We can compare the effectiveness of CTL responses against lytic and budding virus infections. For reference we compare both to the standard model of a budding virus that assumes infected cells have exponentially distributed lifetimes, and immediately following infection become visible to CTL and begin to make virus at a constant rate.

For a meaningful comparison of the three model strategies, we choose parameters such that (i) in the absence of CTL, infected cells have the same expected lifetime and the same net growth rate , and (ii) the duration of the window of opportunity for CTL to detect infected cells before virus is released is equal in the age-structured budding and lytic models. This is the parameter shown in Figure 1.

In the absence of CTL, the expected lifetime of a cell infected with a budding virus is . For a lytic virus it is . In a simple birth-death model , where new infected cells are generated at rate , the expected lifetime of infected cells is and their numbers grow at net rate . We set the growth rate without loss of generality to , and then choose parameters that simultaneously equate the infected cell lifetimes, satisfy equations (19) and (20), and equate the values of for the budding () and lytic strategies. Figure 2 shows the dependence of the net growth rate of infected cells on the CTL density for the different strategies.

We draw three conclusions here. First, if budding and lytic viruses are visible to CTL from the time of infection, both life-history strategies give identical results to the simple birth-death model. If there is any delay in infected cells becoming visible to CTL, the threshold CTL frequency required for immunity increases.

Second, if we control for growth rate, infected cell lifetime and the CTL window of opportunity , a lytic strategy is harder to control than a budding strategy. This is perhaps counterintuitive at first; one might expect that removal of a cell infected with a lytic virus removes all the potentially infective virions and thus would have a more significant impact on slowing the spread of infection than the removal of a continuously-shedding infected cell. However, the result can be understood simply; with these constraints, the cell infected with a lytic virus is visible to CTL for a smaller proportion of its replicative cycle. With equal expected lifetimes and growth rates, the average rate of production of virus per unit time is equal for the two strategies – so on average a CTL killing a lytic-virus-infected cell prevents fewer virions being released than when killing a cell infected with a budding virus.

Our third conclusion is that to make a parameter-independent comparison meaningful requires controlling for the growth rate, cell lifetime and CTL visibility window , as well assuming equal for the two strategies. Relaxing these constraints and allowing the parameters to vary then moves from the general to the specific, and for a given pair of virus life-histories the lytic versus budding conclusion above may be reversed. Given the parameters governing different virus replication strategies, this framework allows us to predict and compare the abilities of CTL to control them.

We assume that the cytopathic effects of SIV begin upon virus shedding (). The growth rate of the infected cell population when susceptible cells are in abundance is given implicitly by equation 19,(21)

In what follows, we assume that both the natural mortality of cells and the contribution of the endogenous CTL response to infected cell death in the first few days of acute infection are negligible. The combined process of presentation of SIV epitopes to naive CTL in local lymph nodes, activation, proliferation and migration of CTL to the infection site is likely to take several days; and similar upslopes of virus load are observed in SIV-infected rhesus macaques in the presence or absence of CD8 responses [19], although CTL likely make a substantial contribution to limiting peak viremia [19]–[22]. We therefore assume that cell death at the site of virus entry in the first week of infection is due mainly to non-CTL mechanisms such as innate immunity and viral cytopathicity, and so we assume in unvaccinated individuals.

Li et al. [23] studied the growth of infected cell numbers and CD4 depletion in the gut lamina propria in SIV infection and showed that local infected cell densities doubled between days 6 and 7 post-inoculation (from ), corresponding to a net exponential growth rate of . The death rate of infected CD4+ T cells could be estimated following depletion of uninfected cells locally, at around 28 days post-infection, and found to be approximately (half-life days). A caveat here is that at this relatively late stage of acute infection this figure very likely includes the contribution of adaptive immune responses to cell death. So it likely overestimates the death rate of infected cells in the first week of infection.

Now suppose that a vaccine can generate a local SIV-specific memory CTL density, . What is the critical value needed to reduce the acute growth rate to zero? Using equation (21) with and the parameter estimates from primary SIV infection above, we can eliminate the unknown quantity and obtain(22)We assume shedding begins post infection [24] and that an infected cell becomes visible to CTL 6–12 hours post infection, depending on the epitope being expressed. We solve (22) numerically to calculate a distribution of estimates of , using flat priors on the uncontrolled growth rate , infected cell death rate , and the CTL window duration . We obtain the median value  = 0.8 day (IQR 0.6–1.1). Thus the CTL elicited by a vaccine must increase the infected cell death rate at least 2-fold (IQR 1.0–4.1).

