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In this paper we propose and analyze an HCV dynamics model taking into consideration the cure of infected hepatocytes and antibody immune response. We incorporate both virus-to-cell and cell-to-cell transmissions into the model. We incorporate a distributed-time delay to describe the time between the HCV or infected cell contacts an uninfected hepatocyte and the emission of new active HCV. We show that the solutions of the proposed model are nonnegative and ultimately bounded. We derive two threshold parameters which fully determine the existence and stability of the three steady states of the model. Using Lyapunov functionals, we established the global stability of the steady states. The theoretical results are confirmed by numerical simulations.

Hepatitis C virus is considered one of the dangerous human viruses that infects the liver and causes the lever cirrhosis. Mathematical modeling and analysis of within-host HCV dynamics have been studied by many authors (see e.g. [

The “cure” of infected cells has been considered in the virus dynamics models in several works (see e.g. [

s ˙ ( t ) = β − δ ^ s ( t ) − α 1 s ( t ) p ( t ) − α 2 s ( t ) y ( t ) + ρ y ( t ) , (1)

y ˙ ( t ) = α 1 s ( t ) p ( t ) + α 2 s ( t ) y ( t ) − ε y ( t ) − ρ y ( t ) , (2)

p ˙ ( t ) = m y ( t ) − γ p ( t ) − q z ( t ) p ( t ) , (3)

z ˙ ( t ) = r z ( t ) p ( t ) − μ z ( t ) , (4)

where, s, y, p and z represent the concentration of uninfected hepatocytes, infected hepatocytes, HCV particles and antibodies, respectively. The uninfected hepatocytes are generated at a constant rate β, die at rate δ ^ s , where δ ^ is the natural death rate constant. The infection rate due to both virus-to-cell and cell-to-cell transmissions is given by α 1 s p + α 2 s y , where α 1 and α 2 are constants. The infected hepatocytes die at rate εy and cure at rate ρy, where ε and ρ are constants. Constant m is the generation rate of the HCV from infected hepatocytes. Antibodies attack the HCV at rate q z p , proliferate at rate r z p and die at rate μz, where q, r and μ are constants.

It is assumed in model (1)-(4) that, the hepatocytes can produce HCV particles once they are contacted by HCV or infected cells. However, there is a time period from the moment of the uninfected hepatocytes that are contacted by the HCV or infected cells and the moment of producing new active HCV particles [

The aim of this paper is to study the qualitative behavior of an HCV dynamics model with antibody immune response. We have incorporated distributed time delay and both virus-to-cell and cell-to-cell transmissions. We derive two threshold parameters and establish the global stability of the three steady states of the model using Lyapunov method.

We propose the following HCV dynamics model with distributed time delay:

s ˙ ( t ) = β − δ ^ s ( t ) − α 1 s ( t ) p ( t ) − α 2 s ( t ) y ( t ) + ρ y ( t ) , (5)

y ˙ ( t ) = α 1 s ( t ) p ( t ) + α 2 s ( t ) y ( t ) − ε y ( t ) − ρ y ( t ) , (6)

p ˙ ( t ) = m ∫ 0 h ρ ( τ ) e − μ 1 τ y ( t − τ ) d τ − γ p ( t ) − q z ( t ) p ( t ) , (7)

z ˙ ( t ) = r z ( t ) p ( t ) − μ z ( t ) . (8)

We assume that, the HCV or infected cell contacts an uninfected hepatocyte at time t − τ , the cell becomes infected at time t, where τ is a distributed parameter over the time interval [ 0, h ] . The factors e − μ 1 τ represents the probability of surviving the hepatocyte during the time delay period, where μ 1 is a constant. ρ ( τ ) is a probability distribution function satisfying ρ ( τ ) > 0 and

∫ 0 h ρ ( τ ) d τ = 1 , ∫ 0 h ρ ( υ ) e ς υ d υ < ∞ ,

where ς and h are positive constants. Let us denote Θ ( τ ) = ρ ( τ ) e − μ 1 τ and F = ∫ 0 h Θ ( τ ) d τ , thus 0 < F ≤ 1 . Let the initial conditions for system (5)-(8) be given as:

s ( η ) = ζ 1 ( η ) , y ( η ) = ζ 2 ( η ) , p ( η ) = ζ 3 ( η ) , z ( η ) = ζ 4 ( η ) , ζ j ( η ) ≥ 0 , η ∈ [ − h , 0 ] , ζ j ∈ C ( [ − h ,0 ] , ℝ ≥ 0 4 ) , j = 1, ⋯ ,4, (9)

where C is the Banach space of continuous functions mapping the interval [ − h ,0 ] into ℝ ≥ 0 4 . Then, the uniqueness of the solution for t > 0 is guaranteed [

In this subsection, we investigate the nonnegativity and boundedness of solutions.

