Modelling the evolution and spread of HIV immune escape mutants

Abstract

During infection with human immunodeficiency virus (HIV), immune pressure from cytotoxic T-lymphocytes (CTLs) selects for viral mutants that confer escape from CTL recognition. These escape variants can be transmitted between individuals where, depending upon their cost to viral fitness and the CTL responses made by the recipient, they may revert. The rates of within-host evolution and their concordant impact upon the rate of spread of escape mutants at the population level are uncertain. Here we present a mathematical model of within-host evolution of escape mutants, transmission of these variants between hosts and subsequent reversion in new hosts. The model is an extension of the well-known SI model of disease transmission and includes three further parameters that describe host immunogenetic heterogeneity and rates of within host viral evolution. We use the model to explain why some escape mutants appear to have stable prevalence whilst others are spreading through the population. Further, we use it to compare diverse datasets on CTL escape, highlighting where different sources agree or disagree on within-host evolutionary rates. The several dozen CTL epitopes we survey from HIV-1 gag, RT and nef reveal a relatively sedate rate of evolution with average rates of escape measured in years and reversion in decades. For many epitopes in HIV, occasional rapid within-host evolution is not reflected in fast evolution at the population level.

Figure 1. A mathematical model of within-host evolution and between-host transmission of escape mutants at a single CTL epitope.

A) The mathematical model in schematic form, where WT and EM denote the wild-type strain and escape mutant strain, respectively. B) Changes in numbers of susceptible () and infected () hosts over time showing the three phases of the epidemic: exponential growth, saturation and equilibrium. C) Changes through time in the proportion of infected hosts with escape at a single CTL epitope for different escape and reversion rates. D and E) Changes through time in the proportion of HLA matched (black lines) and HLA mismatched (red lines) hosts infected with escape at a single CTL epitope for different escape and reversion rates. Different escape rates are compared in D) and different reversion rates are compared in E). The following initial conditions and parameters were used for these plots: X¹(0) = 10⁴, X⁰(0) = ,  = 0.1,  = 0.9,  = 0, p = 0.1, μ = 1/50 years⁻¹, μ+α = 1/10 years⁻¹, βc = 0.3 and B = 10⁵ μ years⁻¹. These parameters yield a basic reproduction number of 3, since for this model R ₀ = βc/(μ+α).

Figure 2. CTL escape and reversion data from the current HIV pandemic.

A) Dataset 1: the evolution of previously described escape mutants in six CTL epitopes. Data from dated B-clade sequences provided in the Los Alamos database. The filled shapes show three epitopes for which the proportion of hosts with escape has remained relatively invariant over the past 20 years. The unfilled shapes show three epitopes for which the proportion of hosts with escape has increased over the last 20 years. B) Dataset 2: cross sectional data describing the proportion of HLA matched and HLA mismatched hosts with described escape mutants in gag, RT and nef. Each dot represents the data for a single CTL epitope (N = 26). Data from 84 chronically infected hosts from Switzerland. C) Dataset 3: escape data from individual case reports described in the literature. Each marker represents the results from one HLA matched host infected with the wild-type epitope at the first sample time. In cases where escape occurred the time between infection and escape is represented by a circle. In cases where escape did not occur the time between infection and the last sample is represented by a triangle. The inferred average time to escape is represented by a horizontal bar. These averages account for data, where available (triangles), from hosts in whom escape mutants do not appear (see Table S3 for details). D) Dataset 3: reversion data from individual case reports described in the literature. Each marker represents the results from one HLA mismatched host infected with a described escape mutant at a particular epitope at the first sample time. The markers are analogous to those described for C). E) Dataset 4: escape data from a longitudinal cohort of 189 acute seroconverters. Estimates are provided for 27 epitopes with previously described escape mutations in gag, RT and nef. These are largely the same epitopes shown in B), though there is some lack of overlap due to the absence of certain data from one or other dataset. N is the number of HLA matched hosts infected with the wild-type epitope at the first sample. In cases where escape occurred, the time between infection and escape is represented by a dot. F) Dataset 4: reversion data from the same longitudinal cohort of individuals. For each epitope N is the number of HLA mismatched hosts infected with an escape mutant at the first sample. In cases where reversion occurred, the time between infection and reversion is represented by a dot.

Figure 3. Observed and inferred escape rates, reversion rates and changes in escape prevalence.

A) A comparison of the mean times to escape inferred from dataset 2, the cross-sectional data (x-axis) and observed in dataset 4 (Figure 2E), the longitudinal cohort study (y-axis). B) A comparison of the mean times to reversion inferred from dataset 2 and observed in dataset 4 (Figure 2F). For A) and B) estimates are provided for epitopes in gag, RT and nef for which escape mutants have been described and for which data are available from both studies. The inferred rates (x-axes) are calculated from dataset 2 using the mathematical model. For A) the observed times to escape (y–axis) are calculated from dataset 4 by considering all HLA-matched hosts who have the wild-type epitope at the first sample. We then sum over all person-years of observation for which an escape mutant is absent and divide by the number of hosts in whom escape mutants emerge. For B) reversion rates from dataset 4 are estimated using an analogous method from all HLA-mismatched hosts who have an escape mutant at the first sample. Note that the epitopes in these graphs are the same as those presented in Figure 2B, except that, epitope FLK is absent from A) and epitope ETF is absent from both A) and B) because the relevant estimates were not available from dataset 4. The data are presented on a linear scale from 0–10 years and on a log scale beyond 10 years. In B) the crosses represent the four epitopes for which we have the least confidence in our inferred reversion rates (see Figure S1). The remaining epitopes are shown as circles. C) A correlation between observed and predicted changes in the escape prevalence of described escape mutants in gag, RT and nef in the population between approximately 1995 and 2005. These are the same 26 epitopes as shown in dataset 2. The observed changes are from sequence data downloaded from the Los Alamos Database (dataset 1, Figures 2A and S3). The predicted changes over the same period are estimated using the mathematical model parameterised by the escape and reversion rates inferred from dataset 2 (x-axes 2A and 2B).