journal of the Egyptian Mathematical Society 000 (2017) 1-7

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Original Article

On a fractional-order model for HBV infection with cure of infected cells

Salman S.M.a'*, Yousef A.M.b

a Faculty of Education, Alexandria University, Alexandria, Egypt b Faculty of Science, South valley University, Qena, Egypt

A R T I C L E I N F 0

Article history: Received 15 April 2017 Revised 29 May 2017 Accepted 28 June 2017 Available online xxx

MSC: 39B05 37N30 37N20

Keywords: Fractional calculus Viral dynamics Hepatitis B virus Equilibrium points Local stability

A B S T R A C T

In this work, a fractional-order model for HBV infection with cure of infected cells is considered. Local asymptotic stability of equilibria is discussed. Numerical simulation using PECE method is presented to illustrate the theoretical analysis. It has been noticed that when introducing a fractional-order derivative to the model, the peak of the infection is reduced, however, the disease takes a longer time to be eradicated.

© 2017 Egyptian Mathematical Society. Production and hosting by Elsevier B.V.

This is an open access article under the CC BY-NC-ND license.

(http://creativecommons.org/licenses/by-nc-nd/40/)

1. Introduction

Over recent years, mathematical models which mimic viral population dynamics in host cells have gained much interest [1,2]. These models improved our understanding of HIV-1 and different viruses, like HBV and HCV [3,4]. Mathematical analysis is important for these models to get a realistic view for the virus dynamics in vivo. Particularly, global stability of a fixed point can enhance our knowledge of virus dynamics. Many models were established for describing the dynamics of HBV infection [5,6]. Such models were modified in [7,8]. Recently, in an HBV model, the infected hepatocytes may be reverted to be uninfected by losing all cccDNA from their nucleus [7].

An HBV infection model of hepatocytes of infected cells which will be cured was introduced by Cruz Vargas-De-Leon [9] and is

= pxz - (v + S)y,

given by:

ddx = k - fix - p xz + Sy,

dy dt dz

dt = a y - Y z'

where x, y and z refer to the densities of hepatocytes which are uninfected, infected, and virions which are disease-free, respectively. Susceptible hepatocytes are supposed to be produced at rate k, die at fx] and infected at pxZ] where p is the rate of infection. The rate of death of infected hepatocytes is vy, ay is the production rate of free virions from infected hepatocytes, and yz is the clearance rate of viral particles. The term Sy appears in the first equation of (1.1), corresponds to the cure-created rate of uninfected hepatocytes.

2. Fractional calculus

* Corresponding author. E-mail addresses: samastars9@gmail.com (S. S.M.), ahmed.hussien1@sci.svu.edu.eg (Y. A.M.).

Indeed, fractional calculus generalizes the classical integer-order differentiation and integration to arbitrary orders [10-12]. A large number of researchers have modeled many real processes with the aid of fractional calculus [13-15]. Due to the applicability of differ-

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S. S.M., Y. A.M./Journal of the Egyptian Mathematical Society 000 (2017) 1-7

ential equations with fractional-order in science and engineering, a lot of applied researchers have been attracted to it. Such equations have been implemented analytically and numerically. In fact, fractional calculus has the ability of performing both fractional-order integration and differentiation. Existence of solutions for differential equations with fractional-order can be found in [12,16].

Many contributions for the developing theorems of the solutions' existence of functional differential equations with fractional-order existed in [11,17,18].

In fractional calculus we can find many definitions of derivatives of fractional-order, however, we will consider the famous Ca-puto derivatives due its advantage on initial value problems which can be summarized in the flexibility of using classical initial conditions without encountering any problem during solvability.

Definition 1. The fractional-order integral of the function f(t), t > 0 can be defined as

I3 f (t ) = j

1 (t - s)3-1

f (s)ds,

where f e R+ , and the fractional-order derivative of ft), t > 0 can be defined as

Da f (t) = In-a Dn f (t) , D = where a e (n - 1, n) .

In addition, consider also the following basic properties. Let

f, y e R+ , a e (0, 1),

• if : L1 ^ L1 , and if fx) e L1 , then ¡Y iff f (x ) = ¡Y+f f (x).

• limf^n if f(x) = ¡n f(x) uniformly on [a, b], n = 1. 2. 3 . . . . ,

where ¡If (x) = ft f (s )ds.

• limf^ o ¡a f (x ) = f (x ) weakly.

• If f(x) is absolutely continuous on [a, b], then lima^ 1 Daf (x) =

df (x) dx '

For the basic features of fractional-order derivatives and integrals, one can see [19-21].

Actually, to find the solution of fractional-order differential equations there are two familiar methods: frequency domain [22] and time domain [23]. Since the frequency domain method is not efficient in observing chaos, the second method is more usable because it is more effective (see [24,25]).