Proposition 1. The solutions of system (5)-(8) with the initial states (9) are nonnegative and ultimately bounded.

Proof. From Equation (5) we have s ˙ | s = 0 = β + ρ y > 0 . Hence, s ( t ) > 0 for all y ≥ 0 . Moreover, for all t ∈ [ 0, h ] we have

y ( t ) = ζ 2 ( 0 ) e − ( ε + ρ ) t + ∫ 0 t e − ( ε + ρ ) ( t − η ) [ α 1 s ( η ) p ( η ) + α 2 s ( η ) y ( η ) ] d η ≥ 0 ,

p ( t ) = ζ 3 ( 0 ) e − ∫ 0 t ( γ + q z ( ξ ) ) d ξ + m ∫ 0 t e − ∫ η t ( γ + q z ( ξ ) ) d ξ ∫ 0 h Θ 2 ( τ ) y ( η − τ ) d τ d η ≥ 0,

z ( t ) = ζ 4 ( 0 ) e − ∫ 0 t ( κ − r p ( ξ ) ) d ξ ≥ 0.

By recursive argument we get y ( t ) ≥ 0 , p ( t ) ≥ 0 , and z ( t ) ≥ 0 , for all t ≥ 0 .

Next, we establish the boundedness of the model’s solutions. The nonnegativity of the model’s solution implies that

s ˙ ( t ) ≤ β − δ ^ s ( t ) + ρ y ( t ) ,

We let Q 1 ( t ) = s ( t ) + y ( t ) , then

Q ˙ 1 ( t ) = β − δ ^ s ( t ) − ε y ( t ) ≤ β − σ 1 ( s ( t ) + y ( t ) ) = β − σ 1 Q 1 ( t ) ,

where σ 1 = m i n { δ ^ , ε } . Hence Q 1 ( t ) ≤ L 1 , if Q 1 ( t ) ≤ L 1 where L 1 = β σ 1 . It follows that s ( t ) ≤ L 1 and y ( t ) ≤ L 1 if s ( 0 ) + y ( 0 ) ≤ L 1 . Moreover, let Q 2 ( t ) = p ( t ) + q r z ( t ) , then

Q ˙ 2 ( t ) = m ∫ 0 h Θ ( τ ) y ( t − τ ) d τ − γ p ( t ) − q μ r z ( t ) ≤ m L 1 F − γ p ( t ) − q μ r z ( t ) ≤ m L 1 − σ 2 ( p ( t ) + q r z ( t ) ) = m L 1 − σ 2 Q 2 ( t ) ,

where σ 2 = m i n { γ , μ } . It follows that, lim sup t → ∞ Q 2 ( t ) ≤ L 2 , where L 2 = m L 1 σ 2 . Since p ( t ) ≥ 0 and z ( t ) ≥ 0 , then lim sup t → ∞ p ( t ) ≤ L 2 and lim sup t → ∞ z ( t ) ≤ L 3 , where L 3 = r q L 2 . Therefore, s ( t ) , y ( t ) , p ( t ) and z ( t ) are ultimately bounded. ,

According to Proposition 1, we can show that the region

Δ = { ( s , y , p , z ) ∈ C 4 : ‖ s ‖ ≤ L 1 , ‖ y ‖ ≤ L 1 , ‖ p ‖ ≤ L 2 , ‖ z ‖ ≤ L 3 } ,

is positively invariant with respect to system (5)-(8).

Lemma 1. For system (5)-(8) there exist two threshold parameters R 0 > 0 , and R 1 z > 0 , such that

1) if R 0 ≤ 1 , then there exists only one steady state Π 0 ,

2) if R 1 z ≤ 1 < R 0 , then there exist only two steady states Π 0 and Π 1 ,

3) if R 0 > 1 and R 1 z > 1 , then there exist three steady states Π 0 , Π 1 and Π 2 .

Proof. Let ( s , y , p , z ) be any steady state satisfying

β − δ ^ s − α 1 s p − α 2 s y + ρ y = 0 , (10)

α 1 s p + α 2 s y − ε y − ρ y = 0 , (11)

m F y − γ p − q z p = 0 , (12)

( r p − μ ) z = 0. (13)

We find that system (10)-(13) admits three steady states.

1) Infection-free steady state Π 0 = ( s 0 , 0 , 0 , 0 ) , where s 0 = β / δ ^ .

2) Chronic-infection steady state without immune response Π 1 = ( s 1 , y 1 , p 1 ,0 ) , where

s 1 = ( ε + ρ ) γ F m α 1 + γ α 2 ,

y 1 = δ ^ s 1 ε ( β ( F m α 1 + γ α 2 ) δ ^ γ ( ε + ρ ) − 1 ) ,

p 1 = F m y 1 γ .

