Turing chemotactic instability in an HIV model

A ternary reaction-diffusion model for early HIV infection dynamics, incorporating logistic growth of target cells, is introduced. According to in vitro and in vivo studies, random movement of target cells, infected cells, and virions and a chemotactic attraction of target cells by cytokines, are included. The research explores the existence of disease-free and coexistence equilibria, conducting linear stability analyses for homogeneous and heterogeneous scenarios. Specifically, conditions for chemotactic-self diffusion instability of the endemic equilibrium are found, indicating that Turing patterns may emerge when the chemotactic effect surpasses a critical threshold. This threshold is lower than in models without logistic growth of target cells, suggesting that the logistic model provides better insights into infection hot spots in the early stages. The location and shape of these patterns, crucial for developing infection control strategies, are investigated using weakly nonlinear analysis and demonstrated through numerical simulations.