As a matter of fact, difficulties arise when trying to find an analytic solution of a differential equation, that is why one searches for numerical methods.

Indeed, the predictor-correctors scheme has been used to simulate many fractional-order differential equations with Caputo fractional derivative such as the fractional-order Chua system, the fractional-order Chen system, the fractional-order Lorenz system, etc. [26].

Recently, it has turned out that applying fractional calculus, rather than integer calculus, to mathematical epidemiology provides a good tool for the description of memory feature which is a hallmark in many biological systems [27]. The property of memory does not take into consideration the involved process history in such models but it also has a great impact on the future of the process [28]. Actually, a differential operator with an integer-order is a local operator, whereas a differential operator with a fractional-order is non local in as it takes into account that the future state do not relay only on the present state but on all of the history of its previous states too.

Now introduce the fractional-order derivatives to the HBV infection model with cells which will be cured after infection given

by (1.1) as follows:

Dax = k - iâx - //xz + Sy, Day = /xz - (v + S)y, Da z = a y - y z,

where a e (0, 1). The major reason of considering a fractional-order model in this paper is that fractional-order differential equations have a relation to systems with memory like systems of the Immune system which creates memory T-cells and B-Cells. Such cells learn from their experience of fighting any threat. So, they can use their expertise to detect and fight the same threat later. On the contrary, integer-order models do not have any information about the memory of neither the hepatocytes nor the free virions in (1.1) (see [29-32]).

To summarize, mathematical models which use ordinary differential equations with integer order have been proven valuable in understanding the dynamics of biological systems. However, the behavior of most biological systems has memory or aftereffects. The modeling of these systems using fractional-order differential equations has more advantages than classical integer-order mathematical modeling, in which such effects are neglected. Consequently, the subject of fractional calculus has gained importance due to its demonstrated applications in numerous diverse and widespread fields of science and engineering. For application of fractional-order derivative, one can see [33-37].

3. Existence of uniformly stable, positive, and eventually bounded solution

Let x1 (t) = x (t) . x2 (t ) = y (t) . x3 (t ) = z(t). f1 x , x2 , x3 ) = X - ¡1x1 - f + Sx2,

f2 (x1 , , ) = f - (V + S)x2,

f3 x , x2 , ) = ax2 - Y*3 .

Let D = {x1 , x2, x3 e R : | x.(t) | < a, t e [0 . T] . i = 1. 2 . 3} . then on D we have:

I f I < k1. I f I < k2 • I f I < k3, I f I < k4, I f I < k5, I ^ I <

k6, I f I < k7 , I f I < k8 , If. I < kg, where k. , i = 1. 2 . . . . , 9 are positive constants. This implies that each of the functions f1 , f2 , f3 satisfies Lipschitz conditions with three arguments x1 , x2 and x3 . Then each of the functions f1 , f2 , f3 are absolutely continuous w.r.t. three arguments x1 , x2 and x3 . Rewrite system (2.1) as follows:

Dax (t) = /1 x, x2 , x3 ). t > 0. and x1 (0) = ^01 , Dax2 (t) = f (x1, x2, x3). t > 0. and x2 (0) = xe , Dax3 (t) = f (x1, x2, x3). t > 0. and (0) = xe .

Note that by a solution of the fractional-order HBV model (2.1) we mean a column vector (x1 , x2 , x3 )' , x 1 , x2 , x3 e C[0, T], T < x>. where C[0, T] denotes the class of continuous functions defined on [0, T] and the "dash" denotes the transpose of the vector.

Theorem 1. The fractional-order model (21) has a unique uniformly Lyapunov stable solution.

Proof. Rewrite system (2.1) in a matrix form:

DaX(t) = F(X(t)). t > 0 . X (0) = X0 ,

where X (t) = (x 1 , x2 , x3 )',

F (X (t)) = ( f1 (x1 (t). x2 (t). x3 (t)). f2 (x 1 (t), x2 (t), x3 (t)). f3 (x1 (t). x2 (t). x3 (t))'.

Now applying Theorem 2.1 [38], we deduce that the fractional-order HBV model (2.1) has a unique solution. In addition, by Theorem 3.2 [38] this solution is uniformly Lyapunov stable. □

S. S.M., Y. A.M./Journal of the Egyptian Mathematical Society 000 (2017) 1-7

Fig. 1. Solution of system (2.1) with a = 0 . 85]

Fig. 2. Solution of system (2.1) with a = 0 . 80.

JID: JOEMS 4

S. S.M., Y. A.M./Journal of the Egyptian Mathematical Society 000 (2017) 1-7

Fig. 3. Solution of system (2.1) with a = 0 . 75 .

Fig. 4. Time series of system (2.1) with different <5, where a = 0 . 85.

