Abstract
In this study, we considered both control strategies -vaccination and quarantine- to solve an epidemiological problem. We used the SVIRS model as our baseline, to further classify infected individuals into classes according to them being symptomatic/asymptomatic and quarantined/non-quarantined. The age since infection is also relevant to describe the population dynamics by detailed features, among which, the age-specific quarantine rate poses some mathematical challenges since the possibility of an infected individual being quarantined exists at any age of infection. Firstly, we provided the model's well-posedness property, and formulated the basic reproduction number R 0 to determine whether the disease dies out or persists. Secondly, we demonstrated how backward bifurcation occurs in the proposed model under a complicated criterion, and explored it to observe the effects of certain parameters on the dynamics. We also investigated, as a special case, the global convergence to an endemic equilibrium using the Lyapunov functionals approach. We conducted some numerical simulations to illustrate the theoretical results and bi-stability due to the occurrence of backward bifurcation.