| Summary: | The first purpose of this article was to establish and analyze system 4 with an abstract function incidence rate under homogeneous Neumann boundary conditions. The system models the dynamics of interactions between pathogens and the host immune system, which has important applications in HIV-1, HCV, flu etc. By the Lyapunov–LaSalle method, we obtained the globally asymptotic stability of the equilibria. Specifically speaking, by introducing the reproductive numbers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>R</mi><mn>0</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>R</mi><mn>1</mn></msub></semantics></math></inline-formula>, we showed that if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>R</mi><mn>0</mn></msub><mo>≤</mo><mn>1</mn></mrow></semantics></math></inline-formula>, then the infection-free equilibrium <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mn>0</mn></msub></semantics></math></inline-formula> is globally asymptotically stable, i.e., the virus is unable to sustain the infection and will become extinct; if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>R</mi><mn>1</mn></msub><mo>≤</mo><mn>1</mn><mo><</mo><msub><mi>R</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula>, then the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mi>T</mi><mi>L</mi></mrow></semantics></math></inline-formula>-inactivated infection equilibrium <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mn>1</mn></msub></semantics></math></inline-formula> is globally asymptotically stable, i.e., the infection becomes chronic but without persistent CTL response; if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>R</mi><mn>1</mn></msub><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mi>T</mi><mi>L</mi></mrow></semantics></math></inline-formula>-activated equilibrium <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mn>2</mn></msub></semantics></math></inline-formula> is globally asymptotically stable, and the infection is chronic with persistent CTL response. Additionally, we also investigate the discretization of the model by using a non-standard finite difference scheme, and our results confirm that the global stability of the equilibria of the continuous model and the discrete model is consistent. Finally, numerical simulations are performed to illustrate the theoretical results. Our model and results are to a certain extent generalizations of and improvements upon the previous results given by Zhu, Wang.
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