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Turing chemotactic instability in an HIV model

Florinda Capone, Roberta De Luca, Vincenzo Luongo

Abstract

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.

Turing chemotactic instability in an HIV model

Abstract

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.

Paper Structure

This paper contains 12 sections, 51 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Key biological aspects and model parameters
  3. Mathematical model and equilibria
  4. Preliminary numerical analysis
  5. Preliminaries to the linear instability analysis
  6. Stability analysis of disease-free equilibrium in the homogeneous case
  7. Stability analysis of the endemic equilibrium in the homogeneous case
  8. Linear stability analysis in the heterogeneous case
  9. Stability analysis of the disease-free equilibrium in the heterogeneous case
  10. Stability analysis for the endemic equilibrium and Turing instability
  11. Weakly nonlinear analysis
  12. Conclusion

Figures (5)

  • Figure 1: Time evolution of $CD4+$ Target cells $u$, $CD4+$ Infected cells $v$, and Virions $w$ (column oriented) at $\tau = 0, 0.3, 1, 1.2$.
  • Figure 2: Time evolution of $CD4+$ Target cells $u$, $CD4+$ Infected cells $v$, and Virions $w$ (column oriented) at $\tau = 0, 40, 50, 150$.
  • Figure 3: Time evolution of $CD4+$ Target cells $u$ at $\tau=0, 40, 50, 150$.
  • Figure 4: Time evolution of $CD4+$ Target cells $u$ at $\tau=10, 40, 80, 125$.
  • Figure 5: Asymptotic behaviour of $d_c$ with respect to $\delta$ for different values of $k$. All the parameters are set as in Sect. 2, while $k=1000$ (dashed line) , $500$ (continuous line). Dotted line represents the AIDS case where $k=180$ and $s=5$.