A stochastic threshold for an epidemic model with Beddington–DeAngelis incidence, delayed loss of immunity and Lévy noise perturbation

A delayed vaccinated epidemic model with nonlinear incidence rate and Lévy jumps

“…Most epidemic models are driven by white noise, and many results have been achieved in this area [11][12][13][14][15][16][17][18][19]. However, under severe environmental perturbations, such as avian influenza, severe acute respiratory syndrome, volcanic eruptions, earthquakes, and hurricanes, the continuity of solutions may be broken; accordingly, a jump process should be introduced to prevent and control diseases [20][21][22][23].…”

Global dynamics of a novel deterministic and stochastic SIR epidemic model with vertical transmission and media coverage

“…For example, one may construct some more realistic but complex models, such as considering the effects of delay, complex network, pulse vaccination and Lévy noise. Some scholars have already done a great deal of work (see [34][35][36][37][38][39][40]). We leave these investigations for future work.…”

Dynamical analysis of a stochastic delayed SIR epidemic model with vertical transmission and vaccination

“…It suffices to prove that P(τ = ∞) = 1, that is P(τ < t) = 0, ∀t > 0, we can see clearly that P(τ < t) ≤ P(τ n < t). We only need to show that lim sup n→∞ P(τ n < t) = 0 for this matter, referring to [35,36], we can take a similar function and use some approaches to prove the theorem. Then, we set a C 2 -function U: R 2 + → R + for all (S(t), I(t)) > 0:…”