An in-host viral model with cure of infected cells and humoral immunity is studied. We prove that the stability is completely determined by the basic reproductive numberR0and show that the infection-free equilibriumE0is globally asymptotically stable if and only ifR0≤1. Moreover, ifR0&gt;1, the infection equilibrium is locally asymptotically stable when the time delayτis small and it loses stability as the length of the time delay increases past a critical valueτ0. Finally, we confirm our analysis by providing several numerical examples.</jats:p

Fucheng Liao

Hui Wang

Rong Wang

Zhixing Hu

Journal of Applied Mathematics

Crossref

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013, Article ID 102757, 5 pageshttp://dx.doi.org/10.1155/2013/102757Research ArticleStability Analysis of an In-Host Viral Model with Cure ofInfected Cells and Humoral ImmunityHui Wang, Rong Wang, Zhixing Hu, and Fucheng LiaoDepartment of Applied Mathematics, University of Science and Technology Beijing, Beijing 100083, ChinaCorrespondence should be addressed to Hui Wang; bkdwh@163.comReceived 29 August 2013; Accepted 4 December 2013Academic Editor: Junjie WeiCopyright © 2013 Hui Wang et al. 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.An in-host viral model with cure of infected cells and humoral immunity is studied. We prove that the stability is completelydetermined by the basic reproductive number 𝑅0and show that the infection-free equilibrium 𝐸0is globally asymptotically stableif and only if 𝑅0≤ 1. Moreover, if 𝑅0> 1, the infection equilibrium is locally asymptotically stable when the time delay 𝜏 is smalland it loses stability as the length of the time delay increases past a critical value 𝜏0. Finally, we confirm our analysis by providingseveral numerical examples.1. IntroductionThehumoral immunity is a kind of immunologicmechanismwhich uses B lymphocytes to produce antigen to preventvirus and there are evidences to prove that the humoralimmunity ismore effective than the cell-mediated immune insome infections such asmalaria infection [1–3].Many authorspresent and develop mathematical systems for the humoralimmunity [4–7]. And the cure of virus is also importantespecially in HBV models [8, 9]. In the present paper, weanalyze an in-host viral model with humoral immunity andintracellular delay, and we incorporate a “cure” of infectedcells into it. We propose the following 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) ,(1)where 𝑇, 𝐼, 𝑉, and 𝐵 represent the uninfected cells, theinfected cells, the virus, and the B cells, respectively. Λ and𝑑 are assumed as the birth rate and death rate of uninfectedcells. 