Effects of neutralizing antibodies on escape from CD8+ T-cell responses in HIV-1 infection

Despite substantial advances in our knowledge of immune responses against HIV-1 and of its evolution within the host, it remains unclear why control of the virus eventually breaks down. Here, we present a new theoretical framework for the infection dynamics of HIV-1 that combines antibody and CD8+ T-cell responses, notably taking into account their different lifespans. Several apparent paradoxes in HIV pathogenesis and genetics of host susceptibility can be reconciled within this framework by assigning a crucial role to antibody responses in the control of viraemia. We argue that, although escape from or progressive loss of quality of CD8+ T-cell responses can accelerate disease progression, the underlying cause of the breakdown of virus control is the loss of antibody induction due to depletion of CD4+ T cells. Furthermore, strong antibody responses can prevent CD8+ T-cell escape from occurring for an extended period, even in the presence of highly efficacious CD8+ T-cell responses.

: Realistic HIV-1 dynamics can be generated under a combination of shortlived CD8+ T cell and long-lived specific antibodies, with short-lived cross-reactive antibodies promoting sequential dominance of variants. Shown here are the effects of CD8+ T cell and specific antibody lifespan on (i) ratio of set-point viral load to final (AIDS) viral load (a-c), (ii) standard deviation in set-point viral load (d-f) and (iii) single strain dominance (g-i) with lifespan of partially cross-reactive response increasing from 10 to 100 to 300 days from left to right column. Other parameters and initial conditions identical to Figure 2 in main text. Single strain dominance can be quantified by the measure ε by comparing the relative prevalence of the two most common antigenic variants within single epidemics, and then averaging across extended periods of time (see Recker et al, 2007 Proc Natl Acad Sci USA 104: 7711-7716). 3. Sensitivity of setpoint viraemia to potency of specific antibody and CD8+ T cell responses Figure S3: Setpoint viraemia is more sensitive to potency of Nab responses than it is to the potency of CTL responses. Setpoint viraemia was calculated by running the model in the absence of immune decay for 8 years, and taking the average viraemia for years 6-8 (Other parameters: ρ = 8; 1/µu = 10 days; 1/µw = 100 days; 1/µz = 1000 days; γ = 1; φ(0) = 1; η = ξ = ω = 3.2*10 -5 ; α = 0 days -1 ; n=4, m=3).

Mathematical analysis
We are able to explain the basic behaviour of the model by analysing the system in the absence of variability in the virus population. Under these circumstances, the equations can be reduced to: This reduced system admits two steady states within (v, z, u) space: C0 = (0, 0, 0) and C* = (v*, z*, u*) given by: The eigenvalues of the Jacobian matrix are the roots of the polynomial: Thus for C0 we find: and hence C0 is a saddle point.
Substituting for v*, z* and u* in C* reveals that the eigenvalues satisfy: Each coefficient in this expression is positive, thus each eigenvalue negative and C* stable.
We can further interpret v* (i.e. the steady state viral load in the absence of immune decay) as the set-point viral load.
One of the chief assumptions of this model is that the NAb response has much greater longevity than the CD8+ response; v* is thus much more sensitive to variation in antibody induction and potency than it is to CD8+ induction and potency as shown in Figure S5. From Figure S1a, it is evident that while ≫ 0, v*(φ) is roughly constant, but experiences a sharp decline as → 0; thus these few assumptions alone can generate the 3-phase pattern of viraemia associated with HIV infection (Figure 2a).
Assuming that the rate of decay of φ is proportional to total viral load (equation 5) does not alter the basic behaviour of the model, but allows us to recover the known relationship between set-point viraemia and time to AIDS. Linking φ to viral load in this manner also causes it to decline significantly in the early stages of infection (see Fig 2a in  Notably, our model is able to capture such dynamics without including any saturating terms or limitations in resource availability such as the availability of CD4+ T cells for infection. Rather, the achievement of set-point and ultimate progression to AIDS both follow from a balance of immune responses. The inclusion of such limiting terms would be unlikely to alter the qualitative dynamics of the model, but may act to quantitatively affect the magnitude of peak viraemia in acute infection.