Clearly Π 1 exists if

β ( F m α 1 + γ α 2 ) δ ^ γ ( ε + ρ ) > 1.

Let us define

R 0 = β ( F m α 1 + γ α 2 ) δ ^ γ ( ε + ρ ) ,

In terms of R 0 , we can write the steady state components for Π 1 as:

s 1 = s 0 R 0 , y 1 = δ ^ s 1 ε ( R 0 − 1 ) ,

p 1 = F m δ ^ s 1 γ ε ( R 0 − 1 ) .

3) Chronic-infection steady state with humoral immune Π 2 = ( s 2 , y 2 , p 2 , z 2 ) , where

s 2 = r y 2 ( ε + ρ ) μ α 1 + r y 2 α 2 , y 2 = − B + B 2 − 4 A C 2 A , p 2 = μ r , z 2 = γ q ( r m F y 2 μ γ − 1 ) . (14)

where

A = r α 2 ε , B = μ ε α 1 − r β α 2 + r δ ^ ( ε + ρ ) , C = − β μ α 1 . (15)

We note that Π 2 exists when r F 2 m y 2 μ γ > 1 . Now we define

R 1 z = r F m y 2 μ γ = F m y 2 p 2 γ . (16)

Then z 2 = γ q ( R 1 z − 1 ) . We define the basic reproduction number for the

humoral immune response R H u m which comes from the limiting (linearized) z-dynamics near z = 0 as:

R H u m z = p 1 p 2

Lemma 2 1) if R 1 z < 1 , then R H u m z < 1 ,

2) if R 1 z > 1 , then R H u m z > 1 ,

3) if R 1 z = 1 then R H u m z = 1 .

Proof. 1) Let R 1 z < 1 , then from Equation (16) we have y 2 < γ p 2 m F , and then using Equation (14) we get

− B + B 2 − 4 A C 2 A < γ p 2 F m ,

that leads to

( 2 A γ p 2 F m + B ) 2 − ( B 2 − 4 A C ) > 0.

Using Equation (15), we can get

4 α 2 ε 2 μ 2 γ ( F m α 1 + γ α 2 ) m 2 F 2 ( 1 − R H u m z ) > 0

then

R H u m z = r F m s 1 ( R 0 − 1 ) δ ^ μ ε γ < 1.

then R H u m z < 1 . Similarly, one can proof 2) and 3) ,.

The following theorems investigate the global stability of the steady states of system (5)-(8). Let us define the function H : ( 0, ∞ ) → [ 0, ∞ ) as H ( l ) = l − 1 − ln l . Denote ( s , y , p , z ) = ( s ( t ) , y ( t ) , p ( t ) , z ( t ) ) .

Theorem 1. Suppose that R 0 ≤ 1 , then the infection-free steady state Π 0 is globally asymptotically stable (GAS).

Proof. Constructing a Lyapunov functional

L 0 ( s , y , p , z ) = s 0 H ( s s 0 ) + y + α 1 s 0 γ p + q α 1 s 0 r γ z + ρ 2 ( δ ^ + ε ) s 0 [ ( s − s 0 ) + y ] 2 + m α 1 s 0 γ ∫ 0 h Θ ( τ ) ∫ t − τ t y ( η ) d η d τ .

We calculate d L 0 d t along the solutions of model (5)-(8) as:

d L 0 d t = ( 1 − s 0 s ) ( β − δ ^ s − α 1 s p − α 2 s y + ρ y ) + α 1 s p + α 2 s y − ε y − ρ y + α 1 s 0 γ ( m ∫ 0 h Θ ( τ ) y ( t − τ ) d τ − γ p − q z p ) + q α 1 s 0 r γ ( r z p − μ z ) + ρ ( δ ^ + ε ) s 0 [ ( s − s 0 ) + y ] ( β − δ ^ s − ε y ) + m α 1 s 0 γ ∫ 0 h Θ ( τ ) [ y − y ( t − τ ) ] d τ . (17)

Collecting terms of Equation (17) and using β = δ ^ s 0 we obtain

d L 0 d t = ( 1 − s 0 s ) ( δ ^ s 0 − δ ^ s ) + α 2 s 0 y + ( 1 − s 0 s ) ρ y − ( ε + ρ ) y + α 1 s 0 F γ m y − q α 1 s 0 r γ μ z − ρ ( δ ^ + ε ) s 0 [ ( s − s 0 ) + y ] ( δ ^ ( s − s 0 ) + ε y ) . (18)

We note that

( 1 − s 0 s ) ρ y = − ρ y s s 0 ( s − s 0 ) 2 + ρ y s 0 ( s − s 0 ) .

