| Summary: | In this paper, we propose and study a Middle East respiratory syndrome coronavirus (MERS-CoV) infection model with cytotoxic T lymphocyte (CTL) immune response and intracellular delay. This model includes five compartments: uninfected cells, infected cells, viruses, dipeptidyl peptidase 4 (DPP4), and CTL immune cells. We obtained an immunity-inactivated reproduction number <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 an immunity-activated reproduction number <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>. By analyzing the distributions of roots of the corresponding characteristic equations, the local stability results of the infection-free equilibrium, the immunity-inactivated equilibrium, and the immunity-activated equilibrium were obtained. Moreover, by constructing suitable Lyapunov functionals and combining LaSalle’s invariance principle and Barbalat’s lemma, some sufficient conditions for the global stability of the three types of equilibria were obtained. It was found that the infection-free equilibrium is globally asymptotically stable 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> and unstable 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>; the immunity-inactivated equilibrium is locally asymptotically stable 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><mo>></mo><msub><mi>R</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula> and globally asymptotically stable 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><mo>></mo><msub><mi>R</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula> and condition (H1) holds, but unstable 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>; and the immunity-activated equilibrium is locally asymptotically stable 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> and is globally asymptotically stable 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> and condition (H1) holds.
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