Mathematics in Applied Sciences and Engineering https://ojs.lib.uwo.ca/mase Volume 2, Number 2, June 2021, pp.123-133 https://doi.org/10.5206/mase/13822 GLOBAL PROPERTIES OF A VIRUS DYNAMICS MODEL WITH SELF-PROLIFERATION OF CTLS CUICUI JIANG, HUAN KONG, GUOHONG ZHANG*, AND KAIFA WANG* Abstract. A viral infection model with self-proliferation of cytotoxic T lymphocytes (CTLs) is pro- posed and its global dynamics is obtained. When the per capita self-proliferation rate of CTLs is sufficient large, an infection-free but immunity-activated equilibrium always exists and is globally asymptotically stable if the basic reproduction number of virus is less than a threshold value, which means that the immune effect still exists though virus be eliminated. Qualitative numerical simula- tions further indicate that the increase of per capita self-proliferation rate may lead to more severe infection outcome, which may provide insight into the failure of immune therapy. 1. Introduction Outbreaks of viral infection have become a major global health concern. Different kinds of virus, such as hepatitis B virus (HBV), hepatitis C virus (HCV), human immunodeficiency virus (HIV), Ebola Virus and Zika Virus, have been associated with severe outcomes. A great deal of effort has been put toward to understand the life cycle of these virus. With the development of biomedical research, mathematical models also play an increasingly important role to provide insights into virus infection and dynamics, as well as on how an infection can be reduced or even eradicated. For example, on HBV infection, Nowak et al. [13] first proposed a basic three-dimensional viral infection model within-host. Note that immune responses play a critical part in the process of viral infections. Nowak et al. [14] further proposed the following four-dimensional system with the cytotoxic T lymphocytes (CTLs) population based on the basic model:   x′ = s−βxv −d1x, y′ = βxv −d2y −pyz, v′ = ky −uv, z′ = cyz −d3z. (1.1) Here x(t),y(t),v(t) and z(t) represent uninfected target cells, infected cells, free virus and CTLs, re- spectively. Uninfected cells are produced at a constant rate s, die at rate d1x, and become infected at rate βxv. Infected cells are produced at rate βxv and die at rate d2y. Free virus are produced from infected cells at rate ky and die at rate uv. CTLs are produced at rate cyz due to the stimulation of infected cells, and die at rate d3z. Infected cells are eliminated by CTLs at rate pyz. After that, based on the basic models, many studies were carried out to analysis of the dynamics of various virus infection within-host, such as [1, 9, 12, 15, 16, 17, 22, 23, 25] and the reference therein. Received by the editors 5 April 2021; revised 13 May 2021; 19 May 2021 accepted; published online 24 May 2021. 2000 Mathematics Subject Classification. Primary 92D30; Secondary 34K20, 34K25. Key words and phrases. Viral infection; Self-proliferation of CTLs; Lyapunov function; Global stability. Cuicui Jiang’s research was supported by NSFC grant 11901576 and the Youth Foundation of Army Medical University 2017XQN05. Guohong Zhang’s research was partially supported by an NSFC grant 11871403. Kaifa Wang’s research was partially supported by an NSFC grant 11771448. 