In this work, we investigate the stability of an SIR epidemic model with a generalized nonlinear incidence rate and distributed delay. The model also includes vaccination term and general treatment function, which are the two principal control measurements to reduce the disease burden. Using the Lyapunov functions, we show that the disease-free equilibrium state is globally asymptotically stable if $\mathcal{R}_{0} \leq 1 $R0≤1 , where $\mathcal{R}_{0} $R0 is the basic reproduction number. On the other hand, the disease-endemic equilibrium is globally asymptotically stable when $\mathcal{R}_{0} > 1 $R0>1 . For a specific type of treatment and incidence functions, our analysis shows the success of the vaccination strategy, as well as the treatment depends on the initial size of the susceptible population. Moreover, we discuss, numerically, the behavior of the basic reproduction number with respect to vaccination and treatment parameters.

To describe the main propagation of the COVID-19 and has to find the control for the rapid spread of this viral disease in real life, in current manuscript a discrete form of the SEIR model is discussed. The main aim of this is to describe the viral disease in simplest way and the basic properties that are related with the nature of curves for susceptible and infected individuals are discussed here. The elementary numerical examples are given by using the real data of India and Algeria.

Fluid viscosity is considered as constant in several boundary layer analyses, but this fluid property can change remarkably when the temperature difference exists in the boundary layer. The Prandtl number and Schmidt number can also change significantly as the fluid viscosity changes depending on temperature. Therefore, this framework is exploring the consequences of varying viscosity and varying Prandtl number on Falkner–Skan flow of Williamson nanofluid over a wedge, plate and stagnation point. The Buongiorno nanofluid model has been employed to manifest the fluid transport properties of the Williamson nanofluid. Similarity transformations are utilized to transform the governing equations into ordinary differential equation and solved numerically using Runge–Kutta (RK) Fehlberg method. Williamson fluid velocity, temperature, concentration, skin friction factor, rate of heat transfer and rate of mass transfer are investigated with emerging parameters, and the outcomes are presented graphically. Computed results manifest that the Williamson nanofluid expresses the opposite nature in velocity and temperature for higher values of Weissenberg number parameter. Positive values of variable viscosity parameter diminish the significance of variable Prandtl number and variable Schmidt number in the boundary layer. Furthermore, it is noticed that the Williamson nanofluid temperature is higher over a plate compared with wedge and stagnation point cases.