Research Article Stability Analysis of an In-Host Viral Model with Cure of Infected Cells and Humoral Immunity

Transcription

1 Appled Mathematcs Volume 203, Artcle ID 02757, 5 pages Research Artcle Stablty Analyss of an In-Host Vral Model wth Cure of Infected Cells and Humoral Immunty Hu Wang, Rong Wang, Zhxng Hu, and Fucheng Lao Department of Appled Mathematcs, Unversty of Scence and Technology Bejng, Bejng 00083, Chna Correspondence should be addressed to Hu Wang; bkdwh@63.com Receved 29 August 203; Accepted 4 December 203 Academc Edtor: Junje We Copyrght 203 Hu Wang et al. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. An n-host vral model wth cure of nfected cells and humoral mmunty s studed. We prove that the stablty s completely determned by the basc reproductve number R 0 and show that the nfecton-free equlbrum E 0 s globally asymptotcally stable f and only f R 0.Moreover,fR 0 >, the nfecton equlbrum s locally asymptotcally stable when the tme delay τ s small and t loses stablty as the length of the tme delay ncreases past a crtcal value τ 0. Fnally, we confrm our analyss by provdng several numercal examples.. Introducton The humoral mmunty s a knd of mmunologc mechansm whchusesblymphocytestoproduceantgentoprevent vrus and there are evdences to prove that the humoral mmunty s more effectve than the cell-medated mmune n some nfectons such as malara nfecton [ 3]. Many authors present and develop mathematcal systems for the humoral mmunty [4 7].Andthecureofvrussalsomportant especally n HBV models [8, 9]. In the present paper, we analyze an n-host vral model wth humoral mmunty and ntracellular delay, and we ncorporate a cure of nfected cells nto t. We propose the followng system: T (t) =Λ βt(t) V (t) dt(t) +bi(t), I (t) =βt(t) V (t) (a+b) I (t), V (t) =ki(t) uv(t) qb(t) V (t), B (t) =gv(t τ) cb(t), where T, I, V, and B represent the unnfected cells, the nfected cells, the vrus, and the B cells, respectvely. Λ and d are assumed as the brth rate and death rate of unnfected cells. β s the nfecton rate and ki(t) represents the number of free vrus whch s produced durng the average nfected cell lfe span. a s the death rate of nfected cells and u () represents the death rate of vrus. g and c represent the brth rate and death rate of B cells. The B cells neutralzaton rate s represented by q. The followng form s taken as the ntal condtons: T (θ) =φ (θ), I (θ) =φ 2 (θ), V (θ) =φ 3 (θ), B (θ) =φ 4 (θ) φ (θ) 0, θ ( τ, 0), φ (0) >0 ( =, 2, 3, 4), where (φ (θ), φ 2 (θ), φ 3 (θ), φ 4 (θ)) C([ τ, 0], R 4 +0 ), the space of contnuous functons mappng the nterval (, 0] nto R 4 +0,and (2) R 4 +0 ={(x,x 2,x 3,x 4 ) x 0,=,2,3,4}. (3) The organzaton of ths paper s as follows. In the next secton, we wll fnd threshold parameters R 0 of system () and t determnes the exstence of the equlbrums. In Secton 3, by structurng sutable Lyapunov functonals and usng LaSalle s nvarance prncple we attan the global stablty of the unnfected equlbrum f R 0.InSecton4, we consder the stablty of the nfected equlbrum and

