THRESHOLD ASYMPTOTIC DYNAMICS FOR A SPATIAL AGE-DEPENDENT CELL-TO-CELL TRANSMISSION MODEL WITH NONLOCAL DISPERSE

This research is concerned to study the global threshold dynam-ics of a hybrid viral infection model, with the assumption that all parameters are space dependent. The considered hybrid model contains three principal equations, the first is an ordinary differential equation that represents the un-infected cells, an age-structured equation modeled by the transport equation for infected-cells, and a diffusive equation that studies the evolution of the virus. The challenging mathematical aspect of this research lies in the fact that the model is partially degenerate and the solution map is not compact. Also, understanding the global threshold dynamics in terms of this partial de-generacy, mostly with age-structured equation and nonlocal diffusion is the ultimate challenging point of this research. The existence and the uniqueness of a global solution are proved. Further, the existence of a global compact attractor is shown. Moreover, the basic reproduction number 910 is identi-fied for the model with its threshold role: If 910 < 1, the infection-free-steady state is globally asymptotically stable; for 910 > 1, the solution of the studied model is uniformly persistent and the pathogen persists in a unique positive steady state, which it is shown using the Lyapunov approach. The results are supported by some graphical representations for illustrating the finding.