Global Stability of a MERS-CoV Infection Model with CTL Immune Response and Intracellular Delay
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), an...
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| Format: | Article |
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MDPI AG
2023-02-01
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| Online Access: | https://www.mdpi.com/2227-7390/11/4/1066 |
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| author | Tuersunjiang Keyoumu Wanbiao Ma Ke Guo |
| author_facet | Tuersunjiang Keyoumu Wanbiao Ma Ke Guo |
| author_sort | Tuersunjiang Keyoumu |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-11T08:28:51Z |
| format | Article |
| id | doaj.art-e9e685049aa74269a6b10e24c841258e |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T08:28:51Z |
| publishDate | 2023-02-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj.art-e9e685049aa74269a6b10e24c841258e2023-11-16T21:57:52ZengMDPI AGMathematics2227-73902023-02-01114106610.3390/math11041066Global Stability of a MERS-CoV Infection Model with CTL Immune Response and Intracellular DelayTuersunjiang Keyoumu0Wanbiao Ma1Ke Guo2School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, ChinaSchool of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, ChinaSchool of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, ChinaIn 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.https://www.mdpi.com/2227-7390/11/4/1066MERS-CoV infectionCTL immune responseglobal stabilityLyapunov functionalsintracellular delay |
| spellingShingle | Tuersunjiang Keyoumu Wanbiao Ma Ke Guo Global Stability of a MERS-CoV Infection Model with CTL Immune Response and Intracellular Delay Mathematics MERS-CoV infection CTL immune response global stability Lyapunov functionals intracellular delay |
| title | Global Stability of a MERS-CoV Infection Model with CTL Immune Response and Intracellular Delay |
| title_full | Global Stability of a MERS-CoV Infection Model with CTL Immune Response and Intracellular Delay |
| title_fullStr | Global Stability of a MERS-CoV Infection Model with CTL Immune Response and Intracellular Delay |
| title_full_unstemmed | Global Stability of a MERS-CoV Infection Model with CTL Immune Response and Intracellular Delay |
| title_short | Global Stability of a MERS-CoV Infection Model with CTL Immune Response and Intracellular Delay |
| title_sort | global stability of a mers cov infection model with ctl immune response and intracellular delay |
| topic | MERS-CoV infection CTL immune response global stability Lyapunov functionals intracellular delay |
| url | https://www.mdpi.com/2227-7390/11/4/1066 |
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