We can assess the importance of modeling virus epitope dynamics by comparing these estimates of to those of the standard model in which . To control virus growth with a CTL response directed against an epitope first visible 6 h post infection ( h) requires an increase in the critical CTL density of 14% (IQR 12–15) over the standard model prediction; a response directed against an epitope visible at 12 h requires an increase of 33% [28]–[38].

The quantity is the minimum killing rate required of one epitope-specific response to control infection. However, a vaccine-induced CTL response may target multiple epitopes. Each specific response will be able to recognise an infected cell beginning a time after it becomes infected, and will then contribute a killing rate . Each may then be required to contribute less than to the infected cell death rate.

Our estimate of the critical killing rate does not depend on the time required for reactivation and/or migration of tissue resident of SIV-specific CTL to the site of infection. It is also independent of inoculum size, provided as required by the assumptions of the model. This independence arises because all rates of growth and death are linear in infected cell numbers. In fact, a delay in the onset of CTL activity allows the growing population of infected cells to approach the steady-state age distribution required for our result to hold. Nevertheless, an early-acting CTL response must act rapidly enough to avert the systemic spread of the virus; thus unsurprisingly it is important to induce a response that is as large and fast-acting as possible.

To put the estimate of into perspective, we can compare it to estimates of the total contribution of CTL to infected cell death late in the acute phase of SIV infection. Ganusov and de Boer [25] calculated bounds of for killing by CTL specific for the KP9 epitope of Gag near the peak of acute infection of rhesus macques with SIV. Mandl et al. [26] obtained a comparable estimate of for the response to Tat epitopes. Since there are likely multiple epitopes being targeted, these figures are lower bounds on the total CTL-mediated killing rate. This rate is then much larger than our estimate of the total CTL killing rate required at the entry site.

These observations suggest that the challenge CTL face in controlling the spread of virus after systemic dissemination is far more severe than the one they face at the infection site. There are several possible reasons for this. First, infected cells are spatially localised. Second, resting CD4 T cells are likely the primary target cell population early in infection, and these cells produce virus at a substantially lower rate than the activated CD4 T cells that are the major source of virus in the disseminated acute phase [27]. Third, progressive exhaustion of CTL likely reduces per-CTL surveillance or killing rates later in infection.

As a step further, knowing the required vaccine-induced cell death rate immediately gives us a lower bound on if we know the surveillance rate – the rate that a single CTL moves between potential targets – at the entry site. The value of is currently unknown, although we can place broad bounds on it using estimates from other in vivo model systems. Direct estimates have been derived from killing of non-replicating cells pulsed with LCMV-derived peptides, in mouse spleens [8], [9], yielding in the range per CTL; again for LMCV in mice using longitudinal blood measurements [12], giving a higher estimates of in the spleen in the range for acute infection, and in chronic infection; and for polyoma virus [13] using peptide-pulsed splenocytes, in acute infection and in chronic infection. (Notably, these studies consistently conclude that the per-cell rate of surveillance and killing of CTL is reduced in chronic infection). Using bounds of cells/minute, yields critical CTL densities in the range to . Despite this uncertainty, which arises from our deliberately conservative bounds on , these results suggest the hopeful message that the required CTL densities are not excessively large. Polyoma virus [13], Influenza (Seddon, B., unpublished observations) and LCMV can all induce memory CTL densities among total splenocytes of at least 1%, and with the obvious caveat of differing anatomical locations, this density is at least two orders of magnitude larger than the predicted minimum densities of mucosa-resident SIV-specific memory CTL required for protection.

We have shown that we expect mass-action kinetics to hold if populations are well-mixed, and either the E∶T ratio is high or handling times are much shorter than cell-cell surveillance times. In early SIV infection, the validity of the well mixed and/or the high E∶T assumptions will depend jointly on the degrees to which infected cells and CTL are clustered or spread diffusely across the tissue. If both populations are well mixed, we can make a rough estimate of E∶T early in SIV infection if CTL are at the predicted critical density. We estimate 3000–10000 cells (of all types) per , assuming close packing with a center-to-center spacing . CTL at a density of will then be present at , so if the density of infected cells is smaller than this. Li et al. [23] measure the density of SIV RNA+ CD4 T cells at day 6 post infection to be approximately with a doubling time of approximately 1 day. Using the well-mixed assumption, we would then expect , and so the model to be valid, until at least day 3 or 4 post infection.

