A Non-Standard Finite Difference Scheme for a Diffusive HIV-1 Infection Model with Immune Response and Intracellular Delay
In this paper, we propose and study a diffusive HIV infection model with infected cells delay, virus mature delay, abstract function incidence rate and a virus diffusion term. By introducing the reproductive numbers for viral infection <inline-formula><math xmlns="http://www.w3.org/199...
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MDPI AG
2022-03-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/11/3/129 |
| _version_ | 1797447106450948096 |
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| author | Xiao-Lan Liu Cheng-Cheng Zhu |
| author_facet | Xiao-Lan Liu Cheng-Cheng Zhu |
| author_sort | Xiao-Lan Liu |
| collection | DOAJ |
| description | In this paper, we propose and study a diffusive HIV infection model with infected cells delay, virus mature delay, abstract function incidence rate and a virus diffusion term. By introducing the reproductive numbers for viral infection <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 for <i>CTL</i> immune response 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>, we show that <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> act as threshold parameter for the existence and stability of equilibria. 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 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, and the viruses are cleared; 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>, the <i>CTL</i>-inactivated 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, and the infection becomes chronic but without persistent <i>CTL</i> 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 <i>CTL</i>-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 <i>CTL</i> response. Next, we study the dynamic of the discreted system of our model by using non-standard finite difference scheme. We find that the global stability of the equilibria of the continuous model and the discrete model is not always consistent. That is, 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>, or <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>, the global stability of the two kinds model is consistent. However, 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 global stability of the two kinds model is not consistent. Finally, numerical simulations are carried out to illustrate the theoretical results and show the effects of diffusion factors on the time-delay virus model. |
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| institution | Directory Open Access Journal |
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| last_indexed | 2024-03-09T13:51:03Z |
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| spelling | doaj.art-5a0e41c0d45549deb48472b0dbcaf7782023-11-30T20:50:18ZengMDPI AGAxioms2075-16802022-03-0111312910.3390/axioms11030129A Non-Standard Finite Difference Scheme for a Diffusive HIV-1 Infection Model with Immune Response and Intracellular DelayXiao-Lan Liu0Cheng-Cheng Zhu1School of Arts and Science, Suqian University, Suqian 223800, ChinaSchool of Science, Jiangnan University, Wuxi 214122, ChinaIn this paper, we propose and study a diffusive HIV infection model with infected cells delay, virus mature delay, abstract function incidence rate and a virus diffusion term. By introducing the reproductive numbers for viral infection <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 for <i>CTL</i> immune response 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>, we show that <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> act as threshold parameter for the existence and stability of equilibria. 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 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, and the viruses are cleared; 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>, the <i>CTL</i>-inactivated 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, and the infection becomes chronic but without persistent <i>CTL</i> 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 <i>CTL</i>-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 <i>CTL</i> response. Next, we study the dynamic of the discreted system of our model by using non-standard finite difference scheme. We find that the global stability of the equilibria of the continuous model and the discrete model is not always consistent. That is, 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>, or <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>, the global stability of the two kinds model is consistent. However, 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 global stability of the two kinds model is not consistent. Finally, numerical simulations are carried out to illustrate the theoretical results and show the effects of diffusion factors on the time-delay virus model.https://www.mdpi.com/2075-1680/11/3/129basic reproduction numberequilibriumglobal stabilityimmune responsenonstandard finite difference schemenumerical simulation |
| spellingShingle | Xiao-Lan Liu Cheng-Cheng Zhu A Non-Standard Finite Difference Scheme for a Diffusive HIV-1 Infection Model with Immune Response and Intracellular Delay Axioms basic reproduction number equilibrium global stability immune response nonstandard finite difference scheme numerical simulation |
| title | A Non-Standard Finite Difference Scheme for a Diffusive HIV-1 Infection Model with Immune Response and Intracellular Delay |
| title_full | A Non-Standard Finite Difference Scheme for a Diffusive HIV-1 Infection Model with Immune Response and Intracellular Delay |
| title_fullStr | A Non-Standard Finite Difference Scheme for a Diffusive HIV-1 Infection Model with Immune Response and Intracellular Delay |
| title_full_unstemmed | A Non-Standard Finite Difference Scheme for a Diffusive HIV-1 Infection Model with Immune Response and Intracellular Delay |
| title_short | A Non-Standard Finite Difference Scheme for a Diffusive HIV-1 Infection Model with Immune Response and Intracellular Delay |
| title_sort | non standard finite difference scheme for a diffusive hiv 1 infection model with immune response and intracellular delay |
| topic | basic reproduction number equilibrium global stability immune response nonstandard finite difference scheme numerical simulation |
| url | https://www.mdpi.com/2075-1680/11/3/129 |
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