Dynamics of an HIV pathogenesis model with CTL immune response and two saturated rates

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1 ISSN England UK World Journal of Modelling and Simulation Vol. (24) No. 3 pp Dynamics of an HIV pathogenesis model with CTL immune response and two saturated rates Youssef Tait Khalid Hattaf 2 Noura Yousfi 3 Laoratory Analysis Modeling and Simulation Morocco 2 Department of Mathematics and Computer Science Faculty of Sciences Ben Msik Morocco 3 University Hassan II Mohammedia P. O Box 7955 Sidi Othman Casalanca Morocco (Received Novemer 23 Accepted May 8 24) Astract. In this paper we propose an HIV infection model with CTL immune response which includes the cure of infected cells and the effect of immune impairment caused y HIV infection. Both the infection transmission process and the proliferation of CTL immune response are modeled y two saturated functions. Moreover the qualitative analysis of the model is investigated and numerical simulations are given to illustrate our theoretical results. Keywords: HIV infection CTL immune response saturated rate staility Introduction Human immunodeficiency virus (HIV) is a virus that causes acquired immunodeficiency syndrome (AIDS). According to the World Health Organization (WHO) more than 35.3 million people living with HIV 2.5 million new infections and.6 million AIDS-related deaths in 22 []. Therefore mathematical models have een used in order to understand the dynamics of HIV infection [2 ]. In this paper we consider the asic model presented y Nowak and Bangham in [2] this model contains four variales that are uninfected cells (T ) infected cells (I) free virus (V ) and Cytotoxic T Lymphocytes (CTL) cells (Z). The model is given y the following nonlinear system of differential equations: dt di dv dz = s dt βv T = βv T δi piz = NδI µv = ciz Z where T () = T I() = I V () = V and Z() = Z are given. Susceptile host cells CD4 + T-cells (T ) are produced at a rate s die at a rate dt and ecome infected y virus at a rate βv T. Infected cells die at a rate δi and are killed y the CTL response at a rate piz. Free virus is produced y infected cells at a rate NδI and decays at a rate µv. CTLs expand in response to viral antigen derived from infected cells at a rate ciz and decay in the asence of antigenic stimulation at a rate Z. Note that the rate of infection in Eq. () is assumed to e ilinear in the virus V and the uninfected target cells T which is not reasonale to descrie the HIV infection. Hence we replace this ilinear form () The authors would like to thank the editor and the anonymous referees for their valuale comments and remarks which have led to the improvement of the quality of this paper. Corresponding author. address: taityoussef2@gmail.com. Pulished y World Academic Press World Academic Union

2 26 Y. Tait & K. Hattaf & N. Yousfi: Dynamics of an HIV pathogenesis model y a saturated mass action presented y Song and Neumann in [2]. Furthermore the proliferation of CTL response in Eq. () is ilinear in I and Z. However in the presence of immune impairment effects caused y HIV infection CTL proliferation is reduced [3]. On the other hand the cure of infected cells y noncytolytic elimination of the cccdna in their nucleus was omitted in Eq. (). Recently this cure of infected cells is considered y several works [ ]. For these three reasons we extend the asic model () to following model given y dt di = βv T +av dv = s dt βv T +av + ρi = NδI µv dz = ciz +αi Z (δ + ρ)i piz where ρ is the cure rate of infected cells and α represents the immune impairment rate. When a = ρ = α = we otain the asic model (). The aim of this work is to study the dynamical ehavior of our Eq. (2) which includes the effect of immune impairment caused y HIV infection and the cure of infected cells. The rest of the paper is organized as follows. Section 2 illustrates some properties of the model s solutions. The model analysis and simulation are given in section 3. Discussion and conclusion of this research are presented in section 4. (2) 2 Positivity and oundedness of solutions In this section we estalish the positivity and oundedness of solutions of Eq. (2) ecause this model descries the evolution of a cell population. Hence the cell densities should remain non-negative and ounded. These properties imply gloal existence of the solutions. In addition for iological reasons we assume that the initial data for system (2) satisfy : T I V Z. Proposition 2. All solutions starting from non-negative initial conditions exist for all t > and remain ounded and nonnegative. Moreover we have i) T (t) T () + s δ ii) V (t) V () + Nδ µ I iii) Z(t) Z() + c p [max(; 2 d )T () + I() + max( s s d ) + max(; δ ) I ] where T (t) = T (t) + I(t) represents the T-cells and δ = min(d δ). Proof. For positivity we show that any solution starting in non negative orthant R 4 + = {(T I V Z) R 4 : T I V E } remains there forever. In fact (T (t) I(t) V (t) Z(t)) R 4 + we have T T = = s + ρi I I= = βv T +av V V = = NδI Ż Z= = Hence positivity of all solutions initiating in R 4 + is guaranteed. Now we prove that the solutions are ounded. As T = s dt δi pz we deduce that T (t) T ()e δ t + s δ ( e δ t ) (3) since e δ t and e δ t thus i). Now we show ii). The equation V = NδI µv implies that V (t) = V ()e µt + Nδ t I(ξ)e (ξ t)µ dξ (4) WJMS for contriution: sumit@wjms.org.uk

