-fractional operator of order 0 < α i < 1. The interaction between S and Q θ Quarantine rate of susceptible

The total population N (t) = S(t) + I(t) + R(t) + Q(t). Then

The dynamics of the fractional COVID-19 model will be studied in a biologically feasible set denoted by Ω = (S, Q, I, R) ∈ R 4 + : N (t) ≤ Λ/δ . 5 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

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In this section, we prove the existence of the system of solutions by applying the fixedpoint theorem.

Let We use the following notations for the right side of the COVID-19 models

by applying the definition of fractional integral to the Equation (1) and using the properties of fractional calculus we get

To prove the following theorems, we will assume that S(t) ≤ C 1 , Q(t) ≤ C 2 , I(t) ≤ C 3 and R(t) ≤ C 4 , where C i , i = 1, . . . , 4, are some positive constants. Denote

Theorem 3. Assume that the following inequality is verified

then the kernels G 1 , G 2 , G 3 and G 4 fulfill Lipschitz conditions, furthermore are contraction mappings.

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Proof. Let S 1 and S 2 be two functions of susceptible idividuals, then we have

where L 1 is defined in (4) . Similar results for the kernels G 2 , G 3 , G 4 , and G 5 can be obtained using {Q 1 , Q 2 } , {I 1 , I 2 } and {R 1 , R 2 } respectively, as follows

where L 2 , L 3 and L 4 are expressed in (4) . Therefore, the Lipschitz conditions are fulfilled for G 2 , G 3 , and G 4 . In addition, since 0 ≤ max{L 1 , L 2 , L 3 , L 4 } < 1, the kernels are contractions.

Theorem 4. Assume that the following inequality fulfill:

then the COVID-19 model with ABC-fractional operator (2) has a unique solution.

Proof. The proof of this theorem is based on the fixed point method and using Picard operator. We consider the following notations

Keeping in mind the above notation, system (2) may be expressed as

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The copyright holder for this preprint this version posted July 23, 2021. ; https://doi.org/10.1101/2021.07.18.21260711 doi: medRxiv preprint Now using (3) and according to the above notation we get the following fractional integral equation

define

Now using the metric on C (J, B 1 , B 2 , B 3 , B 4 ) induced by the norm given as

We define Picard's operator as

Now let's try to prove that P is a contraction.

by using the triangular inequality and the Lipschitz condition we will find

If condition (6) fulfill then P is a contraction on C (J, B 1 , B 2 , B 3 , B 4 ) . Then by Banach fixed point theorem P has a unique fixed point. Hence, the fractional COVID-19 model (2) has a unique continuous solution.

In this section we determine the equilibria of the COVID-19 models, for that we solve :

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The copyright holder for this preprint this version posted July 23, 2021. ; https://doi.org/10.1101/2021.07.18.21260711 doi: medRxiv preprint then the disease-free equilibrium is given by

Now let's determine the expression of basic reproduction number denoted by R 0 , by using the method presented in [36] , we get

Furthermore, the system (2) has a unique endemic steady state

such as

,

where the value of I * is the positive solution of the following algebraic equation

with

It remains to show that the solution of (17) is real and positive. We note

Let the discriminant of (17) be ∆, so that ∆ = B 2 − 4AC.We conclude the following results for the endemic equilibrium.

Theorem 5. The system (2) has a unique endemic steady state whenever R 0 > 1 given by

where the expressions of S * , Q * and R * are given in (16) and

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In this part we prove the local asymptotic stability of the equilibria of the commensurate model of (2) , that is to say in the case where the equations have the same fractional order α ∈ (0, 1). Since R(t) is not present in first three equations, then we can write the system (2) as

where

To show the local asymptotic stability of Atangana-Baleanu-Caputo model, it suffices to show that all the eigenvalues λ 1 , λ 2 and λ 3 of the jacobian matrix evaluated at the equilibrium point satisfy the condition

this result is mentioned in [37, Lemma 3.1.] and other references [38, 39] .

is locally asymptotically and if R 0 > 1, then F * = (S * , E * , I * ) is locally asymptotically stable.

