Global stability of a HBV infection model with saturated infection rates and time delay

Weijuan Pang, Zhixing Hu and Fucheng Liao

Department of Applied Mathematics, University of Science and Technology Beijing, Beijing, 100083, P. R. China

Corresponding author: Weijuan Pang, E-mail address:ustbpwj@163.com

Keywords: HBV, saturated infection rate, Lyapunov functional, global stability.

Abstract.This paper investigates the global stability of a viral infection model of HBV infection of

hepatocytes with saturated infection rate and intracellular delay. we obtain if the basic reproductive

number is less than or equal to one, the infection-free equilibrium is globally asymptotically stable.

If it’s greater than one, we obtain the sufficient conditions for the global stability of the infected

equilibrium.

Introduction

Mathematical models have been proven to be valuable in understanding the dynamics of HIV,

HBV and HCV infection. Recently, there have been lots of papers on virus dynamics within host,

some include an immune response [1, 5], and others don't contain an immune response [2-4]. Being

different from the epidemic compartment models, virus models concentrate on the disease dynamics

within an infected individual.

In 2012, Cruz [6] introduces a viral infection model of HBV infection of hepatocytes with

“cure” of infected cells and intracellular delay. Although the incidence rate in most HBV models is

bilinear, actual infection rates are probably not strictly linear in each variable. Therefore, it is

reasonable for us to assume that the infection rate in saturated mass action / (1 )p qxv vβ α+ , where

p , q and α are positive constants [7].

In this paper, we shall investigate model with saturated infection rate ( 1).p q= = The model is

written in the following form:

d ( ) ( )( ) ( ) ( ),

d 1 ( )

d ( ) ( )( ) ( ) ( ),

d 1 ( )

d( ) ( )e ( ),

d

q

x t v tx t b dx t y tt v t

x t v ty t y tt v t

v t y t v tt

τ

β δα

β µ δα

σ τ γ−

= − − + + = − + +

= − −

(1.1)

with the initial conditions

1 2 3( ) ( ), ( ) ( ), ( ) ( ),

( ) 0, [ ,0], (0) 0, ([ , 0], ), 1, 2,3.i i i

x y v

C R i

θ ϕ θ θ ϕ θ θ ϕ θϕ θ θ τ ϕ ϕ τ +

= = =

≥ ∈ − > ∈ − = (1.2)

where ( )x t , ( )y t and ( )v t represent the number of the uninfected hepatocytes, the

productively infected hepatocytes and virus particles at time t , respectively. The uninfected

hepatocytes are generated at the rate b , die at the rate d and become infected hepatocytes by

virus at the rate ( ) ( )x t v tβ . µ is the death rate of the infected hepatocytes. σ is the virion

production rate for infected cell and γ is the clearance rate of free virion. δ denotes the rate at

which uninfected hepatocytes are created through “cure”, τ denotes the time necessary for the

newly produced virion by the infected hepatocytes to the mature and infectious particles, e qτ− is

the probability of survival of immature virions.

By the fundamental theory of functional differential equations, system (1.1) has a unique positive

solution ( ( ), ( ), ( ))x t y t v t , satisfying initial conditions (1.2).

Advanced Materials Research Vols. 791-793 (2013) pp 1314-1317Online available since 2013/Sep/04 at www.scientific.net© (2013) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMR.791-793.1314

All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP,www.ttp.net. (ID: 129.186.1.55, Iowa State University, Ames, United States of America-21/09/13,07:05:56)

The existence of equilibria and local stability

Define the basic reproduction number as [ ]0 e / ( )qR b dτβσ γ µ δ−= + , then it’s not difficult to

obtain the following conclusion.

Theorem 2.1 The system (1.1) always possesses uninfected equilibrium 0 ( ,0,0)b

Ed

. If 0 1R > ,

then there exists a unique infected equilibrium * * * *( , , )E x y v , where

( ) ( )

** * *

0 0

0

( ), ( 1), ( 1).

( ) ( )

b y b dx y R v R

d d R d

µ β µ δα µ δ µβ α µ δ µβ

− += = − = −+ + + +

Theorem 2.2 If 0 1R < , then the uninfected equilibrium 0E is locally asymptotically stable; If

0 1R > , 0E is unstable.

