Permanence and extinction in a nonautonomous discrete SIRVS epidemic model with vaccination

A Consistent Discrete Version of a Nonautonomous SIRVS Model

“…In this section, we will extend the results obtained for the model with the usual mass action incidence in [35] to our generalized family of models. Namely, suitable thresholds are defined and conditions for persistence and extinction of the disease are obtained.…”

“…For a family of models containing the previous discrete-time model, we achieved results on the persistence and the extinction of the disease (Theorems 3 and 4). They contain the results of Zhang [35] as a particular case. Our threshold conditions depend on the parameters of the model and of the incidence function derivative, with respect to the infectives, computed on some disease-free solution.…”

“…As a corollary of our results, we consider the periodic case where we have a unique number that establishes the boundary between the regions of permanence and extinction. Although the proofs of our results are inspired in [35], some difficulties must be dealt with. In particular, it was necessary to understand the right conditions to impose to the incidence functions in order to overcome the technical difficulties.…”

“…In this subsection we do some simulation to illustrate our results. To begin, we compare our model (1) with mass action incidence (ϕ(S, I) = SI and ψ(V, I) = V I) with Zhang's model [35]. We use the following set of parameters: φ(h) = h+0.2h 2 , Λ = 0.5, µ(t) = γ(t) = δ(t) = 0.3, α(t) = 0.05, η = 0.05, p = 2/3 and β(t) = σ(t) = b(1 + 0.3 cos(tπ/2)).…”

“…In [35], a discrete non-autonomous epidemic model with vaccination and mass action incidence was obtained by Mickens method. We emphasize that, in the particular mass-action case, our model is not exactly similar to the model in [35], although Mickens rules were considered in both. We briefly compare computationally these two slightly different models in Section 5.…”