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As a widely used tool modeling some processes and systems in a variety of fields, fractional delay differential equations (FDDEs) with higher order have attracted much attention of the scientific community for years. Motivated by Yao et al. (2022), in which a single term has been done, we are much more interested in the stability analysis for multi-term FDDEs. In addition to the widely used Laplace transform method and decoupling technique for the characteristic equation, a region embedding technique is first introduced to handle the multiple fractional exponents. The existing results are generalized to multi-term FDDEs with higher order and the damping term of the classical integer-order delay differential equation is extended to fractional calculus. Numerical simulations for FDDEs and time-fractional telegraph equations with time delay are presented to illustrate the efficiency and validity of our results. • A novel condition for the stability is expressed in an algebraical criterion. • The Laplace transform method and a method of region embedding are used. • Our results extend the stability results in Yao et al. (2022) to the multi-term FDDEs. • The attenuation phenomenon of the damping term is generalized to FDDEs. [ABSTRACT FROM AUTHOR]

Among many other factors that affect the preventive interventions to any infectious disease, not reporting timely in a hospital is also one of the catastrophic behavior of human beings in any society. Similarly, masses who do not report make it difficult for healthcare researchers to measure the actual data and develop prevention strategies, accordingly. Therefore, there is a critical need to structure a potential epidemic model with the unreported class of individuals. This novel idea is deliberated in this paper to study the profiles of the epidemic model of virulent diseases due to the individuals that report timely and those who don't report in hospitals for any reason. Mathematically, a system of seven equations is taken into consideration, which describes the susceptible individuals, exposed, people who do not report to the hospital and those who report to the hospital, and individuals who are quarantined, infected, and recovered. So, with the consideration of new compartments, the conventional SIR epidemic model expands to SEU R R P QIR. The innovative design is made more realistic by utilizing proportional fractional-order differential equations with time delay. A special simplified expansion of this derivative reduces its computational cost and produces the results with fractional index, which helps to predict each fractional change. In addition, an optimal control methodology is also carried out to analyze the effectiveness of the awareness campaign in shifting the unreported individuals to the reported class, with optimal cost function for the unreported cases. Discussions are supported through the very recent deadly pandemic as an example to conclude the practical advantage of the model. The sensitivity analysis of basic reproduction numbers based on effective awareness campaigns is also the part of this study, which infers public awareness campaigns may be devised to motivate and guide such individuals to approach any healthcare center or a hospital. • Compartments for unreported and reported cases in the epidemic model. • Analysis through system of fractional order differential equations with time lags. • Realistic profiles of disease dynamics at different states. • Awareness campaign as optimal control approach to restrain unreported class. [ABSTRACT FROM AUTHOR]

• In this article we have studied the asymptotic stability and mean square stability of fractional-order (1,2] stochastic differential equations (SDEs). • We have considered the family of SDEs with variable delay in state. • For obtaining main results we used Banach fixed point theorem and imposed Lipschitz condition on nonlinearity. • Results are advanced and weighted enough as a contribution to stability theory of fractional SDEs. In this article, we discuss the asymptotic stability and mean square stability of stochastic differential equations of fractional-order 1 < α ≤ 2. We have considered the family of stochastic differential equations with variable delay in the state. For proving our main results, we apply the Banach fixed point theorem and imposed the Lipschitz condition on nonlinearity. Finally, we present an example to illustrate the obtained theory. [ABSTRACT FROM AUTHOR]

• Existence and exponential boundedness of solution of fractional neutral delay differential equations are investigated. • Explicit solutions of linear equations are derived in terms of generalized Mittag–Leffler functions. • Stability properties are analyzed. • Numerical simulations illustrate the asymptotic behavior. This paper discusses some properties of solutions to fractional neutral delay differential equations. By combining a new weighted norm, the Banach fixed point theorem and an elegant technique for extending solutions, results on existence, uniqueness, and growth rate of global solutions under a mild Lipschitz continuous condition of the vector field are first established. Be means of the Laplace transform the solution of some delay fractional neutral differential equations are derived in terms of three-parameter Mittag–Leffler functions; their stability properties are hence studied by using use Rouché's theorem to describe the position of poles of the characteristic polynomials and the final value theorem to detect the asymptotic behavior. By means of numerical simulations the theoretical findings on the asymptotic behavior are verified. [ABSTRACT FROM AUTHOR]