123 124 CUICUI JIANG, HUAN KONG, GUOHONG ZHANG*, AND KAIFA WANG* Recent studies on the production mechanism of immune cells show that its self-proliferation cannot be neglected besides the stimulation of infected cells in [7]. Thus, to understand the effect of self- proliferation, based on the system (1.1) and [7], we propose the following new virus infection model:  x′ = s−βxv −d1x, y′ = βxv −d2y −pyz, v′ = ky −uv, z′ = cyz + rz(1 − z m ) −d3z. (1.2) Here the logistic proliferation term rz(1 − z/m) describes the self-proliferation of CTLs, in which parameter r denotes a per capita self-proliferation rate, and m means the capacity of CTLs population. When r = 0, i.e., without self-proliferation of CTLs, (1.2) has been completely analyzed in [10] if there is not explicit dynamics of free virus under a plausible quasi steady-state assumption. To explore the effects of the recruitment of immune responses on virus infection, the main contribution of the present paper is to obtain the complete global properties of (1.2) when r > 0. 2. Global dynamics analysis Since we are interested in the dynamics of viral infection, and not the initial processes of infection, we assume that the initial condition of (1.2) has the form x(0) > 0,y(0) > 0,v(0) > 0 and z(0) > 0. Based on the initial conditions, it is easy to show that the solutions of system (1.2) are non-negative and ultimately bounded. The equilibria of (1.2) are the solutions of the following algebraic equations:  s−βxv −d1x = 0, βxv −d2y −pyz = 0, ky −µv = 0, cyz + rz(1 − z m ) −d3z = 0. (2.1) Clearly, system (1.2) always has infection-free equilibrium E0 = ( s d1 , 0, 0, 0). According to the definition and algorithm of the basic reproduction number of virus in [4], we can obtain the basic reproduction number of virus R0 = βsk d1d2µ . Using the fourth equation of (2.1), we have z = 0 or z = m r (r −d3 + cy). (2.2) When z = 0, based on the first three equations of (2.1), it is easy to obtain that the immunity-inactivated infection equilibrium E1 = ( s d1R0 , d1µ(R0−1) βk , d1(R0−1) β , 0) always exists if R0 > 1. In addition, using the third and first equation of (2.1), we have v = k µ y and x = s βv + d1 = sµ βky + d1µ . (2.3) After that, using the second equation and (2.3), we have βk µ xy −d2y −pyz = 0. (2.4) Thus, when y = 0, using (2.2), we know that an infection-free but immunity-activated equilibrium E2 = ( s d1 , 0, 0, m(r−d3) r ) will appear if r > d3. Otherwise, using (2.2), (2.3) and (2.4), we have f(y) ≡ βk µ x−d2 −pz = βks βky + d1µ −d2 − mp r (r −d3 + cy) = 0. Clearly, f(+∞) < 0 and function f(y) is monotonically decreasing since f′(y) < 0 always valid. As a result, f(y) = 0 has a unique positive root if f(0) = βks d1µ −d2 − mp r (r −d3) = d2[R0 − 1 − mp rd2 (r −d3)] > 0, GLOBAL PROPERTIES OF A VIRUS DYNAMICS MODEL WITH SELF-PROLIFERATION OF CTLS 125 i.e., R0 > 1 + mp rd2 (r −d3). When r < d3, according to (2.2), in order to keep the positive of z, we need f( d3 −r c ) = βksc βk(d3 −r) + d1µ −d2 > 0 is valid, i.e., R0 > 1 + βk(d3−r) cd1µ . In summary, we have the following proposition. Proposition 2.1. The following hold. (i) If R0 > 1, the immunity-inactivated infection equilibrium E1 always exists. Especially, an infection-free but immunity-activated equilibrium E2 will appear if r > d3. (ii) Suppose that 0 ≤ r ≤ d3. If R0 > 1 + βk(d3−r) cd1µ , system (1.2) has a unique immunity-activated infection equilibrium E3 = (x3,y3,v3,z3), where x3 = sµ βky3 + d1µ , v3 = k µ y3, z3 = m r (r −d3 + cy3), and y3 is the unique positive root of f(y) = 0 in this case. (iii) Suppose that r > d3. If R0 > 1 + pm(r−d3) rd2 , system (1.2) has a unique immunity-activated infection equilibrium E4 = (x4,y4,v4,z4), where x4 = sµ βky4 + d1µ , v4 = k µ y4, z4 = m r (r −d3 + cy4), and y4 is the unique positive root of f(y) = 0 in this case. In order to obtain the stability of above mentioned equilibria, we first give the Jacobian matrix J of system (1.2) at (x,y,v,z), J =   −βv −d1 0 −βx 0 βv −d2 −pz βx −py 0 k −µ 0 0 cz 0 cy + r −d3 − 2rzm   . (2.5) So we have the following results. Theorem 2.2. The following hold. (i) When 0 ≤ r ≤ d3, the infection-free equilibrium E0 is globally asymptotically stable if R0 < 1, and it is unstable when R0 > 1. (ii) When r > d3, the infection-free equilibrium E0 is always unstable. Proof. According to (2.5), we have the characteristic equation of system (1.2) at E0 (λ + d1)(λ− (r −d3))H0(λ) = 0, (2.6) where H0(λ) = λ 2 + (d2 + µ)λ + d2µ(1−R0). It is easy to show that λ1 = −d1 < 0 and λ2 = r−d3 are the roots of (2.6). Further, we can get all roots of H0(λ) are negative real part if R0 < 1, and there is one positive real root if R0 > 1. (i) When 0 ≤ r < d3, we have λ2 < 0. As a result, the infection-free equilibrium E0 is locally asymptotically stable if R0 < 1 and 0 ≤ r < d3, and is unstable if R0 > 1. When r = d3, λ2 = 0. Thus, the center manifold is a curve tangent to the z-axis. In this case, settling a transformation 126 CUICUI JIANG, HUAN KONG, GUOHONG ZHANG*, AND KAIFA WANG* x̃ = x−x0, ỹ = y, ṽ = v, z̃ = z, we know (1.2) becomes  x′ = −βxv − βs d1 v −d1x, y′ = βxv + βs d1 y −d2y −pyz, v′ = ky −µv, z′ = cyz − r m z2, (2.7) where we substitute x,y,v,z for x̃, ỹ, ṽ, z̃. To obtain the approximative expression of the center manifold, we set x = k2z 2 + k3z 3 + O(z3), y = n2z 2 + n3z 3 + O(z3), v = b2z 2 + b3z 3 + O(z3), (2.8) and obtain dx dt = [2k2z + 3k3z 2 + O(z2)] dz dt , dy dt = [2n2z + 3n3z 2 + O(z2)] dz dt , dv dt = [2b2z + 3b3z 2 + O(z2)] dz dt . (2.9) Substituting (2.7) and (2.8) into (2.9), we have  − ( βs d1 b2 + d1k2)z 2 − ( βs d1 b3 + d1k3 − 2r m k2)z 3 + O(z3) = 0, − ( βs d1 n2 + d2n2)z 2 − ( βs d1 n3 + d2n3 − 2r m n2)z 3 + O(z3) = 0, (kn2 −µb2)z2 + (kn3 −µb3 + 2r m b2)z 3 + O(z3) = 0. (2.10) Comparing the coefficients of z and z2 in (2.10), we have k2 = k3 = n2 = n3 = b2 = b3 = 0. As a result, substituting (2.8) into the last equation of (2.7), we have dz dt = − r m z2 + O(z3). (2.11) Thus, the zero point z = 0 of (2.11) is locally asymptotically stable, then E0 is locally asymptotically stable if R0 < 1 and r = d3. Let L0 = x−x0 −x0 ln x x0 + y + d2 k v + p c z. Taking the time derivative of L0 along the solution of system (1.2), we have L′0 =(1 − x0 x )(s−βxv −d1x) + βxv −d2y −pyz + d2 k (ky −µv) + p c ( cyz + rz(1 − z m ) −d3z ) =d1x0(2 − x x0 − x0 x ) + d2µ k (R0 − 1)v − prz2 cm + p(r −d3)z c ≤ 0 GLOBAL PROPERTIES OF A VIRUS DYNAMICS MODEL WITH SELF-PROLIFERATION OF CTLS 127 if R0 < 1 and 0 ≤ r ≤ d3, and L′0 = 0 only if x = x0,v = 0 and z = 0 simultaneously, i.e., the maximal invariant subset in {(x,y,v,z) : L′0|(1.2) = 0} is the singleton {E0}. As a result, E0 is globally asymptotically