3 World Journal of Modelling and Simulation Vol. (24) No. 3 pp then V (t) V () + Nδ µ I ( e µt ). Sine e µt we deduce ii). Finally we show iii). The equation Ż = ciz +αi Z implies that Since we get Ż Z ciz. ciz = c p [s ( T + dt ) + ( I + δi)] Z(t) [ c p (T () + I() s ) + Z()] e t + c p {s + t If d and δ we have If d and δ we have Z(t) Z() + c p If d and δ we have If d and δ we have Z(t) Z() + c p From Eq. (5)-(9) we deduce iii). [ ( d)t (ξ) + ( δ)i(ξ) ] e (ξ t) dξ T (t) I(t) }. (5) Z(t) Z() + c p Z(t) Z() + c p [ s + T () + I() ]. (6) [ ( s + T () + I() + δ ) ] I. (7) [ ( s d + 2 d ) ] T () + I(). (8) [ ( s d + 2 d ) ( T () + I() + δ ) ] I. (9) 3 Analysis of the model In this section we show that there exists a disease free equilirium point and two infection equilirium points we study the staility of these equilirium points. 3. Staility of free equiliria By a simple calculation system (2) has always one disease-free equilirium E f = ( s d ) corresponding to the maximal level of healthy CD4 + T-cells. Therefore the asic reproduction numer of (2) is given y R = The Jacoian matrix of the system (2) at an aritrary point is given y βnδs dµ(δ + ρ). () Proposition 3. J = d βv +av ρ βt (+av ) 2 βv βt +av (δ + ρ) pz pi (+av ) 2 Nδ µ. () cz ci (+αi) 2 +αi WJMS for suscription: info@wjms.org.uk

4 28 Y. Tait & K. Hattaf & N. Yousfi: Dynamics of an HIV pathogenesis model. If R < then the disease free equilirium E f is locally asymptotically stale. 2. If R > then E f is unstale. Proof. The Jacoian matrix evaluated at E f is J Ef = The characteristic polynomial of J Ef is d ρ βs d βs (δ + ρ) d Nδ µ P Ef (ξ) = (ξ + d)(ξ + )[ξ 2 + (δ + ρ + µ)ξ + (δ + ρ)µ( R )] then the eigenvalues of the matrix J Ef are ξ = d ξ 2 = ξ 3 = (δ + ρ + µ) (δ + ρ + µ) 2 4(δ + ρ)µ( R ) 2 ξ 4 = (δ + ρ + µ) + (δ + ρ + µ) 2 4(δ + ρ)µ( R ) 2. (2) it is clear that ξ ξ 2 and ξ 3 are negative. Moreover ξ 4 is negative when R < thus E f is locally asymptotically stale. Fig.. The infection dies out when R < as shown in the case of the equilirium E f. For this simulation we choose s = 5 β =.24 d =.2 δ =.5 p =. N = 5 µ = 3 ρ =. a =. α =. c=.3 and =.2 WJMS for contriution: sumit@wjms.org.uk

