Global Dynamics of a Stochastic Viral Infection Model with Latently Infected Cells

In this paper, we study the global dynamics of a stochastic viral infection model with humoral immunity and Holling type II response functions. The existence and uniqueness of non-negative global solutions are derived. Stationary ergodic distribution of positive solutions is investigated. The solution fluctuates around the equilibrium of the deterministic case, resulting in the disease persisting stochastically. The extinction conditions are also determined. To verify the accuracy of the results, numerical simulations were carried out using the Euler–Maruyama scheme. White noise’s intensity plays a key role in treating viral infectious diseases. The small intensity of white noises can maintain the existence of a stationary distribution, while the large intensity of white noises is beneficial to the extinction of the virus

Dynamics of Tumor-Immune System with Random Noise

With deterministic differential equations, we can understand the dynamics of tumor-immune interactions. Cancer-immune interactions can, however, be greatly disrupted by random factors, such as physiological rhythms, environmental factors, and cell-to-cell communication. The present study introduces a stochastic differential model in infectious diseases and immunology of the dynamics of a tumor-immune system with random noise. Stationary ergodic distribution of positive solutions to the system is investigated in which the solution fluctuates around the equilibrium of the deterministic case and causes the disease to persist stochastically. In some conditions, it may be possible to attain infection-free status, where diseases die out exponentially with a probability of one. Some numerical simulations are conducted with the Euler–Maruyama scheme in order to verify the results. White noise intensity is a key factor in treating infectious diseases

Fractional Stochastic Differential Equations with Hilfer Fractional Derivative: Poisson Jumps and Optimal Control

In this work, we consider a class of fractional stochastic differential system with Hilfer fractional derivative and Poisson jumps in Hilbert space. We study the existence and uniqueness of mild solutions of such a class of fractional stochastic system, using successive approximation theory, stochastic analysis techniques, and fractional calculus. Further, we study the existence of optimal control pairs for the system, using general mild conditions of cost functional. Finally, we provide an example to illustrate the obtained results

Existence of solutions and approximate controllability of impulsive fractional stochastic differential systems with infinite delay and Poisson jumps

summary:The paper is motivated by the study of interesting models from economics and the natural sciences where the underlying randomness contains jumps. Stochastic differential equations with Poisson jumps have become very popular in modeling the phenomena arising in the field of financial mathematics, where the jump processes are widely used to describe the asset and commodity price dynamics. This paper addresses the issue of approximate controllability of impulsive fractional stochastic differential systems with infinite delay and Poisson jumps in Hilbert spaces under the assumption that the corresponding linear system is approximately controllable. The existence of mild solutions of the fractional dynamical system is proved by using the Banach contraction principle and Krasnoselskii's fixed-point theorem. More precisely, sufficient conditions for the controllability results are established by using fractional calculations, sectorial operator theory and stochastic analysis techniques. Finally, examples are provided to illustrate the applications of the main results