Journal of the Egyptian Mathematical Society (2014) xxx, xxx–xxx

Egyptian Mathematical Society

Journal of the Egyptian Mathematical Society

www.etms-eg.orgwww.elsevier.com/locate/joems

ORIGINAL ARTICLE

Global stability of a virus dynamics model with cure rate

and absorption

Khalid Hattaf a,b,*, Noura Yousfi a

a Department of Mathematics and Computer Science, Faculty of Sciences Ben M’sik, Hassan II University,P.O. Box 7955, Sidi Othman, Casablanca, Moroccob Centre Regional des Metiers de l’Education et de la Formation (CRMEF), Derb Ghalef, Casablanca, Morocco

Received 2 October 2013; revised 13 December 2013; accepted 24 December 2013

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KEYWORDS

Virus dynamics;

Compound matrices;

Global stability

Corresponding author at: Ce

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orocco. Tel.: +212 06644078

mail address: k.hattaf@yaho

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Abstract In this paper, we investigate a mathematical model which takes account the cure of

infected cells and the loss of viral particles due to the absorption into uninfected cells. The global

stability of the model is determined by using the direct Lyapunov method for disease-free equilib-

rium, and the geometrical approach for chronic infection equilibrium.

2010 MATHEMATICS SUBJECT CLASSIFICATION: 34D20; 34D23; 37N25; 92D30

ª 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society.

1. Introduction

The aim of this work is to study the dynamical behavior of thefollowing model describing the interaction between the suscep-

tible host cells ðxÞ, infected cells ðyÞ and free virus ðvÞ, thismodel is formulated by the following nonlinear system ofdifferential equations:

ional des Metiers de l’Educa-

Derb Ghalef, Casablanca,

Hattaf).

tian Mathematical Society.

g by Elsevier

. Hattaf, N. Yousfi, Global stabical Society (2014), http://dx.d

ing by Elsevier B.V. on behalf of E

2.010

_x ¼ k� dx� fðx; y; vÞvþ qy;

_y ¼ fðx; y; vÞv� ðaþ qÞy;_v ¼ ky� uv� ifðx; y; vÞv;

ð1Þ

where the susceptible host cells are produced at a rate k, die ata rate dx and become infected by virus at a rate fðx; y; vÞv.Infected cells may be killed because of viral or immune effects,or they may be lost by noncytolytic elimination of the cccDNAin their nucleus. The loss rate of infected cells is given by aþ q,where a is the death rate of infected cells and q is the reversionrate into the uninfected state. The term qy into first equation of(1) gives a measure of the uninfected cells which are created

through ‘‘cure’’, per unit time. Recently, this cure of infectedcells is considered by several works [1–6]. Finally, free virusis produced by infected cells at a rate ky, decays at a rate uv

and the parameter i takes only the values 0 or 1. When i ¼ 0corresponds to the system treated by Hattaf et al. in [6], andi ¼ 1 takes account the loss of viral particles when it entersthe target cells. Note that, when a pathogen enters an

ility of a virus dynamics model with cure rate and absorption,oi.org/10.1016/j.joems.2013.12.010

gyptian Mathematical Society.

2 K. Hattaf, N. Yousfi

uninfected cell, the number of pathogens in the blood de-

creases by one. This is called the absorption effect, which isconsidered in [9–11] and is ignored by many authors such as[1–8]. As in [6–8], we assume that the function fðx; y; vÞ is

continuously differentiable in the interior of R3þ and satisfies:

fð0; y; vÞ ¼ 0; for all y P 0 and v P 0; ðH1Þ

@f

@xðx; y; vÞ > 0; for all x > 0; y P 0 and v P 0; ðH2Þ

@f

@yðx; y; vÞ 6 0 and

@f

@vðx; y; vÞ 6 0; 8 x; y; v P 0: ðH3Þ

The rest of our paper is organized as follows. Section 2 dealswith some preliminary results concerning positivity and bound-

edness of solutions, basic reproduction number and existence ofequilibria. In Section 3, we discuss the stability of equilibria.The paper ends with some applications in Section 4.