stable based on the LaSalle’s invariance principle. (ii) When r > d3, the eigenvalue λ2 = r − d3 > 0. So the infection-free equilibrium E0 is always unstable. � Theorem 2.3. Suppose that immunity-inactivated infection equilibrium E1 exists, i.e., R0 > 1. (i) When 0 ≤ r < d3, E1 is globally asymptotically stable if 1 < R0 < 1 + βk(d3−r) cd1µ , and it is unstable when R0 > 1 + βk(d3−r) cd1µ . (ii) When r ≥ d3, E1 is always unstable. Proof. Let x1 = s d1R0 ,y1 = d1µ(R0−1) βk ,v1 = d1(R0−1) β . According to (2.5), we have the following charac- teristic equation of system (1.2) at E1 (λ− cy1 −r + d3)H1(λ) = 0, where H1(λ) = λ 3 + a1λ 2 + a2λ + a3, and a1 = d2 + µ + d1R0 > 0, a2 = d1(d2 + µ)R0 > 0, a3 = d1d2µ(R0 − 1) > 0. Here we use βv1 = d1(R0 − 1),βkx1 = d2µ. Further, we have a1a2 −a3 = R20d 2 1(d2 + µ) + d1d2µ(R0 + 1) + R0d1(d 2 2 + µ 2) > 0. Clearly, λ1 = r − d3 + cy1 = r − d3 + cd1µ(R0−1) βk is an eigenvalue at E1 of system (1.2), and the real parts of H1(λ) are negative according to Routh-Hurwitz criterion. (i) When 0 ≤ r < d3, we know λ1 < 0 if R0 < 1 + βk(d3−r) cd1µ , i.e., E1 is locally asymptotically stable in this case. Otherwise, it is unstable. Let L1 = x−x1 −x1 ln x x1 + y −y1 −y1 ln y y1 + βx1v1 ky1 (v −v1 −v1 ln v v1 ) + p c z. Taking the time derivative of L2 along the solution of system (1.2), and using s = βx1v1 +d1x1,βx1v1 = d2y1 and ky1 = µv1, we have L′1 =(1 − x1 x )(s−βxv −d1x) + (1 − y1 y ) ( βxv −d2y −pyz ) + (1 − v1 v ) βx1v1 ky1 (ky −µv) + p c ( cyz + rz(1 − z m ) −d3z ) =d1x1 ( 2 − x1 x − x x1 ) + βx1v1 ( 3 − x1 x − y1xv yx1v1 − v1y vy1 ) + pd1µ βk ( R0 − 1 − βk(d3 −r) cd1µ ) z − prz2 cm ≤ 0 if R0 < 1 + βk(d3−r) cd1µ , and L′1 = 0 only if x = x1, y y1 = v v1 and z = 0. In this case, it is easy to obtain that the maximal invariant subset in {(x,y,v,z) : L′1|(1.2) = 0} is the singleton {E1}. As a result, E1 is globally asymptotically stable based on the LaSalle’s invariance principle. (ii) When r ≥ d3, λ1 = r −d3 + cy1 > 0 is always valid. So E1 is always unstable. � Theorem 2.4. Suppose that r > d3. The infection-free but immunity-activated equilibrium E2 is globally asymptotically stable if R0 < 1 + pm(r−d3) rd2 and unstable if R0 > 1 + pm(r−d3) rd2 . 128 CUICUI JIANG, HUAN KONG, GUOHONG ZHANG*, AND KAIFA WANG* Proof. According to (2.5), we have the following characteristic equation of system(1.2) at E2 (λ + d1)(λ + r −d3)H3(λ) = 0, where H3(λ) = λ 2 + (d2 + µ + pz2)λ + d2µ(1 + pm(r−d3) rd2 −R0). It is easy to obtain that λ1 = −d1 < 0,λ2 = −r + d3 < 0 are the eigenvalues at E2 of system(1.2), and all roots of H3(λ) are negative real part if R0 < 1 + pm(r−d3) rd2 . Thus, E2 is locally asymptotically stable if R0 < 1 + pm(r−d3) rd2 . Otherwise, it is unstable. Let L2 = x−x2 −x2 ln x x2 + y + d2 k R0v + p c (z −z2 −z2 ln z z2 ), where x2 = s d1 ,z2 = m(r−d3) r . Taking the time derivative of L2 along the solution of system (1.2), we have L′2 =(1 − x2 x )(s−βxv −d1x) + βxv −d2y −pyz + d2 k R0(ky −µv) + p c (1 − z2 z ) ( cyz + rz(1 − z m ) −d3z ) =d1x2(2 − x x2 − x2 x ) + d2(R0 − 1 − pm(r −d3) rd2 )y − rp cm (z −z2)2 ≤ 0 if R0 < 1 + pm(r−d3) rd2 , and L′2 = 0 only if x = x2,y = 0 and z = z2, i.e., the maximal invariant subset in {(x,y,v,z) : L′2|(1.2) = 0} is the singleton {E2}. As a result, E2 is globally asymptotically stable based on