Persistence and stability in an SVIR epidemic model with relapse on timescales
In this paper, an Susceptible-Vaccinated-Infected-Recovered (SVIR) epidemic model incorporating relapse dynamics on a timescale was studied. Using the dynamic inequalities: $ \mathsf{S}(\mathtt{r})\le \alpha^\mathsf{U}/(\alpha^\mathsf{L}+ \gamma^\mathsf{L})+\epsilon, $ $ \mathsf{V}(\mathtt{r})\le \g...
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2025-02-01
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| author | Sabbavarapu Nageswara Rao Mahammad Khuddush Ahmed H. Msmali Ali H. Hakami |
| author_facet | Sabbavarapu Nageswara Rao Mahammad Khuddush Ahmed H. Msmali Ali H. Hakami |
| author_sort | Sabbavarapu Nageswara Rao |
| collection | DOAJ |
| description | In this paper, an Susceptible-Vaccinated-Infected-Recovered (SVIR) epidemic model incorporating relapse dynamics on a timescale was studied. Using the dynamic inequalities: $ \mathsf{S}(\mathtt{r})\le \alpha^\mathsf{U}/(\alpha^\mathsf{L}+ \gamma^\mathsf{L})+\epsilon, $ $ \mathsf{V}(\mathtt{r})\le \gamma^\mathsf{U}\ell_{11}/(\alpha^\mathsf{L} + \delta_1^\mathsf{L})+\epsilon, $ $ \mathsf{I}(\mathtt{r})\le\alpha^\mathsf{U}/\alpha^\mathsf{L}+\epsilon, $ $ \mathsf{R}(\mathtt{r})\le(\delta_1^\mathsf{U}\ell_{12}+\delta^\mathsf{U}\ell_{13})/(\alpha^\mathsf{L}+\mathsf{d}^\mathsf{L})+\epsilon, $ $ \mathsf{S}(\mathtt{r})\ge \alpha^\mathsf{L}/(\alpha^\mathsf{U} +\beta^\mathsf{U}\ell_1+ \gamma^\mathsf{U})+\epsilon, $ $ \mathsf{V}(\mathtt{r})\ge \gamma^\mathsf{L}\ell_0/(\alpha^\mathsf{U}+ \beta_1^\mathsf{U}\ell_1+\delta_1^\mathsf{U})+\epsilon, $ $ \mathsf{I}(\mathtt{r})\ge\mathsf{d}^\mathsf{L}\ell_{03}/(\delta^\mathsf{U}+\alpha^\mathsf{U})+\epsilon, $ $ \mathsf{R}(\mathtt{r})\ge\delta_1^\mathsf{L}\ell_{02}/(\alpha^\mathsf{U}+\mathsf{d}^\mathsf{U})+\epsilon, $ and constructing an appropriate Lyapunov functional, sufficient conditions were determined to guarantee the permanence of the system. Additionally, the existence, uniqueness, and uniform asymptotic stability of globally attractive, almost periodic positive solutions were derived. Furthermore, an in-depth analysis highlighted the significance of relapse dynamics. Numerical simulations were included to validate the system's permanence, demonstrating that the disease persists under certain conditions. These simulations revealed that vaccination and relapse dynamics played a crucial role in controlling the epidemic. Specifically, as long as the infected population remained smaller than the susceptible population, the infection was controlled, keeping both the infected and recovered populations low. Their oscillatory behavior suggested that periodic vaccinations may be key to stabilizing disease dynamics. This study underscored the applicability of the proposed model in providing a robust theoretical foundation for understanding and managing the spread of infectious diseases. |
| format | Article |
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| publishDate | 2025-02-01 |
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| spelling | doaj-art-004e6385cfbb47b3bfcbff82757d757a2025-08-20T03:17:09ZengAIMS PressAIMS Mathematics2473-69882025-02-011024173420410.3934/math.2025194Persistence and stability in an SVIR epidemic model with relapse on timescalesSabbavarapu Nageswara Rao0Mahammad Khuddush1Ahmed H. Msmali2Ali H. Hakami3Department of Mathematics, College of Science, Jazan University, P.O. Box 114, Jazan, 45142, Saudi ArabiaDepartment of Mathematics, LearningMate, Straive, SPi Technologies Pvt. Ltd., Visakhapatnam, 