2. Preliminaries

In this section, we establish the positivity and boundedness ofsolutions, basic reproduction number and existence of equilibria.

2.1. Positive invariance and boundedness

Theorem 2.1. The octant R3þ ¼ fðx; y; vÞ 2 R3 : x P 0; y

P 0; v P 0g is positively invariant with respect (1). Moreover,all solutions of (1) are uniformly bounded in the compact subsetC ¼ ðx; y; vÞ 2 R3

þ : xþ y 6 kd ; v 6 kk

ud

� �, where d ¼ minfa; dg.

Proof. The positive invariance of the positive orthant is trivial.

It remains to show that the system (1) is uniformly bounded.Let xðtÞ; yðtÞ; vðtÞð Þ be any solution with positive initial condi-tions ðx0; y0; v0Þ. Adding the first two equations of the system

(1) gives, ddtðxþ yÞ ¼ k� dx� ay 6 k� dðxþ yÞ, with

d ¼ minfa; dg. Then we obtain that lim supt!1ðxþ yÞ 6 kd.

On the other hand, from the third equation of the system, it

is easy to see that lim supt!1v 6kkud. Hence, all solutions of

the system (1) which start in R3þ are eventually confined in

the region C. This completes the proof. h

2.2. Basic reproduction number and equilibria

By a simple calculation, system (1) has always one disease-freeequilibrium Ef

kd; 0; 0

� �. Therefore, the basic reproduction num-

ber of (1) is given by

R0 ¼ðk� ðaþ qÞiÞf k

d; 0; 0

� �uðaþ qÞ : ð2Þ

Using the same technique in [6], we deduce that there exists aunique endemic equilibrium when R0 > 1. Hence, we have thefollowing result.

Theorem 2.2.

(i) If R0 6 1, then the system (1) has a unique disease-freeequilibrium of the form Ef

kd ; 0; 0� �

.

Please cite this article in press as: K. Hattaf, N. Yousfi, Global stabJournal of the Egyptian Mathematical Society (2014), http://dx.d

(ii) If R0 > 1, the disease-free equilibrium is still present and

the system (1) has a unique chronic infection equilibriumof the form E�ðx�; y�; v�Þ with x� 2 0; k

d

� �; y� > 0 and

v� > 0.

3. Local and global stability of equilibria

The Jacobian matrix of (1) at an arbitrary point is given by

J ¼

�d� @f@xv � @f

@yvþ q � @f

@vv� f

@f@xv @f

@yv� ðaþ qÞ @f

@vvþ f

�i @f@xv k� i @f

@yv �u� i fþ @f

@vv

� �

0BB@

1CCA: ð3Þ

Based on Jacobine matrix approach by evaluating (3) at Ef andE�, we can obtain the following results.

Theorem 3.1. The disease-free equilibrium Ef is locally asymp-

totically stable if R0 < 1 and it is unstable if R0 > 1.

Theorem 3.2. Suppose that R0 > 1. If i ¼ 0 or if i ¼ 1 and thefunction f satisfies the following hypothesis

fðx; y; vÞ þ v@f

@v

� �P 0; for all x; y; v P 0; ðH4Þ

then the chronic infection equilibrium E� is locally asymptoti-cally stable.

Remark 3.3. The assumption (H4) is verified by different typesof the incidence rate including the mass action, the standard

incidence, the saturation incidence, Beddington-DeAngelisincidence function, Crowley-Martin incidence function andthe more generalized incidence function proposed by Hattaf

el al. (see Section 5 in [8]).

Based on the following Lyapunov functional

VðtÞ ¼ kaþq yðtÞ þ vðtÞ, it is not hard to establish the following

theorem.

Theorem 3.4. Ef is globally asymptotically stable in C if a P dand R0 6 1.

In order to establish the global stability of the chronic infec-tion equilibrium E� when R0 > 1, we need first to show the fol-lowing lemma.

Lemma 3.5. If R0 > 1, the system (1) is uniformly persistent.