We also note that the limits of applicability of mass-action models to killing assays are still ill-defined. Ganusov et al. [28] demonstrated that for killing of LCMV-pulsed cells in the mouse spleen, a mass-action model appears to hold for E∶T ratios as low as 0.1.

If infected cells are tightly clustered and sparsely infiltrated by CTL, one would expect the rate of killing by CTL to be limited by the handling time once a cluster has been located. In the well-mixed deterministic model, the total rate of loss of infected cells will then be linear in CTL densities and independent of infected cell numbers (equation (10)). In this regime, the model predicts CTL will ultimately fail to control the infection, assuming susceptible cells are abundant and accessible. These assumptions may not hold, however. The density of susceptible cells in healthy tissue is an upper limit to infected cell densities in very early infection, before the influx of SIV-specific CD4 cells that provide new targets. Resting CD4 T cells are present at a density of 100–200 mm in the vaginal mucosa (using immunohistochemical staining of biopsies taken from healthy rhesus macaques – Gordon S., Franchini G. (NCI), unpublished data), meaning susceptible cells make up 1–5% of the total cell population in the lamina propria. Li et al. [23] measure mean peak infected cell densities of 60 cells ; so if infected cells are clustered in patches, the local availability of susceptible cells may be a factor affecting the rate of virus spread.

All of these uncertainties emphasise that more precise estimates for and need to be obtained with assays in tissues whose spatial organisation more closely reflects typical HIV/SIV infection sites in oral, rectal or vaginal mucosa, and using cells infected with live replicating virus rather than peptide-pulsed targets.

There are inevitably many qualifications to this result that are specific to SIV, as well as more general issues that we present in the Discussion. Several will tend to increase the estimate of the critical frequency. First, as noted above, our estimate of the infected cell death rate may be too high, as it is taken from data at the infection site 28 days-post infection when the primary CTL and antibody responses are likely present. We incorporate our uncertainty in this death rate by using a wide spread of plausible values. Second, in the early growth phase of natural primary infection, selective pressure on the mutating virus exerted by the developing CTL response is expected to be low. A vaccine-induced memory CTL population will increase this pressure. To minimise this effect, broad coverage of virus epitopes to both early and conserved proteins is required. Third, we neglect the longer-term effect of the early generation of latently infected CD4 T cells that escape CTL detection, for which there is some evidence in acute infection [29].

Other factors will act to reduce our estimate. For example, in addition to cytolysis, CTL secrete soluble factors that may make a substantial contribution to CTL-mediated protection in both the acute [30] and chronic [31], [32] phases. Our model includes cytolytic effects only and so may overestimate the number of memory CTL required to control infection. In addition, a vaccine is likely to induce specific antibody that will further decrease the rate of transmission of virus from cell to cell, lower the growth rate of infected cell numbers and so lower the estimate of CTL frequencies necessary to reduce growth to zero.

Finally, it is worth noting the difficulties in connecting rates of surveillance and killing by a single CTL in a given tissue to estimates of the total-body contribution of CTL to infected cell death. The problems stem from the dimensionality of rate constants and the implicit averaging of the effects of CTL in different anatomical locations. For example, Wick et al [33] estimated the rate at which individual CTL kill using blood from HIV-infected individuals. In their terminology, and are the densities of infected cells and specific CTL in units of cells per unit volume of blood, and the rate of loss of due to CTL is . First, with this choice of units the parameter has units of and is no longer interpretable as a rate of surveillance (see Text S1); rather, is the rate of loss of infected cells in blood per blood CTL. Second, using their estimate of and measurements of , Wick et al. conclude that each CTL in blood kills approximately 0.7 infected cells per day. However, the loss of infected cells in blood is the also the result of killing of infected cells by CTL in spleen and other tissues; it is not clear how obtained from blood measurements relates to per-CTL killing rates in other anatomical locations.