+
+
Modelling the influence of activation-induced apoptosis of CD4 and CD8 T-cells on the immune system response of a HIV infected patient # Belmudes ,
∗ Stan ,
# Fonteneau ,
+ Zeggwagh ,
Guy-Bart Florence Raphaël Frederic + † # Marie-Anne Lefebvre , Christian Michelet and Damien Ernst
University of Cambridge (United Kingdom), #University of Liège (Belgium), +Ecole Supérieure d’Electricité (France), †Hôpital Pontchaillou, Rennes (France)
Based on the HIV infection dynamics model proposed by Adams et al. in [1], we propose an extended model represented by a set of nonlinear Ordinary Differential Equations (ODEs) that aims at incorporating the influence of activation-induced apoptosis of CD4+ and CD8+ T-cells on the immune system response of HIV infected patients. Through this model we study the influence of this phenomenon on the time evolution of specific cell populations such as plasma concentrations of HIV copies, or blood concentrations of CD4+ and CD8+ T-cells. In particular, this study shows that depending on its intensity, the apoptosis phenomenon can either favor or mitigate the long-term evolution of the HIV infection.
Introduction
Apoptosis-compliant model for the HIV infection dynamics ∗ ˙ T1 = λ1 − d1T1 − k1V T1−aT1T1 T1 T˙2 = λ2 − d2T2 − k2V T2 ∗ ∗ ∗ ˙ T1 = k1V T1 − δT1 − m1ET1 ∗ ∗ ∗ ˙ T2 = k2V T2 − δT2 − m2ET2 ∗ ∗ ˙ V = NT δ (T + T ) − cV
(7) (8) (9) (10) (11)
2
1
− (ρ1k1T1 + ρ2k2T2) V ∗ ∗ b (T + T ) E 1 2 E˙ = λE + ∗ E (12) ∗ (T1 + T2 ) + Kb dE (T1∗ + T2∗) ∗ − ∗ E − δE E−aE T1 E ∗ (T1 + T2 ) + Kd
−5
Time evolution of the state variables for a =10 T
1
6
−4
and a =0
Time evolution of the state variables for a =10
E
T
10
T
V 4
10
* 1
T
3
10
2
10
T*
2
T2
1
10
E
V T*
1
4
10
aT
3
(in ml/(cells x day))
4
5
6 −4
x 10
1
Fig. 5: Two-parameter continuation of the saddle-node bifurcation point LP1 corresponding to (aT1 , aE ) = (3.87418 × 10−5, 0). A CUSP bifurcation point appears −4 −4 at (aT1 , aE ) = 4.838 × 10 , 1.956 × 10 .
5
x 10 E 4
10
CUSP Bifurcation Point
3
10
6
V T2
2
10
T*
1
1
T*
2
0
0
100
200
300 Time in days
400
500
10
600
Stable eq. point (eq. point 1)
7
10
0
10
0
100
200
300 Time in days
400
500
600
5
7 4 6 3
T
E
1
2
10
LP1
Stable eq. point (eq. point 2 −−"non−healthy")
Time evolution of the state variables for a =10−2 and a =0 6
Number of cells per ml of blood or plasma
3.5
1
−4
x 10
3
a
E
a 2.5
2
1 1.5
(in ml/(cells x day))
1
0.5
0
T
(in ml/(cells x day))
1
0
T2
3
10
V 2
10
Fig. 6: Concentrations of non-infected CD4+ T-cells (T1) corresponding to the equilibrium point to which the patient’s state converges when starting from the primo-infection initial condition.
E
*
T1
* 2
T
0
10
0
100
200
300 Time in days
400
500
600
Max. value (138510) is reached
3000
−5
at a =3.87418× 10 T
1
2500 Settling−time in days
3
4
10
1
2000
1500
1000
500
0.1
0.2
0.3
5
10
−4
x 10
2 T
10
0
5 4
5
10
Steady−state of variable T1 (in number of cells per ml of blood)
Number of cells per ml of blood or plasma
10
2
10 Number of cells per ml of blood or plasma
Number of cells per ml of blood or plasma
10
0.4 0.5 0.6 aT (in ml/(cells x day))
0.7
0.8
0.9
1 −4
x 10
Fig. 3: Evolution of the settling-time (minimum time required for the state variables to be within an infinite-norm distance of 1 percent of their asymptotic value) as a function of the apoptosis parameter aT1 with aE =0.
1
1
E
(6)
5
0 0
1
1
T
1 stable equilibrium point (eq. point 1)
5
0
Time evolution of the state variables
LP1
T
1
5
+
6
1
10
(1) (2) (3) (4) (5)
10
3 equilibrium points: * 2 stable (eq. points 1 and 2); * 1 unstable
and a =0
1
6
3500
: number of non-infected (infected) CD4 Tlymphocytes (in cells/ml) T2 (T2∗) : number of non-infected (infected) macrophages (in cells/ml) V : number of free HIVs (in virions/ml) E : number of HIV-specific cytotoxic CD8+ T-cells (in cells/ml)
−4
Analysis of the apoptosis-compliant model
1
T1 (T1∗)
CUSP bifurcation point 1
Settling−time as a function of aT
T˙1 = λ1 − d1T1 − k1V T1 T˙2 = λ2 − d2T2 − k2V T2 ∗ ∗ ∗ ˙ T1 = k1V T1 − δT1 − m1ET1 ∗ ∗ ∗ ˙ T2 = k2V T2 − δT2 − m2ET2 ∗ ∗ ˙ V = NT δ (T1 + T2 ) − cV − (ρ1k1T1 + ρ2k2T2) V ∗ ∗ b (T + T ) E 1 2 E˙ = λE + ∗ E (T1 + T2∗) + Kb dE (T1∗ + T2∗) E − δ E − ∗ E (T1 + T2∗) + Kd
LP2
−4
Fig. 2: Time-evolution of the state variables of the model (7)-(12) for aT1 = 10−5 , aT1 = 10−4, and aT1 = 10−2 starting from the primo-infection initial condition with aE =0. The apoptosis phenomenon can either favor or mitigate the long-term evolution of the HIV infection.