Proof: Denote 0 / ,x b d= the characteristic equation of system (1.1) at 0E is of the form

2

1 2 3( )( e ) 0,d a a a λτλ λ λ −+ + + + = (2.1)

where 0

1 2 3 0, ( ), e ( ) .qa a a x Rτγ µ δ γ µ δ σβ γ µ δ−= + + = + = − = − +

Equation (2.1) has a negative root dλ = − . All the other roots of (2.1) are determined by the

equation 2

1 2 3e 0.a a a λτλ λ −+ + + = (2.2) When 0,τ = (2.2) will becomes 2

1 2 3 0.a a aλ λ+ + + =

Clearly, 1 0a > and 2 3 0( )(1 ).a a Rγ µ δ+ = + − so 0 1R < , the uninfected equilibrium 0E is

locally asymptotic stable when 0.τ = if 0,τ > assume ( 0)iw w > is a root of (2.2), we have 4 2 2 2 2

1 2 2 3( 2 ) 0. w a a w a a+ − + − = 2 2 2

1 22 ( ) 0,a a γ µ δ− = + + > 2 2 2 2

2 3 0 0( ) (1 )(1 ).a a R Rγ µ δ− = + + −

If 0 1,R < 2 2

2 3 0,a a− > equation (2.2) has no pure imaginary root. Hence if 0 1,R < the

uninfected equilibrium 0E is locally asymptotically stable for all 0τ ≥ .

If 0 1,R > let 2

1 2 3( ) ef a a a λτλ λ λ −= + + + , then

2 3 0(0) ( )(1 ) 0, lim ( ) .f a a R fλ

γ µ δ λ→∞

= + = + − < = +∞

Hence (2.2) has necessarily a positive root, the uninfected equilibrium 0E is unstable if 0 1.R >

Theorem 2.3 If 0 1,R > the infected equilibrium *E is locally asymptotically stable.

Proof: At * * * *( , , ),E x y v the characteristic equation of system (1.1) is

3 2

1 2 3 1 2( )e 0,b b b c c λτλ λ λ λ −+ + + + + = (2.3)

where * *

1 2* *, ( ) ( ) ( ),

1 1

v vb d b d d

v v

β βµ δ γ µ δ γ µ δ γ µ γα α

= + + + + = + + + + + ++ +

* * *

3 1 2* * 2 * 2( ) , e , e .

1 (1 ) (1 )

q qv x xb d c c d

v v v

τ τβ β βγ µ δ µγ σ σα α α

− −= + + = − = −+ + +

When 0,τ = the equation (2.3) becomes 3 2

1 2 1 3 2( ) 0,b b c b cλ λ λ+ + + + + = (2.4)

Obviously * *

1 2 1 3 2 2 1* *( ) ( ) ( ( ) ( )) ( )( )

1 1

v vb b c b c d d d b c

v v

β βµ δ γ µ γ δ γα α

+ − + = + + + + + + + ++ +

*

* *

1( ( ) (1 )( ) ) 0,

1 1

vd d

v v

βµ µ δ γ γ µ δ γα α

+ + + + + − + >+ +

which implies that all the roots of equation (2.4) have negative real part.

When 0,τ > Suppose that ( 0)iw wλ = > is a root of (2.3). we have

6 2 4 2 2 2 2 2

1 2 2 1 3 1 3 2( 2 ) ( 2 ) 0,w b b w b bb c w b c+ − + − − + − = (2.5)

Where

2* * *

2 2 2 2 2

1 2 * * *

22 2 2 0,

1 1 1

v v vb b d d

v v v

β β δβµ δµ δ γα α α

− = + + + + + + + > + + +

2 2 2 2 2 2 2 * *

2 1 3 12 ( ) 2( ) / (1 )b b b c d d d d v vµ δ γ µ µδ δγ β α− − = + + + + + +

Advanced Materials Research Vols. 791-793 1315

*

2 2 2 2 2 2 2

* * 2

1( )( ) ( ) 1 0,

1 (1 )

vd

v v

βγ µ γ µ δ γα α

+ + + + + − ≥ + +

* *

3 2 * * 2( ) e 0

1 (1 )

qv xb c d d

v v

τβ βγ µ δ µγ σα α

−− = + + + >+ +

, 2 2

3 2 3 2 3 2( )( ) 0.b c b c b c− = + − >

Similar to the above method, we get the infected equilibrium *E is locally asymptotically

stable.