Since we are dealing with a biological model, it does not make sense to have negative solutions. The following theorem prove that the state variables x(t), y(t), and z( t) are all positive.

Theorem 2. Assume that the parameters X, n, f, 5, v, a, and y are all positive and real and denote R+ = {X e R3 |X > 0} and let X(t) = (x(t).y(t). z(t))'. Then for any X(0) > 0, the solution X(t) of (21) on t > 0 will remain in R+.

Proof. We will prove this theorem by contradiction. Suppose there exists t* > 0 at which one of the elements of the solution will be zero and until all elements of the solution will be positive. Let a e (0, 1), there are three possibilities:

(i) If x(t * ) = 0 holds, then y( /) > x(t) > 0 on [0, t*j.

0, z(t) > 0 when t e [0, t*) and

Let m1 = min/e] 0 , t*] z(t)] c1 = f + pm1 . The first equation of (2.1) gives:

Dax(t) > c1x(t)] t e [0 ] t*] ] hence

x(t) > x(0)Ea(-c1/a)] t e [0 ] t*] ]

where Ea(t) = r(ka+1) is the Mittag Leffler function. Since x(0) > 0, then x(£J) > 0 which is a contradiction. (ii) If y(t* ) = 0 holds, then x(t) > 0, z(t) > 0 when t e [0, t*] and y(t) > 0 when t e [0, t*). From the second equation of system (2.1) we have

Day(t) > -(v + S)y(t)* t e [0 * t*] *

S. S.M., Y. A.M./Journal of the Egyptian Mathematical Society 000 (2017) 1-7

Fig. 5. Time series of system (2.1) with different <5, where a = 0 . 75.

which implies that

y (t) > y(0)Ea(-(v + S)ta). t e [0 . t*].

Since y(0) > 0, one has y(£J) > 0 which is a contradiction. (iii) If z(t*) = 0 holds, then x( t) > 0, y(t) > 0 when t e [0, t*] and z(/) > 0 when t e [0, tJ ). The third equation of (2.1) gives

Daz(t) > -yz(t), t e [0 . t*] .

which implies

z(t) > z(0)Ea(-Yta). t e [0 . t*] .

Since z(0) > 0, one has z(t*) > 0 which is a contradiction. Therefore, the solution of system (2.1) will remain in R+ . Adding the first two equations in (2.1), for the biological justification, we suppose that | < v.

Da{x(t) + y(t)} = k - ix - vy < k - i(x + y). (3.1)

which implies that

x (t) + y(t) < [x(0) + y(0)]Ea(-it) + i[1 -Ea(-lt)] . (3.2) Hence, we have lim sup [x(t) + y(t)] < I. The last equation

of (2.1) then leads to

Daz(t) < I - yz,

which in turn implies sup z(t) < Ik □

4. Equilibrium points and their asymptotic stability

For any infectious disease, the most important issue is its ability to invade a population. Many epidemiological models have a disease-free equilibrium (DFE) where the disease is not present in the population. These models usually have a parameter, which is the basic reproduction number R0 . If R0 < 1. then the DFE is locally asymptotically stable, and the disease will not spread, but if R0 > 1. then the DFE is unstable and the disease will spread all over the population. The basic reproduction number for system (2.1) is given by:

Yl(v + S).

System (2.1) has two equilibria, namely:

The disease-free equilibrium (DFE) E0 = (0 . 0) . and if

R0 > 1 , there is an endemic equilibrium (EE) E * = (

Yl(v+/) ka a/v ' vy

I(v+/)

Y(v+/) k

a/ ' v

Next, we discuss local stability analysis of the two equilibria of (2.1).

Proposition 1. E0 is locally asymptotically stable if 0 < R0 < 1 and is unstable if R0 > 1.

Proof. Calculate the Jacobian matrix of (2.1) evaluated at E0 to obtain:

-I S J = ( 0 -(v + S) 0 a

The characteristic equation reads: P (A) = A3 + (¡1 + v + 5 + y)A2 + v + 5 + y) + Y(< + v)

-af)A + Y1(5 + v) - afk = 0. (4.1)

which has three eigenvalues: A1 =

-(Y + V + 5)1 + y/^2(y + V + 5)2 - + Y1V - fka)

-(Y + v + S)i -ji2(y + v + S)2 - 4i(ySi + Yiv - //ka)

If i(y + S + v)2 < 4(y$i + Yiv - /ka)/ then A2 and A3 are two complex conjugates. The asymptotic stability of E0 can be guaranteed if the following sufficient condition is satisfied [39]:

\arg(Ai) I > a j, i = 1 , 2 / 3/

If 0 < R0 < 1. then the above three characteristic roots will have negative real parts. Thus, E0 is locally asymptotically stable.

