Dynamical analysis of a stochastic non-autonomous SVIR model with multiple stages of vaccination

Abstract

In this paper, we analyze the dynamics of a new proposed stochastic non-autonomous SVIR model, with an emphasis on multiple stages of vaccination, due to the vaccine ineffectiveness. The parameters of the model are allowed to depend on time, to incorporate the seasonal variation. Furthermore, the vaccinated population is divided into three subpopulations, each one representing a different stage. For the proposed model, we prove the mathematical and biological well-posedness. That is, the existence of a unique global almost surely positive solution. Moreover, we establish conditions under which the disease vanishes or persists. Furthermore, based on stochastic stability theory and by constructing a suitable new Lyapunov function, we provide a condition under which the model admits a non-trivial periodic solution. The established theoretical results along with the performed numerical simulations exhibit the effect of the different stages of vaccination along with the stochastic Gaussian noise on the dynamics of the studied population.

Asymptotic behaviour of a non‐autonomous multispecies Holling type II model with a complex type of noises

“…Nevertheless, these autonomous systems, including stochastic prey–predator model with Holling type II, (1) and (2), often struggle to accurately represent fluctuating ecosystems influenced by seasonal variations, temperature and humidity, as these factors introduce dynamic parameter variations. Thus, some authors have introduced stochastic non‐autonomous models with simple linear diffusion components to capture this phenomenon (Branicki & Uda, 2021; Du, 2014; Ji & Jiang, 2017; Jiang et al, 2008; Jiang et al, 2017; Li & Mao, 2009; Li & Shuai, 2010; Liu, 2015; Mehdaoui et al, 2023; Sengupta & Das, 2019; Zhang et al, 2017). However, most of these models concentrate on introducing stochasticity through linear environmental noise affecting the populations or the functional response, failing to account for the diverse impact of different noises on the dynamical system.…”

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

“…Motivated by such a fact, the extension of deterministic models to the stochastic case by adding different types of noise has been established by many researchers. In the context of Gaussian white noise, we refer, for instance, to earlier studies [11][12][13][14][15][16][17][18], in which the authors considered a Gaussian white noise in the disease transmission rate, while we refer to other works [19][20][21][22][23][24][25], in which a multiplicative Gaussian white noise was chosen. One limitation of the previous type of noise is its incapacity to incorporate the sudden severe changes within the studied population; such changes are exhibited for instance by earthquakes, hurricanes and volcanoes [26].…”