Analysis of a fractional-order model for acute and chronic hepatitis-B transmission with Mittag-Leffler kernels

Definition 2.1

The one-parameter ( E a ) and two-parameter ( E a , b ) M-L functions in terms of the power series are defined as follows:

and

where the function Γ ( w ) is defined as Γ ( w ) = ∫ 0 ∞ u w − 1 e − u d u = ( w − 1 ) ! , ω > 0 .

Definition 2.2

The Atangana-Baleanu fractional derivative in Caputo sense of a function ξ ( t ) ∈ H 1 ( 0 , c ) , c > 0 with χ ∈ ( 0,1 ] is defined as follows:

where

and B ( χ ) = 1 − χ + χ Γ ( χ ) satisfying

for ξ ( t ) ∈ C [ b 1 , b 2 ] and the Lipschitz condition

Definition 2.3

The Atangana-Baleanu fractional integral of a function is ξ ( t ) ∈ H 1 ( 0 , c ) , with c > 0 is defined as follows:

Lemma 3.1

The epidemiologically feasible region of the model (4) is follows:

Proof

The dynamics of the system’s entire human population in (4) is as follows:

By applying the Laplace transform to (6), we obtain

where k = χ ( τ χ + μ χ ) B ( χ ) + ( 1 − χ ) ( τ χ + μ χ ) and B ( χ ) = 1 − χ + χ Γ ( χ ) . By using an asymptotic characteristic of the M-L function [11,21], we observe that N ( t ) ≤ ρ χ ( τ χ + μ χ ) as t → ∞ , and thus, N ( t ) is bounded, and therefore, S ( t ) , A ( t ) , C ( t ) , and R ( t ) are bounded. Hence, Ω is a positive invariant set of system (4).□

Theorem 4.1

The HBV free equilibrium (DFE) point E 0 of the proposed fractional order model (4) is locally asymptotically stable (LAS) if ℛ 0 < 1 and unstable if ℛ 0 > 1 .

Proof

The Jacobian matrix of system (4) at E 0 is expressed as follows:

Therefore, by the Routh-Hurwitz stability conditions for fractional-order systems [4], the necessary and sufficient condition is

for various fractional order models. Therefore, the disease-free equilibrium of system (4) is asymptotically stable if all of the eigenvalues w i , i = 1 , … , 4 of J ( E 0 ) satisfy condition (13). Hence, a sufficient condition for the local asymptotic stability of the equilibrium points is that the eigenvalues w i , i = 1 , … , 4 , of the Jacobian matrix J ( E 0 ) satisfy the condition ∣ a r g ( w i ) ∣ > κ π 2 . This confirms that fractional-order differential equations are, at least, as stable as their integer-order counterparts. The characteristic equation of J ( E 0 ) is expressed as follows:

where P = ( γ χ + ω χ + μ χ ) + ( δ χ + ε χ + μ χ ) , and Q = ( γ χ + ω χ + μ χ ) ( δ χ + ε χ + μ χ ) ( 1 − ℛ 0 ) . For ℛ 0 < 1 , the quadratic equation ( ( λ χ ) 2 + λ χ P + Q ) = 0 has all terms positive, and thus, its roots must all be negative. Using the Routh-Hurwitz criteria for fractional-order models, all the eigenvalues have negative real parts and satisfy the condition given by (13). Therefore, all the eigenvalues have negative real parts if ℛ 0 < 1 . This completes the proof.□

Theorem 4.2

Given χ ∈ ( 0 , 1 ] and ℛ 0 ≤ 1 . Thus, the DFE of system (4) is globally asymptotically stable (GAS) on a positively invariant region Ω .

Proof

Consider a suitable Lyapunov function of the form

By applying the Atangana-Baleanu Caputo derivative on (15), we obtain

Exploiting equations (4) and (15) gives

Hence, D 0 , t χ A B C L ( t ) ≤ 0 for ℛ 0 ≤ 0 , and D 0 , t χ A B C L ( t ) = 0 if and only if S ( t ) = S ( 0 ) and A ( t ) = C ( t ) = R ( t ) = 0 . Thus, the largest compact invariant set is { ( S , A , C , R ) ∈ R 4 : A B C D t χ L ( t ) = 0 } . □

Theorem 4.3

The HBV EE point E * of the proposed fractional order model (4) is LAS if ℛ 0 > 1 and is unstable if ℛ 0 < 1 .

Proof

In a similar way as in Theorem 4.1, the Jacobian matrix can be written as follows:

Therefore, by the Routh-Hurwitz stability conditions for fractional order systems [4,22], the necessary and sufficient condition for stability is

for various fractional order models. Therefore, the EE of system (4) is asymptotically stable if all the eigenvalues w i , i = 1 , … , 4 of J ∣ * ∣ ( E * ) satisfy condition (19). Hence, a sufficient condition for the local asymptotic stability of the equilibrium points is that the eigenvalues w i , i = 1 , … , 4 of the Jacobian matrix satisfy the condition ∣ arg ( w i ) ∣ > κ π 2 . This confirms that fractional order differential equations are, at least, as stable as their integer-order counterparts. The characteristic equation of J ∣ * ∣ ( E * ) is expressed as follows:

The aforementioned equation gives one negative eigenvalue λ 1 χ = − μ χ . The other eigenvalues can be obtained from the following equation:

where

It is clear that B , D , E > 0 whenever ℛ 0 > 1 . Also,

Therefore, by Routh-Hurwitz criteria, all roots of (21) have negative real parts if and only if ℛ 0 > 1 . Therefore, E * is LAS when ever ℛ 0 > 1 . □

Theorem 4.4

The DEE point E * of the proposed fractional-order model (4) is GAS whenever ℛ 0 ≥ 1 .

Proof

To study the global stability of E * for (4), we propose the following Lyapunov function:

Now, we calculate the derivative with respect to time of (31) and then using model (9), we obtain

If we put Φ 1 = ( τ χ + μ χ ) , Φ 2 = ( γ χ + ω χ + μ χ ) , Φ 3 = ( δ χ + ε χ + μ χ ) , and Φ 4 = μ χ . Then, after simple arrangements, we obtain

Since the right-hand side of system (26) has a negative sign, the derivative on right-hand side is less than or equal to zero, i.e., D 0,1 χ A B C L * ≤ 0 . Substituting S = S * , A = A * , C = C * , and R = R * in (26), D 0,1 χ A B C L * yields zero, i.e., D 0,1 χ A B C L * = 0 . Therefore, the largest invariant set in { S ( t ) , A ( t ) , C ( t ) , R ( t ) ∈ R 4 : D 0,1 χ A B C L * = 0 } is the singleton invariant set E * , where E * is the DEE point. Then, by applying invariant principle of LaSalle [26], it implies that E * is globally asymptotically stable.□