Therefore

d L 0 d t = − δ ^ ( s − s 0 ) 2 s + α 2 s 0 y − ρ y s s 0 ( s − s 0 ) 2 + ρ y s 0 ( s − s 0 ) − ( ε + ρ ) y + α 1 s 0 F γ m y − q α 1 s 0 r γ μ z − ρ δ ^ ( s − s 0 ) 2 ( δ ^ + ε ) s 0 − ρ ε ( s − s 0 ) y ( δ ^ + ε ) s 0 − ρ δ ^ ( s − s 0 ) y ( δ ^ + ε ) s 0 − ρ ε y 2 ( δ ^ + ε ) s 0

= − ( δ ^ s 0 + ρ y + ρ δ ^ s δ ^ + ε ) ( s − s 0 ) 2 s s 0 − ρ ε y 2 ( δ ^ + ε ) s 0 − q α 1 s 0 r γ μ z + ( ε + ρ ) ( ( m α 1 F + γ α 2 ) s 0 γ ( ε + ρ ) − 1 ) y = − ( δ ^ s 0 + ρ y + ρ δ ^ s δ ^ + ε ) ( s − s 0 ) 2 s s 0 − ρ ε y 2 ( δ ^ + ε ) s 0 − q α 1 s 0 r γ μ z + ( ε + ρ ) ( R 0 − 1 ) y .

Since R 0 ≤ 1 , then d L 0 d t ≤ 0 for all s , y , p , z > 0 . Moreover d L 0 d t = 0 if and only if s ( t ) = s ( 0 ) , y ( t ) = z ( t ) = 0 . Let Γ 0 = { ( s , y , p , z ) : d L 0 d t = 0 } and Γ 0

be the largest invariant subset of Γ 0 . The solution of system (5)-(8) tend to Γ 0 . For each element of Γ 0 we have y ( t ) = 0 , then y ˙ ( t ) and Equation (6) we get

y ( t ) = 0 = α 1 s 0 p (t)

Then p ( t ) = 0 . It follows that Γ 0 contains a single point that is { Γ 0 } . Appling LaSalle’s invariance principle (LIP), we get that

Theorem 2. Suppose that

Proof. Let us define a function

Calculating

Collecting terms of Equation (19), we get

Applying condition of equilibrum

we get

thus

We note that

Then

Consider the following equalities

Simplify Equation (20) and let

Equation (22) can be rewrite as:

We note that

From Lemma 2 we have

Theorem 3. Suppose that

Proof. Define a function

Calculating

Collecting terms of Equation (24) and applying the equilibrium conditions for

we get

We note that

Using equalities (21) in case

Equation (25) can be simplified as:

We note that,

This section is devoted to performing some numerical simulations for model (5)-(8). Let us choose

where

Moreover,

Hence, model (5)-(8), becomes

For model (26)-(29), the threshold parameters are given by:

where y_{2} is given by Equation (14). Model (26)-(29) will be solved using the values of the parameters listed in

Now we investigate our theoretical results given in Theorem 1-3. We consider the following two cases:

Case I: Effect of α, μ and h on the asymptotic behaviors of steady states:

In this case, we have chosen three different initial conditions for model (26)-(29) as follows:

Initial-1:

Initial-2:

Initial-3:

Further, we fix the value of

Scenario 1:

Notation | Value | Notation | Value | Notation | Value | Notation | Value |
---|---|---|---|---|---|---|---|

10 | 0.01 | q | 0.1 | 0.1 | |||

0.01 | 0.5 | r | Varied | ||||

Varied | m | 10 | 0.1 | ||||

0.0001 | 3 | Varied |

Scenario 2:

persistence of the HCV particles but with inactive antibody immune response.

Scenario 3:

Case II: Effect of the time delays on the free HCV particles dynamics:

Let us take the initial conditions (Initial-2). We choose the values

From

The steady states | |||
---|---|---|---|

0.0 | 6.73 | 3.47 | |

8 | 3.13 | 1.57 | |

15 | 1.65 | 0.77 | |

23 | 0.85 | 0.35 |

decreases the concentration of infected hepatocytes, free HCV particles, antibodies, and increases the population of uninfected hepatocytes. Therefore, the steady states of the system will eventually stabilized around the healthy state

Elaiw, A.M., Ghaleb, S.A. and Hobiny, A. (2018) Effect of Time Delay and Antibodies on HCV Dynamics with Cure Rate and Two Routes of Infection. Journal of Applied Mathematics and Physics, 6, 1120-1138. https://doi.org/10.4236/jamp.2018.65096