𝛽 is the infection rate and 𝑘𝐼(𝑡) represents the numberof free virus which is produced during the average infectedcell life span. 𝑎 is the death rate of infected cells and 𝑢represents the death rate of virus. 𝑔 and 𝑐 represent the birthrate and death rate of B cells. The B cells neutralization rate isrepresented by 𝑞.The following form is taken as the initial conditions:𝑇 (𝜃) = 𝜑1(𝜃) ,𝐼 (𝜃) = 𝜑2(𝜃) ,𝑉 (𝜃) = 𝜑3 (𝜃) ,𝐵 (𝜃) = 𝜑4(𝜃)𝜑𝑖 (𝜃) ≥ 0, 𝜃 ∈ (−𝜏, 0) , 𝜑𝑖 (0) > 0 (𝑖 = 1, 2, 3, 4) ,(2)where (𝜑1(𝜃), 𝜑2(𝜃), 𝜑3(𝜃), 𝜑4(𝜃)) ∈ C([−𝜏, 0],R4+0), the spaceof continuous functions mapping the interval (−∞, 0] intoR4+0, andR4+0= {(𝑥1, 𝑥2, 𝑥3, 𝑥4) | 𝑥𝑖≥ 0, 𝑖 = 1, 2, 3, 4} . (3)The organization of this paper is as follows. In the nextsection, we will find threshold parameters 𝑅0of system(1) and it determines the existence of the equilibriums.In Section 3, by structuring suitable Lyapunov functionalsand using LaSalle’s invariance principle we attain the globalstability of the uninfected equilibrium if 𝑅0≤ 1. In Section 4,we consider the stability of the infected equilibrium and2 Journal of Applied Mathematicsthe occurrence of local Hopf bifurcation. In Section 5, wepresent the numerical simulations to illustrate our results.Finally, we offer concluding remarks in the last section.2. Existence of EquilibriumWe can easily find that system (1) always has an uninfectedequilibrium 𝐸0= (𝑇0, 0, 0, 0) = ((Λ/𝑑), 0, 0, 0).Denote𝑅0=Λ𝑘𝛽𝑢𝑑 (𝑎 + 𝑏). (4)We call 𝑅0the basic reproductive number. It is easy to provethat if 𝑅0> 1, model (1) exist an infected equilibrium𝐸∗(𝑇∗, 𝐼∗, 𝑉∗, 𝐵∗), where𝑉∗=(√(𝛽𝑐𝑢𝑎 − (𝑎 + 𝑏) 𝑑𝑔𝑞)2+ 4Λ𝑔𝑞𝑎𝑐𝑘𝛽2−𝛽𝑐𝑢𝑎 − (𝑎 + 𝑏) 𝑑𝑔𝑞)× (2𝑎𝛽𝑔𝑞)−1,𝑇∗=𝑔𝑞 (𝑎 + 𝑏)𝑘𝛽𝑐𝑉∗+𝑢 (𝑎 + 𝑏)𝑘𝛽,𝐵∗=𝑔𝑐𝑉∗,𝐼∗=𝑔𝑞𝑘𝑐𝑉∗2+𝑢𝑘𝑉∗.(5)3. Global Attraction ofInfection-Free EquilibriumTheorem 1. The infection-free equilibrium 𝐸0of system (1) isglobally attractive if 𝑅0≤ 1.Proof. Let (𝑇(𝑡), 𝐼(𝑡), 𝑉(𝑡), 𝐵(𝑡)) be any positive solution ofsystem (1) with initial conditions (2).Define𝐿 (𝑡) =𝑔𝑘2𝑞 (𝑎 + 𝑏)⋅(𝑇 − 𝑇0)2𝑇0+𝑏𝑔𝑘2𝑞 (𝑎 + 𝑏) (𝑎 + 𝑑)⋅(𝑇 − 𝑇0+ 𝐼)2𝑇0+𝑔𝑘𝑞 (𝑎 + 𝑏)𝐼 +𝑔𝑞𝑉 +12𝐵2+ 𝑔𝐵∫𝑡𝑡−𝜏𝑉 (𝜉) 𝑑𝜉.(6)Calculating the derivative of 𝐿(𝑡) along positive solutions ofsystem (1) and notingΛ = 𝑑𝑇0and𝑇𝑉(𝑇−𝑇0) = 𝑉(𝑇 − 𝑇0)2+𝑇0𝑉(𝑇 − 𝑇0), it follows that?̇? = − (𝑔𝑘𝛽𝑞 (𝑎 + 𝑏)𝑉 +𝑑𝑔𝑘 (𝑎 + 𝑑) + 𝑏𝑑𝑔𝑘𝑞 (𝑎 + 𝑏) (𝑎 + 𝑑))×(𝑇 − 𝑇0)2𝑇0−𝑎𝑏𝑔𝑘𝑞 (𝑎 + 𝑏) (𝑎 + 𝑑)𝐼2− 𝑐𝐵2+𝑔𝑢𝑞(𝑅0− 1)𝑉.