The present study is related to the design of a new mathematical model based on the Lane-Emden pantograph delay differential equation. The new model is obtained by using the sense of delay differential equation and standard Lane-Emden second-order equation. For the numerical solutions of the designed model, a well-known Bernoulli collocation method is implemented. In order to check the perfection and exactness of the designed model, three different nonlinear examples have been solved by using the Bernoulli collocation scheme. Furthermore, the comparison of the numerical results obtained by the Bernoulli collocation scheme with the exact solutions is also presented. Moreover, some numerical tables and the graphs of absolute error are plotted using different values of N for all problems.

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In this study, a novel reaction–diffusion model for the spread of the new coronavirus (COVID-19) is investigated. The model is a spatial extension of the recent COVID-19 SEIR model with nonlinear incidence rates by taking into account the effects of random movements of individuals from different compartments in their environments. The equilibrium points of the new system are found for both diffusive and non-diffusive models, where a detailed stability analysis is conducted for them. Moreover, the stability regions in the space of parameters are attained for each equilibrium point for both cases of the model and the effects of parameters are explored. A numerical verification for the proposed model using a finite difference-based method is illustrated along with their consistency, stability and proving the positivity of the acquired solutions. The obtained results reveal that the random motion of individuals has significant impact on the observed dynamics and steady-state stability of the spread of the virus which helps in presenting some strategies for the better control of it.

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