Global dynamics of a time-delayed malaria model with asymptomatic infections and standard incidence rate
A time-delayed model of malaria transmission with asymptomatic infections and standard incidence rate is presented and its basic reproduction number $ {R}_{0} $ is calculated. We focus on the global dynamics of the model with respect to $ {R}_{0} $. If and only if $ {R}_{0} > 1 $, the model e...
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| Language: | English |
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AIMS Press
2023-04-01
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| Series: | Electronic Research Archive |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/era.2023179?viewType=HTML |
| _version_ | 1797808488570683392 |
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| author | Songbai Guo Xin Yang Zuohuan Zheng |
| author_facet | Songbai Guo Xin Yang Zuohuan Zheng |
| author_sort | Songbai Guo |
| collection | DOAJ |
| description | A time-delayed model of malaria transmission with asymptomatic infections and standard incidence rate is presented and its basic reproduction number $ {R}_{0} $ is calculated. We focus on the global dynamics of the model with respect to $ {R}_{0} $. If and only if $ {R}_{0} > 1 $, the model exists a unique malaria-infected equilibrium $ E^{\ast} $, whereas it always possesses the malaria-free equilibrium $ E_{0} $. We first prove the local stability of the equilibria $ E_0 $ and $ E^{\ast} $ by using proof by contradiction and the properties of complex modulus. Secondly, by utilizing the Lyapunov functional method and the limiting system of the model with some novel details, we show that the equilibrium $ {E}_{0} $ is globally asymptotically stable (GAS) when $ {R}_{0} < 1 $, globally attractive (GA) when $ {R}_{0} = 1 $ and unstable when $ {R}_{0} > 1 $; the equilibrium $ E^{\ast} $ is GAS if and only if $ {R}_{0} > 1 $. In particular, in order to obtain global attractivity of the equilibrium $ E^{\ast} $, we demonstrate the weak persistence of the system for $ {R}_{0} > 1 $. Our results imply that malaria will gradually disappear if $ {R}_{0}\leq1 $ and persistently exist if $ {R}_{0} > 1 $. |
| first_indexed | 2024-03-13T06:38:16Z |
| format | Article |
| id | doaj.art-93b81364715a4707bd0f37688ddcbe16 |
| institution | Directory Open Access Journal |
| issn | 2688-1594 |
| language | English |
| last_indexed | 2024-03-13T06:38:16Z |
| publishDate | 2023-04-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Electronic Research Archive |
| spelling | doaj.art-93b81364715a4707bd0f37688ddcbe162023-06-09T01:16:37ZengAIMS PressElectronic Research Archive2688-15942023-04-013163534355110.3934/era.2023179Global dynamics of a time-delayed malaria model with asymptomatic infections and standard incidence rateSongbai Guo 0Xin Yang1Zuohuan Zheng21. School of Science, Beijing University of Civil Engineering and Architecture, Beijing 102616, China1. School of Science, Beijing University of Civil Engineering and Architecture, Beijing 102616, China2. School of Mathematics and Statistics, Hainan Normal University, Haikou 571158, China 3. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China 4. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, ChinaA time-delayed model of malaria transmission with asymptomatic infections and standard incidence rate is presented and its basic reproduction number $ {R}_{0} $ is calculated. We focus on the global dynamics of the model with respect to $ {R}_{0} $. If and only if $ {R}_{0} > 1 $, the model exists a unique malaria-infected equilibrium $ E^{\ast} $, whereas it always possesses the malaria-free equilibrium $ E_{0} $. We first prove the local stability of the equilibria $ E_0 $ and $ E^{\ast} $ by using proof by contradiction and the properties of complex modulus. Secondly, by utilizing the Lyapunov functional method and the limiting system of the model with some novel details, we show that the equilibrium $ {E}_{0} $ is globally asymptotically stable (GAS) when $ {R}_{0} < 1 $, globally attractive (GA) when $ {R}_{0} = 1 $ and unstable when $ {R}_{0} > 1 $; the equilibrium $ E^{\ast} $ is GAS if and only if $ {R}_{0} > 1 $. In particular, in order to obtain global attractivity of the equilibrium $ E^{\ast} $, we demonstrate the weak persistence of the system for $ {R}_{0} > 1 $. Our results imply that malaria will gradually disappear if $ {R}_{0}\leq1 $ and persistently exist if $ {R}_{0} > 1 $.https://www.aimspress.com/article/doi/10.3934/era.2023179?viewType=HTMLmalaria transmission modeltime delaylyapunov functionalglobal dynamics |
| spellingShingle | Songbai Guo Xin Yang Zuohuan Zheng Global dynamics of a time-delayed malaria model with asymptomatic infections and standard incidence rate Electronic Research Archive malaria transmission model time delay lyapunov functional global dynamics |
| title | Global dynamics of a time-delayed malaria model with asymptomatic infections and standard incidence rate |
| title_full | Global dynamics of a time-delayed malaria model with asymptomatic infections and standard incidence rate |
| title_fullStr | Global dynamics of a time-delayed malaria model with asymptomatic infections and standard incidence rate |
| title_full_unstemmed | Global dynamics of a time-delayed malaria model with asymptomatic infections and standard incidence rate |
| title_short | Global dynamics of a time-delayed malaria model with asymptomatic infections and standard incidence rate |
| title_sort | global dynamics of a time delayed malaria model with asymptomatic infections and standard incidence rate |
| topic | malaria transmission model time delay lyapunov functional global dynamics |
| url | https://www.aimspress.com/article/doi/10.3934/era.2023179?viewType=HTML |
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