Background Experimentally brought to light by Russell and hypothetically explained by Korteweg–de Vries, the KDV equation has drawn the attention of several mathematicians and physicists because of its extreme substantial structure in describing nonlinear evolution equations governing the propagation of weakly dispersive and nonlinear waves. Due to the prevalent nature and application of solitary waves in nonlinear dynamics, we discuss the soliton solution and application of the fractional-order Korteweg–de Vries (KDV) equation using a new analytical approach named the “Modified initial guess homotopy perturbation.” Results We established the proposed technique by coupling a power series function of arbitrary order with the renown homotopy perturbation method. The convergence of the method is proved using the Banach fixed point theorem. The methodology was demonstrated with a generalized KDV equation, and we applied it to solve linear and nonlinear fractional-order Korteweg–de Vries equations, which are in Caputo sense. The method’s applicability and effectiveness were established as a feasible series of arbitrary orders that accelerate quickly to the exact solution at an integer order and are obtained as solutions. Numerical simulations were conducted to investigate the effect of Caputo fractional-order derivatives in the dispersion and propagation of water waves by varying the order $$\alpha$$ α on the $$[0,1]$$ [ 0 , 1 ] interval. Comparative analysis of the simulation results, which were presented graphically and discussed, reveals that the degree of freedom of the Caputo fractional-order derivative is vital to controlling the magnitude of environmental hazards associated with water waves when adjusted. Conclusion The proposed method is recommended for obtaining convergent series solutions to fractional-order partial differential equations. We suggested that applied mathematicians and physicists investigate this work to better understand the impact of the degree of freedom posed by Caputo fractional-order derivatives in wave dispersion and propagation, as physical applications can help divert wave-related environmental hazards.

The importance of the saturation term in an SEIR (Susceptible, Exposed, Infected, and Recovered) epidemic model was examined in this article. To estimate the basic reproduction number (R0), examine the stabilities and run numerical simulations on the model, the next generation matrix, the Lyapunov function and Runge-Kutta techniques were used. The numerical simulation results reveal that, the saturation term has a significant influence in the model’s susceptible and infected compartments. However, as demonstrated by the simulation results, saturation term has a greater influence on vulnerable people than on infected people. As a result, greater sensitization programs through seminars, media, and awareness will be more beneficial to the vulnerable class than the afflicted class during disease eradication.

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Background The COVID-19 pandemic has put the world's survival in jeopardy. Although the virus has been contained in certain parts of the world after causing so much grief, the risk of it emerging in the future should not be overlooked because its existence cannot be shown to be completely eradicated. Results This study investigates the impact of vaccination, therapeutic actions, and compliance rate of individuals to physical limitations in a newly developed SEIQR mathematical model of COVID-19. A qualitative investigation was conducted on the mathematical model, which included validating its positivity, existence, uniqueness, and boundedness. The disease-free and endemic equilibria were found, and the basic reproduction number was derived and utilized to examine the mathematical model's local and global stability. The mathematical model's sensitivity index was calculated equally, and the homotopy perturbation method was utilized to derive the estimated result of each compartment of the model. Numerical simulation carried out using Maple 18 software reveals that the COVID-19 virus's prevalence might be lowered if the actions proposed in this study are applied. Conclusion It is the collective responsibility of all individuals to fight for the survival of the human race against COVID-19. We urged that all persons, including the government, researchers, and health-care personnel, use the findings of this research to remove the presence of the dangerous COVID-19 virus.

In this paper, a study of preventive measures capable of curbing the spread of COVID 19pandemic to avoid its second wave was carried out. The existence and uniqueness of theproposed mathematical model is assured, the basic reproduction number is established, thelocal and global stability of the disease free equilibrium are well obtained and the variationaliteration method is applied to solve the mathematical model. Numerical simulation of theincluded control parameters are carried out. The obtained results and outcomes are presentedgraphically. It was revealed that enlightenment to vaccination awareness should be encouragedas vaccination is a good strategy of capturing the spread of the disease. Keywords: Covid-19, Basic Reproduction Number, Local stability, Global Stability,variational Iteration Method