A dynamically consistent nonstandard finite difference scheme for a generalized SEIR epidemic model

This work is devoted to the proposal and analysis of a new mathematical study of the transmission dynamics of infectious diseases. First, a generalized SEIR epidemic model is presented that uses general nonlinear incidence rates to describe the 'psychological' effect. Then, a rigorous mathematical analysis is performed for the proposed SEIR model. We establish positivity and boundedness, calculate the basic reproduction number, determine possible equilibrium points (disease-free and endemic), and investigate their asymptotic stability properties of the SEIR model. The obtained results improve and extend a SEIR model constructed in a recent work; moreover, the proposed model is useful for studying the COVID-19 epidemic in particular and other infectious diseases in general. For the purpose of numerical simulation, the Mickens method is applied to construct a dynamically consistent non-standard finite difference (NSFD) model for the proposed SEIR epidemic model. The constructed NSFD scheme is able to provide reliable approximations that not only preserve the dynamic properties of the SEIR model for all values of the step size, but also are easy to implement. Finally, a series of illustrative numerical experiments are performed to support the theoretical findings and confirm the advantages of the NSFD scheme over some well-known standard methods.