A mathematical model for a disease outbreak considering waning-immunity class with nonlinear incidence and recovery rates

Abstract

In the spread of infectious diseases, intervention levels play a crucial role in shaping interactions between healthy and infected individuals, leading to a nonlinear transmission process. Additionally, the availability of medical resources limits the recovery rate of infected patients, adding further nonlinear dynamics to the healing process. Our research introduces novelty by combining nonlinear incidence and recovery rates alongside waning immunity in an epidemic model. We present a modified SIRW-type model, examining the epidemic problem with these factors. Through analysis, we explore conditions for non-endemic and co-existing cases based on the basic reproduction ratio. The local stability of equilibria is verified using the Routh-Hurwitz criteria, while global stability is assessed using Lyapunov functions for each equilibrium. Furthermore, we investigate bifurcations around both non-endemic and co-existing equilibria. Numerically, we give some simulations to support our analytical findings.

Keywords

• SIRW–type model
• Basic reproduction ratio
• Waning immunity
• Nonlinear incidence rate
• Nonlinear recovery rate
• Lyapunov functions