Dynamical analysis of diphtheria epidemic model with natural immunity rate on exposed individuals

Abstract

Diphtheria disease remains endemic in some countries due to the lack of immunization coverage, such as Indonesia, where health care are something need to be pursued and optimized. In the last three years, there are still some areas in Indonesia that its complete basic immunization coverage are less than 50%. Apart from being caused by low immunization coverage, other factors as nutritional adequacy, environmental hygiene and virulence levels of the disease also influence the occurrence of diphtheria outbreaks. In this study, we propose a mathematical model that considers natural immunity rate of the exposed individuals in the spread of diphtheria. This study aims to understand the dynamics of the proposed model. The behavior of the system is known by analysing the stability of its equilibrium points. Numerical simulations using Maple are also carried out to illustrate the dynamics of the system. The stability analysis and numerical simulations show that the disease-free equilibrium point is stable if ?0 < 1, whereas the endemic equilibrium point is only feasible and stable if ?0 > 1. The results shown that complete basic immunization coverage and natural immunity rate of the population affect the basic reproduction number.