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In this paper, we study two-dimensional, three-dimensional monotonic and nonmonotonic immune responses in viral infection systems. Our results show that the viral infection systems with monotonic immune response has no bistability appear. However, the systems with nonmonotonic immune response has bistability appear under some conditions. For immune intensity, we got two important thresholds, post-treatment control threshold and elite control threshold. When immune intensity is less than post-treatment control threshold, the virus will be rebound. The virus will be under control when immune intensity is larger than elite control threshold. While between the two thresholds is a bistable interval. When immune intensity is in the bistable interval, the system can have bistability appear. Select the rate of immune cells stimulated by the viruses as a bifurcation parameter for nonmonotonic immune responses, we prove the system exhibits saddle-node bifurcation and transcritical bifurcation.

Monotonic and nonmonotonic immune responses in viral infection systems