Behavioral response of population on transmissibility and saturation incidence of deadly pandemic through fractional order dynamical system.

The world entered in another wave of the SARS-CoV-2 due to non-compliance of standard operating procedures appropriately, initiated by respective governments. Apparently, measures like using face masks and social distancing were not observed by populace that ultimately worsens the situation. The behavioral response of the population induces a change in the dynamical outcomes of the pandemic, which is documented in this paper for all intents and purposes. The innovative perception is executed through a compartmental model with the incorporation of fractional calculus and saturation incident rate. In the first instance, the epidemiological model is designed with proportional fractional definition considering the compartmental individuals of susceptible, social distancing, exposed, quarantined, infected, isolated and recovered populations. By virtue of proportional fractional derivative, effective dynamical outcomes of equilibrium states and basic reproduction number are successfully elaborated with memory effect. The expansion of this derivative greatly simplifies the model to integer order while remaining in the fractional context. Subsequently, the memory effects on the asymptotic profiles are demonstrated through various graphical plots and tabulated values. In addition, the inclusion of saturation incident rate further explains the transmissibility of infection for different behavior of susceptible individuals. Mathematically, the results are also validated through comparative analysis of values with the solutions attained from fractional fourth order Runge-Kutta method (FRK4).

Dynamics of fractional order nonlinear system: A realistic perception with neutrosophic fuzzy number and Allee effect.

Introduction: The fusion of fractional order differential equations and fuzzy numbers has been widely used in modelling different engineering and applied sciences problems. In addition to these, the Allee effect, which is of high importance in field of biology and ecology, has also shown great contribution among other fields of sciences to study the correlation between density and the mean fitness of the subject. Objectives: The present paper is intended to measure uncertain dynamics of an economy by restructuring the Cobb-Douglas paradigm of the renowned Solow-Swan model. The purpose of study is further boosted innovatively by subsuming the perception of logistic growth with Allee effect in the dynamics of physical capital and labor force. Methods: Fractional order derivative and neutrosophic fuzzy (NF) theory are applied on the parameters of the Cobb-Douglas equation. Distinctively, cogitating fractional order derivative to study the change at each fractional stage; single-valued triangular neutrosophic fuzzy numbers (SVTNFN) to cope the uncertain situations; logistic growth function with Allee effect to analyze the factors in natural way, are the significant and novel features of this endeavor. Results: The incorporation of the aforementioned theories and effects in the Cobb-Douglas equation, resulted in producing maximum sustainable capital investment and maximum capacity of labor force. The solutions in intervals located different possible solutions for different membership degrees, which accumulated the uncertain circumstances of a country. Conclusion: Explicitly, these notions add new facts and figures not only in the dynamical study of capital and labor, which has been overlooked in classical models, but also left the door open for discussion and implementation on classical models of different fields.