Proof. The Jacobian matrix J(F 0 ) for the fractional model given by (18) , at the equilibrium F 0 is given by:

We find the eigenvalues by solving the following characteristic equation

We obtain the equation

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The eigenvalues of matrix J(F 0 ) are

If R 0 < 1 then λ 1 , λ 2 and λ 3 are real negative roots. Then

Hence the disease free equilibrium of the fractional model (18) is locally asymptotically stable. Now, we prove the local stability of the endemic equilibrium point. We consider the Jacobian matrix J(F * ) for the fractional COVID-19 model given by (18) , at the equilibrium F * is given by

The eigenvalues of this matrix can be obtained by solving the flowing characteristic equation

Therefore, we find the following characteristic equation:

where

,

, c 3 = (δ 2 + δγ + δθ + γθ)λcI * + (γ + δ) cβλI * 2 1 + σI *

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its discriminant, the following lemma completes the proof of theorem.

Lemma 7. Asume that R 0 > 1 and one of the following conditions is satisfied: 

To develop approximation schemes for the fractional model, we will use the Fractional version of Euler Method [41] . To applied this method, using same notation of section 4 and we consider the incommensurate system of ABC-fractional differential equations

We consider t 0 = 0 < t 1 < . . . t k . . . < t N = T an uniform time discretization of an interval [0, T ] in N + 1 points. Let h = T N is the time step size. We denote Z i k as the numerical approximation of Z i (t k ).

Using AB-fractional integral, we have

after using the fractional Euler method [41] , we obtain

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The copyright holder for this preprint this version posted July 23, 2021. where the coefficients C i k+1,j , j = 0, . . . , k, are given by C i k+1,j = −(k − j) α + (k − j + 1) α i for k = 0, . . . , N − 1 and i = 1, 2, 3, 4.

According to [41] , the numerical scheme (27) for the fractional model (2) is conditionally stable. Now we will give a theorem for estimating the error between exact and numerical solution of the proposed model. Theorem 8. Let Z(t) and (Z k ) k be respectively the exact and approximate solutions of the fractional incommensurate model (2) . If 0 ≤ max {L 1 , L 3 } < 1, where L 1 and L 3 are given in (4), then we have

where C is an independent positive constant free of k and h.

Proof. The proof is a direct conclusion of [41, Theorem 4].

The model calibration problem seeks to estimate the model parameters which, to some extent, to make the error between the observed values and the predicted values as small as possible, we use for that the Least Squares method to produce the values of the parameters and the fractional multi-order. Some model parameters can be known by these real values, on the other hand some parameters are unknown and need to be estimated, the same for the initial conditions of the COVID-19 model. We consider the model solution Z(t) = (S(t), Q(t), I(t), R(t)), depending on the vector of parameters ξ = (β, c, λ, θ, δ, γ, σ, Λ), the initial conditions Z 0 = (S 0 , Q 0 , I 0 , R 0 ) and the fractional orders α = (α 1 , α 2 , α 3 , α 4 , α 5 ). The initial conditions S 0 , I 0 and R 0 are known from data of COVID-19, but Q 0 is unknown so it is necessary to estimate its value.

The vector of parameters belongs to a set given by

where l b and u b are of lower and upper vector values for the model parameters respectively. Let the vector Θ ∈ R N of the observation data at given times t k , k = 1, . . . N and a prediction vector Ψ(ξ, Q 0 , α), the calibration aims at finding a vector (ξ * , Q * 0 , α * ) such that

The aim is to minimize the objective function 13 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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subject to (ξ, Q 0 , α) ∈ Θ and Equations (2) to obtain the best estimate of parameters and the vector of fractional orders α. Now we consider the real data of COVID-19 for Mexico (that are found on the WHO web-page) to estimate the model parameters using the least squares method. In the first time we estimate the parameters for the classical model (1), second for the commensurate model which the four equations of the fractional model have the same order and finally the incommensurate model (2) when each equation has a different order .