The Model of Adams et al.
1 stable equilibrium point (eq. point 2)
at (aT =4.838× 10 ,aE=1.956× 10 )
ml . aT1 and aE are expressed in cells×day
Human Immunodeficiency Virus (HIV) is a retrovirus that may lead to the lethal Acquired Immune Deficiency Syndrome (AIDS). Progression from HIV infection to AIDS is primarily due to an extensive depletion of CD4+ T-cells. T-cell loss may be due to direct destruction by the virus (direct virus-induced cytolysis) or to defective T-cell generation. In 1991, apoptosis, also called programmed cell death, has been suggested as another mechanism responsible for T-cell depletion during the evolution of HIV-1 infection and an extensive body of recent literature is supporting this hypothesis. To the best of our knowledge, no mathematical model has yet tried to explain the influence of the activation-induced apoptosis phenomenon on the HIV infection dynamics. We propose here a modification of the model proposed by Adams et al. in [1]. This modification aims at modelling the activation-induced apoptosis phenomenon and at analyzing its influence on the HIV infection dynamics.
−4
x 10
T1 (in cells/ml)
Abstract
aE (in ml/(cells x day))
∗
Bifurcation diagram
x 10
9
8 stable eq. point (eq. point 1) 7
6
5
4 unstable eq. point 3 saddle−node bifurcation point (LP ) 1 at a =3.874× 10−5 (a =0) T E
2
1
Discussion Using a combination of numerical simulations and bifurcation analysis, we found that for some ranges of values of theapoptosis parameters, these activation-induced apoptosis phenomena had non-linear effects that could be beneficial to the immune system during the HIV infection. On the other hand, when the magnitude of the apoptosis parameters becomes too large, this potential beneficial effect disappears and activation-induced apoptosis mechanisms were then found to aggravate the HIV infection. Furthermore, since the HIV infection worsens when these activation-induced apoptosis rates become too large, one could also relate the progression of the HIV infection to AIDS to a change of magnitude in these rates. These findings need to be taken with caution since they are dependent on several modelling assumptions that would certainly require careful experimental validation. These results could potentially help in designing new anti-HIV therapies based on a drug-mediated regulation of the activation-induced apoptosis factors (such as gp120) in HIV infected patients. These therapies could be based on the injection of some specific interleukins to HIV positive patients, such as for example IL-2, IL-7 and IL-15 [2, 3, 5], although the role of interleukins on the immune system of HIV-infected patients/macaques is still a controversial issue since other studies (see e.g. [4]) have shown that they could have a detrimental effect.
stable eq. point (eq. point 2 −− "non−healthy") 3
10
1 E
2
10
0
0.1
0.2
0.3
0.4 0.5 0.6 0.7 Parameter aT (in ml/(cells x day)) 1
T*
0.8
0.9
1 −4
x 10
References
2
1
10
T
2
0
10
0
100
200
300 Time in days
400
500
600
Fig. 1: Time evolution of the state variables of the model (1)-(6) starting from the primo-infection initial condition (T1, T2, T1∗, T2∗, V, E) = (106, 3198, 0, 0, 1, 10).
Fig. 4: Bifurcation diagram of the equilibrium concentrations of non-infected CD4+ T-cells (T1eq ) when the bifurcation parameter aT1 varies from 0 to 10−4. A saddle-node bifurcation point (LP1) exists at aT1 = 3.874×10−5. Only the infected equilibrium points (equilibrium points 2, 3 and 4) are represented and aE is chosen equal to 0.
[1] B.M. Adams, H.T. Banks, Hee-Dae Kwon, and H.T. Tran. Dynamic multidrug therapies for HIV: Optimal and STI control approaches. Mathematical Biosciences and Engineering, 1(2):223–241, September 2004. [2] B. Ahr, V. Robert-Hebmann, C. Devaux, and M. Biard-Piechaczyk. Apoptosis of uninfected cells induced by HIV envelope glycoproteins. Retrovirology, 1:12, 2004. PMID: 15214962. [3] S. Beq, J.F. Delfraissy, and J. Theze. Interleukin-7 (IL-7): immune function, involvement in the pathogenesis of HIV infection and therapeutic potential. Eur. Cytokine Netw., 15(4):279–289, December 2004. [4] C. Fluur, A. De Milito, T.J. Fry, N. Vivar, L. Eidsmo, A. Atlas, C. Federici, P. Matarrese, M. Logozzi, E. Rajnavolgyi, C.L. Mackall, S. Fais, F. Chiodi, and B. Rethi. Potential role for IL-7 in Fas-mediated T cell apoptosis during HIV infection. The Journal of Immunology, 178:5340–5350, April 2007. [5] L. Vassena, M. Proschan, A.S. Fauci, and P. Lusso. Interleukin 7 reduces the levels of spontaneous apoptosis in CD4+ and CD8+ T cells from HIV-1-infected individuals. Proceedings of the National Academy of Sciences of the United States of America, 104(7):2355–60, February 2007.
