Fig. 4

Modeling HIV-1 evolution under immune selection. a Schematic diagram of the HIV-1 selection model. The wild-type infected cell (black) produces \(R_{0}\) offspring up to generation \(g_{s1}\) and afterwards produces \(sR_{0}\) offspring due to immune selection (\(sR_{0} < 1\)). One mutant-type infected cell (red) starts to produce \(R_{1}\) secondary infected cells at generation \(g_{s1}\). At the viral set point (\(g \ge g_{s}\)) the reproductive ratio of the mutant-type infected cells becomes equal to 1. b The total HIV-1 infected cell population level (blue line) as a function of time post infection. The infected cell population consists of the wild-type (black) and mutant-type infected cells (red). The model parameters are \(R_{0} = 6\), \(sR_{0} = 0.78\), \(R_{1} = 1.5\), \(g_{s1} = 15\) (30 days) and \(g_{s} = 50\) (100 days). The HIV-1 single cycle mutation rate is \(2.16 \times 10^{ - 5}\) per base per generation [33] and the generation time is 2 days [34]. The wild-type infected cell population grows exponentially and starts to decline at the emergence of selection pressure, approximately 1 month post infection [24, 28]. Subsequently, a mutant-type infected cell arises and replicates exponentially. c The coalescent time distribution of wild–wild virus pairs (black bars) and mutant–mutant pairs (red bars). The coalescent time distribution of wild-mutant pairs is the same as that of wild–wild pairs (black bars). The wild–wild and wild-mutant pairs most likely coalesce at the transmission point, while the most recent common ancestor of a mutant-type virus pair is the mutant-founder virus that appears at generation \(g_{s1}\), i.e. 30 days post infection. When the wild-type virus population size is comparable with the mutant-type population size at 132 days post infection, the coalescent probability of any random viral pair including wild–wild, wild-mutant, and mutant–mutant pairs (blue bars) peaks at the transmission point (generation 0). d Diversity dynamics of the one-mutant model (red line), two-mutant model (blue line), and neutral model (dotted black line). The model parameters for the two-mutant model are R ₀ = 6, sR ₀ = 0.78, R ₁ = 1.5, R ₂ = 1.5, g _s1 = 15 (30 days), g _s2 = 16 (32 days), and g _s  = 50 (100 days). The diversity dynamics of the two-mutant model more closely resemble the neutral model’s diversity dynamics than the one-mutant model’s