2 2 Appled Mathematcs the occurrence of local Hopf bfurcaton. In Secton 5, we present the numercal smulatons to llustrate our results. Fnally, we offer concludng remarks n the last secton. 2. Exstence of Equlbrum We can easly fnd that system () alwayshasanunnfected equlbrum E 0 =(T 0, 0, 0, 0) = ((Λ/d), 0, 0, 0). Denote R 0 = Λkβ ud (a+b). (4) We call R 0 the basc reproductve number. It s easy to prove that f R 0 >,model() exst an nfected equlbrum E (T,I,V,B ),where V =( (βcua (a+b) dgq) 2 + 4Λgqackβ 2 βcua (a+b) dgq) (2aβgq), T = gq (a+b) V + kβc B = g c V, I = gq kc V 2 + u k V. 3. Global Attracton of Infecton-Free Equlbrum u (a+b), kβ Theorem. The nfecton-free equlbrum E 0 of system () s globally attractve f R 0. Proof. Let (T(t), I(t), V(t), B(t)) beanypostvesolutonof system () wth ntal condtons (2). Defne L (t) = gk 2q (a+b) (T T 0) 2 T bgk 2q (a+b)(a+d) (T T 0 +I) 2 T 0 gk q (a+b) I+g q V+ 2 B2 t +gb V (ξ) dξ. t τ (5) (6) Calculatng the dervatve of L(t) along postve solutons of system () and notng Λ=dT 0 and TV(T T 0 )=V(T T 0 ) 2 + T 0 V(T T 0 ),tfollowsthat L= ( gkβ (a+d) +bdgk V+dgk ) q (a+b) q (a+b)(a+d) (T T 0) 2 T 0 cb 2 + gu q (R 0 )V. abgk q (a+b)(a+d) I2 If R 0, t follows from (7) thatl 0.Thesolutonsare lmted to the largest nvarant subset M of { L(t) = 0}. Its clear that L(t) = 0 f and only f (T,I,V,B) = (T 0,I 0,V 0,B 0 ). So we can obtan the global attracton of E 0 by usng LaSalle s nvarance prncple. Ths completes the proof. 4. Permanence of the System for R 0 > Theorem 2. If R 0 >, the nfected equlbrum E of system () s locally asymptotcally stable when τ=0. Proof. The characterstc equaton assocated wth system () at E s λ 4 +m 3 λ 3 +m 2 λ 2 +m λ+m 0 +e λτ (n 2 λ 2 +n λ+n 0 )=0. Notng that T = ((gq(a + b))/(kβc))v + ((u(a + b))/(kβ)), B =(g/c)v,wecanobtan m = If τ=0,(8)becomes m 3 =a+b+c+d+u+qb +βv, m 2 =bd+a(d+βv ) +(u+qb )(d+βv ) +c(a+b+d+u+qb +βv ), (a+c) gq βv 2 c + (au + cu + ac) βv +cd(a+u+b), m 0 =c(auβv +aqβv B ), n 2 =gqv, n =gqv (a+b+d+βv ), n 0 =dgq(a+b) V +agqβv 2. (7) (8) (9) λ 4 +m 3 λ 3 +(m 2 +n 2 )λ 2 +(m +n )λ+m 0 +n 0 =0. (0) Thenwecanobtanm 0 +n 0 >0, m +n >0, m 2 +n 2 >0,and m 3 >0. It follows from Routh-Hurwtz crteron that all roots of (0) have negatve real parts when τ=0. Ths competes the proof.

