A diffusion-based HIV model with inflammatory cytokines and adaptive immune impairment

HIV continues to pose a critical threat to global public health, contributing to a high number of deaths worldwide. The virus predominantly attacks CD4 + T lymphocytes, which are essential for coordinating immune responses. A progressive decline in these cells is a hallmark of HIV pathogenesis. Recent research has underscored the role of inflammatory cytokines in promoting viral spread and exacerbating immune dysfunction. This study presents a spatially structured model of HIV infection incorporating the role of inflammatory cytokines. The model consists of six interacting components: healthy CD4 + T cells, HIV-infected cells, inflammatory cytokines, free viral particles, cytotoxic T lymphocytes (CTLs), and antibodies. It accounts for both cell-free (virus-to-cell) and direct (cell-to-cell) modes of transmission. The model also captures the suppression of adaptive immune responses involving CTLs and B cells. Motivated by recent findings that immune and infected cells, as well as viruses, may migrate from high to low concentration areas, we introduce diffusion terms to represent spatial movement, resulting in a system of nonlinear partial differential equations. We first establish the model's mathematical well-posedness by proving the existence and boundedness of global solutions. A basic reproduction number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> is derived, serving as a threshold parameter that governs the stability of two equilibria: the HIV-free equilibrium ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mi mathvariant="script">F</mml:mi><mml:mi mathvariant="script">E</mml:mi></mml:mrow></mml:math> ) and the HIV-persistent equilibrium ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:mi mathvariant="script">PE</mml:mi></mml:mrow></mml:math> ). By constructing suitable Lyapunov functions and applying LaSalle's invariance principle, we demonstrate that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mrow><mml:mi mathvariant="script">F</mml:mi><mml:mi mathvariant="script">E</mml:mi></mml:mrow></mml:math> is globally asymptotically stable when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>≤</mml:mo><mml:mn>1</mml:mn></mml:math> , while <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mrow><mml:mi mathvariant="script">PE</mml:mi></mml:mrow></mml:math> becomes globally stable if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>&gt;</mml:mo><mml:mn>1</mml:mn></mml:math> . Numerical simulations are performed to validate the analytical results, and a sensitivity analysis of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M8"><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> is carried out to evaluate the impact of critical model parameters.