Dynamical analysis for delayed virus infection models with cell-to-cell transmission and density-dependent diffusion

In this paper,certain delayed virus dynamical models with cell-to-cell infection and density-dependent diffusion are investigated.For the viral model with a single strain,we have proved the well-posedness and studied the global stabilities of equilibria by defining the basic reproductive number R_(0) and structuring proper Lyapunov functional.Moreover,we found that the infection-free equilibrium is globally asymptotically stable if R_(0)<1,and the infection equilibrium is globally asymptotically stable if R_(0)>1.For the multi-strain model,we found that all viral strains coexist if the corresponding basic reproductive number R^(e)_(j)>1,while virus will extinct if R^(e)_(j)<1.As a result,we found that delay and the density-dependent diffusion does not influence the global stability of the model with cell-to-cell infection and homogeneous Neumann boundary conditions.

Dynamical analysis of tumor-immune-help T cells system

In this paper,we construct a mathematical model to investigate the interaction between the tumor cells,the immune cells and the helper T cells(HTCs).We perform math-ematical analysis to reveal the stability of the equilibria of the model.In our model,the HTCs are stimulated by the identification of the presence of tumor antigens.Our investigation implies that the presence of tumor antigens may inhibit the existence of high steady state of tumor cells,which leads to the elimination of the bistable behavior of the tumor-immune system,i.e.the equilibrium corresponding to the high steady state of tumor cells is destabilized.Choosing immune intensity c as bifurcation parameter,there exists saddle-node bifurcation.Besides,there exists a critical value C*,at which a Hopf bifurcation occurs.The stability and direction of Hopf bifurcation are discussed.