Backward bifurcation, basic reinfection number and robustness of a SEIRE epidemic model with reinfection

TL;DR

This paper introduces a SEIRE epidemic model incorporating reinfection, revealing conditions for backward bifurcation and bistability, and emphasizes the importance of the basic reinfection number as a new threshold for disease control.

Contribution

It develops a novel SEIRE model with reinfection, identifies new thresholds including the basic reinfection number, and introduces the concept of robustness for bistable epidemic systems.

Findings

Backward bifurcation occurs when R0 is between Rc and 1.

Disease cannot be eradicated if R0 exceeds Rc.

Numerical simulations confirm theoretical results.

Abstract

Recent evidences show that individuals who recovered from COVID-19 can be reinfected. However, this phenomenon has rarely been studied using mathematical models. In this paper, we propose a SEIRE epidemic model to describe the spread of the epidemic with reinfection. We obtain the important thresholds $R_0$ (the basic reproduction number) and Rc (a threshold less than one). Our investigations show that when $R_0 > 1$, the system has an endemic equilibrium, which is globally asymptotically stable. When $R_c < R_0 < 1$, the epidemic system exhibits bistable dynamics. That is, the system has backward bifurcation and the disease cannot be eradicated. In order to eradicate the disease, we must ensure that the basic reproduction number $R_0$ is less than $R_c$. The basic reinfection number is obtained to measure the reinfection force, which turns out to be a new tipping point for disease dynamics. We also give definition of robustness, a new concept to measure the difficulty of completely eliminating the disease for a bistable epidemic system. Numerical simulations are carried out to verify the conclusions.