global properties of virus dynamics models with multitarget cells and discrete time delays

Hindawi Publishing CorporationDiscrete Dynamics in Nature and Society Volume 2011, Article ID 201274, 19 pages doi:10.1155/2011/201274 Research Article Global Properties of Virus Dynamic

Hindawi Publishing Corporation

Discrete Dynamics in Nature and Society

Volume 2011, Article ID 201274, 19 pages

doi:10.1155/2011/201274

Research Article

Global Properties of Virus Dynamics Models with Multitarget Cells and Discrete-Time Delays

1 Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O Box 80203,

Jeddah 21589, Saudi Arabia

2 Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, Egypt

Correspondence should be addressed to A M Elaiw,a m elaiw@yahoo.com

Received 8 July 2011; Accepted 16 October 2011

Academic Editor: Yong Zhou

Copyrightq 2011 A M Elaiw and M A Alghamdi This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

We propose a class of virus dynamics models with multitarget cells and multiple intracellular delays and study their global properties The first model is a 5-dimensional system of nonlinear delay differential equations DDEs that describes the interaction of the virus with two classes of target cells The second model is a2n  1-dimensional system of nonlinear DDEs that describes the dynamics of the virus, n classes of uninfected target cells, and n classes of infected target cells.

The third model generalizes the second one by assuming that the incidence rate of infection is given by saturation functional response Two types of discrete time delays are incorporated into these models to describei the latent period between the time the target cell is contacted by the virus particle and the time the virus enters the cell,ii the latent period between the time the virus has penetrated into a cell and the time of the emission of infectiousmature virus particles Lyapunov functionals are constructed to establish the global asymptotic stability of the uninfected and infected steady states of these models We have proven that if the basic reproduction number

R0is less than unity, then the uninfected steady state is globally asymptotically stable, and if R0> 1

or if the infected steady state exists, then the infected steady state is globally asymptotically stable

1 Introduction

Nowadays, various types of viruses infect the human body and cause serious and dangerous diseases Mathematical modeling and model analysis of virus dynamics have attracted the interests of mathematicians during the recent years, due to their importance in understanding the associated characteristics of the virus dynamics and guiding in developing efficient anti-viral drug therapies Several mathematical models have been proposed in the literature to

2 Discrete Dynamics in Nature and Society

by a system of nonlinear ordinary differential equations ODEs Others are given by a system

of nonlinear delay differential equations DDEs to account the intracellular time delays The

and given by

where xt, yt, and vt represent the populations of uninfected target cells, infected cells, and free virus particles at time t, respectively Here, λ represents the rate of which new target cells are generated from sources within the body, d is the death rate constant, and β is the

and shows that they die with rate constant a The virus particles are produced by the in-fected cells with rate constant p, and are removed from the system with rate constant c The parameter τ accounts for the time between viral entry into the target cell and the production

of new virus particles The recruitment of virus-producing cells at time t is given by the num-ber of cells that were newly infected cells at time t − τ and are still alive at time t The

of infected cells but not yet virus-producing cells

A great effort has been made in developing various mathematical models of viral in-fections with discrete or distributed delays and studying their basic and global properties, such as positive invariance properties, boundedness of the model solutions and stability

dy-namics model Most of the existing models are based on the assumption that the virus attacks

HBV Since the interactions of some types of viruses inside the human body is not very clear and complicated, therefore, the virus may attack more than one class of target cells Hence, virus dynamics models describing the interaction of the virus with more than one class of

models with two target cells and investigated the global asymptotic stability of their steady

The purpose of this paper is to propose a class of virus dynamics models with multi-target cells and establish the global stability of their steady states The first model considers the interaction of the virus with two classes of target cells In the second model, we assume

that the virus attacks n classes of target cells The third model generalizes the second one by

assuming that the infection rate is given by saturation functional response We incorporate

particles The global stability of these models is established using Lyapunov functionals,

Discrete Dynamics in Nature and Society 3

steady state exists, then the infected steady state is GAS for all time delays

2 Virus Dynamics Model with Two Target Cells and Delays

In this section, we introduce a mathematical model of virus infection with two classes of

particle and the contacting virus enters the cells The recruitment of virus-producing cells at

2.1 Initial Conditions

2.6

4 Discrete Dynamics in Nature and Society

2.2 Nonnegativity and Boundedness of Solutions

Proposition 2.1 Let x1t, y1t, x2t, y2t, vt be any solution of 2.1–2.5 satisfying the

initial conditions2.6, then x1t, y1t, x2t, y2t, and vt are all nonnegative for t ≥ 0 and

ultimately bounded.