the LaSalle’s invariance principle. � Theorem 2.5. The following hold. (i) When 0 ≤ r ≤ d3, the immunity-activated infection equilibrium E3 is globally asymptotically stable as long as it appears, i.e., R0 > 1 + βk(d3−r) cd1µ . (ii) When r > d3, the immunity-activated infection equilibrium E4 is globally asymptotically stable as long as it appears, i.e., R0 > 1 + pm(r−d3) rd2 . Proof. For the sake of description, the equilibria E3 and E4 are uniformly denoted as E∗ = (x∗,y∗,v∗,z∗). First, we discuss the local stability of E∗, according to (2.5), we have J(E∗) =   −d1 −σ 0 −αµk 0 σ −α αµ k −py∗ 0 k −µ 0 0 cz∗ 0 − rmz∗   , where σ = βv∗ > 0,α = d2 + pz∗ > 0, αµ k = βx∗ > 0. The characteristic equation of system (1.2) at E∗ is λ4 + â1λ 3 + â2λ 2 + â3λ + â4 = 0, in which â1 = r m z∗ + α + µ + $ > 0, â2 = cpz∗y∗ + r m z∗(α + µ + $) + $(α + mu) > 0, â3 = cpz∗y∗(µ + $) + r m z∗$(α + µ) + αµσ > 0, â4 = cµpz∗y∗$ + αµσ r m z∗ > 0, $ = d1 + σ. GLOBAL PROPERTIES OF A VIRUS DYNAMICS MODEL WITH SELF-PROLIFERATION OF CTLS 129 After some calculations, we have â1â2 − â3 =cpz∗y∗( r m z∗ + α) + (α + µ + $) r2 m2 z2∗ + r m z∗(α + µ + $) 2 + α(α$ + 2d1µ + µσ + $) + µ$(µ + $) > 0, â3(â1â2 − â3) − â21â4 =µαc 2p2z2∗y 2 ∗ + c 2p2µ r m z3∗y 2 ∗ + c 2p2z2∗y 2 ∗$( r m z∗ + α) + cp r2 m2 z3∗y∗ [ α(2$ + µ) + ($ + µ)2 ] + cp r m z2∗y∗ [ (2$ + µ)α2 + µ(3d1 + 2µ + 4σ) + 2$ 2 ] α + (µ + $)($2 + µ2) ] + cαpz∗y∗ [ ($2 + µσ)α + $($ + d1µ)) ] − cαµ2pσz∗y∗ + [ r3 m3 z3∗ + r2 m2 z2∗(α + µ + $) + r m z∗$(α + µ) ][ $α2 + α($2 + 2d1µ + µσ) + µ$(µ + $) ] + αµσ [ $α2 + ($2 + 2d1µ + µσ)α + µ($ 2 + d1µ) ] + αµ3σ2. Notice that this only has one negative term. Using the first and the last terms, and this negative term, we have αµ(c2p2z2∗y 2 ∗ − cµpσz∗y∗ + µ 2σ2) ≥ αµ(2cpµσz∗y∗ − cµpσz∗y∗) > 0, thus, â3(â1â2− â3)− â21â4 > 0. Moreover, it follows from the Routh-Hurwitz criterion that E∗ is locally asymptotically stable. In order to obtain the global stability of E∗, let L∗ = x−x∗ −x∗ ln x x∗ + y −y∗ −y∗ ln y y∗ + βx∗v∗ ky∗ (v −v∗ −v∗ ln v v∗ ) + p c (z −z∗ −z∗ ln z z∗ ). Taking the time derivative of L∗ along the solution of system(1.2), we get L′∗ =(1 − x∗ x )(s−βxv −d1x) + (1 − y∗ y )(βxv −d2y −pyz) + (1 − v∗ v ) βx∗v∗ ky∗ (ky −µv) + p c (1 − z∗ z ) ( cyz + rz(1 − z m ) −d3z ) . Using s = βx∗v∗ + d1x∗, βx∗v∗ = d2y∗ + py∗z∗, ky∗ = µv∗ and y∗ = rz∗ + m(d3 −r) mc , we have L′∗ =2d1x∗ −d1x−d1x∗ x∗ x + βx∗v∗ + βx∗v −βx∗v∗ x∗ x −βxv y∗ y + βx∗v∗ −py∗z∗ + py∗z + pyz∗ −βx∗v −βx∗v∗ v∗y vy∗ + βx∗v∗ + p c ( rz(1 − z m ) −d3z ) −pyz∗ − p c ( rz∗(1 − z m ) −d3z∗ ) =d1x∗(2 − x∗ x − x x∗ ) + βx∗v∗(3 − x∗ x − xvy∗ x∗v∗y − v∗y vy∗ ) − pr mc (z −z∗)2 ≤ 0, and L′∗ = 0 only if x = x∗, y y∗ = v v∗ and z = z∗. In this case, it is easy to obtain that the maximal invari- ant subset in {(x,y,v,z) : L′∗|(1.2) = 0} is the singleton {E∗}. As a result, E∗ is globally asymptotically stable if it exists based on the LaSalle’s invariance principle. � For the convenience of reading, we summarize the complete global properties of system (1.2) as shown in Figure 1. 130 CUICUI JIANG, HUAN KONG, GUOHONG ZHANG*, AND KAIFA WANG* 1 R 0 E 0 is GAS E 0 is US E 1 is GAS E 3 is GAS E 1 is US E 0 is US 0≤ rd 3 E 0 is USE0 is US E 1 is US E 2 is GAS E 1 is US E 4 is GAS E 2 is US E 0 is US Figure 1. Global properties of system (1.2). Here, E0 = ( s d1 , 0, 0, 0), E1 = ( s d1R0 , d1µ(R0−1) βk , d1(R0−1) β , 0), E2 = ( s d1 , 0, 0, m(r−d3) r ), E3 = (x3,y3,v3,z3) and E4 = (x4,y4,v4,z4) are the equilibria of system (1.2), the expression of E3 and E4 is shown in Proposition 2.1. R1 = 1 + βk(d3−r) cd1µ and R2 = 1 + pm(r−d3) rd2 . 