Global stability and positive recurrence of a stochastic SIS model with Lévy noise perturbation

Focusing on epidemic model in random environments, this paper uses white noise and Lévy noise to model the dynamics of the SIS epidemic model subject to the random changes of the external environment. We show that the jump encourages the extinction of the disease in the population. We first, give a rigorous proof of the global stability of the disease-free equilibrium state. We also establish sufficient conditions for the persistence of the disease. The presented results are demonstrated by numerical simulations.Faculty of Sciences, Ibn Tofail University-Kénitra, Morocc

A Probabilistic SIRI Epidemic Model Incorporating Incidence Capping and Logistic Population Expansion

This study presents a newly developed stochastic SIRI epidemic model, which combines logistic growth with a saturation incidence rate. This research mainly examines the presence and uniqueness of positive solutions within the formulated model. Furthermore, we aim to analyze the long-term performance of the system and provide valuable insights into disease extinction in a population. Our investigation delves into the conditions required for disease extinction, which are crucial in predicting and controlling the spread of deadly diseases. To substantiate our assertions, we have devised a stochastic Lyapunov function, which serves as a robust mathematical framework for demonstrating the presence of a discernible stationary ergodic distribution. This mathematical foundation significantly contributes to the understanding of model behavior. To complement our analytical findings, we conduct numerical simulations, which reinforce our results and provide a comprehensive understanding of the behavior of our proposed model, and open new avenues for future research in this area

Gaussian approximation of the empirical process under random entropy conditions

International audienceWe obtain rates of strong approximation of the empirical process indexed by functions by a Brownian bridge under only random entropy conditions. The results of Berthet and Mason [P. Berthet, D.M. Mason, Revisiting two strong approximation results of Dudley and Philipp, in: High Dimensional Probability, in: IMS Lecture Notes-Monograph Series, vol. 51, 2006, pp. 155-172] under bracketing entropy are extended by combining their method to properties of the empirical entropy. Our results show that one can improve the universal rate v(n) = o(root log log n) from Dudley and Philipp [R.M. Dudley, W. Philipp, Invariance principles for sums of Banach space valued random elements and empirical processes, Z. Wahrsch. Verw. Gebiete 62 (1983) 509-552] into v(n) -> 0 at a logarithmic rate, under a weak random entropy assumption which is close to necessary. As an application the results of Kolchinskii [V.I. Kolchinskii, Komlos-Major-Tusnady approximation for the general empirical process and Haar expansions of classes of functions, J. Theoret. Probab. 7 (1994) 73-118] are revisited when the conditions coming in addition to random entropy are relaxed

Quelques approximations gaussiennes du processus empirique indexé par des fonctions

Nous obtenons des vitesses d'approximation forte du processus empirique par une suite de ponts browniens dans le cadre indexé par des fonctions. Nous travaillons sous des conditions d'entropie aléatoire et adaptons la méthode de Berthet et Mason (2006). Au vu de Giné et Zinn (1984) et Talagrand (1987) notre condition la plus faible est quasiment nécessaire pour la propriété de Donsker, mais garantie néanmoins une vitesse (logn)^(-a) qui améliore significativement la vitesse universelle (loglogn)^1/2 de Dudley et Philipp (1983). Notre condition la plus forte conduit à des vitesses d'approximation gaussienne polynomiales. Nous étudions également le cas où les variables aléatoires sont faiblement dépendantes.We obtain some rates of strong approximation of the function indexed empirical process by a sequence of Brownian bridges. We work under random entropy conditions and adapt the recent technique of Berthet and Mason (2006). In view of Giné and Zinn (1984) and Talagrand (1987) our weakest condition is close to necessary for the Donsker property, but however guaranty a rate (logn)^(-a) which significatively improves the universal (loglogn)^1/2 of Dudley and Philipp (1983). Our strongest condition leads to polynomial rates of Gaussian approximation. We also study the case where randoms variables are weakly dependents.RENNES1-BU Sciences Philo (352382102) / SudocSudocFranceF

Dynamics of hybrid switching diffusions SIRS model

peer reviewedThe main aim of this paper is to study the effect of the environmental noises in the asymptotic properties of a stochastic version of the classical SIRS epidemic model. The model studied here include white noise and telegraph noise modeled by Markovian switching. We obtained conditions for extinction both in probability one and in pth moment. We also established the persistence of disease under different conditions on the intensities of noises, the parameters of the model and the stationary distribution of the Markov chain. The highlight point of our work is that our conditions are sufficient and almost necessary for extinction and persistence of the epidemic. The presented results are demonstrated by numerical simulations

Qualitative Analysis of a SIR Epidemic Model with a Nonlinear Relapse and Incidence Rate Stochastically Perturbed

In this paper, we analyze a stochastic SIR epidemic model in a constant population with a relapse and nonlinear perturbation. First, we illustrate show that the system has a unique global positive solution that belongs to a positively invariant set. Then, we obtain sufficient conditions for the extinction and persistence in the mean. Finally, numerical simulations are carried out to illustrate the theoretical results