Proof From2.1 and 2.3, we have

0d i β i vξdξ  λ i

0

e−

t

η d i β i vξdξ

dη, i 1, 2, 2.7

0

β i x i



η − τ i



v

η − τ i



e −a i t−η dη, i 1, 2,

0



e −n1ω1p1y1



η − ω1





η − ω2



e −ct−η dη,

2.8

˙

˙

2.9

2.3 Steady States

depends on the basic reproduction number given by

R0 e −m1τ1n1ω1p1β1a2x01 e −m2τ2n2ω2 p2β2a1x0

2

Discrete Dynamics in Nature and Society 5

where

R1 e −m1τ1n1ω1p1β1λ1

a1d1c , R2 e −m2τ2n2ω2p2β2λ2

1, 0, x02, 0, 0 which is called uninfected steady state,

E1 x∗

x∗1

⎧

⎪

⎨

⎪

⎩

α5c

α1α5 α2α3, if α4 0,

2α1α4 , if α4/ 0,

x∗2

⎧

⎪

⎨

⎪

⎩

α3c

α1α5 α2α3, if α4 0,

c

α2 α1α5 α2α3− α4c −

y1∗ d1

a1e m1τ1

x0 1

x∗1 − 1



x1∗, y2∗ d2

a2e m2τ2

x0 2

x2∗ − 1



x∗2, v∗ d1

β1

x0 1

x1∗ − 1



,

2.14

where

α1 e −n1ω1m1τ1p1β1

a1 , α2 e −n2ω2m2τ2p2β2

a2 , α3 λ2β1,

α4 β1d2− β2d1, α5 λ1β2.

2.15

2.4 Global Stability

In this section, we prove the global stability of the uninfected and infected steady states of

It is clear that Hz ≥ 0 for any z > 0 and H has the global minimum H1 0.

6 Discrete Dynamics in Nature and Society

Theorem 2.2 i If R0≤ 1, then E0is GAS for any τ1, τ2, ω1, ω2≥ 0.

Proof. i We consider a Lyapunov functional

where

W11 e −m1τ1x01H x1

x01



x20





p1e n1ω1v,

W12 e −m1τ1

0

0

W13 a1

0

0

2.18

dW11

dt e −m1τ1 1−x01

x1



x2





p1e n1ω1˙v,

dW12

dt e −m1τ1

0

d

0

d

0

d

0

d

β1x1v − β1x1t − τ1vt − τ1 γe −m2τ2

β2x2v − β2x2t − τ2vt − τ2.

2.19

dW13

dt a1





It follows that

dW1

dt e −m1τ1 1−x01

x1





λ1− d1x1− β1x1v

 γ



e −m2τ2 1−x02

x2





λ2− d2x2− β2x2v



p1e n1ω1

Discrete Dynamics in Nature and Society 7

β1x1v − β1x1t − τ1vt − τ1 γe −m2τ2

β2x2v − β2x2t − τ2vt − τ2







x1 −x1

x01





x2 −x2

x02



p1 e n1ω1v



x1 −x1

x01





x2 −x2

x02



2.21

Since the arithmetical mean is greater than or equal to the geometrical mean, then the first

1, x2 x0

v 0, ˙v 0 From 2.5 we drive that

1, x2 x0

τ1, τ2, ω1, ω2≥ 0

ii Define a Lyapunov functional as

W2 e −m1τ1x∗1H x1

x∗1



1H y1

y∗1



 γ



e −m2τ2x∗2H



x2

x2∗



2H



y2

y∗2



p1e n1ω1v∗H  v

v∗



0

x∗1v∗



dθ

0

H



x∗2v∗



dθ  a1y∗1

0

y1∗



dθ

0

H



y2∗



dθ.

