Analysis of a viral infection model with immune impairment and cure rate

Transcription

1 Analysis of a viral infecion model wih immune impairmen and cure rae Xuewei Shi Jianwen Jia School of Mahemaics and Compuer Science, Shanxi Normal Universiy, Shanxi, Linfen,1, P.R. China Absrac: In his paper, he dynamics behavior of a delayed viral infecion model wih immune impairmen and cure rae is sudied. I is shown ha here exiss hree equilibria. By analyzing he characerisic equaions, he local sabiliy of he infecion-free equilibrium and he immune-exhaused equilibrium of he model are esablished. In he following, he sabiliy of he posiive equilibrium is sudied. Furhermore, we invesigae he exisence of Hopf bifurcaion by using a delay as a bifurcaion parameer. Finally, numerical simulaions are carried ou o explain he mahemaical conclusions. Keywords Viral infecion; Immune impairmen; Cure rae; Hopf bifurcaion; Sabiliy 1 Inroducion Mahemaical models of epidemic and viral dynamics have araced much aenion in recen years. A proper model may play a significan role in he developmen of a beer undersanding of he disease and he various drug herapy sraegics used agains i. There has been much ineres in mahemaical modeling of viral dynamics wihin-hos. During he process of viral infecion, as soon a virus invades hos cells, Cyooxic T Lymphocye (CTL) cells which aack infeced cells play an imporan role in responding o he aggression. In order o invesigae he role of he populaion dynamics of viral infecion wih CTL response, Nowak and Bangham [1] consruced a mahemaical model ha describing he basic dynamics of he ineracion beween acivaed CD + T cells x(), infeced CD + T cells y(), viruses v() and immune cells z(). There are hree kinds of models ha describe he ineracion beween a virus populaion y() and he number of CTL cells z() [,, ]: a self-regulaing CTL response z = c bz, a linear CTL response z = cy bz, and a nonlinear CTL response z = cyz bz. Besides, Barholdy e al. [] and Wodarz e al. [6] found ha he urnover of free virus is much faser han ha of infeced cells, which allowed hem o make a quasi-seady sae assumpion: v () =, ha is, he amoun of free virus is simply proporional o he number of infeced cells. Furhermore, Time delays should be considered in viral This work is suppored by Naural Science Foundaion of Shanxi province (111-). Corresponding auhors. addresses: jiajw.8@16.com (J. Jia) 1

2 models, especially he delay beween viral appearance and he producion of new immune paricles. A delayed viral infecion model wih immune response is given by Xie e al. [7] x () = s dx() βx()y(), y () = βx()y() ay() py()z(), (1.1) z () = cy( τ)z( τ) bz(), where acivaed CD + T cells are produced a a consan rae s, decay a a rae d, and can become infeced a a rae ha is proporional o he number of infeced CD + T cells y() wih a ransmission rae consan β. The infeced CD + T cells are assumed o decay a he rae of a and kill a a rae p by he CTL responses. The CTL responses proliferae a a rae proporional o he number of infeced CD + T cells a a previous ime cy( τ)z( τ) and decay a a rae b. Furhermore, τ is he ime delay of CTL responses. However, here are numerous experimenal resuls suggesing ha he virus generaes muans which escape from specific immune responses [8, 9]. For example, during he course of HIV-1 infecion, he modulaion of dendriic cells by HIV infecion is a key aspec in viral pahogenesis. I conribues o viral evasion from immuniy because he dysfuncion of dendriic cells engenders some impairmen effecs for CTL inducemen [1], and Regoes e al. provided some models wih immune impairmen. Iwasa e al. [11] discovered he exisence of so-called Risky hreshold and Immunodeficiency hreshold on he impairmen rae. The former implies ha immune sysem may collapse when he impairmen rae of HIV exceeds he hreshold value. The laer implies ha he immune sysem always collapses when he impairmen rae exceeds he value. Wang e al. [1] considers he following DDE model wih an immune impairmen erm myz. x () = s dx() βx()y(), y () = βx()y() ay() py()z(), (1.) z () = cy( τ) bz() my()z(), where m denoes he immune impairmen rae. In addiion, he consideraion of cure rae of infeced cells is anoher innovaion in he modeling for viral dynamics. Recenly, his cure of infeced cells is considered by several works [1, 1]. Noe ha he significance of cure rae for he viral infecion model, on he basis of he sysem (1.), in his paper, we invesigae he viral model as follows: x () = s dx() βx()y() + ρy(), y () = βx()y() (a + ρ)y() py()z(), (1.) z () = cy( τ)z( τ) bz() my()z(), where he sae variables x, y, z and he parameers s, d, β, a, p, c, b and m have he same biological meanings as in he model (1.1) and (1.). ρ is he rae of cure, we assume ha each infeced cell rever o he uninfeced sae a a rae ρ. Suppose c > m in his paper.

