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The extinction and the persistence of the population of the harmful sheep blowfly (Lucilia cuprina) are discussed in this paper through a stochastic mathematical model. Using appropriate Lyapunov functionals, the extinction of these flies depends on the time to oviposition and the time between generations. The oviposition period must be sufficiently short. Another species of blowfly, called Lucilia sericata, also attacks sheep in many areas. Both blowflies are ectoparasites of warm-blooded vertebrates, particularly domestic sheep. These two blowflies are related to sharing the same mitochondrial DNA sequences, although the two species are distinct. A sufficiently small egg-laying delay then leads to the rapid extinction of both blowfly populations.

In this paper, a new mathematical model that describes the dynamics of the within-host COVID-19 epidemic is formulated. We show the stochastic dynamics of Target-Latent-Infected-Virus free within the human body with discrete delay and noise. Positivity and uniqueness of the solutions are established. Our study shows the extinction and persistence of the disease inside the human body through the stability analysis of the disease-free equilibrium $$E_0$$ E 0 and the endemic equilibrium $$E^*$$ E ∗ , respectively. Moreover, we show the impact of delay tactics and noise on the extinction of the disease. The most interesting result is even if the deterministic system is inevitably pandemic at a specific point, extinction will become possible in the stochastic version of our model.

This manuscript is involved in the study of stability of the solutions of functional differential equations (FDEs) with random coefficients and/or stochastic terms. We focus on the study of different types of stability of random/stochastic functional systems, specifically, stochastic delay differential equations (SDDEs). Introducing appropriate Lyapunov functionals enables us to investigate the necessary conditions for stochastic stability, asymptotic stochastic stability, asymptotic mean square stability, mean square exponential stability, global exponential mean square stability, and practical uniform exponential stability. Some examples with numerical simulations are presented to strengthen the theoretical results. Using our theoretical study, important aspects of epidemiological and ecological mathematical models can be revealed. In ecology, the dynamics of Nicholson’s blowflies equation is studied. Conditions of stochastic stability and stochastic global exponential stability of the equilibrium point at which the blowflies become extinct are investigated. In finance, the dynamics of the Black–Scholes market model driven by a Brownian motion with random variable coefficients and time delay is also studied.

This paper investigates the dynamics behavior of the mathematical model of Human Immunodeficiency Virus (HIV) influenced by stochastic perturbations. This paper is involved in exploring the disease-free equilibrium’s stochastic global exponential stability with a basic reproductive number $$R_{0} < 1$$ . Necessary and sufficient conditions for stochastic global exponential stability of the disease-free equilibrium $$E^{0}$$ of the nonlinear HIV stochastic system are derived. This can be accomplished by the exponential analysis of the global mean square stability of trivial equilibrium of the corresponding linear system. Finally, we provide areas of stability of $$E^{0}$$ and numerical simulations to confirm the analytical results by using the fundamental Euler–Maruyama (EM) algorithm.

This work deals with the construction of finite difference solutions of random advection Cauchy type partial differential equation containing uncertainty through the coefficient of the velocity. Under appropriate hypothesis on the velocity random variable, we establish that the constructed random finite difference solution is mean square consistent and mean square stable over the whole real line. In addition, the main statistical functions, such as the mean, of the approximate solution stochastic process generated by truncation of the exact finite difference solution are given. Finally, we apply the proposed technique to several illustrative examples which show our discussing for the mean square stability.

This work is concerned with the discussion of the numerical approximation for random Cauchy transport model in one dimension. The random (forward time, backward space) finite difference scheme is used to find the stochastic solution. The impression of the consistency and the random von-Neumann stability technique under the mean square sense are studied. Using some examples, we can support our main objective of this model statistically.

This paper studies the solutions of variational methods for random ordinary (partial) differential equations in L2−space. These methods are called Galerkin method, Petrov-Galerkin method, Least-Squares method and Collocation method. Some basic properties of these methods where applying on random problems will be shown throughout some numerical examples.