3 Appled Mathematcs 3 Let λ=v andthensubsttutetnto(8). Separatng the real and magnary parts, we wll gan Denote V 4 m 2 V 2 +m 0 =(n 2 V 2 n 0 ) cos Vτ n V sn Vτ, m 3 V 3 m V =(n 2 V 2 n 0 ) sn Vτ+n V cos Vτ. Squarng and addng ()yeld We have that V 8 +(m 2 3 2m 2) V 6 +(m 2 2 2m m 3 +2m 0 n 2 2 ) V4 +(m 2 2m 0m 2 n 2 +2n 0n 2 ) V 2 +(m 2 0 n2 0 )=0. G (x) =x 4 +(m 2 3 2m 2)x 3 +(m 2 2 2m m 3 +2m 0 n 2 2 )x2 +(m 2 2m 0m 2 n 2 +2n 0n 2 )x +(m 2 0 n2 0 ). G (x) =4x 3 +3(m 2 3 2m 2)x 2 +2(m 2 2 2m m 3 +2m 0 n 2 2 )x +(m 2 2m 0m 2 n 2 +2n 0n 2 ). () (2) (3) (4) Hence f G(x) = 0 has a postve root x=v 2,thenλ=±V are a couple of purely magnary characterstc roots. Moreover, for a real practce model wth fully known coeffcents, the exactly four roots of G(x) = 0 canbenumercallycalculated wth the help of computatonal software such as Matlab. Let x ( k, k 4)be the postve roots of G(x) = 0 and V = x. Solvng ()wthrespecttoτ,wecanobtan τ (j) = V arccos (( n 2V 6 +( n 0 +n m 3 n 2 m 2 ) V 4 n 2 V2 +(n 2 V 2 n 0 ) 2 ) + V 2jπ, +( (n 0m 2 n m +n 2 m 0 ) V 2 n 0m 0 n 2 V2 +(n 2 V 2 n 0 ) 2 )) where k; k 4;j=0,,2,...Let (5) τ 0 = mn {τ (j) k;j=0,,2,...}. (6) Ths means that τ 0 stheleastvalueofτ whch can be used to make the characterstc equaton have purely magnary roots. Theorem 3. G (V 2 ) and Re[dλ/dτ] have the same sgn. Proof. Let the characterstc equaton be n the followng form: We can obvously know from (8)that f (λ) +g(λ) e λτ =0. (7) f (λ) =λ 4 +m 3 λ 3 +m 2 λ 2 +m λ+m 0, g (λ) =n 2 λ 2 +n λ+n 0. (8) Calculatng the dervatve of (7) wthrespecttoτ, wecan obtan that dλ dτ (f (λ) +g (λ) e λτ g(λ) e λτ τ) [ dλ dτ ] Therefore, Re[ dλ dτ ] Obvously, =g(λ) e λτ λ, = f (λ) +g (λ) e λτ g (λ) e λτ λ τ λ = f (V )+g (V )e V τ g(v )e V τ V = Re [ f (V )+g (V )e V τ g(v )e V τ V ] f (V = Re [ ) V ( f(v )) + g (V ) V g(v ) ] τ V. = Re [ ( f (V ) f(v ) + g (V ) g(v ) V f(v ) f(v ) g(v ) g(v ) )] = Re [ V g (V ) g(v ) f (V ) f(v ) f(v ) 2 ]. sgn (Re[ dλ dτ ] ) (9) (20) = sgn (Im (g (V ) g(v ) f (V ) f(v ))). (2) And after some calculatons, we can get Im (g (V ) g(v ) f (V ) f(v )) =4V 7 +3( 2m 2 +m 2 3 ) V5 +2 (m 2 2 2m m 3 +2m 0 n 2 2 ) V3 +(m 2 2m 0m 2 n 2 +2n 0n 2 ) V. (22)

4 4 Appled Mathematcs.6.4 V I 30 Fgure : Equlbrum E 0 s globally asymptotcally stable f R 0 < E 0 70 T 80 B I 0.2 Fgure 2: Equlbrum E s locally asymptotcally stable when τ s very small. 2 E T 3 4 Therefore, G (V 2 ) and Re[dλ/dτ] Ths completes the proof. havethesamesgn. Applyng Theorems 2 and 3 and the Hopf bfurcaton theorem for functonal dfferental equaton [0], we derve the exstence of a Hopf bfurcaton as stated n Theorem 4. Theorem 4. Suppose that G(x) = 0 has at least one smple postve root and τ 0 s defned n (6). Then there s a Hopf bfurcaton for the system () asspassesupwardsthroughτ 0 leadng to a perodc soluton that bfurcates from E,where τ (j) = V arccos (( n 2V 6 +( n 0 +n m 3 n 2 m 2 ) V 4 n 2 V2 +(n 2 V 2 n 0 ) 2 ) +( (n 0m 2 n m +n 2 m 0 ) V 2 n 0m 0 n 2 V2 +(n 2 V 2 n 0 ) 2 )) + V 2jπ, k; j=0,,2, Numercal Examples (23) Example. For system (), consder all parameters as follows: a = 0., b = 0., β = 0., c = 0.00, d = 0.2, g = 0., k = 0., λ=, q = 0.00, u=0,andτ=0.inthscase,we can obtan that R 0 = < ; the global attracton of the nfecton-free equlbrum E 0 s llustrated by Fgure. Example 2. Consder all parameters of system () as follows: a = , b = , β = , c = , d = 0.929, g = , k = 2.645, λ = , q = 3.738,and u =.3978.Inthscase,wecanobtanthatR 0 = >. () When τ = 0.75, E s locally asymptotcally stable and the stablty s llustrated by Fgure B I 0.2 Fgure 3: Delay can destablze E andleadtohopfbfurcaton when τ ncreases past a crtcal value τ 0. (2) When τ = 0.9, the delay can destablze E and lead to Hopf bfurcaton. We wll show that n Fgure Dscussons In ths paper, we consder an n-host vral model humoral mmunty. In addton, a cure of nfected cells s ncorporated. Frstly, t has been shown that f the basc reproductve number s less than unty, the nfecton-free equlbrum E 0 s globally attractve, and the tme delay has no effect on the dynamcs of the system. Then, a detaled analyss on the local asymptotc stablty of the nfecton equlbrum E of the model s carred out. If E s feasble, the basc reproducton number of the vrus s greater than unty; E s globally attractve for any tme delay under some parameter condtons. By takng the dscrete tme delay as a bfurcaton parameter, t s shown that ths system undergoes a sequence E 2 T 3 4