3 The iniial condiions for sysem (1.) ake he form x(θ) = ϕ 1 (θ), y(θ) = ϕ (θ), z(θ) = ϕ (θ), ϕ i (θ), θ [ τ, ), ϕ i () > (i = 1,, ), (1.) where (ϕ 1 (θ), ϕ (θ), ϕ (θ)) C([ τ, ], R+ ), we denoe by C = C([ τ, ], R + ) he Banach space of coninuous funcions mapping he inerval [ τ, ] ino R+ wih he opology of uniform convergence, where R+ = {(x 1, x, x ) x i, i = 1,, }. In his paper, our focus is o sudy dynamics of model (1.). In he nex Secion, he condiions for exisence of equilibria and he boundedness of soluions are derived. In Secion, by analyzing he corresponding characerisic equaions, we sudy he local asympoic sabiliy of he infecionfree equilibrium and he immune-exhaused equilibrium of sysem (1.). Meanwhile, we discuss he global sabiliy of he infecion-free equilibrium by means of suiable Lyapunov funcional and LaSalle invariance principle in his secion. We analyze he local sabiliy of he posiive equilibrium and he exisence of he Hopf bifurcaion occurring a he posiive equilibrium in Secion. Finally, some numerical simulaions are given in Secion. The exisence of equilibria and he boundedness of soluions In his secion, we sudy he equilibrium of sysem (1.). Denoe R = βs (a + ρ)d, R 1 = β[(c m)s + ρb] (a + ρ)[(c m)d + βb], where R and R 1 are called he basic reproducion raio and he immune response reproducive number of model (1.), respecively. I is easy o show he following conclusions. Theorem.1 (1) E (x,, ), where x = s d. If R < 1 and R 1 < 1, sysem (1.) only has an infecion-free equilibrium () If R > 1 and R 1 < 1, in addiion o he infecion-free equilibrium E, sysem (1.) has anoher equilibrium E 1 (x 1, y 1, ), which corresponding o he survival of free virus and he exincion of CTL. I is called as immune-exhaused equilibrium, where x 1 = a+ρ β and y 1 = sβ (a+ρ)d aβ. () If R > 1 and R 1 > 1, in addiion o he infecion-free equilibrium E, sysem (1.) has anoher infeced equilibrium E (x, y, z ), which corresponding o he survival of free virus and CTL, where x = (c m)s+bρ (c m)d+bβ, y = b c m and z = βx (a+ρ) p. Nex, we prove all soluions of sysem (1.) are posiive and ulimaely bounded. Theorem. Under he above iniial condiions (1.), all soluions of sysem (1.) are posiive and here exiss a posiive consan M such ha each soluion saisfies x() < M, y() < M, z() < M afer sufficienly large ime. Proof. By he second and hird equaions of sysem (1.) we have y() = y()e (βx(θ) (a+ρ) pz(θ))dθ and z() = z()e (b+my(θ))dθ respecively. By (1.), obviously y() > and z() > for all >. Furher, we show x() >. If no, hen we le 1 > be he firs ime such ha x() =. By he firs equaion of sysem (1.), we have x ( 1 ) = s + ρy( 1 ) >. Tha means x() < for ( 1 ε, 1 ),