2.23 Differentiating with respect to time yields

dW2

dt e −m1τ1



x1



λ1− d1x1− β1x1v





y1





8 Discrete Dynamics in Nature and Society

 γ



e −m2τ2



x2



λ2− d2x2− β2x2v





y2





p1e n1ω1



v





β1x1v − β1x1t − τ1vt − τ1  β1x∗1v∗ln



x1v





β2x2v − β2x2t − τ2vt − τ2  β2x∗2v∗ln



x2v





1ln



y1





2ln



y2





x1



y1  a1y1∗

 γ



e −m2τ2



x2



y2



v − a1c

p1 e n1ω1v  a1c

p1 e n1ω1v∗



x1v





x2v





y1





y2



.

2.24

λ1 d1x1∗ β1x∗1v∗, λ2 d2x∗2 β2x∗2v∗, a1y1∗e m1τ1 β1x∗1v∗,

we obtain

dW2

dt e −m1τ1



d1x∗1 a1y∗1e m1τ1− d1x1−x∗1

x1



d1x∗1 a1y1∗e m1τ1− d1x1



v∗



∗

y1x∗1v∗  2a1y∗1

 γ



e −m2τ2

d2x2∗ a2y2∗e m2τ2− d2x2



x2



d2x∗2 a2y∗2e m2τ2− d2x2



Discrete Dynamics in Nature and Society 9

v∗



∗

y2x∗2v∗  2a2y∗2



vy1∗ − γa2y2∗v

vy∗2 −a1c

p1 e n1ω1v∗ v

v∗





x1v





x2v





y1





y2



x1∗ −x1∗

x1



∗ 1

x1 − a1y1∗y

∗

y1x∗1v∗

vy1∗  3a1y∗1 a1y∗1ln



x1vy1





x∗2 −x∗2

x2



∗ 2

x2 − a2y∗2y

∗

y2x∗2v∗

vy2∗  3a2y∗2 a2y∗2ln



x2vy2



p1e n1ω1

e −n1ω1p1y∗1 e −n2ω2p2y∗2− cv∗ v

v∗.

2.26

ln



x1vy1

 ln



x1∗

x1



y1∗v



y1x∗1v∗



,

ln



x2vy2

 ln



x2∗

x2



 ln



y∗2v



y2x2∗v∗



,

2.27

dW2

dt e −m1τ1d1x1∗ 2−x1

x∗1 −x∗1

x1





x∗2 −x∗2

x2





H

1

x1



y∗1v



y1x∗1v∗





H



x∗2

x2



 H



y2∗v



 H



y2x∗2v∗



.

2.28

10 Discrete Dynamics in Nature and Society Since the arithmetical mean is greater than or equal to the geometrical mean, then the first

1, y∗1, x∗2, y∗2, v∗ > 0,

x1 x∗

1, x2 x∗

2, v v∗, and H 0, that is,

3 Basic Virus Dynamics Model with Multitarget Cells and Delays

In this section, we propose a virus dynamics model which describes the interaction of the

incorporated into the model The model is a generalization of those of one class of target cells

˙x i λ i − d i x i − β i x i v, i 1, , n,

˙v

n



i 1

3.1

class i, respectively, v is the population of the virus particles All the parameters of the model

have the same biological meaning as given in the previous section

3.2

  is



Similar to the previous section, the nonnegativity and the boundedness of the solutions

Discrete Dynamics in Nature and Society 11

3.1 Steady States

1, , x0

n , y0

1, , y0

n , v0,

i λ i /d i , y0

1, , x∗n , y1∗, , y n∗, v∗ The coordinates of the infected steady state, if they exist, satisfy the equalities:

λ i d i x∗i  β i x i∗v∗, i 1, , n, 3.3

a i y∗i e −m i τ i β i x i∗v∗, i 1, , n, 3.4

cv∗ n

i 1

R0 n

i 1

R i n

i 1

e −m i τ i n i ω iβ i p i λ i

only with the target cells of class i.

