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A nonlinear mathematical model of COVID-19 containing asymptomatic as well as symptomatic classes of infected individuals is considered and examined in the current paper. The largest eigenvalue of the next-generation matrix known as the reproductive number is obtained for the model, and serves as an epidemic indicator. To better understand the dynamic behavior of the continuous model, the unconditionally stable nonstandard finite difference (NSFD) scheme is constructed. The aim of developing the NSFD scheme for differential equations is its dynamic reliability, which means discretizing the continuous model that retains important dynamic properties such as positivity of solutions and its convergence to equilibria of the continuous model for all finite step sizes. The Schur–Cohn criterion is used to address the local stability of disease-free and endemic equilibria for the NSFD scheme; however, global stability is determined by using Lyapunov function theory. We perform numerical simulations using various values of some key parameters to see more characteristics of the state variables and to support our theoretical findings. The numerical simulations confirm that the discrete NSFD scheme maintains all the dynamic features of the continuous model. [ABSTRACT FROM AUTHOR]

The hepatitis B infection is a global epidemic disease which is a huge risk to the public health. In this paper, the transmission dynamics of hepatitis B deterministic model are presented and studied. The basic reproduction number is attained and by applying it, the local as well as global stability of disease-free and endemic equilibria of continuous hepatitis B deterministic model are discussed. To better understand the dynamics of the disease, the discrete nonstandard finite difference (NSFD) scheme is produced for the continuous model. Different criteria are employed to check the local and global stability of disease-free and endemic equilibria for the NSFD scheme. Our findings demonstrate that the NSFD scheme is convergent for all step sizes and consequently reasonable in all respect for the continuous deterministic epidemic model. All the aforementioned properties and their effects are also proved numerically at each stage to show their mathematical as well as biological feasibility. The theoretical and numerical findings used in this paper can be employed as a helpful tool for predicting the transmission of other infectious diseases. [ABSTRACT FROM AUTHOR]

In the present paper, the SIR model with nonlinear recovery and Monod type equation as incidence rates is proposed and analyzed. The expression for basic reproduction number is obtained which plays a main role in the stability of disease-free and endemic equilibria. The nonstandard finite difference (NSFD) scheme is constructed for the model and the denominator function is chosen such that the suggested scheme ensures solutions boundedness. It is shown that the NSFD scheme does not depend on the step size and gives better results in all respects. To prove the local stability of disease-free equilibrium point, the Jacobean method is used; however, Schur–Cohn conditions are applied to discuss the local stability of the endemic equilibrium point for the discrete NSFD scheme. The Enatsu criterion and Lyapunov function are employed to prove the global stability of disease-free and endemic equilibria. Numerical simulations are also presented to discuss the advantages of NSFD scheme as well as to strengthen the theoretical results. Numerical simulations specify that the NSFD scheme preserves the important properties of the continuous model. Consequently, they can produce estimates which are entirely according to the solutions of the model. [ABSTRACT FROM AUTHOR]