key: cord-0920470-n7phsdch
authors: Karaman, Bahar
title: The global stability investigation of the mathematical design of a fractional-order HBV infection
date: 2022-03-31
journal: J Appl Math Comput
DOI: 10.1007/s12190-022-01721-2
sha: 81bd5cbe8580a8302baf9298054d309854ea7afc
doc_id: 920470
cord_uid: n7phsdch

This work presents approximate solutions of a fractional-order design for hepatitis B virus infection. The numerical solution of the system is given by using an implicit fractional linear multi-step method of the second order. Here, Caputo fractional derivative is considered for fractional derivative. Basic theoretical properties are discussed. We prove the global stability analysis of the fractional-order model. Numerical simulations are demonstrated to display our theoretical results. This current study is to reveal that the order of the fractional derivative [Formula: see text] does not affect the regular state’s stability concerning both theoretical and numerical results. Besides, if the fractional-order [Formula: see text] increases, the solutions converge more rapidly to the regular states. Finally, we note that this study can provide beneficial outcomes for understanding and estimating the dissipation of distinct epidemics.

In the current paper, we will concern numerical solutions of fractional-order mathematical design HBV infection which is formulated as [24] :

in here D β denotes the CFD and β ∈ (0, 1). Also, P, R, S denotes the densities of hepatocytes that are uninfected, infected, and disease-free virions, respectively. It is assumed that the birth rate of sensitive hepatocytes is at the constant rate λ 1 , the death rate constant is μ, and parameter λ 2 is the infection rate constant. η refers to the rate of death of infected hepatocytes, the output rate of free virions from infected hepatocytes is σ , the recovery rate of viral particles is γ . Infected hepatocytes are cured by noncytolytic stages at a constant rate δ per cell [28] . The readers can read more details about the proposed model (1) in [24] . This model (1) is a generalization of an integer-order which is depicted by Su et. al. [28] .

The main aim of this research is to propose, analyze, and simulate the HBV model using fractional calculus. The model yet has not been solved numerically by using the fractional trapezoidal method FTM. Numerical solutions are obtained under the MATLAB environment using the code f lmm2 by R. Garrappa, see details [29, 30] . This code contains optionally three methods, but we will choose FTM to obtain the approximate solutions of the mentioned design. To the best of our knowledge, no one yet has also examined the analysis of the global stability for this present epidemiological model. Thus, we will investigate the global stability analysis of the fractional structure by using the Lyapunov stability theorem. Another point that is worthy of being emphasized is that the order of the fractional derivative β does not affect the regular state's stability concerning both theoretical and numerical results. Besides, if the fractional-order β increases, the solutions converge more rapidly to the regular states. That's why when we compare to the other existing results in the available literature, we would also like to say that this study will contribute to the literature.

The designation of the current study is as follows: We will first present some principal concepts of fractional calculus in Sect. 2. The existence and uniqueness of the proposed system will be discussed in Sect. 3. In the next section, the description of the FLMS techniques for the system of fractional differential equations (SFDEs) will be constructed. Then, we will carry out the global stability analysis. Finally, we will illustrate the numerical experiments in Sect. 6 and end the study with a short outcome that will be dedicated in Sect. 7.

The part gives some fundamental definitions. Firstly, we want to give the definitions of CFD and Riemann-Liouville integral (RLI). Then we want to recall the notions of Mittag-Leffler function (MLF).

Let g ∈ C n and t, β ∈ R. Then the CFD D β g(t) of order β is introduced by

where the function (.) is named a Gamma function.

CFD was introduced by the Italian mathematician Caputo in 1967 [31] . It is so useful and flexibility for resolving the initial value problems with classical initial conditions.

The RLI for a function g ∈ L 1 ([0, T ]), (L 1 represents the set of Lebesque integrable functions), of order β ∈ (0, 1) is introduced as

provide that the integral exists on the right hand side. Notice that also D β is a left inverse of the RLI, namely D β J β g = g, but not its right inverse [32] since

The book by Diethelm [32] can be examined by the readers for more informations of above results.

Definition 3 [33] The function with one-parameter is introduced as

in here α > 0. The MLF with two parameter is constructed as follows

The MLF was described by Mittag-Leffler in 1903 [33] . Since the MLF frequently is used to compute the solutions of FDE, it is of great importance in fractional calculus.

The textbooks [32, 34, 35] are recommended to readers for more comprehensive introduction.

in here 0 < β ≤ 1.

The equilibrium points of the given system (7) are computed when g(t, u) = 0.

Theorem 1 [37] If all eigenvalues λ j of the Jacobian matrix J = ∂ g ∂u assessed at the equilibrium points fulfill | arg(λ j ) |> απ 2 , then the points are called local asymptotical stability.

We state the Arithmetic-Geometric means inequality for demonstrating the fractional derivative of Lyapunov function is non-positive.

Lemma 1 [38] Let v 1 , v 2 , . . . , v n be positive real numbers. Then

Besides, exact equality only occurs when v 1 = v 2 = · · · = v n .