The resulting fitted curves for reported infected and recovered cases of COVID-19 in Mexico is shown in Figure 2 for the classical model, in Figure 3 for the commensurate model and in Figure 4 for the incommensurate model, where the best fitted values of the parameters are depicted in Table 3 and the estimated values of the fractional orders of the two fractional models are given in Table 4 . Furthermore, the initial conditions are given in the table 2. From Table 3 we notice that our models give a good approximation of COVID-19 real data in Mexico compared to the classical model and more better than the case in which the orders of the four model equations have the same fractional order. The error of least square for the 14 .

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The copyright holder for this preprint this version posted July 23, 2021. ; commensurate and incommensurate models are respectively 3.2931e+2 and 3.2901e+2, this means that the incommensurate model with ABC-fractional derivative gives a better fit of the real data compared to the commensurate model with ABC-fractional derivative. In general the ABC-fractional derivative gives results that are closer to the real data compared to the classical model.

When a virus spreads quickly it is necessary to look for methods to control the virus, that is, seek to increase or decrease some parameters which have an influence in the transmission of the virus. Sensitivity analysis is one way to determine the importance of each parameter for disease transmission.

Precisely, we use the definition of the sensitivity index which gives the significance of the variable associated with each parameter of the model to reduce the disease, the sensitivity index is defined as in [42] .

Definition 9. The sensitivity index of R 0 with respect to x, defined by

The sign of each index makes it possible to know whether the parameter increases (positive sign) or decreases (negative sign) the value of R 0 , Using the expression of R 0 given in (15) to calculate the sensitivity index for R 0 with respect to the parameters of the model, we will find

,

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The copyright holder for this preprint this version posted July 23, 2021. ; https://doi.org/10.1101/2021.07. 18.21260711 doi: medRxiv preprint Note that R 0 does not depend on σ so Γ R 0 σ = 0. 

< 0 which means that an increment in Λ, c, β, λ and δ will cause R 0 to increase, while a decrement in the recovery rate of infected γ or the quarantine rate of susceptible θ will cause R 0 to decrease.

Based on numerical fractional Euler method derived in section (6) , we present the graphical results of the proposed incommensurate fractional model (2) in Figures 5 and 6 to analyze the influence of fractional order. We have considered the initial values given in Table 2 and the estimated parameters given in Table 3 .

The obtained figures show the importance of choosing the fractional order in the 2019-nCOV model, as well as choosing a different order in each equation. From Figures 5-6 , we see that any slight change in the fractional orders gives a change in the dynamics of the infected people and this indicates the effectiveness of this type of derivation. It can also be used to predict the number of people infected with COVID-19 better than the classical model and the fractional commensurate model (see Table 3 ).

In this chapter we have studied an incommensurate fractional model of COVID-19 with Atangana-Baleanu fractional derivative, we have proved the existence together with the uniqueness of the solution using the fixed-point theorem. We have determined the equilibrium points of the proposed model and the expression of base reproduction number R 0 as well as the local stability of each equilibrium point. The numerical fractional version of Euler method is used to approximate the solutions of the fractional model. We simulated the models with five different fractional order vector and we compared the results obtained. The model efficiency is shown by comparing the error values between the COVID-19 data in Mexico and the values predicted by the three models studied. The best fit for COVID-19 16 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted July 23, 2021. data in Mexico is established when using incommensurate model with Atangana-Baleanu operator. The Least Squares method is used to determine the best fit to real data and estimate fractional order. Finally, the chapter gives an example of the effectiveness of using the Atangana-Baleanu operators in real problems and the importance of incommensurate propriety in the fractional model.

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. CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. 

. CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. 

. CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. 

. CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted July 23, 2021. ; https://doi.org/10.1101/2021.07.18.21260711 doi: medRxiv preprint

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