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Applying the local fractional integrals, a generalized identity involving the local second-order differentiable mappings is first developed in this paper. A series of fractal integral inequalities pertaining to Simpson type, for the mappings whose local second-order derivatives are generalized [Formula: see text]-convex in absolute value at some power, are then deduced by the discovered identity. Finally, from an application perspective, a range of fractal outcomes with regard to [Formula: see text]-type special means, Simpson numerical integrations, midpoint numerical integrations and wave equations are presented, correspondingly.

An actively controlled Susceptible-Infected-Susceptible (actSIS) contagion model is presented for studying epidemic dynamics with continuous-time feedback control of infection rates. Our work is inspired by the observation that epidemics can be controlled through decentralized disease-control strategies such as quarantining, sheltering in place, social distancing, etc., where individuals actively modify their contact rates with others in response to observations of infection levels in the population. Accounting for a time lag in observations and categorizing individuals into distinct sub-populations based on their risk profiles, we show that the actSIS model manifests qualitatively different features as compared with the SIS model. In a homogeneous population of risk-averters, the endemic equilibrium is always reduced, although the transient infection level can exhibit overshoot or undershoot. In a homogeneous population of risk-tolerating individuals, the system exhibits bistability, which can also lead to reduced infection. For a heterogeneous population comprised of risk-tolerators and risk-averters, we prove conditions on model parameters for the existence of a Hopf bifurcation and sustained oscillations in the infected population.

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