We will display the properties of the solution for the proposed design (1). The following outcomes will be used for displaying the solutions are non-negative and bounded, respectively.

The next lemma is significant to indicate the uniform boundedness of the solution.

where 0 < β ≤ 1, α, ξ ∈ R and α = 0, and t 0 ≥ 0 is the initial time. Then

The following theorems show the existence and uniqueness of non-negative and the uniform bounded solutions, respectively. The existence and uniqueness of nonnegative solutions are demonstrated by Zhou and Sun [24] .

There is a unique solution to fractional order model (1) with initial conditions (2) and the solution will remain in R +

The following result indicates the solutions of the system (1) are uniformly bounded.

Proof We will utilize a strategy given in [40] . Let's define the function

According to Lemma 5 and Corollary 6 in [41] , by taking t → ∞, we get 0 ≤ H (t) ≤ 2λ 1 α . Consequently, the solutions of the sysyem (1) starting in R + 3 are uniformly bounded within the region 1 

The elementary techniques such as one-step and multi-step methods are utilized for obtaining the approximate solutions of ordinary differential equations. In the onestep method, using only one approximation of the solution at the preceding stage can be calculated the solution. But, we use more previously obtained approximations to calculate the solution in the multi-step methods.

In recent years, with the widespread use of fractional differential equations, many researchers have worked to obtain analytical and numerical solutions to such differential equations. The First pioneering study is proposed by Lubich [42] . He proposed a reliable and effective methodology for FLMS. This strategy is, which a generalization of the classical linear multi-step methods, a powerful strategy for a SFDEs. Now, we will focus on the initial value problem for a SFDEs in the following as 

The integral formulation (9) is certainly convenient, as it yields using the theoretical and numerical results already available for the Volterra integral equations class to study and solve FDE.

The main idea in FLMS technique is the approximation of the RLI (3) in terms of the convolution quadrature

on uniform grids t n = t 0 + nh, h > 0, and in which convolution and starting quadrature weights w n, j , ω n− j do not depend on h. We note that starting weights w n, j are very significant in the first part integration interval to overcome the possibly singular character of integrand function at t 0 . We would also like to say that the convolution quadrature weights ω n− j are the major piece of the quadrature rule and qualify the certain FLMS methods [30] . Now, let us remember the first and second characteristic polynomials ρ(ζ ) and σ (ζ ), which are defined as follows ρ(ζ ) = ρ 0 ζ k + ρ 1 ζ k−1 + · · · + ρ k and σ (ζ ) = σ 0 ζ k + σ 1 ζ k−1 + · · · + σ k for a linear multistep method. A generating function is obtained as the coefficients of the formal power series

for the linear multistep method. In order to generate a quadrature rule for fractional problems (8) by evaluating the convolution weights as the coefficients in the formal power series of the fractional-order power of the generating function

This method named as FLMS methods, when applied to (8) read as

The convergence properties are given in the following studies [42, 43] . The readers can be read more detailed information for the computation of the weights by investigating the studies [30, [42] [43] [44] .

In this study, we will use the fractional trapezoidal rule, which is one of the FLMS methods to solve the numerically proposed model. This rule is the generalization of the standard trapezoidal rule to fractional differential equations. Firstly, let us consider its classical formulation for ordinary differential equations

with characteristic polynomials ρ(ζ ) = ζ − 1 and σ (ζ ) = (ζ + 1)/2 and generating function

Evaluating the weights ω n in FLMS (12) as the coefficients in the formal power series (11) is one of the major difficulties. Despite of establishing some sophisticated algorithms for manipulating formal power series [45] , for most methods an efficient tool is the J. C. P. Miller formula stated by the following theorem [46] .

Let ϕ(ζ ) = 1+ ∞ n=1 a n ζ n be a formal power series. Then for any β ∈ C,

where coefficients ν (β) n can be recursively evaluated as

In the most general case, this formula allows to evaluate the first N coefficients of (ω(ζ )) β with a number of operations proportional to N 2 . In the study [30] , the author advises the twice application of the Miller formula to (1 + ζ ) β and (1 − ζ ) −β and thus evaluate the coefficient of their product by an Fast Fourier transform algorithm, with a number of operations proportional to 3N log 2 4N when N 2 [45] .

We will focus on the global stability analysis of the mentioned design (1) equilibria. But first, we will give the results of the local asymptotical stability of the equilibrium points. The authors in [24] demonstrated the basic reproductive number. It is obtained as R 0 = λ 2 (σ −η−δ) γ (η+δ) . This number tells us that the predicted number of secondary cases produced, in a completely susceptible population, by a typical infective individual [47] . It is also so important tool mathematically because of indicates the spread of the disease. The disease-free equilibrium point E * = λ 1 μ , 0, 0 and endemic equilibrium point

are found in [24] . Then, they obtained Theorem 3.1, and Proposition 3.1-3.2 in [24] in order to prove local asymptotical stability of the given fractional system (1). If R 0 < 1, the disease-free equilibrium is local stability which means that the disease does not spread. When R 0 > 1, the disease perennially exists in the society, and then transforms an epidemic. When the reproductive number passes the unity, the endemic equilibrium is local stability.