530002, Andhra Pradesh, IndiaDepartment of Mathematics, College of Science, Jazan University, P.O. Box 114, Jazan, 45142, Saudi ArabiaDepartment of Mathematics, College of Science, Jazan University, P.O. Box 114, Jazan, 45142, Saudi ArabiaIn this paper, an Susceptible-Vaccinated-Infected-Recovered (SVIR) epidemic model incorporating relapse dynamics on a timescale was studied. Using the dynamic inequalities: $ \mathsf{S}(\mathtt{r})\le \alpha^\mathsf{U}/(\alpha^\mathsf{L}+ \gamma^\mathsf{L})+\epsilon, $ $ \mathsf{V}(\mathtt{r})\le \gamma^\mathsf{U}\ell_{11}/(\alpha^\mathsf{L} + \delta_1^\mathsf{L})+\epsilon, $ $ \mathsf{I}(\mathtt{r})\le\alpha^\mathsf{U}/\alpha^\mathsf{L}+\epsilon, $ $ \mathsf{R}(\mathtt{r})\le(\delta_1^\mathsf{U}\ell_{12}+\delta^\mathsf{U}\ell_{13})/(\alpha^\mathsf{L}+\mathsf{d}^\mathsf{L})+\epsilon, $ $ \mathsf{S}(\mathtt{r})\ge \alpha^\mathsf{L}/(\alpha^\mathsf{U} +\beta^\mathsf{U}\ell_1+ \gamma^\mathsf{U})+\epsilon, $ $ \mathsf{V}(\mathtt{r})\ge \gamma^\mathsf{L}\ell_0/(\alpha^\mathsf{U}+ \beta_1^\mathsf{U}\ell_1+\delta_1^\mathsf{U})+\epsilon, $ $ \mathsf{I}(\mathtt{r})\ge\mathsf{d}^\mathsf{L}\ell_{03}/(\delta^\mathsf{U}+\alpha^\mathsf{U})+\epsilon, $ $ \mathsf{R}(\mathtt{r})\ge\delta_1^\mathsf{L}\ell_{02}/(\alpha^\mathsf{U}+\mathsf{d}^\mathsf{U})+\epsilon, $ and constructing an appropriate Lyapunov functional, sufficient conditions were determined to guarantee the permanence of the system. Additionally, the existence, uniqueness, and uniform asymptotic stability of globally attractive, almost periodic positive solutions were derived. Furthermore, an in-depth analysis highlighted the significance of relapse dynamics. Numerical simulations were included to validate the system's permanence, demonstrating that the disease persists under certain conditions. These simulations revealed that vaccination and relapse dynamics played a crucial role in controlling the epidemic. Specifically, as long as the infected population remained smaller than the susceptible population, the infection was controlled, keeping both the infected and recovered populations low. Their oscillatory behavior suggested that periodic vaccinations may be key to stabilizing disease dynamics. This study underscored the applicability of the proposed model in providing a robust theoretical foundation for understanding and managing the spread of infectious diseases.https://www.aimspress.com/article/doi/10.3934/math.2025194svir modelepidemic modeltimescalealmost periodic positive solution |
| spellingShingle | Sabbavarapu Nageswara Rao Mahammad Khuddush Ahmed H. Msmali Ali H. Hakami Persistence and stability in an SVIR epidemic model with relapse on timescales AIMS Mathematics svir model epidemic model timescale almost periodic positive solution |
| title | Persistence and stability in an SVIR epidemic model with relapse on timescales |
| title_full | Persistence and stability in an SVIR epidemic model with relapse on timescales |
| title_fullStr | Persistence and stability in an SVIR epidemic model with relapse on timescales |
| title_full_unstemmed | Persistence and stability in an SVIR epidemic model with relapse on timescales |
| title_short | Persistence and stability in an SVIR epidemic model with relapse on timescales |
| title_sort | persistence and stability in an svir epidemic model with relapse on timescales |
| topic | svir model epidemic model timescale almost periodic positive solution |
| url | https://www.aimspress.com/article/doi/10.3934/math.2025194 |
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