Global asymptotical stability

Theorem 3.1 If 0 1,R ≤ then the uninfected equilibrium 0E is globally asymptotically stable.

Proof: Define a Lyapunov functional

( )20 2 0

1 0 0 0

1 ( )( ) ( ( ) ) ( ( ) ) ( ) ( ) ( ) ( ) ( )d .

2 2( ) e qV t x t x x t x y t y t v t y t u u

x d x

τ

τδ µ δ µ δ

µ σ −

+= − + − + + + + + −+ ∫

By the relations 0b dx= and 0 0 2 0 0( ) ( )( ) ( )( ) ( )( ),x t v t x x v t x x x v t x x− = − + − we have

( )

0 2 0 2 0 01

0 0 0

0 2 0 2

0

0 2

0

d ( ) ( )( ( ) ) ( ( ) ) ( ( ) ) ( ( ) ) ( )

d (1 ( )) 1 ( )

( ) ( ) ( )( ( ) ) ( )( ( ) ) ( ) ( ) ( )

( ) 1 ( ) e

( ( ) )( )

1 ( )

q

V d v t v tx t x x t x x t x x t x y t

t x x v t v t x

x t v td x t x d x t x y t y t v t

d x v t

v d x t xd

v d x

τ

β β δα α

δ β γ µ δµ µµ α σ

β δ δµα µ

−

= − − − − − − + −+ +

+− − + + − + + −+ +

−= − + + −+ +

02 2

00

( )( ) ( ) (1 ) ( ),

( ) 1 ( ) e q

xy t v t R v t

d x v t ταβ γ µ δ

µ α σ −

+− − −+ +

from which we obtain 1d0

d

V

t≤ holds for all ( ), ( ), ( ) 0x t y t v t > when 0 1.R ≤ By LaSalle’s

invariance principle, equilibrium 0E of system (1.1) is globally asymptotically stable.

Theorem 3.2 If 0

( )1

( )R

d

β µ δβδ α µ δ

+< <− +

, then the infected equilibrium *E is globally

asymptotically stable.

Proof: Define a Lyapunov functional * *

2 21 22*( ) ( ) ( ),

1

x vV t V t V t

v

βα

= ++

where ( )2* * * *

21 * *

( )( ) ( ) ln ( ( ) ) ( ( ) )

2( )

x tV t x t x x x t x y t y

x d x

δµ

= − − + − + − +

* *

* * * *

* * * *

( ) e ( )( ) ln ( ) ln ,

(1 )

qy t x v v ty t y y v t v v

y y v v

τβσ α

+ − − + − − +

* *

22 * *0

1 ( )( ) ( ) ln d .

y t uV t y t u y y u

y y

τ −= − − −

∫

In order to achieve simplification in next calculation, we note that the infected equilibrium *E

meets the equalities * *

* *

*,

1

x vb dx y

v

β δα

= + −+

* *

* *( ) ,

(1 )

x v

y v

βµ δα

+ =+

*

*

e,

q y

v

τσγ−

= * *,b dx yµ= +

we have * 2

* * * 221

* *

* * *

* * * *

* * * * * *

* * * * * *

d ( ) ( ( ) )( ( ) ) ( ( ) )

d ( ) ( ) ( )

1 ( ) 1 ( ) ( )(1 )

1 1 1 ( )

( ) 1 ( ) ( ) ( ) ( )(

1 ( ) 1 ( ) ( )

V dx t x t xdx y y t y t y

t d x t x d x

x v v t v v t v t

v v v t v v

x v x y t v x t y v t y t y t v

v x t y v t x y t v y y

δ δµδ δµ µ

β α αα α α

β α τ τα α

−= − − + + − −+ +

+ +− − − ++ + +

+ − −− + + − ++ + *

1 ( )4).