A most significant concern for the FDE is about that the global stability of the solution. To our knowledge, the global stability of the disease-free equilibrium point E * and endemic equilibrium point E * * of the present design (1) has not been proved yet. Now, we will use the extended Barbalat's lemma and Lyapunov functions to demonstrate the stability of fractional systems. To prove the global stability of the disease-free equilibrium and endemic equilibrium points, motivated by the studies in [23, 25, 27] .

Theorem 4 [25] If ν : R → R is a uniformly continuous function on [t 0 , ∞) and J β ν s ≤ N , ∀t > t 0 with β ∈ (0, 1), s and N are positive constants, then ν(t) → 0 as t → ∞.

We will utilize the given Lemma to find Lyapunov candidate functions to demonstrate the stability of SFDEs.

Lemma 5 [26] Assume that z(t) ∈ R + is a continuous function. Then for any time t ≥ t 0 ,

Now, let's give the following Lemma which denotes the extended Volterra-type Lyapunov function to SFDEs through an inequality for approximating the CFD of the function. This Lemma is described by Leon [27] .

Lemma 6 [27] Assume that z(t) ∈ R + is a continuous function. Then for any time t ≥ t 0 , ∈ (0, 1) . (15) The next outcome indicates the solutions of model (1) are uniformly continuous. The proof is done in a similar argument as in [23] . Proof We describe a Lyapunov function V : {(P, R, T ) ∈ : P > 0} → R such as

By using Lemmas 5 and 6, we have

By using the Lemma (7) and the assumption of inf P(t) > 0 t≥0 , we have the uniform continuity of (P−P 0 ) 2 P , R 2 and P S P+R . According to Theorem 4, we get (P−P 0 ) 2 P → 0, R 2 → 0 and P S P+R → 0 as t → ∞. We have P → P 0 , R → 0 and S → 0 as t → ∞. Therefore, lim t→∞ (P, R, S) = (P 0 , 0, 0) independent of the initial data in the interior of . This result indicates that E * is globally asymptotically stable in the interior of .

The endemic equilibrium point E * * of the system (1) is globally asymptotically stable on the interior of if R 0 > 1.

We assume that R 0 > 1, so that the concerned endemic equilibrium exists. Now, we will define a Lyapunov function W : {(P, R, T ) ∈ : P > 0} → R such as

By using Lemma 6, we have

It can be displayed from (1) the endemic steady state,

We will use the above relation in (16) , then we get

By using Lemma 1, since the arithmetic mean is greater than geometric mean it follows that

Hence, we can say that N = {E * * } = {(P 1 , R 1 , S 1 )}. Afterwards, by the Lyapunov-LaSalle invariance principle, the model is globally asymtotically stable at E * * when R 0 > 1.

This current section gives the numerical simulations for the present system (1). Also, our theoretical results are validated by the some numerical simulations. The numerical simulations of different values of β are illustrated in Figs. 1 and 2. We would like to say that the spreading of the infection during the first 1500 days converges toward the disease-free equilibrium point E * = (2.5 × 10 9 , 0, 0). In this step, R 0 = 0.4952 < 1 implies our theoretical result about the stability of E * . In the Fig. 2 , the numerical solutions converge to the endemic equilibrium point E * * = (2.1077 × 10 8 , 1.0923 × 10 9 , 4.1651 × 10 10 ). In this case, we have R 0 = 6.1824 > 1. Thus this situation supports the stability result of E * * . It is observed that from the numerical outcomes, the order of the fractional derivative β has no affect on the stability of the two equilibria. It should be noted that for higher values of β, which describes the long memory behavior, the solutions converge more quickly to the regular states. Besides, to R(t), we can see the smallest values to β imply a wider period infectiously, so the disease takes a long time to be eradicated. This property is significant from the health point of view because it reflects a long period in which the infected individuals can affect the health system. 

This work proposes the numerical solutions of a fractional-order design of HBV infection by using the fractional trapezoidal formula. The mathematical representation of the fractional model is demonstrated by using the Caputo fractional derivative. Since the extension of Barbalat's lemma to the fractional situation is an effective tool to examine the asymptotic stability analysis of the fractional dynamical systems, we have used this lemma to prove the global stability of equilibrium points are discussed by composing a suitable Lyapunov function. Then, we have shown that the global asymptotical stability of the equilibrium points E * and E * * for R 0 < 1, and R 0 > 1, respectively. All obtained numerical experiments are illustrated by graphs. Another point that is worthy of being emphasized is that the order of the fractional derivative β does not affect the regular state's stability concerning both theoretical and numerical results. Besides, if the fractional-order β increases, the solutions converge more rapidly to the regular states. As a concluding remark, we note that this study can provide beneficial outcomes for understanding and estimating the dissipation of distinct epidemics.

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