Qualitative Analysis of a COVID-19 Mathematical Model with a Discrete Time Delay
The aim of this paper is to investigate the qualitative behavior of a mathematical model of the COVID-19 pandemic. The constructed SAIRS-type mathematical model is based on nonlinear delay differential equations. The discrete-time delay is introduced in the model in order to take into account the la...
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2024-12-01
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| author | Abraham J. Arenas Gilberto González-Parra Miguel Saenz Saenz |
| author_facet | Abraham J. Arenas Gilberto González-Parra Miguel Saenz Saenz |
| author_sort | Abraham J. Arenas |
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| description | The aim of this paper is to investigate the qualitative behavior of a mathematical model of the COVID-19 pandemic. The constructed SAIRS-type mathematical model is based on nonlinear delay differential equations. The discrete-time delay is introduced in the model in order to take into account the latent stage where the individuals already have the virus but cannot yet infect others. This aspect is a crucial part of this work since other models assume exponential transition for this stage, which can be unrealistic. We study the qualitative dynamics of the model by performing global and local stability analysis. We compute the basic reproduction number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="script">R</mi><mn>0</mn><mi>d</mi></msubsup></semantics></math></inline-formula>, which depends on the time delay and determines the stability of the two steady states. We also compare the qualitative dynamics of the delayed model with the model without time delay. For global stability, we design two suitable Lyapunov functions that show that under some scenarios the disease persists whenever <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">R</mi><mn>0</mn><mi>d</mi></msubsup><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>. Otherwise, the solution approaches the disease-free equilibrium point. We present a few numerical examples that support the theoretical analysis and the methodology. Finally, a discussion about the main results and future directions of research is presented. |
| format | Article |
| id | doaj-art-8d81130c3d3e4a4c8573a1ed05dadf3d |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-12-01 |
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| series | Mathematics |
| spelling | doaj-art-8d81130c3d3e4a4c8573a1ed05dadf3d2025-01-10T13:18:18ZengMDPI AGMathematics2227-73902024-12-0113112010.3390/math13010120Qualitative Analysis of a COVID-19 Mathematical Model with a Discrete Time DelayAbraham J. Arenas0Gilberto González-Parra1Miguel Saenz Saenz2Departamento de Matematicas y Estadistica, Universidad de Cordoba, Monteria 230002, ColombiaDepartment of Mathematics, New Mexico Tech, Socorro, NM 87801, USADepartamento de Matematicas y Estadistica, Universidad de Cordoba, Monteria 230002, ColombiaThe aim of this paper is to investigate the qualitative behavior of a mathematical model of the COVID-19 pandemic. The constructed SAIRS-type mathematical model is based on nonlinear delay differential equations. The discrete-time delay is introduced in the model in order to take into account the latent stage where the individuals already have the virus but cannot yet infect others. This aspect is a crucial part of this work since other models assume exponential transition for this stage, which can be unrealistic. We study the qualitative dynamics of the model by performing global and local stability analysis. We compute the basic reproduction number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="script">R</mi><mn>0</mn><mi>d</mi></msubsup></semantics></math></inline-formula>, which depends on the time delay and determines the stability of the two steady states. We also compare the qualitative dynamics of the delayed model with the model without time delay. For global stability, we design two suitable Lyapunov functions that show that under some scenarios the disease persists whenever <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">R</mi><mn>0</mn><mi>d</mi></msubsup><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>. Otherwise, the solution approaches the disease-free equilibrium point. We present a few numerical examples that support the theoretical analysis and the methodology. Finally, a discussion about the main results and future directions of research is presented.https://www.mdpi.com/2227-7390/13/1/120delay differential equationsstability analysisLyapunov functionnumerical simulation |
| spellingShingle | Abraham J. Arenas Gilberto González-Parra Miguel Saenz Saenz Qualitative Analysis of a COVID-19 Mathematical Model with a Discrete Time Delay Mathematics delay differential equations stability analysis Lyapunov function numerical simulation |
| title | Qualitative Analysis of a COVID-19 Mathematical Model with a Discrete Time Delay |
| title_full | Qualitative Analysis of a COVID-19 Mathematical Model with a Discrete Time Delay |
| title_fullStr | Qualitative Analysis of a COVID-19 Mathematical Model with a Discrete Time Delay |
| title_full_unstemmed | Qualitative Analysis of a COVID-19 Mathematical Model with a Discrete Time Delay |
| title_short | Qualitative Analysis of a COVID-19 Mathematical Model with a Discrete Time Delay |
| title_sort | qualitative analysis of a covid 19 mathematical model with a discrete time delay |
| topic | delay differential equations stability analysis Lyapunov function numerical simulation |
| url | https://www.mdpi.com/2227-7390/13/1/120 |
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