Abstract
A SIR epidemic model that describes the dynamics of childhood disease with a saturated incidence rate and vaccination program at a constant rate was investigated. For the continuous model we first show its basic properties, namely, the non-negativity and boundedness of solutions. Then we investigate the existence and both local and global stability of the equilibrium points. It was found that the existence and stability properties of equilibrium points fully determined the basic reproduction number. We also propose and analyze a discrete-time analogue of the continuous childhood diseases by applying a nonstandard finite difference method. It is shown that our discrete model preserves the dynamical properties of the corresponding continuous model, such as the positivity solutions, the population conservation law, the existence of equilibrium points and their global stability properties.
Highlights
One of the major issues in public health problems is childhood diseases, which can spread rapidly among children age 5 and below due to frequent contact with their peers at school or elsewhere
The SIR epidemic model of childhood diseases with saturated incidence rate and constant vaccination strategy is proposed and the dynamics are investigated in this paper
It is shown that the solutions of the constructed discrete nonstandard finite difference (NSFD) model are always non-negative and satisfy the exact population conservation law
Summary
One of the major issues in public health problems is childhood diseases (measles, mumps, influenza, smallpox, chickenpox, Rubella, Polio, etc.), which can spread rapidly among children age 5 and below due to frequent contact with their peers at school or elsewhere. The SIR epidemic model of childhood diseases with saturated incidence rate and constant vaccination strategy is dS bSI. From a mathematical point of view, Makinde [1] has provided a qualitative analysis for the SIR model with a bi-linear incidence rate He applied the Adomian decomposition method to obtain an analytical approximation to its solutions. Cui et al [20] constructed a NSFD scheme for the SIR epidemic model of childhood diseases with bi-linear incidence rate and showed that the resulting discrete system is dynamically consistent with the original continuous model. The SIR epidemic model of childhood diseases with saturated incidence rate and constant vaccination strategy is proposed and the dynamics are investigated in this paper. The dynamics of the discrete model are compared to the continuous model to check their consistency
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