Dynamics of a nonlinear epidemic transmission model incorporating a class of hospitalized individuals: a qualitative analysis and simulation

Abstract

This study aims to develop a novel mathematical epidemic compartmental model that includes a compartment or class for individuals who become infected and experience severe illness due to the infection. These individuals require hospitalization and the use of specialized medical equipment, such as ventilators, ICU beds, etc, during an outbreak. This compartment is referred to as the ‘hospitalized population compartment’ throughout this study. Additionally, the model incorporates a saturated incidence rate for new infection cases and the hospitalization rate for individuals severely affected by the infection, intending to create a more realistic scenario of the dynamics of disease transmission. The model is developed by integrating a compartment for hospitalized individuals into the standard susceptible-infected-recovered compartmental model and is subsequently mathematically analyzed for qualitative behavior. In this model, the saturated hospitalization rate reflects that the number of severely infected individuals who can be hospitalized is limited at any given time due to constraints in sufficient hospital infrastructure availability. The incidence rate of susceptibles becoming infected is modeled using the Holling Type II functional form, which incorporates inhibitory effects observed within the population. The study analyzes the mathematical model for two types of equilibria: the disease-free equilibrium (DFE) and the endemic equilibrium (EE). To investigate the stability of both equilibria, the basic reproduction number, , is calculated using the next-generation matrix method. The findings indicate that when , the DFE is locally asymptotically stable. Conversely, when , the DFE becomes unstable, leading to the emergence of a positive EE. Additionally, the study explores the occurrence of forward and backward transcritical bifurcations under specific conditions when . Furthermore, the study delves into both the local and global stability behaviors of the EE. Numerical simulations of the model are also performed to support the theoretical findings.