(7)If 𝑅0≤ 1, it follows from (7) that ?̇? ≤ 0. The solutions arelimited to the largest invariant subset 𝑀 of {?̇?(𝑡) = 0}. It isclear that ?̇?(𝑡) = 0 if and only if (𝑇, 𝐼, 𝑉, 𝐵) = (𝑇0, 𝐼0, 𝑉0, 𝐵0).So we can obtain the global attraction of 𝐸0by using LaSalle’sinvariance principle. This completes the proof.4. Permanence of the System for 𝑅0> 1Theorem 2. If 𝑅0> 1, the infected equilibrium 𝐸∗ of system(1) is locally asymptotically stable when 𝜏 = 0.Proof. The characteristic equation associated with system (1)at 𝐸∗ is𝜆4+ 𝑚3𝜆3+ 𝑚2𝜆2+ 𝑚1𝜆 + 𝑚0+ 𝑒−𝜆𝜏(𝑛2𝜆2+ 𝑛1𝜆 + 𝑛0) = 0.(8)Noting that 𝑇∗ = ((𝑔𝑞(𝑎 + 𝑏))/(𝑘𝛽𝑐))𝑉∗ + ((𝑢(𝑎 + 𝑏))/(𝑘𝛽)),𝐵∗= (𝑔/𝑐)𝑉∗, we can obtain𝑚3= 𝑎 + 𝑏 + 𝑐 + 𝑑 + 𝑢 + 𝑞𝐵∗+ 𝛽𝑉∗,𝑚2= 𝑏𝑑 + 𝑎 (𝑑 + 𝛽𝑉∗)+ (𝑢 + 𝑞𝐵∗) (𝑑 + 𝛽𝑉∗)+ 𝑐 (𝑎 + 𝑏 + 𝑑 + 𝑢 + 𝑞𝐵∗+ 𝛽𝑉∗) ,𝑚1=(𝑎 + 𝑐) 𝑔𝑞𝑐𝛽𝑉∗2+ (𝑎𝑢 + 𝑐𝑢 + 𝑎𝑐) 𝛽𝑉∗+ 𝑐𝑑 (𝑎 + 𝑢 + 𝑏) ,𝑚0= 𝑐 (𝑎𝑢𝛽𝑉∗+ 𝑎𝑞𝛽𝑉∗𝐵∗) ,𝑛2= 𝑔𝑞𝑉∗,𝑛1= 𝑔𝑞𝑉∗(𝑎 + 𝑏 + 𝑑 + 𝛽𝑉∗) ,𝑛0= 𝑑𝑔𝑞 (𝑎 + 𝑏)𝑉∗+ 𝑎𝑔𝑞𝛽𝑉∗2.(9)If 𝜏 = 0, (8) becomes𝜆4+ 𝑚3𝜆3+ (𝑚2+ 𝑛2) 𝜆2+ (𝑚1+ 𝑛1) 𝜆 + 𝑚0+ 𝑛0= 0.(10)Then we can obtain𝑚0+𝑛0> 0,𝑚1+𝑛1> 0,𝑚2+𝑛2> 0, and𝑚3> 0. It follows fromRouth-Hurwitz criterion that all rootsof (10) have negative real parts when 𝜏 = 0.This competes theproof.Journal of Applied Mathematics 3Let 𝜆 = 𝑖V and then substitute it into (8). Separating thereal and imaginary parts, we will gainV4− 𝑚2V2+ 𝑚0= (𝑛2V2− 𝑛0) cos V𝜏 − 𝑛1V sin V𝜏,𝑚3V3− 𝑚1V = (𝑛2V2− 𝑛0) sin V𝜏 + 𝑛1V cos V𝜏.(11)Squaring and adding (11) yieldV8+ (𝑚23− 2𝑚2) V6+ (𝑚22− 2𝑚1𝑚3+ 2𝑚0− 𝑛22) V4+ (𝑚21− 2𝑚0𝑚2− 𝑛21+ 2𝑛0𝑛2) V2+ (𝑚20− 𝑛20) = 0.(12)Denote𝐺 (𝑥) = 𝑥4+ (𝑚23− 2𝑚2) 𝑥3+ (𝑚22− 2𝑚1𝑚3+ 2𝑚0− 𝑛22) 𝑥2+ (𝑚21− 2𝑚0𝑚2− 𝑛21+ 2𝑛0𝑛2) 𝑥+ (𝑚20− 𝑛20) .(13)We have that𝐺󸀠(𝑥) = 4𝑥3+ 3 (𝑚23− 2𝑚2) 𝑥2+ 2 (𝑚22− 2𝑚1𝑚3+ 2𝑚0− 𝑛22) 𝑥+ (𝑚21− 2𝑚0𝑚2− 𝑛21+ 2𝑛0𝑛2) .(14)Hence if𝐺(𝑥) = 0 has a positive root 𝑥 = V2, then 𝜆 = ±𝑖V area couple of purely imaginary characteristic roots. Moreover,for a real practice model with fully known coefficients, theexactly four roots of 𝐺(𝑥) = 0 can be numerically calculatedwith the help of computational software such as Matlab. Let𝑥𝑖(1 ≤ 𝑖 ≤ 𝑘, 1 ≤ 𝑘 ≤ 4) be the positive roots of𝐺(𝑥) = 0 andV𝑖= √𝑥𝑖.Solving (11) with respect to 𝜏, we can obtain𝜏(𝑗)𝑖=1V𝑖arccos((𝑛2V6𝑖+ (−𝑛0+ 𝑛1𝑚3− 𝑛2𝑚2) V4𝑖𝑛21V2𝑖+ (𝑛2V2𝑖− 𝑛0)2)+((𝑛0𝑚2−𝑛1𝑚1+𝑛2𝑚0) V2𝑖−𝑛0𝑚0𝑛21V2𝑖+ (𝑛2V2𝑖− 𝑛0)2))+1V𝑖⋅ 2𝑗𝜋,(15)where 1 ≤ 𝑖 ≤ 𝑘; 1 ≤ 𝑘 ≤ 4; 𝑗 = 0, 1, 2, . . .. Let𝜏0= min {𝜏(𝑗)𝑖| 1 ≤ 𝑖 ≤ 𝑘; 𝑗 = 0, 1, 2, . . .} . (16)This means that 𝜏0is the least value of 𝜏 which can be used tomake the characteristic equation have purely imaginary roots.Theorem 3. 