5 Appled Mathematcs 5 of Hopf bfurcatons and stablty swtches are observed by usng smulatons. Acknowledgments Ths work was supported by Natonal Natural Scence Foundaton of Chna ( and 0703), the Basc Scentfc Research Foundaton of Central Unversty (FRF-BR--048B and FRF-BR-2-004), and the Basc Theory Research Foundaton for Engneerng Research Insttute of USTB (YJ202-00). References [] M. Nowak and C. Bangham, Populaton dynamcs of mmune responses to persstent vruses, Scence, vol. 272, no. 5258, pp , 996. [2] H. Zhu and X. Zou, Dynamcs of a HIV- nfecton model wth cell-medated mmune response and ntracellular delay, Dscrete and Contnuous Dynamcal Systems B,vol.2,no.2,pp , [3] X. Wang and Y. Tao, Lyapunov functon and global propertes of vrus dynamcs wth CTL mmune response, Internatonal Bomathematcs,vol.,no.4,pp ,2008. [4]R.M.Anderson,R.M.May,andS.Gupta, Non-lnearphenomena n host-paraste nteractons, Parastology,vol.99,pp. S59 S79, 989. [5] A. Murase, T. Sasak, and T. Kajwara, Stablty analyss of pathogen-mmune nteracton dynamcs, Mathematcal Bology,vol.5,no.3,pp ,2005. [6] D.Wodarz,R.M.May,andM.A.Nowak, Theroleofantgenndependent persstence of memory cytotoxc T lymphocytes, Internatonal Immunology,vol.2,no.4,pp ,2000. [7] C. Chyaka, W. Garra, and S. Dube, Modellng mmune response and drug therapy n human malara nfecton, Computatonal and Mathematcal Methods n Medcne,vol.9,no.2, pp.43 63,2008. [8] C. Vargas-De-León, Stablty analyss of a model for HBV nfecton wth cure of nfected cells and ntracellular delay, Appled Mathematcs and Computaton,vol.29,no.,pp , 202. [9] K. Wang, A. Fan, and A. Torres, Global propertes of an mproved hepatts B vrus model, Nonlnear Analyss: Real World Applcatons, vol., no. 4, pp , 200. [0] J. Hale, Theory of Functonal Dfferental Equatons, Sprnger, New York, NY, USA, 977.

6 Advances n Operatons Research Advances n Decson Scences Appled Mathematcs Algebra Probablty and Statstcs The Scentfc World Journal Internatonal Dfferental Equatons Submt your manuscrpts at Internatonal Advances n Combnatorcs Mathematcal Physcs Complex Analyss Internatonal Mathematcs and Mathematcal Scences Mathematcal Problems n Engneerng Mathematcs Dscrete Mathematcs Dscrete Dynamcs n Nature and Socety Functon Spaces Abstract and Appled Analyss Internatonal Stochastc Analyss Optmzaton