Analysis of Equilibria and Connecting Orbits in a Nonlinear Viral Infection Model

Extensive modeling studies on viral infection have significantly improved immunological insights into the dynamics of host responses to infectious agents and helped to design new avenues for experimentation. Various dynamical behaviors have been found in existing models, in particular, the global stability of the boundary equilibrium or the positive equilibrium, as well as different types of bifurcations. However, limited studies have been performed on the connection of invariant sets when global stability results no longer hold. This motivates the current study through considering the dynamics of a viral infection model, with new features that nonmonotonic functional responses are contained in the cytotoxic T lymphocytes (CTL) growth rate and incidence rate. The well-posedness of the four-dimensional differential equation model is established, and the model is further reduced into a three-dimensional nonmonotone system. The reduced system is demonstrated to admit three types of equilibria that represent different states of viral infection. The local stability and global stability of these equilibria are established under some suitable conditions. The coexistence of two positive equilibria poses challenges to model analysis, which is addressed through an algebraic-topological invariant, the Conley index. The index provides a topological description of the local dynamics around each equilibrium. With the aid of connection matrices, nontrivial invariant sets are detected, and the existence of connecting orbits between these invariant sets are determined. Further numerical simulations are conducted to supplement and verify the analytical results. It is shown that the model exhibits the rich dynamical phenomenon including bistability and periodic solutions due to diverse nonmonotonicities. Global dynamics from local stability analysis in the current study extensively extend and improve some existing studies on virus dynamics models.