If R0 > 1. then at least one eigenvalue will be a positive real root. Thus, E0 is unstable and the endemic equilibrium E* emerges. Biologically, the local asymptotic stability of the DFE indicates that the infection by the virus of hepatitis B never persists. □

As a matter of fact, the region of stability of the fractional-order model is greater than the region of stability of the integer-order model.

The characteristic equation (4.1) can be rewritten as:

P (A) = A3 + a1 A2 + a2 A + a3 = 0 . where:

a1 = ¡x + v + 5 + y,

S. S.M., Y. A.M./Journal of the Egyptian Mathematical Society 000 (2017) 1-7

Fig. 6. The infected hepatocytes for different a with <5 = 0 .502.

a2 = // + 5 + Y) + Y (5 + v) - /

a3 = y/(S + v) - oftk.

The discriminant of P(A) is given by:

D(P) = 18aia2as + (a1 a2 )2 - 4as (a! )3 - 4(a2 )3 - 27(a3)2.

If the discriminant of P( A)D ( P) is positive, then Routh-Hurwitz conditions [40,41]are the necessary and sufficient conditions for (4.3), i.e.,

a1 > 0 . a3 > 0 . a1a2 > a3 if D(P) > 0.

Next, we discuss the stability of E* for R0 > 1. The Jacobian matrix at E* is given by

-ftko . //S

vy ' V J = [?(./% - ^

Y(/+J) /

) -(/ + S) e^Or1

\arg(s.)| > a — , i = 1. 2 . 3 .

The discriminant of G(s) is given by

1 b1 b 2 b3 0

0 1 b 1 b2 b3

D (G ) = - 3 2b1 b 2 0 0

0 3 2b1 b2 0

0 0 3 2b 1 b2

Then the characteristic equation of the linearized system of (2.1) at E* is

G (s) = s3 + b 1s2 + b2s + b3, where

Bko 5/

b1 = ----— + / + S + y,

b = ftko - Sftok + Sftok + Yftok - //Sy

2 YV V V V '

SBko //S2 y

b3 = Y/S ---+ L-1-.

According to the stability conditions in [39] and [42] we have the following proposition.

Proposition 2. E/ is locally asymptotic stable if all eigenvalues of J(E/) satisfy

That is, D(G) = 18b1 b2b3 + (b1 b2)2 - 4 b3fa3 - 4b3 - 27b2 .

Now considering the stability conditions in [43], the following proposition can be stated.

Proposition 3.

(i) If Ro > 3 then E* is locally asymptotically stable.

(ii) If D(G) > 0, and the conditions of Routh-Hurwitz are satisfied, so,

b3 > 0 , b3 > 0 , b3 b2 > b3 ,

then E* is locally asymptotically stable.

(iii) If D(G) < 0, b3 > 0 , b2 > 0 , b3b2 = b3, a s (0, 1), then E* is locally asymptotically stable.

(iv) If D(G) < 0, b3 > 0, b2 > 0, b3 > 0, a s (0 . 5 , 33 )* then E* is locally asymptotically stable.

(v) If D(G) < 0, b3 < 0, b2 < 0, a > 2 , then E* is unstable.

5. Numerical methods and results

In this section, a numerical solution of system (2.1) is given to verify the correctness of the theoretical analysis in Section 3. The PECE method is applied [44]. We used the same values of the parameters mentioned in [9], that is, k = 5 x 105 cel Is. (mL.d)-1,

o = 6 . 24d

¡1 = 0. 003d -1 , $ = 4 x 10-10 ml. (copies.d * 0 . 65d-1 , v = 0 . 1d-1 , x (0) = 1. 7 x 108 cel Is. (ml )-1 , y (0) = 0 . z(0) = 400 copies. (ml )-1 . In order to illustrate the effect of the fractional-order parameter a on the infection, we considered different values of it, that is, a s [0.75, 0.85].

The disease-free equilibrium E0 = ( 0. 0) = (1.6667 x 108 , 0 . 0) is unstable when R0 > 1 as seen in Figs. 1-3 for different S and asymptotically stable when R0 < 1 as seen in Figs. 4 and 5. As it is quite clear from Fig. 6, in the fractional-order case, the peak of the infection is reduced, however, the disease takes a longer time to be eradicated.

S. S.M., Y. A.M./Journal of the Egyptian Mathematical Society 000 (2017) 1-7

6. Conclusion and future work

In this work, a model for HBV infection with a fractional-order derivative is introduced as a generalization of an integer-order model. The positivity of the solution is proved. Local asymptotic stability of fixed point is discussed. Numerical simulations are performed to confirm the analysis by applying PECE method. Since fractional-order models possess memory, FODE gives us a more realistic way to model viral dynamics. Indeed, the global asymptotic behavior of FODE is still open since the chain rule is not valid there and this will be our future work.

Acknowledgment

The authors would like to thank the anonymous referees for their valuable comments which improved the manuscript.

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