3. Numerical simulations Although the complete global properties of system (1.2) have been obtained in Figure 1, it is noted that the immunity-activated infection equilibrium E3 or E4 is related to the parameters of self- proliferation of CTLs (r and m) from Proposition 2.1. When E3 or E4 is globally asymptotically stable, the infected cells (y3 or y4) and corresponding proportion of infected cells y3 x3+y3 or y4 x4+y4 are often used to describe the severity of the infection. In this section, we give numerical simulations to investigate the effect of self-proliferation of CTLs, and explore the potential significance in clinical practice. First, we fix s = 1.0 × 104 ml−1 · day−1, β = 3.0 × 10−4 ml−1 · day−1, p = 0.5 ml−1 · day−1, c = 9.6 × 10−6 ml−1 · day−1, d1 = 0.01 day−1, d2 = 1.0 day−1, d3 = 0.035 day−1, k = 8 virions/cell, µ = 2.4 day−1, which are within the similar ranges as those ones employed in [5, 12, 17, 22]. Then, let parameters r and m change to qualitatively explore their influence on the values of E3 or E4, and the corresponding proportion of infected cells. Figure 2. Simulations of E3 and the corresponding proportion of y3 when 0 ≤ r ≤ d3. Panel (A)-(E) are the simulated surface with two parameters changing. GLOBAL PROPERTIES OF A VIRUS DYNAMICS MODEL WITH SELF-PROLIFERATION OF CTLS 131 When r ∈ [0.0, 0.035] and m ∈ [100, 1000], i.e., 0 ≤ r ≤ d3, Figure 2 shows the simulated surface of E3 and the corresponding proportion of y3 changing with parameters r and m. Along with the increase of parameter r, although the infected cells (y3) and virus load (v3) will decrease (Figure 2B and Figure 2C), the proportion of infected cells ( y3 x3+y3 ) is gradually increase (Figure 2E). In particular, the immune cells (z3) also decrease with the increase of r (Figure 2D). These qualitatively indicate that the severity of infection may increase with the increase of r in case of 0 ≤ r ≤ d3. Along with the increase of parameter m, Figure 2A and Figure 2D show that uninfected cells (x3) and immune cells (z3) will increase gradually, while infected cells (y3), virus load (v3) and the proportion of infected cells ( y3 x3+y3 ) are decrease gradually. These qualitatively indicate that increasing parameter m can reduce the severity of virus infection. Figure 3. Simulations of E4 and the corresponding proportion of y4 when 0 ≤ r ≤ d3. Panel (A)-(E) are the simulated surface with two parameters changing. When r ∈ [0.0351, 0.07] and m ∈ [100, 1000], i.e., r > d3, Figure 3 shows the simulated surface of E4 and the corresponding proportion of y4 changing with parameters r and m. Comparing Figure 3 with Figure 2, we can find that all components of the immunity-activated infection equilibrium E4 will increase with the increase of parameter r, including the proportion of infected cells ( y4 x4+y4 ). Along with the increase of parameter m, the variation of all components is similar to that in Figure 2. 