3.2 Global Stability

In the following theorem, the global stability of the uninfected and infected steady states of

Theorem 3.1 i If R0≤ 1, then E0is GAS for any τ i , ω i ≥ 0, i 1, , n.

Proof i Define a Lyapunov functional W1as follows:

W1 n

i 1

γ i



e −m i τ i x0i H x i

x i0



0

0



p1e n1ω1v,

3.7

satisfies

dW1

dt n

i 1

γ i



e −m i τ i 1−x

0

i

x i





λ i − d i x i − β i x i v

β i x i v − β i x i t − τ i vt − τ i a i





12 Discrete Dynamics in Nature and Society

p1e n1ω1

n



i 1



i 1

e −m i τ i γ i λ i



x0

i

x i



p1 e n1ω1

n



i 1

e −m i τ i n i ω ip i β i x0

i

a i c − 1



v

i 1

e −m i τ i γ i λ i



x0

i

x i



3.8

Since the arithmetical mean is greater than or equal to the geometrical mean, then the first

y i , v > 0 Similar to the previous section, one can show that the maximal compact invariant set

in{dW1/dt 0} is the singleton {E0} when R0 ≤ 1 The global stability of E0follows from LaSalle’s invariance principle

W2 n

i 1

γ i



e −m i τ i x∗i H x i

x i∗



i H y i

y i∗



0

x∗i v∗



dθ

0

y∗i



dθ



p1e n1ω1v∗H  v

v∗



.

3.9

Differentiating with respect to time yields

dW2

dt n

i 1

γ i



e −m i τ i



x i



λ i − d i x i − β i x i v





y i





β i x i v − β i x i t − τ i vt − τ i e −m i τ i β i x∗i v∗ln



x i v







y i



p1



v

n



i 1



i 1

γ i



e −m i τ i



λ i − d i x i−λ i x∗i

x i  d i x∗i  β i x∗i v



y i  a i y i∗



x i v





y i



p1 v − v

∗

v

n



i 1

p1 v∗.

3.10

Discrete Dynamics in Nature and Society 13

a1ce n1ω1

p1 v∗ a1e n1ω1

p1

n



i 1

e −n i ω i p i y i∗ n

i 1

we obtain

dW2

dt n

i 1

γ i



e −m i τ i



d i x i∗ e m i τ i a i y∗i − d i x i−x∗i

x i



d i x∗i  e m i τ i a i y∗i



∗

y i x∗i v∗  2a i y∗i − a i y∗i v

vy∗i



x i v





y i



i 1

e −m i τ i γ i β i x i∗−a1ce n1ω1

p1



v

i 1

γ i



e −m i τ i d i x∗i 2− x i∗

x i − x i

x i∗



∗

i

x i − a i y i∗v

vy∗i

∗

y i x∗i v∗  3a i y∗i  a i y i∗ln



x i vy i



p1

n



i 1

e −n i ω i p i y∗i − cv∗



v

v∗.

3.12

equa-lity:

ln



x i vy i

 ln

i

x i



vy i∗



y i x∗i v∗



,

3.13

dW2

dt n

i 1

γ i



e −m i τ i d i x∗i 2−x∗i

x i − x i

x∗i



i

x i



vy∗i



y i x∗i v∗



.

