Modeling the evolutionary dynamics of quantitative antiretroviral resistance in HIV-1 infection

This paper develops a rigorous mathematical framework capable of linking invasion thresholds, trait-dependent fitness, and evolutionary stability. We propose an integro-differential within-host HIV-1 model in which the level of antiretroviral resistance is represented as a continuous (quantitative) trait structuring infected cells and viral particles. The model incorporates key biological mechanisms, including intracellular delay, cytotoxic T lymphocyte (CTL) immune responses, and both virus-to-cell and cell-to-cell transmission pathways. From a theoretical perspective, we establish the well-posedness of the model and prove its dissipativity and asymptotic compactness. Using perturbation and spectral methods, we characterize the basic reproduction number, R₀, as the spectral radius of an associated next-generation operator. We further derive a direct connection between R₀ and a resistance-dependent fitness function Θ, thereby linking epidemiological invasion criteria to the adaptive landscape governing resistance evolution. We prove that the infection-free equilibrium is globally asymptotically stable when R₀<1, whereas uniform persistence occurs when R₀>1. Our analysis reveals a fundamental evolutionary principle: viral persistence is a necessary condition for evolutionary selection. In particular, the sign of max_xΘ(x)-1 determines whether adaptive evolution can occur, while the shape of Θ determines the location of evolutionary attractors. Numerical simulations further highlight three qualitatively distinct regimes. When treatment suppresses the maximal invasion fitness below unity, viral extinction occurs before adaptive structuring can emerge. When treatment is only partially effective, viral persistence coexists with directional selection toward resistant phenotypes, generating stable evolutionary attractors at elevated resistance levels. In contrast, in the absence of therapy, resistance-associated fitness costs dominate and selection favors low-resistance strains. Finally, we show that the long-term evolutionary outcome depends not only on the fitness landscape but also on the structure of the mutation process. Under symmetric mutation kernels, evolutionary attractors coincide with fitness optima, whereas directional mutation biases can substantially shift the dominant phenotype away from the fitness-maximizing resistance level. These results demonstrate how therapeutic pressure and mutation jointly reshape the adaptive landscape and determine the emergence, persistence, and evolutionary endpoint of drug-resistant HIV populations.