( ) 1

v t

v t v

αα

++ −+

* * *

22

* * * * *

d ( ) ( ) ( ) ( ) ( )ln ln ln

d ( ) ( ) ( )

V y t y t y t v x t y v t x

t y y y v t x y t v x t

τ τ− −= − + + + +

1316 Chemical and Mechanical Engineering, Information Technologies

Combining the two part of the Lyapunov functional 2 ( )V t , we obtain * 2 * * * *

* * * 22

* * *

* * * 2 * * * *

* * * * * *

* *

*

d ( ) ( ( ) )( ( ) ) ( ( ) ) ( 1 ln )

d ( ) ( ) ( ) 1 ( ) ( )

( ( ) ) ( ) ( )( 1 ln )

1 (1 )(1 ( )) 1 ( ) ( )

1(

1

V dx t x t x x v x xdx y y t y t y

t d x t x d x v x t x t

x v v t v x v y t v y t v

v v v v t v y v t y v t

x v

v

δ δµ βδ δµ µ α

β α β τ τα α α α

βα

−= − − + + − − − − −+ + +

− − −− − − −+ + + +

+−+

* * *

* * * * *

( ) ( ) 1 ( ) ( ) ( )ln 2)

1 ( ) ( ) 1 ( )

v x t y v t v t x t y v t

v t x y t v v x y t v

α αα α

++ − −+ +

Noticed that * * * *

* * * * * * *

1 ( ) ( ) 1 ( ) ( ) ( ) ( ) ( )2 2 ln

1 ( ) ( ) 1 ( ) ( )

v x t y v t v t x t y v t x t y v t

v t x y t v v x y t v x y t v

α αα α

+ ++ ≥ ≥ ++ +

. Hence, we suppose

that * * 0,dx yδ− > By the LaSalle’s invariance principle, the uninfected equilibrium *E of

system (1.1) is globally asymptotically stable.

Substituting the expression of * *,x y into * * 0,dx yδ− > we get 0

( ),

( )R

d

β µ δβδ α µ δ

+<− +

hence if 0

( )1

( )R

d

β µ δβδ α µ δ

+< <− +

, the infected equilibrium *E is globally asymptotically stable.

Conclusion

In this paper, we consider a viral infection model of HBV infection of hepatocytes with a class

of saturated infection rate and time delay. In our analysis, For 0 1,R ≤ we establish the global

asymptotic stability of the uninfected equilibrium in Theorem 3.1 and for 0 1,R > we obtain

sufficient conditions on which the uninfected equilibrium is globally asymptotically stable. Further

analysis and proofs on the global stability of the uninfected equilibrium are needed and this problem

will be concerned in our further studies.

Acknowledgments

This work was supported by National Natural Science Foundation of China (61174209,

11071013), the Basic Scientific Research Foundation of Central University (FRF-BR-11-048B,

FRF-BR-12-004), and the Basic Theory Research Foundation for Engineering Research Institute of

USTB (YJ2012-001).

References

[1] C. Bartholdy, J. P. Christensen, D. Wodarz, A. R. Thomsen, Persistent virus infection despite

chronic cytotoxic T-lymphocyte activation in gamma interferon-deficient mice infected with

lymphocytic chrori-omeningitis virus, J. Virol.74 (2000), 10304-10311.

[2] S. Bonhoeffer, R. M. May, G. M. Shaw, M. A. Nowak, Virus dynamics and drug therapy, Proc.

Natl. Acad. Sci. USA, 94 (1997), 6971-6976.

[3] A. V. M. Herz, S. Bonhoeffer, R. M. Anderson, R. M. May, M. A. Nowak, Viral dynamics in

vivo: Limitations on estimates of intracellular delay and virus decay, Proc. Natl. Acad. Sci. USA, 93

(1996), 7247-7251.

[4] A. Korobeinikov, Global properties of basic virus dynamics models, Bulletin of Mathematical

Biology, 66 (2004), 879-883.

[5] W. M. Liu, Nonlinear oscillations in models of immune responses to persistent viruses,

Theoretical Population Biology , 52 (1997), 224-230.

[6] Cruz Vargas-De-Leon, Stability analysis of a model for HBV infection with cure of infected

cells and intracellular delay, Applied Mathematics and Computation, 219 (2012), 389-398.

[7] Xinyu Song, Avidan U. Neumann,Global stability and periodic solution of the viral dynamics, J.

Math. Anal. Appl., 329 (2007), 281-297.

Advanced Materials Research Vols. 791-793 1317

Chemical and Mechanical Engineering, Information Technologies 10.4028/www.scientific.net/AMR.791-793 Global Stability of a HBV Infection Model with Saturated Infection Rates and Time Delay 10.4028/www.scientific.net/AMR.791-793.1314