Monkeypox 2022, a new re-emerging disease, is caused by the Monkeypox virus. Structurally, this virus is related to the smallpox virus and infects the host in a similar way; however, the symptoms of Monkeypox are more severe. In this research work, a mathematical model for understanding the dynamics of Monkeypox 2022 is suggested that takes into account two modes of transmission: horizontal human dissemination and cross-infection between animals and humans. Due to lack of substantial knowledge about the virus diffusion and the effect of external perturbations, the model is extended to the probabilistic formulation with Lévy jumps. The proposed model is a two block compartmental system that requires the form of Itô-Lévy stochastic differential equations. Based on some assumptions and nonstandard analytical techniques, two principal asymptotic properties are proved: the eradication and continuation in the mean of Monkeypox 2022. The outcomes of the study reveals that the dynamical behavior of the proposed Monkeypox 2022 system is chiefly governed by some parameters that are precisely correlated with the noise intensities. To support the obtained theoretical finding, examples based on numerical simulations and real data are presented at the end of the study. The numerical simulations also exhibit the impact of the innovative adopted mathematical techniques on the findings of this work.

The aim of this paper is to generalize the nonlinear incidence rate of a stochastic SIRS (susceptible-infected-recovered-susceptible) epidemic model. Our basic model was enriched with the hypotheses of vertical transmission and transfer from infected to susceptible individuals, to approach the reality. Our analysis showed that the model is well-posed. Under some conditions imposed on the intensity of the white noise perturbation, the global stability of the system is proven. Furthermore, the threshold of our model which determines the extinction and persistence of the disease is established. Numerical examples are realized to prove the rigor of our theoretical results.

This article deals with a Caputo fractional-order viral model that incorporates the non-cytolytic immune hypothesis and the mechanism of viral replication inhibition. Firstly, we establish the existence, uniqueness, non-negativity, and boundedness of the solutions of the proposed viral model. Then, we point out that our model has the following three equilibrium points: equilibrium point without virus, equilibrium state without immune system, and equilibrium point activated by immunity with humoral feedback. By presenting two critical quantities, the asymptotic stability of all said steady points is examined. Finally, we examine the finesse of our results by highlighting the impact of fractional derivatives on the stability of the corresponding steady points.

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.