5 World Journal of Modelling and Simulation Vol. (24) No. 3 pp Infection steady states In this section we focus on the existence and staility of the no-disease-free disease steady states that we are going to name : infection steady states. In fact it is easily verified that the system (2) has two of them: E = (T I V ) where T = s ans+µ d ans+µr I = s µ(r ) δ ans+µr V = Ns(R ) ans+µr ; E 2 = (T 2 I 2 V 2 Z 2 ) where T 2 = (anδρ)i2 2 + (ansδ + µρ)i 2 + µs Nδ(ad + β)i 2 + µd I 2 = c α V 2 = NδI 2 µ Z 2 = Nδ[adρ + δ(ad + β)]i 2 + (βnsδ dµ(ρ + δ)) ). p(nδ(ad + β)i 2 + µd) From the iological point of view the point E represents asent of CTL immune response and E 2 represents HIV chronic of disease with CTL response. In order to study the local staility of the points E and E 2 we define the following numers : D = cs δ (3) µδr D = D δ(ans + µr ) + αµs(r ) (4) H = R +. D f (5) The numer D represents the asic defence rate and H is the half harmonic mean of R and D. The importance of these parameters is related the following results. Theorem 3.2. If R < then the point E does not exists and E = E f when R =. 2. If R > and H < then E is locally asymptotically stale. 3. If R > and H > then E is unstale. Proof. It easy to verified that if R < then the point E does not exists and E = E f when R =. We assume that R > the Jacoian matrix at E is d βv +av ρ βt (+av ) 2 βv βt J = +av (δ + ρ) (+av pi ) 2 Nδ µ. ci +αi The characteristic equation associated with J E is given y ( ξ + ci ) (ξ 3 + a ξ 2 + a 2 ξ + a 3 ) = + αi where a = d + δ + µ + ρ + βv + av βv av a 2 = (δ + µ + ρ)d + (µ + δ) + µ(δ + ρ) + av + av av βv a 3 = µd(δ + ρ) + µδ. + av + av WJMS for suscription: info@wjms.org.uk

6 22 Y. Tait & K. Hattaf & N. Yousfi: Dynamics of an HIV pathogenesis model Then ci +αi = D f (H ) H is an eigenvalue of J E. The sign of the eigenvalue D f (H ) H is negative if H <.Then the staility of E is determined y the distriution of the roots of the equation ξ 3 +a ξ 2 +a 2 ξ+a 3 = Using the same technique as that defined in [4] y K. Hattaf and N. Yousfi and from the Routh-Hurwitz Theorem given in [7] the equation ξ 3 + a ξ 2 + a 2 ξ + a 3 = has all its roots negative real parts if R >. Then E is locally asymptotically stale when R > and H <. If H > the eigenvalue D f (H ) H is not negative then E is unstale. As shown in Fig. 2 the situation where R > and H < is characterized y the fast decline of the CTLs cells and the persistence of the infection when time goes y. By a simple calculation we can show that if R > then the condition of local asymptotic staility of E is equivalent to R D < ( R )(δ(ans + µr ) + αµs(r ) ) µδr which means that if the asic defense numer y CTLs response is elow the threshold ( R R )( δ(ans+µr )+αµs(r ) µδr ) then immune response cannot keep up with the infection and it eventually vanishes. Theorem 3.3. If α > c or H < then the point E 2 does not exists and E 2 = E when H =. 2. If α < c and H > then E 2 is locally asymptotically stale. Proof. We notice that the condition α < c and H > is equivalent to I 2 < I. Then it easy to verified that the point E 2 does not exists if H < or α > c and E 2 = E when H =. We assume that α < c and H > the Jacoian matrix at E 2 is J E2 = d C ρ C 2 C ρ δ pz 2 C 2 pi Nδ µ ξ cz 2 (+αi 2 ) 2 where C = βv 2 +av 2 and C 2 = βt 2 (+av 2 ) 2. The characteristic equation associated with J E2 is given y where (ξ 4 + ξ ξ ξ + 4 ) = (6) = d + C + µ + ρ + δ + pz 2 2 = µ(d + C + ρ) + (d + C )(δ + pz 2 ) + p ( + αi 2 ) Z 2 + dρ 3 = µdρ + µc (δ + pz 2 ) + p ( + αi 2 ) Z 2(µ + d + C ) 4 = p ( + αi 2 ) Z 2µ(d + C ). From the Routh-Hurwitz Theorem given in [7] the eigenvalues of the aove matrix have negative real parts when H >. Consequently E 2 is locally asymptotically stale when H >. As shown in Fig. 3 the situation where α < c and H > is characterized y an increase CD4 + cells numer and a light decrease of the virus load. y a simple calculation we show that under the condition α < c E 2 is locally asymptotically stale is equivalent to WJMS for contriution: sumit@wjms.org.uk