𝐺󸀠(V2𝑖) and Re[𝑑𝜆/𝑑𝜏]−1𝜏=𝜏(𝑗)𝑖have the same sign.Proof. Let the characteristic equation be in the followingform:𝑓 (𝜆) + 𝑔 (𝜆) 𝑒−𝜆𝜏= 0. (17)We can obviously know from (8) that𝑓 (𝜆) = 𝜆4+ 𝑚3𝜆3+ 𝑚2𝜆2+ 𝑚1𝜆 + 𝑚0,𝑔 (𝜆) = 𝑛2𝜆2+ 𝑛1𝜆 + 𝑛0.(18)Calculating the derivative of (17) with respect to 𝜏, we canobtain that𝑑𝜆𝑑𝜏(𝑓󸀠(𝜆) + 𝑔󸀠(𝜆) 𝑒−𝜆𝜏− 𝑔 (𝜆) 𝑒−𝜆𝜏⋅ 𝜏)= 𝑔 (𝜆) 𝑒−𝜆𝜏⋅ 𝜆,[𝑑𝜆𝑑𝜏]−1𝜏=𝜏(𝑗)𝑖=𝑓󸀠(𝜆) + 𝑔󸀠(𝜆) 𝑒−𝜆𝜏𝑔 (𝜆) e−𝜆𝜏 ⋅ 𝜆−𝜏𝜆=𝑓󸀠(𝑖V𝑖) + 𝑔󸀠(𝑖V𝑖) 𝑒−𝑖V𝑖𝜏𝑔 (𝑖V𝑖) 𝑒−𝑖V𝑖𝜏 ⋅ 𝑖V𝑖−𝜏𝑖V𝑖.(19)Therefore,Re[𝑑𝜆𝑑𝜏]−1𝜏=𝜏(𝑗)𝑖= Re[𝑓󸀠(𝑖V𝑖) + 𝑔󸀠(𝑖V𝑖) 𝑒−𝑖V𝑖𝜏𝑔 (𝑖V𝑖) 𝑒−𝑖V𝑖𝜏 ⋅ 𝑖V𝑖]= Re[𝑓󸀠(𝑖V𝑖)𝑖V𝑖⋅ (−𝑓 (𝑖V𝑖))+𝑔󸀠(𝑖V𝑖)𝑖V𝑖⋅ 𝑔 (𝑖V𝑖)]= Re[ 1𝑖V𝑖⋅ (−𝑓󸀠(𝑖V𝑖) ⋅ 𝑓 (𝑖V𝑖)𝑓 (𝑖V𝑖) ⋅ 𝑓 (𝑖V𝑖)+𝑔󸀠(𝑖V𝑖) ⋅ 𝑔 (𝑖V𝑖)𝑔 (𝑖V𝑖) ⋅ 𝑔 (𝑖V𝑖))]= Re[ 1𝑖V𝑖⋅𝑔󸀠(𝑖V𝑖) ⋅ 𝑔 (𝑖V𝑖) − 𝑓󸀠(𝑖V𝑖) ⋅ 𝑓 (𝑖V𝑖)󵄨󵄨󵄨󵄨𝑓 (𝑖V𝑖)󵄨󵄨󵄨󵄨2] .(20)Obviously,sign(Re[𝑑𝜆𝑑𝜏]−1𝜏=𝜏(𝑗)𝑖)= sign (Im (𝑔󸀠 (𝑖V𝑖) ⋅ 𝑔 (𝑖V𝑖) − 𝑓󸀠(𝑖V𝑖) ⋅ 𝑓 (𝑖V𝑖))) .(21)And after some calculations, we can getIm (𝑔󸀠 (𝑖V𝑖) ⋅ 𝑔 (𝑖V𝑖) − 𝑓󸀠(𝑖V𝑖) ⋅ 𝑓 (𝑖V𝑖))= 4V7𝑖+ 3 (−2𝑚2+ 𝑚23) V5𝑖+ 2 ⋅ (𝑚22− 2𝑚1𝑚3+ 2𝑚0− 𝑛22) V3𝑖+ (𝑚21− 2𝑚0𝑚2− 𝑛21+ 2𝑛0𝑛2) V𝑖.(22)4 Journal of Applied Mathematics506070800102030405060700510152025303540E0TIVFigure 1: Equilibrium 𝐸0is globally asymptotically stable if 𝑅0< 1.Therefore, 𝐺󸀠(V2𝑖) and Re[𝑑𝜆/𝑑𝜏]−1𝜏=𝜏(𝑗)𝑖have the same sign.This completes the proof.Applying Theorems 2 and 3 and the Hopf bifurcationtheorem for functional differential equation [10], we derivethe existence of a Hopf bifurcation as stated inTheorem 4.Theorem 4. Suppose that 𝐺(𝑥) = 0 has at least one simplepositive root and 𝜏0is defined in (16). Then there is a Hopfbifurcation for the system (1) as s passes upwards through 𝜏0leading to a periodic solution that bifurcates from 𝐸∗, where𝜏(𝑗)𝑖=1V𝑖arccos ((𝑛2V6𝑖+ (−𝑛0+ 𝑛1𝑚3− 𝑛2𝑚2) V4𝑖𝑛21V2𝑖+ (𝑛2V2𝑖− 𝑛0)2)+ ((𝑛0𝑚2−𝑛1𝑚1+𝑛2𝑚0) V2𝑖−𝑛0𝑚0𝑛21V2𝑖+ (𝑛2V2𝑖− 𝑛0)2))+1V𝑖⋅ 2𝑗𝜋, 1 ≤ 𝑖 ≤ 𝑘; 𝑗 = 0, 1, 2, . . . .(23)5. Numerical ExamplesExample 1. For system (1), consider all parameters as follows:𝑎 = 0.1, 𝑏 = 0.1, 𝛽 = 0.1, 𝑐 = 0.001, 𝑑 = 0.2, 𝑔 = 0.1,𝑘 = 0.1, 𝜆 = 1, 𝑞 = 0.001, 𝑢 = 10, and 𝜏 = 10. In this case, wecan obtain that 𝑅0= 0.375 < 1; the global attraction of theinfection-free equilibrium 𝐸0is illustrated by Figure 1.Example 2. Consider all parameters of system (1) as follows:𝑎 = 2.41979, 𝑏 = 2.80472, 𝛽 = 2.34292, 𝑐 = 2.2057, 𝑑 =0.9129, 𝑔 = 4.2074, 𝑘 = 2.6415, 𝜆 = 4.60059, 𝑞 = 3.738, and𝑢 = 1.3978. In this case, we can obtain that 𝑅0= 4.2709 > 1.