4. Discussion In this paper, a viral infection model with self-proliferation of CTLs is proposed and its global dynamic behavior is obtained. From Figure 1, comparing with [10], we can find the dynamic behavior of (1.2) will not change if the per capita self-proliferation rate of CTLs is insufficient, i.e., 0 < r < d3. However, when r = d3, the immunity-inactivated infection equilibrium E1 is always unstable if it appears. Especially, when r > d3, a new steady state (named as infection-free but immunity-activated equilibrium E2) appears and is globally asymptotically stable if the basic reproduction number is less than a threshold, which means that the immune effect still exists though virus be eliminated. This is consistent with the clinical practice of virus infection because immune cells are usually not depleted after a patient recovers. In fact, memory T cells can be maintained in lymphoid and nonlymphoid organs through self-renewal [3], including central memory T cells and effector memory T cells [18]. In particular, recent study on HBV also find that HBV specific TNF-α CD4 T cells may be in the early stage of differentiation rather than depletion of T cells [19], which suggests that the stability of immunity-inactivated infection equilibrium E1 may be impossible in clinical practice. On the effect of self-proliferation of immune cells, qualitative numerical simulations (Figure 2 and Fig- ure 3) indicate that although there are different shape mode under different intensity of self-proliferation, 132 CUICUI JIANG, HUAN KONG, GUOHONG ZHANG*, AND KAIFA WANG* the increase of per capita self-proliferation rate (r) can worsen the infection, while the increase of the capacity of CTLs (m) can reduce the severity of infection. Thus, inappropriate intensity of per capita self-proliferation rate may lead to more severe infection outcome, which is similar to the effect of cova- lently closed circular DNA (cccDNA) self-amplification rate in HBV infection [8]. These may provide insight into the failure of immune therapy [2]. Recently, under a plausible quasi steady-state assumption, [6] ignored the direct effect of virus load, but introduced the delayed activation effect of immune cells, and the dynamics of the corresponding virus model was studied. Compared with the results of [6], we can find that quai steady-state assumption cannot affect the dynamic behavior, which is consistent with the conclusion of [20]. Note that the spatial migration of virus particles is an inherent characteristic of virus infection within-host [21, 28]. 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Jin, Traveling waves for a reaction-diffusion-advection predator-prey model, Nonlinear Anal. Real 36 (2017), 203-232. [28] T. Zhang, W. Wang and K. Wang, Minimal wave speed of a bacterial colony model, Appl. Math. Model. 40 (2016), 10419-10436. Department of Mathematics, College of Basic Medicine, Army Medical University, Chongqing 400038, P. R. China E-mail address: jiangcc16@foxmail.com School of Mathematics and Statistics, Southwest University, Chongqing 400715, P. R. China E-mail address: 1349966363@qq.com Co-corresponding author, School of Mathematics and Statistics, Southwest University, Chongqing 400715, P. R. China E-mail address: zgh711@swu.edu.cn Co-corresponding author, School of Mathematics and Statistics, Southwest University, Chongqing 400715, P. R. China E-mail address: kfwang72@163.com