7 World Journal of Modelling and Simulation Vol. (24) No. 3 pp Fig. 2. The virus persists in the asence of the CTLS response as shown in the case of the equilirium E. For this simulation we choose s = 5 β =.24 d =.2 δ =.5 p =. N = 2 µ = 3 ρ =. a =. α =. c =.3 and =.2 which correspond to R > and H <. R D > ( R )(δ(ans + µr ) + αµs(r ) ) µδr which means that if the asic defense numer y CTL response is aove the threshold ( R ) then immune response CTL can reduce the concentration of virus. R )( δ(ans+µr )+αµs(r ) µδr 4 Discussion and conclusion In this work we gave a rigorous local staility analysis of a new mathematical model descriing the interactions etween human imunodeficiency virus (HIV) infection of CD4+ T-cells and cytotoxic T lymphocytes ( CTL ) immune response. The disease free equilirium is locally asymptotically stale if the asic infection reproduction numer satisfies R <. For R > the staility of the two endemic equilirium points is dependent on the asic defence numer y CTLs response D. Indeed If R D < ( R )(δ(ans + µr ) + αµs(r ) ) µδr then E is locally asymptotically stale while E 2 is is locally asymptotically stale when R D > ( R )(δ(ans + µr ) + αµs(r ) ). µδr From the R formula we notice that R is independent of the parameters of the activation of CTL that are c α and p. which means that CTL do not permit to eliminate the virus. However they can reduce the concentration of viral load and increase the concentration of CD4+ healthy cells this can e seen when comparing the components of the viral load and those of the concentration of CD4+ efore and after the WJMS for suscription: info@wjms.org.uk

8 222 Y. Tait & K. Hattaf & N. Yousfi: Dynamics of an HIV pathogenesis model Fig. 3. an increase of the CD4 + cells and light decrease of the virus load as shown in the case of the equilirium E 2. For this simulation we choose s = β =.24 d =.2 δ =.5 p =. N = 2µ = 3 ρ =. a =. α =. c=.3 and =.2 which correspond to H > and α < c. activation of CTL that is to say under the condition H >. In fact By simple calculations show that: T < T 2 and V 2 < V where T T 2 V and V 2 are given aove. So the cellular immune can control the load of virus. References [] World Health Organization. Gloal Health Sector Strategy on HIV/AIDS Availale at: Accessed on August [2] M. Nowak C. Bangham. Population dynamics of Immune Responses to Persitent Viruses. Science : [3] K. Hattaf N. Yousfi. Two optimal treatments of HIV infection model. World Journal of Modelling and Simulation 22 8: [4] K. Hattaf N. Yousfi. A delay differential equation model of HIV with therapy and cure rate. International Journal of Nonlinear Science 2 2: [5] K. Hattaf N. Yousfi. Dynamics of HIV infection model with therapy and cure rate. International Journal of Tomography and Statistics 2 6(): [6] K. Hattaf N. Yousfi. Optimal control of a delayed HIV infection model with immune response using an efficient numerical method. ISRN Biomathematics 22 doi:.542/22/2524: 7. [7] B. EL Boukari K. Hattaf N. Yousfi. Modeling the therapy of HIV infection with CTL response and cure rate. International Journal of Ecological Economics and Statistics (): 7. [8] D. Li W. Ma. Asymptotic properties of an HIV- infection model with time delay. J. Math. Anal. Appl : [9] R. Xi. Gloal staility of an HIV- infection model with saturation infection and intracellular delay. Journal of Mathematical Analysis and Applications 2 375: [] X. Zhou X. Song X. Shi. A differential equation model of HIV infection of CD4 + T-cells with cure rate. Journal of Mathematical Analysis and Applications (2): WJMS for contriution: sumit@wjms.org.uk

9 World Journal of Modelling and Simulation Vol. (24) No. 3 pp [] R. Ouifki G. Witten. A model of HIV- infection with HAART therapy and intracellular delay. Discrete Cont.Dyn- B 27 8: [2] X. Song A. Neumann. Gloal staility and periodic solution of the viral dynamics. J. Math. Anal. Appl : [3] S. Iwami T. Miura S. Nakaoka et al. Immune impairment in HIV infection: Existence of risky and immunodeficiency thresholds. Journal of Theoretical Biology 29 26: [4] K. Hattaf N. Yousfi A. Tridane. Mathematical analysis of a virus dynamics model with general incidence rate and cure rate. Nonlinear Anal. RWA 22 3: [5] K. Hattaf N. Yousfi. Gloal staility of a virus dynamics model with cure rate and asorption. Journal of the Egyptian Mathematical Society 24. [6] C. Lv L. Huang Z. Yuan. Gloal staility for an HIV- infection model with Beddington-DeAngelis incidence rate and CTL immune response. Communications in Nonlinear Science and Numerical Simulation 24 9: 2 27 [7] I.S. Gradshteyn I. Ryzhik Routh-Hurwitz Theorem. sixth ed. in: Tales of Integrals Series and Products. Academic Press San Diego CA 2. WJMS for suscription: info@wjms.org.uk