4 where ε is an arbirarily small posiive consan. This leads o a conradicion. I follows ha x() is always posiive. Nex, we prove he ulimae boundedness of soluions of (1.). Le N() = x() + y() + p cz( + τ). Since all soluions of (1.) are posiive, we have N () = s dx() ay() p s dx() ay() pb cz( + τ) < s qn. pb pm y()z() cz( + τ) c y( + τ)z( + τ) where q = min{d, a, b}. Therefore, N() < s q + ε for all large, ε is an arbirarily small posiive consan. Thus, here exiss some posiive consan M such ha x() < M, y() < M, z() < M afer sufficienly large ime. The sabiliy analysis of infecion-free equilibrium and he immuneexhaused equilibrium In he following, we will discuss he local sabiliy of infecion-free equilibrium and he immuneexhaused equilibrium for sysem (1.) by analyzing he corresponding characerisic equaions. For simpliciy, Le E = (x, y, z) is he arbirarily equilibrium of sysem(1.), hen he characerisic equaion abou E is given by λ + d + βy βx ρ βy λ βx + (a + ρ) + pz py =. (.) mz cze λτ λ + b + my cye λτ Theorem.1 If R < 1 and R 1 < 1, hen he infecion-free equilibrium E (x,, ) is locally asympoically sable, and E is unsable if R > 1. Proof. By (.), he characerisic equaion abou E is (λ + d)(λ + b)(λ βs d + a + ρ) =. (.1) I is clear ha Eq. (.1) have eigenvalues λ 1 = d <, λ = b < and λ = βs d If R < 1, hen λ <, herefore E is locally asympoically sable. If R > 1, hen λ >, so E is unsable. (a + ρ). Theorem. If R > 1 and R 1 < 1, hen he immune-exhaused equilibrium E 1 (x 1, y 1, ) is locally asympoically sable, and E 1 is unsable if R > 1 and R 1 > 1. Proof. By (.), he characerisic equaion abou E 1 is [λ + (d + βy 1 )λ + aβy 1 ](λ + b + my 1 cy 1 e λτ ) =. I is easy o see ha λ 1 + λ = (d + βy 1 ) <, deermined by he following equaion λ 1 λ = aβy 1 >, hen λ 1 <, λ <, and λ is f(λ) λ + b + my 1 cy 1 e λτ =. (.)

5 If R > 1, R 1 < 1 and τ =, hen λ = (c m)y 1 b = (c m) sβ (a+ρ)d aβ b < (c m) sβ dβx (βx ρ)β b =. Therefore, if τ =, R > 1 and R 1 < 1, E 1 is locally asympoically sable. If τ >, le iω(ω > ) is a soluion of Eq. (.). On subsiuing λ = iω ino Eq. (.) and separaing real and imaginary pars, we have iω = icy 1 sin(ωτ), b + my 1 = cy 1 cos(ωτ). Squaring and adding he wo equaions of (.), we derive ha ω = [(c m)y 1 b][(c + m)y 1 + b]. From R > 1 and R 1 < 1, we have (c m)y 1 b <. This leads o a conradicion. So, Eq. (.) has no pure imaginary roo. Therefore, E 1 is locally asympoically sable for all τ. If R > 1 and R 1 > 1, hen f() = b (c m)y 1 <, lim λ f(λ) = +. Hence, Eq. (.) has a leas one posiive real roo. So he immune-exhaused equilibrium E 1 is unsable. Nex, we sudy he global sabiliy of infecion-free equilibrium E of sysem (1.). Theorem. If R < 1, R 1 < 1 and d < a, d < b, hen he infecion-free equilibrium E (x,, ) is globally asympoically sable. Proof. Define Lyapunov funcional V () = y() + p c m z() + cp y(θ)z(θ)dθ. c m τ Taking derivaive of V () along he posiive soluions of he sysem (1.), we ge V () = < ( sβ d (βx() (a + ρ))y() + ( cp c m mp c m p)y()z() (a + ρ))y(). pb c m z() Obviously, V () = if and only if y() =, by he Lasalle invariance principle we have lim y() =. Hence, he limi equaion of sysem (1.) is (.) x () = s dx(). (.) Noe ha he equilibrium of Eq. (.) is x = s d which is globally asympoically sable. Thus, by he limi equaion we ge E (x,, ) is globally aracive. I is proved ha E is locally asympoically sable by he Theorem.1 when R < 1 and R 1 < 1. Therefore, E is globally asympoically sable. The sabiliy of infeced equilibrium and Hopf bifurcaion The characerisic equaion of sysem (1.) abou E is given by P (λ, τ) + Q(λ, τ)e λτ =, (.1)

