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• A higher order stochastically perturbed HIV/AIDS model with differential infectivity and amelioration is studied. • We derive sufficient criteria for the existence and uniqueness of an ergodic stationary distribution. • The existence of a stationary distribution indicates that the infectious disease will be prevalent. • We make up adequate criteria for complete eradication and wiping out of the infectious disease. In this paper, we study the dynamical behavior of a higher order stochastically perturbed HIV/AIDS model with differential infectivity and amelioration. We derive sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by establishing a series of suitable Lyapunov functions. In a biological point of view, the existence of a stationary distribution indicates that the infectious disease will be prevalent and persistent in the population. Moreover, we make up adequate criteria for complete eradication and wiping out of the infectious disease. Finally, we introduce some numerical simulations to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

• A higher order stochastically perturbed SIRI epidemic model with relapse and media coverage is studied. • We get sufficient criteria for the existence and uniqueness of an ergodic stationary distribution. • We make up adequate conditions for complete eradication and wiping out of the infectious disease. • The existence of a stationary distribution implies that the disease will prevail and persist in the long term. This paper is intended to explore a higher order stochastically perturbed SIRI epidemic model with relapse and media coverage. Firstly, we derive sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by establishing a suitable stochastic Lyapunov function. Then we obtain adequate conditions for complete eradication and wiping out of the infectious disease. In a biological interpretation, the existence of a stationary distribution implies that the disease will prevail and persist in the long term. Finally, the theoretical results are illustrated by computer simulations, including two examples based on real-life disease. [ABSTRACT FROM AUTHOR]