Proof. This lemma follows from a uniform persistence result,Theorem 4.3 in [12]. To show that system (1) satisfies all theconditions of Theorem 4.3 in [12] if R0 > 1, we chooseX ¼ R3 and the set E ¼ C. The maximal invariant set M on

the boundary @C is the singleton Ef and is isolated. By Theo-rem 4.3 in [12], we can see that the uniform persistence of sys-tem (1) is equivalent to the unstability of the disease-free

equilibrium Ef. Hence, by Theorem 3.1, we know if R0 > 1,the system (1) is uniform persistence. h

Next, we establish a set of conditions which are sufficientfor the global stability of the chronic infection equilibriumE�. According to Lemma 3.5, we know if R0 > 1, the system

(1) is uniform persistence. Hence, there exists a compact

ility of a virus dynamics model with cure rate and absorption,oi.org/10.1016/j.joems.2013.12.010

Global stability of a virus dynamics model 3

absorbing set K � C [15]. Along each solution ðxðtÞ; yðtÞ; vðtÞÞof (1) such that X0 ¼ ðxð0Þ; yð0Þ; vð0ÞÞ 2 K, we put

�p1 ¼ lim supt!1

supX02K

1

t

Z t

0

� @f@y

yðsÞ � @f@v

vðsÞ� �

ds;

�q1 ¼ lim supt!1

supX02K

1

t

Z t

0

@f

@xyðsÞ � @f

@yyðsÞ � @f

@vvðsÞ

� �ds:

Theorem 3.6. Assume R0 > 1 and (H4) hold. Ifmaxfkyv � i @f@y y; i

@f@x yg ¼

kyv � i @f@y y and i�p1 < d or if i�q1 < d, then

E� is globally asymptotically stable.

Proof. To investigate the global stability of E�, we apply the

geometrical approach developed by Li and Muldowney in[13]. The second additive compound matrix of the Jacobianmatrix J, given by (3), is defined by

J½2� ¼j11 þ j22 j23 �j13

j32 j11 þ j33 j12

�j31 j21 j22 þ j33

0B@

1CA; ð4Þ

where jkl is the ðk; lÞth entry of J. Let P ¼ diag 1; yv; yv

� �. Then

PfP�1 ¼ diag 0;

_y

y� _v

v;

_y

y� _v

v

� �;

where matrix Pf is obtained by replacing each entry pij of P by

its derivative in the direction of solution of (1). In addition, wehave

B ¼ PfP�1 þ PJ½2�P�1 ¼

B11 B12

B21 B22

� �;

where

B11 ¼ �ðaþ dþ qÞ � @f

@xvþ @f

@yv;

B12 ¼ vy

@f@vvþ fðx; y; vÞ

� �vy

@f@vvþ fðx; y; vÞ

� �� ;

B21 ¼k� i @f

@yv

� yv

i @f@xy

0@

1A; B22 ¼

b11 b12

b21 b22

� �;

which

b11 ¼_y

y� _v

v� u� d� @f

@xv� i fþ @f

@vv

� �; b12 ¼ q� @f

@yv;

b21 ¼@f

@xv; b22 ¼

_y

y� _v

v� a� q� uþ @f

@yv� i fþ @f

@vv

� �:

Let ðw1;w2;w3Þ denote the vector in R3, choose a norm in R3 asjw1;w2;w3j ¼ maxfjw1j; jw2j þ jw3jg and let l be the Lozinskiimeasure with respect to this norm. Then we have the following

estimate, see [14]:

lðBÞ 6 supfg1; g2g; ð5Þ

where g1 ¼ l1ðB11Þ þ jB12j and g2 ¼ jB21j þ l1ðB22Þ, here l1

denotes the Lozinskii measure with respect to l1 vector norm,

jB12j and jB21j are matrix norms with respect to l1 norm. More-over, we have

g1 ¼_y

y� dþ v2

y

@f

@v� @f

@xvþ @f

@yv 6

_y

y� d: ð6Þ

Please cite this article in press as: K. Hattaf, N. Yousfi, Global stabJournal of the Egyptian Mathematical Society (2014), http://dx.d

