Mathematical epidemiology that describes the complex dynamics on social networks has become increasingly popular. However, a few methods have tackled the problem of coupling network topology with complex incidence mechanisms. Here, we propose a simplicial susceptible-infected-recovered-susceptible (SIRS) model to investigate the epidemic spreading via combining the network higher-order structure with a nonlinear incidence rate. A network-based social system is reshaped to a simplicial complex, in which the spreading or infection occurs with nonlinear reinforcement characterized by the simplex dimensions. Compared with the previous simplicial susceptible-infected-susceptible (SIS) models, the proposed SIRS model can not only capture the discontinuous transition and the bistability of a complex system but also capture the periodic phenomenon of epidemic outbreaks. More significantly, the two thresholds associated with the bistable region and the critical value of the reinforcement factor are derived. We further analyze the stability of equilibrium points of the proposed model and obtain the condition of existence of the bistable states and limit cycles. This work expands the simplicial SIS models to SIRS models and sheds light on a novel perspective of combining the higher-order structure of complex systems with nonlinear incidence rates.

In this paper a class of uncertain linear dynamical systems called fuzzy fractional linear dynamical systems is investigated. The aim is to find control inputs to keep the states of the fuzzy fractional dynamical systems near the zero in an optimal manner. The optimality criterion is in a form of a granular fuzzy integral whose integrand is a quadratic function of the state variables and control inputs. The fuzzy fractional dynamical system is described using Fuzzy Fractional Differential Equations (FFDEs). In order to achieve the aim, an effective approach for solving FFDEs should be at disposal. Due to some restrictions imposed by the previous approaches dealing with FFDEs, a new approach is proposed. The proposed approach is based on the granular derivative and the so-called relative-distance-measure fuzzy interval arithmetic. New definitions of fuzzy fractional derivatives and integral called left and right granular Riemann-Liouville fuzzy fractional derivatives, left and right granular Caputo fuzzy fractional derivatives, and the left and right granular fuzzy fractional integral are also presented. In addition, the concepts of granular fuzzy partial derivative and granular fuzzy chain rule are introduced. By the approximations of the granular fuzzy fractional integral and the granular Caputo fuzzy fractional derivative, the approximation solution to the FFDEs is obtained. Consequently, based on the new concepts and theorems, the solution to the fuzzy fractional quadratic regulator problem is given by a theorem. The paper closes with an example of regulating the motion of Boeing 747 in longitudinal direction with the presence of uncertainty in the initial conditions and the coefficients.

This paper is devoted to make a framework for studying a class of uncertain differential equations called Z-differential equations. In order to achieve the purpose, we first introduce four basic operations on $\rm Z^{+}$ -numbers based on semi-granular function. Then, the limit and continuity concepts of a Z-number-valued function are given, under a definition of a metric on the space of $\rm Z^{+}$ -numbers. Moreover, the concepts of Z-differentiability, Z-integral, and Z-Laplace transform of a Z-number-valued function are introduced. In addition, by giving some theories proved in this paper, a basis for calculus - Z-calculus - is established. We further give theories based on which existence and uniqueness of Z-differential equations are investigated. A conceptual unity between Z-differential equations and $\rm Z^{+}$ -numbers is also shown. The conceptual unity demonstrates that a Z-differential equation may be expressed as a bimodal differential equation combining a fuzzy differential equation and a random differential equation. Moreover, the concept of a bimodal cut called (s, μ)-cut is introduced and its relation to other new concepts such as acceptable time and acceptable information area is explained. Using an example, the application of Z-differential equations in medicine is clarified. It is demonstrated that Z-differential equations outperform fuzzy differential equations in making a decision under uncertainty.