6 where P (λ, τ) = λ + b 1 λ + b λ + b, Q(λ, τ) = b λ + b λ + b 6, and b 1 = b + d + (m + β)y, b = (b + my )(d + βy ) mpy z + β x y ρβy, b = (β x y ρβy )(b + my ) mpy z (d + βy ), b = cy, b = cpy z cy (d + βy ), b 6 = cpy z (d + βy ) cβ x y + cρβy. When R 1 > 1 and τ =, Eq. (.1) becomes λ + (b 1 + b )λ + (b + b )λ + b + b 6 =. (.) By direcly calculaing, we have βx ρ >, λ + Aλ + Bλ + C =, where A = b 1 + b = b + d + (m + β)y cy = d + βy >, B = b + b = (b + my )(d + βy ) mpy z + β x y ρβy + cpy z cy (d + βy ) = bpz + (βx ρ)βy >. C = b + b 6 = (β x y ρβy )(b + my ) mpy z (d + βy ) + cpy z (d + βy ) cβ x y + cρβy = bpz (d + βy ) >. AB C = (d + βy )[bpz + βy (βx ρ)] bpz (d + βy ) = βy (βx ρ)(d + βy ) >. According o Rouh-Hurwiz crierion, we know ha all roos of (.) have negaive real par. Hence, he infeced equilibrium E (x, y, z ) is locally asympoically sable when τ =. Therefore, we have he following Theorem. Theorem.1 If τ =, R > 1 and R 1 > 1, hen he infeced equilibrium E (x, y, z ) is locally asympoical sable. In order o invesigae he locally asympoically sable of E for τ >, we will apply he following resul ha has been proposed by Boese (199) [1]. Lemma.1 Consider Eq. (.1), where P (λ) and Q(λ) are analyic funcions in Reλ > and saisfy he following condiions: (1) P (λ) and Q(λ) have no common imaginary roo; () P ( iy) = P (iy), Q( iy) = Q(iy) for real y; () P () + Q() ; () lim sup{ Q(λ) P (λ) : λ, Reλ } < 1; () F (y) P ( iy) Q( iy) for real y has a mos a finie number of real zeros. Then he following saemens are rue (a) If F (y) = has no posiive roos, hen no sabiliy swich may occur. (b) If F (y) = has a leas one posiive roo and each of hem is simple, hen as τ increases, a finie number of sabiliy swiches may occur, and evenually he considered equaion becomes unsable. In he following, we check Eq. (.1) saisfying Lemma.1. Obviously, P (iω) + Q(iω) = (b 1 + b )ω + b + b + iω(ω + b + b ). 6

7 This implies condiion (1) of Lemma.1 is saisfied. And condiion () is rue, for We have P ( iω) = iω b 1 ω + ib ω + b = P (iω), Q( iω) = b ω + ib ω + b 6. P () + Q() = b + b 6 >, lim sup Q(λ) λ P (λ) = b λ + b λ + b 6 λ + b 1 λ = < 1. + b λ + b Condiion () and () are also rue. To deermine he sabiliy of he equilibrium E, he exisence of posiive roos of he following equaion needs o be analyzed F (ω) P ( iω) Q( iω) = ω 6 + a 1 ω + a ω + a =, (.) where a 1 = b 1 b b, a = b b 1b b + b b 6, a = b b 6. Le h = ω, we denoe G(h) h + a 1 h + a h + a =. (.) Then G (h) = h + a 1 h + a =. (.) Thus Eq. (.) has wo real roos h 1 = a 1+ and h = a 1, where = a 1 a. Now, we inroduce he following resuls which were proved by Song and Yuan [16]. Lemma. For he polynomial Eq. (.), we have he following resuls: (1) If a <, hen (.) has a leas one posiive roo. () If a, and, hen (.) has no posiive roo. () If a, and >, hen (.) has posiive roo if and only if h 1 = a 1 > and G(h 1 ). From Lemma., we know ha if a, and = a 1 a, hen Eq. (.) has no posiive roo. If a,a 1 > and a >, hen Eq. (.) also has no posiive roo. Hence, we can conclude he following heorem. Theorem. Suppose ha R > 1, R 1 > 1, if a 1 >, a > and a, or a and, hen he infeced equilibrium E (x, y, z ) is locally asympoically sable for τ >. In he following, we sudy he Hopf bifurcaion of he infeced equilibrium. We assume ha for some τ >, iω(ω > ) is a roo of (.1). On subsiuing λ = iω ino Eq. (.1) and separaing real and imaginary pars, we have b b 1 ω = (b 6 b ω ) cos ωτ b ω sin ωτ, b ω ω = (b 6 b ω ) sin ωτ b ω cos ωτ. Squaring and adding he wo equaions of (.6), hen ω saisfies Eq.(.). Le h = ω, hen h saisfies Eq.(.). From Lemma., we know ha if a 1 >, a > and a <, hen Eq. (.) has a leas one posiive roo. Under he condiion, if h, hen G (h) >. Tha is o say, G(h) is monoone increasing in h [, + ). Thus, if a <, here is a unique posiive ω saisfying (.), i.e. he characerisic Eq. (.6) 7

