We propose a general model to investigate the joint impact of viral diffusion and cell-to-cell transmission on viral dynamics. The mathematical challenge lies in the fact that the model system is partially degenerate and the solution map is not compact. While the simpler cases with only indirect transmission mode or weak cell-to-cell transmission mode have been extensively studied in the literature, it remains an open problem to understand the local and global dynamics of fully coupled viral infection model with partial degeneracy. In this paper, we identify the basic reproduction number as the spectral radius of the sum of two linear operators corresponding to direct and indirect transmission modes. It is well-known that viral mobility may induce infection in low-risk regions. However, as diffusion coefficient increases, we prove that the basic reproduction number actually decreases, which indicates that faster viral movements may result in a lower level of viral infection. By an innovative construction of Lyapunov functionals, we further demonstrate that the basic reproduction number is the threshold parameter which determines global picture of viral dynamics. In addition to the traditional dichonomy results of extinction and persistence as obtained in earlier works for many simpler models, we are able to prove global asymptotic stability of infection-free steady state and global attractiveness (as well as uniqueness) of chronic-infection steady state, depending on whether the basic reproduction number is smaller or greater than one. Numerical simulation supports our theoretical results and suggests an interesting phenomenon: boundary layer and internal layer may occur when the diffusion parameter tends to zero.

我们提出了一个通用模型来研究病毒扩散和细胞间传递对病毒动力学的联合影响。数学上的挑战在于以下事实：模型系统部分退化，求解图不紧凑。尽管在文献中已对仅具有间接传播模式或弱细胞间传播模式的较简单病例进行了广泛研究，但了解具有部分简并性的完全耦合病毒感染模型的局部和全局动力学仍然是一个悬而未决的问题。在本文中，我们将基本再现数确定为对应于直接传输模式和间接传输模式的两个线性算子之和的频谱半径。众所周知，病毒的流动性可能会在低风险地区引起感染。但是，随着扩散系数的增加，我们证明基本繁殖数量实际上减少了，这表明更快的病毒运动可能导致更低水平的病毒感染。通过Lyapunov功能的创新构造，我们进一步证明了基本繁殖数是确定病毒动力学全局图的阈值参数。除了在早期的工作中针对许多更简单的模型获得的传统的灭绝和持久性二分法结果之外，我们还能够证明无感染稳态的全局渐近稳定性和慢性感染稳态的全局吸引力（以及唯一性） ，取决于基本复制数是小于还是大于1。数值模拟支持我们的理论结果并提出了一个有趣的现象：