3.14

arithmetical mean is greater than or equal to the geometrical mean and H ≥ 0 Clearly,

14 Discrete Dynamics in Nature and Society

4 Virus Dynamics Model with Saturation Infection Rate

In this section, we proposed a virus dynamics model which describes the interaction of the

virus with n classes of target cells taking into account the saturation infection rate and

multi-ple intracellular delays:

1 α i v t , i 1, , n,

i 1

4.1

4.1 Steady States

1, , x0

n , y01, , y0

n , v0,

E1x∗

1, , x n∗, y1∗, , y n∗, v∗ The coordinates of the infected steady state, if they exist, satisfy the equalities:

λ i d i x i∗ β i x∗i v∗

1 α i v∗, i 1, , n,

a i y i∗ e −m i τ i β i x i∗v∗

1 α i v∗, i 1, , n,

cv∗ n

i 1

e −n i ω i p i y∗i

4.2

4.2 Global Stability

In this section, we study the global stability of the uninfected and infected steady states of

Theorem 4.1 i If R0≤ 1, then E0is GAS for any τ i , ω i ≥ 0, i 1, , n.

Discrete Dynamics in Nature and Society 15

Proof i Define a Lyapunov functional W1as follows:

W1 n

i 1

γ i



e −m i τ i x0i H x i

x0

i



0

0



p1 v,

4.3

dW1

dt n

i 1

γ i



e −m i τ i 1− x i0

x i



λ i − d i x i− β i x i v





β i x i v







p1

n



i 1



i 1

γ i e −m i τ i



λ i − d i x i − λ i

x0i

x i  d i x0

i  β i x0i v



p1 v

i 1

γ i e −m i τ i λ i



x0i − x i0

x i



p1 v  a1ce

n1ω1

p1

n



i 1

e −m i τ i n i ω ip i β i x0i v

i 1

γ i e −m i τ i λ i



x0i − x

0

i

x i



p1 v  a1ce

n1ω1

p1

n



i 1

R i v

i 1

γ i e −m i τ i λ i



x0

i

x i



p1

n



i 1

R i α i v2

4.4

W2 n

i 1

γ i



e −m i τ i x i∗H x i

x∗i



i H y i

y∗i



0



dθ

0

y i∗



dθ



p1 v∗H  v

v∗



.

4.5

16 Discrete Dynamics in Nature and Society Differentiating with respect to time yields

dW2

dt n

i 1

γ i



e −m i τ i



x i



λ i − d i x i− β i x i v







y i







β i x i v









y i



p1



v

n



i 1



i 1

γ i



e −m i τ i



λ i − d i x i−λ i x i∗

x i  d i x∗i  β i x∗i v



y∗i

y i  a i y∗i







y i



p1 v − a1e

n1ω1

p1

v∗ v

n



i 1

p1 v∗.

4.6

a1ce n1ω1

p1 v a1ce

n1ω1

p1 v∗ v

v∗ a1e n1ω1

p1

v

v∗

n



i 1

e −n i ω i p i y∗i v

v∗

n



i 1

we obtain

dW2

dt n

i 1

γ i



e −m i τ i



d i x∗i  e m i τ i a i y∗i − d i x i− x i∗

x i



d i x∗i  e m i τ i a i y i∗



∗

vy i∗





v∗



Discrete Dynamics in Nature and Society 17

i 1

γ i



e −m i τ i d i x i∗ 2−x∗i

x i − x i

x∗i



∗

i

x i  3a i y∗i  a i y i∗



v

v∗



∗

vy∗i





i 1

γ i



e −m i τ i d i x i∗ 2−x∗i

x i − x i

x∗i



∗

i

x i  4a i y∗i − a i y i∗y

∗



v

v∗  1 α i v



vy∗i





.

4.8 Then using the following equalities:

ln



 ln

i

x i



vy∗i



 ln







,

v

v∗  1 α i v

4.9

we obtain

dW2

dt n

i 1

γ i



e −m i τ i d i x i∗ 2−x∗i

x i − x i

x∗i





x∗i

x i



vy i∗



 H







.

4.10

3 Basic Virus Dynamics Model with Multitarget Cells and Delays< /b>

In this section, we propose a virus dynamics model which...

only with the target cells of class i.

3.2 Global Stability

In the following theorem, the global stability of the uninfected and infected steady states of. .. Virus Dynamics Model with Saturation Infection Rate

In this section, we proposed a virus dynamics model which describes the interaction of the

virus with n classes of target