8 (.1) has a pair of purely imaginary roos of he form ±iω. From (.6), we ge he corresponding τ n > such ha he characerisic Eq.(.1) has a pair of purely imaginary roos ±iω, where τ n = 1 ω arccos (b 1ω b )(b 6 b ω ) + b ω (ω b ω ) (b 6 b ω ) + (b ω ) + nπ ω, n =, 1,,. In he following, we will prove ha d(reλ) dτ τ=τn >. This will signify ha here exiss a leas one eigenvalue wih posiive real par for τ > τ n. Differeniaing (.1) wih respec o τ, we ge (λ + b 1 λ + b ) dλ dτ + (b λτ dλ λ + b )e dτ τ(b λ λτ dλ + b λ + b 6 )e dτ λ(b λ + b λ + b 6 )e λτ =. This gives By Calculaion, we have sign{ d(reλ) dτ ( dλ dτ ) 1 = = λ +b 1 λ+b λe λτ (b λ +b λ+b 6 ) + b λ+b λ(b λ +b λ+b 6 ) τ λ λ +b 1 λ b λ (λ +b 1 λ +b λ+b ) + b λ b 6 λ (b λ +b λ+b 6 ) τ λ. } τ=τn = sign{re( dλ dτ ) 1 } λ=iω = sign{ ω6 +(b 1 b b )ω (b b 6 ) } ω 6 +a 1ω a ω [(b 6 b ω ) +(b ω ) ] = sign{ ω [(b 6 b ω ) +(b ω ) ] }. By a 1 = b 1 b b >, a = b b d(reλ) 6 <, we ge dτ τ=τn >. This roo of characerisic Eq.(.1) crosses he imaginary axis from he lef o he righ as τ coninuously varies from a number less han τ n o one greaer han τ n. Therefore, he ransversal condiion holds and he condiions for Hopf bifurcaion are hen saisfied a τ = τ n. Then, from he above analysis, we can conclude he following heorem. Theorem. Assume ha R > 1, R 1 > 1, a 1 >, a >, a < and ω is he firs posiive roo of Eq.(.), hen he infeced equilibrium E (x, y, z ) is locally asympoically sable when τ < τ and unsable when τ > τ, where τ = 1 ω arccos (b 1ω b )(b 6 b ω ) + b ω (ω b ω ) (b 6 b ω ) + (b ω ). Furhermore, when τ = τ, sysem (1.) undergoes a Hopf bifurcaion o periodic soluions a he infeced sae E. Numerical simulaions To demonsrae he heoreical resuls obained in his paper, we will give some numerical simulaions. Example 1. s = ; β =. ; d =.1 ; ρ = ; a = ; p =.1 ; c =. ; b =.1 ; m =.1 and τ =. Through calculaion, we ge R =.7871 < 1 and R 1 = < 1, sysem (1.) only has an infecion-free equilibrium E (,, ). Hence, according o Theorem.1, E is locally asympoically sable for his case (see Fig. 1 (a)-(c) and (d)). Example. s = ; β =. ; d =.1 ; ρ = ; a = ; p =.1 ; c =. ; b =.1 ; m =.1 and τ =. Through calculaion, we ge R = > 1 and R 1 =.987 < 1, hen in addiion 8

9 o he infecion-free equilibrium E, sysem (1.) has anoher equilibrium E 1 (,, ). According o Theorem., he immune-exhaused equilibrium E 1 (,, ) is locally asympoically sable for his case (see Fig. (a)-(c) and (d)). Example. s = ; β =. ; d =. ; ρ = ; a =. ; p =. ; c =. ; b =.9 ; m =. and τ =. Through calculaion, we ge R = 1.6 > 1 and R 1 = 1. > 1, hen sysem (1.) has anoher infeced equilibrium E (8.6, 9, 9.18). Besides, we ge a 1 = 7.1 >, a = 1.81 > and a = 19.9 >, according o Theorem., E is locally asympoically sable for his case (see Fig. (a)-(c) and (d)). Example. s = ; β =. ; d =. ; ρ = 1 ; a = ; p =. ; c =. ; b =.9 ; m =.. Here, R = 1.67 > 1, R 1 = 1.6 > 1, a 1 =.681 >, a =.8 > and a =. <, which saisfy he condiions of Theorem.. Hence, we shall uilize hese values o work ou he criical ime delay preserving sabiliy or no. Through calculaion, we obain he criical ime delay τ =.96. Therefore, he infeced equilibrium E (9, 9,.) remains locally asympoically sable for τ =.1 < τ (see Fig. (a)-(c) and (d)), while for τ = 1 > τ, E is unsable (see Fig. (a)-(c) and (d)). (1a) Time series of x() of he sysem. 1 (1b) Time series of y() of he sysem. 8 6 x() 1 y() (1c) Time series of z() of he sysem. (1d) z() z() y() 1 x() Fig 1.R =.7871 < 1 and R 1 = < 1, he infecion-free equilibrium E (,, ) is locally asympoically sable. 6 Conclusion In his paper, we have sudied a delayed viral infecion model wih immune impairmen and cure rae. There exis hree equilibria: infecion-free equilibrium E (x,, ), immune-exhaused equilibrium E 1 (x 1, y 1, ), and he posiive equilibrium E (x, y, z ). By analyzing he model, we obained he basic reproducion raio R and he immune response reproducive number R 1. If R < 1 and R 1 < 1, E is locally asympoically sable for all τ, and E is also globally asympoically sable when d < a, d < b (see Fig 1). If R > 1 and R 1 < 1, E 1 is locally asympoically sable for all τ (see Fig ). Furhermore, we discuss he sabiliy and he exisence of Hopf bifurcaion of he infeced equilibrium. If R > 1, R 1 > 1, a 1 >, a > and a >, hen E is locally asympoically sable for all τ (see Fig ). However, if a <, as τ increases, sabiliy swiches may occur. There exiss 9

