RESUMEN
Several studies have previously been conducted on the dynamics of probabilistic epidemic models driven by Lévy disorder. All of these works have used the Poisson counting process with finite Lévy measures. However, this scope disregards a considerable category of correlated Lévy jump processes governed by an infinite Lévy measure. In this research, we take into consideration this general framework applied to an epidemic model with a quarantine strategy. Under an appropriate hypothetical setting, we infer the exact threshold value between the ergodicity and the disease disappearance. Our analysis completes the work presented by Privault and Wang (J Nonlinear Sci 31(1):1-28, 2021) and puts forward a novel analytical aspect to deal with other stochastic models in several areas. As a numerical application, we implement the algorithm of Rosinski (Stoch Process Appl 117:677-707, 2007) for tempered stable Lévy processes with an infinite Lévy measure.
RESUMEN
This study concentrates on the analysis of a stochastic SIC epidemic system with an enhanced and general perturbation. Given the intricacy of some impulses caused by external disturbances, we integrate the quadratic Lévy noise into our model. We assort the long-run behavior of a perturbed SIC epidemic model presented in the form of a system of stochastic differential equations driven by second-order jumps. By ameliorating the hypotheses and using some new analytical techniques, we find the exact threshold value between extinction and ergodicity (persistence) of our system. The idea and analysis used in this paper generalize the work of N. T. Dieu et al. (2020), and offer an innovative approach to dealing with other random population models. Comparing our results with those of previous studies reveals that quadratic jump-diffusion has no impact on the threshold value, but it remarkably influences the dynamics of the infection and may worsen the pandemic situation. In order to illustrate this comparison and confirm our analysis, we perform numerical simulations with some real data of COVID-19 in Morocco. Furthermore, we arrive at the following results: (i) the time average of the different classes depends on the intensity of the noise (ii) the quadratic noise has a negative effect on disease duration (iii) the stationary density function of the population abruptly changes its shape at some values of the noise intensity. Mathematics Subject Classification 2020: 34A26; 34A12; 92D30; 37C10; 60H30; 60H10.
RESUMEN
Hepatitis B epidemic was and is still a rich subject that sparks the interest of epidemiological researchers. The dynamics of this epidemic is often modeled by a system with constant parameters. In reality, the parameters associated with the Hepatitis B model are not certain, but the interval in which it belongs to can readily be determined. Our paper focuses on an imprecise Hepatitis B model perturbed by Lévy noise due to unexpected environmental disturbances. This model has a global positive solution. Under an appropriate assumption, we prove the existence of a unique ergodic stationary distribution by using the mutually exclusive possibilities lemma demonstrated by Stettner in 1986. Our main effort is to establish an almost perfect condition for the existence of the stationary distribution. Numerical simulations are introduced to illustrate the analytical results.