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: $\mathsf{S}(\mathtt{r})\le {\alpha }^{\mathsf{U}}/({\alpha }^{\mathsf{L}}+{\gamma }^{\mathsf{L}})+ϵ,$ $\mathsf{V}(\mathtt{r})\le {\gamma }^{\mathsf{U}}{\ell }_{11}/({\alpha }^{\mathsf{L}}+{\delta }_1^{\mathsf{L}})+ϵ,$ $\mathsf{I}(\mathtt{r})\le {\alpha }^{\mathsf{U}}/{\alpha }^{\mathsf{L}}+ϵ,$ $\mathsf{R}(\mathtt{r})\le ({\delta }_1^{\mathsf{U}}{\ell }_{12}+{\delta }^{\mathsf{U}}{\ell }_{13})/({\alpha }^{\mathsf{L}}+{\mathsf{d}}^{\mathsf{L}})+ϵ,$ $\mathsf{S}(\mathtt{r})\ge {\alpha }^{\mathsf{L}}/({\alpha }^{\mathsf{U}}+{\beta }^{\mathsf{U}}{\ell }_1+{\gamma }^{\mathsf{U}})+ϵ,$ $\mathsf{V}(\mathtt{r})\ge {\gamma }^{\mathsf{L}}{\ell }_0/({\alpha }^{\mathsf{U}}+{\beta }_1^{\mathsf{U}}{\ell }_1+{\delta }_1^{\mathsf{U}})+ϵ,$ $\mathsf{I}(\mathtt{r})\ge {\mathsf{d}}^{\mathsf{L}}{\ell }_{03}/({\delta }^{\mathsf{U}}+{\alpha }^{\mathsf{U}})+ϵ,$ $\mathsf{R}(\mathtt{r})\ge {\delta }_1^{\mathsf{L}}{\ell }_{02}/({\alpha }^{\mathsf{U}}+{\mathsf{d}}^{\mathsf{U}})+ϵ,$ 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.