10 1 6 7 (1a) Time series of x() of he sysem. 8 (1b) Time series of y() of he sysem. 7 6 x() y() (1c) Time series of z() of he sysem. (1d) z() 6 z() y() x() Fig.R = > 1 and R 1 =.987 < 1, he immune-exhaused equilibrium E 1 (,, ) is locally asympoically sable. 1 (1a) Time series of x() of he sysem. 9 (1b) Time series of y() of he sysem x() 6 y() x x 1 (1c) Time series of z() of he sysem. (1d) z() z() x 1 y() x() Fig.R = 1.6 > 1, R 1 = 1. > 1, a 1 = 7.1 >, a = 1.81 > and a = 19.9 >, he infeced equilibrium E (8.6, 9, 9.18) is locally asympoically sable for τ. 1 (1a) Time series of x() of he sysem. 9 (1b) Time series of y() of he sysem x() 8 y() (1c) Time series of z() of he sysem. (1d). z(). z() y() 7 8 x() Fig. R = 1.67 > 1, R 1 = 1.6 > 1, a 1 =.681 >, a =.8 > and a =. <, τ =.96, τ =.1 < τ, he infeced equilibrium E (9, 9,.) remains locally asympoically sable. 1

11 (1a) Time series of x() of he sysem. 1 (1b) Time series of y() of he sysem x() y() x x 1 6 (1c) Time series of z() of he sysem. (1d) z() z() x y() 6 x() 8 1 Fig. R = 1.67 > 1, R 1 = 1.6 > 1, a 1 =.681 >, a =.8 > and a =. <, τ =.96, τ = 1 > τ, E (9, 9,.) is unsable a posiive number τ such ha E is locally asympoically sable for τ < τ and unsable for τ > τ (see Fig and Fig ). References [1] M.A. Nowak, C.R.M. Bangham, Populaion dynamics of immune response o persisen viruses, Science 7(1996) [] R.J. De Boer, A.S. Perelson, Towards a general funcion describing T cell proliferaion, J. Theor. Biol. 17(199) [] R.J. De Boer, A.S. Perelson, Targe cell limied and immune conrol models of HIV infecion: a comparison, J. Theor. Biol. 19(1998) 1-1. [] M.A. Nowak, R.M. May, Viral Dynamics, Oxford Universiy Press, Oxford,. -8. [] C. Barholdy, J.P. Chrisensen, D. Wodarz, A.R. Thomsen, Persisen virus infecion despie chronic cyooxic T-lymphocye acivaion in Gamma inerferon-deficien mice infecion wih lymphocyic choriomeningiis virus, J. Virol. 7() [6] D. Wodarz, J.P. Chrisensen, A.R. Thomsen, The imporance of lyic and nonlyic immune response in viral infecions, Trends Immunol. () 19-. [7] Q. Xie, D. Huang, S. Zhang, J. Cao, Analysis of a viral infecion model wih delayed immune response, Appl. Mah. Model. (1) [8] Eric Avila-Vales, Noé Chan-Chí, Gerardo García-Almeida, Analysis of a viral infecion model wih immune impairmen, inracellular delay and general non-linear incidence rae, Chaos, Solions, Fracals. 69(1)

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