(1) When 𝜏 = 0.75,𝐸∗ is locally asymptotically stable andthe stability is illustrated by Figure 2.123 40.20.40.60.810.20.40.60.811.21.41.6E∗TIBFigure 2: Equilibrium 𝐸∗ is locally asymptotically stable when 𝜏 isvery small.12 340.30.40.50.60.70.80.911.11.21.30.20.40.60.81E∗TIBFigure 3: Delay can destabilize 𝐸∗ and lead to Hopf bifurcationwhen 𝜏 increases past a critical value 𝜏0.(2) When 𝜏 = 0.9, the delay can destabilize 𝐸∗ and leadto Hopf bifurcation. We will show that in Figure 3.6. DiscussionsIn this paper, we consider an in-host viral model humoralimmunity. In addition, a “cure” of infected cells is incorpo-rated. Firstly, it has been shown that if the basic reproductivenumber is less than unity, the infection-free equilibrium 𝐸0is globally attractive, and the time delay has no effect onthe dynamics of the system. Then, a detailed analysis onthe local asymptotic stability of the infection equilibrium𝐸∗ of the model is carried out. If 𝐸∗ is feasible, the basicreproduction number of the virus is greater than unity; 𝐸∗ isglobally attractive for any time delay under some parameterconditions. By taking the discrete time delay as a bifurcationparameter, it is shown that this system undergoes a sequenceJournal of Applied Mathematics 5of Hopf bifurcations and stability switches are observed byusing simulations.AcknowledgmentsThis work was supported by National Natural Science Foun-dation of China (61174209 and 11071013), the Basic ScientificResearch Foundation of Central University (FRF-BR-11-048Band FRF-BR-12-004), and the Basic Theory Research Foun-dation for Engineering Research Institute of USTB (YJ2012-001).References[1] M. Nowak and C. Bangham, “Population dynamics of immuneresponses to persistent viruses,” Science, vol. 272, no. 5258, pp.74–79, 1996.[2] H. Zhu and X. 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Stability Analysis of an In-Host Viral Model with Cure of Infected Cells and Humoral Immunity

An in-host viral model with cure of infected cells and humoral immunity is studied. We prove that the stability is completely determined by the basic reproductive number R0 and show that the infection-free equilibrium E0 is globally asymptotically stable if and only if R0≤1. Moreover, if R0>1, the infection equilibrium is locally asymptotically stable when the time delay τ is small and it loses stability as the length of the time delay increases past a critical value τ0. Finally, we confirm our analysis by providing several numerical examples

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

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