and

g2 ¼_y

y� _v

v� dþmax

ky

v� i

@f

@yy; i

@f

@xy

�� u

� i fþ @f@v

v

� �: ð7Þ

If max kyv� i @f

@yy; i @f

@xy

n o¼ ky

v� i @f

@yy, then

g2 ¼_y

y� d� i

@f

@yyþ @f

@vv

� �: ð8Þ

From (5), (6) and (8), we get lðBÞ 6 _yy� d� i @f

@yyþ @f

@vv

� ,

Hence,

�q2 ¼ limt!1

sup supX02K

1

t

Z t

0

lðBÞds 6 �dþ i�p1 < 0:

In general case, we have

g2 ¼_y

y� dþ i

@f

@xy� @f

@yy� @f

@vv

� �: ð9Þ

From (5), (6) and (9), we get lðBÞ 6 _yy� dþ

i @f@xy� @f

@yy� @f

@vv

� . Consequently,

�q2 ¼ limt!1

sup supX02K

1

t

Z t

0

lðBÞds 6 �dþ i�q1 < 0:

By Theorem 3.5 in [13], E� is globally asymptoticallystable. h

From Theorem 3.6, we obtain the following result.

Corollary 3.7. Assume R0 > 1 and (H4) hold. If i ¼ 0, then E� isglobally asymptotically stable.

4. Applications

Here, we give some examples of incidence functions for whichwe apply our theoretical results concerning the global stabilityof E� when R0 > 1.

Example 1. Mass Action when fðx; y; vÞ ¼ bx. In this case, the

hypotheses H1, H2,H3 and H4, are satisfied. In addition,

Z t

0

@f

@xyðsÞ � @f

@yyðsÞ � @f

@vvðsÞ

� �ds ¼

Z t

0

byðsÞds:

Since _xþ _y ¼ k� dx� ay, we have

Z t

0

byðsÞds 6 bkat� b

aðxðtÞ þ yðtÞ � xð0Þ � yð0ÞÞ:

Thus �q1 6bka. By applying Theorem 3.6, we deduce that E� is

globally asymptotically stable if i ¼ 0 or if i ¼ 1 and bk < da.

Example 2. Standard Incidence when fðx; y; vÞ ¼ bxxþy. In the

same, the hypotheses H1, H2, H3 and H4 are satisfied.Moreover,

Z t

0

@f

@xyðsÞ � @f

@yyðsÞ � @f

@vvðsÞ

� �ds ¼

Z t

0

byðsÞxðsÞ þ yðsÞ ds 6 bt:

ility of a virus dynamics model with cure rate and absorption,oi.org/10.1016/j.joems.2013.12.010

4 K. Hattaf, N. Yousfi

Then �q1 6 b. From Theorem 3.6, E� is globally asymptoti-cally stable if i ¼ 0 or if i ¼ 1 and b < d.

Example 3. Saturation Incidence when fðx; y; vÞ ¼ bx1þv. Then

H1,H2,H3 and H4 are satisfied and @f@x yðsÞ ¼

by1þv 6

byv . If we

suppose that b < k, we get max kyv �

@f@y y;

@f@x y

n o¼ ky

v . Further,

Z t

0

� @f@y

yðsÞ � @f@v

vðsÞ� �

ds ¼Z t

0

bxðsÞvðsÞð1þ vðsÞÞ2

ds 6

Z t

0

bxðsÞds:

Since _xþ _y ¼ k� dx� ay, we have

Z t

0

bxðsÞds 6 bkdt� b

dðxðtÞ þ yðtÞ � xð0Þ � yð0ÞÞ:

Hence �p1 6bkd. By Theorem 3.6, we deduce that E� is globally

asymptotically stable if i ¼ 0 or if i ¼ 1 and b < minðk; ddk Þ.

Acknowledgment

The authors would like to thank the anonymous referees and

the editor for their valuable remarks and comments whichhave led to improve the quality of this work.

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