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

Abstract

a b s t r a c tA stochastic susceptible-infectious-recovered epidemic model with nonlinear incidence rate is formulated to discuss the effects of temporary immunity, vaccination, and Lévy jumps on the transmission of diseases. We first determine the existence of a unique global positive solution and a positively invariant set for the stochastic system. Sufficient conditions for extinction and persistence in the mean of the disease are then achieved by constructing suitable Lyapunov functions. Based on the analysis, we conclude that noise intensity and the validity period of vaccination greatly influence the transmission dynamics of the system.

Analysis of a stochastic SVIR model with time‐delayed stages of vaccination and Lévy jumps

“…Taking the law of large numbers for local martingales and Remark 1 into account, we let t → +∞ in (38) to obtain (33). This concludes the proof.…”

“…• The introduction of time delays in the disease incidence rates: We emphasize that the introduction of time delays in the previous works [35][36][37][38][39][40][41][42] was done for SIR and SIQR type-models, in the aim of incorporating the immunity loss and the time elapsed before the vaccine effectiveness wears off. On the other hand, in this paper, we have extended the use of time delays to incorporate other characteristics such as the validation time of the vaccine and the time separating the stages of vaccination.…”

“…Case in point, in order to incorporate the temporary immunity and the vaccine ineffectiveness, many models have been extended and generalized. We give a brief review of the existing literature by mentioning the pioneering results of Liu et al [35,36], Berrhazi et al [37], Fan et al [38], El Fatini and Sekkak [39], El Fatini and Boukanjime [40], and Liu et al [41], in which the conditions of persistence and extinction of the disease for SIR-type epidemic models with different types of noise were established. The same types of results were proved by Liu et al [42] for an SIQR model.…”

A stochastic SIQR epidemic model with Lévy jumps and three-time delays

“…Thus, it is inappropriate and unrealistic to prefer the deterministic over stochastic approach while modeling such epidemics. Based on these facts, numerous researchers take into account the stochastic epidemic modeling driven by Lévy jumps [22] , [23] , [24] , [25] , [26] , [27] , [28] , [29] , [30] . Recently, Koufi et al.…”

Using entropy-driven amplifier circuit response to build nonlinear model under the influence of Lévy jump

“…Lévy jump is often used in infectious disease models [ 39 – 42 ], proving that the influence of Lévy noise can lead to the extinction of diseases. Lévy jumps are also used in epidemic models [ 43 ] and virus dynamics models [ 44 ]. [ 45 ] and [ 46 ] introduce the introduction of Lévy jump in the predator system, and analyze the sufficient conditions for the species’ continued survival and extinction.…”