A New Susceptible-Infectious (SI) Model With Endemic Equilibrium

TL;DR

This paper introduces a new SI epidemic model with two infectious stages, analyzing its stability, positivity, and conditions for endemic equilibrium, providing insights into disease dynamics with multiple infectious forms.

Contribution

The paper develops and analyzes a novel SI model with two infectious groups and transitions, including stability and exact solutions, advancing understanding of complex disease dynamics.

Findings

Positivity of solutions is established.

Stability conditions for various equilibria are derived.

Exact solutions are obtained under specific conditions.

Abstract

The focus of this article is on the dynamics of a new susceptible-infected model which consists of a susceptible group ($S$) and two different infectious groups ($I_1$ and $I_2$). Once infected, an individual becomes a member of one of these infectious groups which have different clinical forms of infection. In addition, during the progress of the illness, an infected individual in group $I_1$ may pass to the infectious group $I_2$ which has a higher mortality rate. In this study, positiveness of the solutions for the model is proved. Stability analysis of species extinction, $I_1$-free equilibrium and endemic equilibrium as well as disease-free equilibrium is studied. Relation between the basic reproduction number of the disease and the basic reproduction number of each infectious stage is examined. The model is investigated under a specific condition and its exact solution is obtained.