A delayed SIR model with general nonlinear incidence rate

Abstract

An SIR epidemic model is investigated and analyzed based on incorporating an incubation time delay and a general nonlinear incidence rate, where the growth of susceptible individuals is governed by the logistic equation. The threshold parameter σ 0 is defined to determine whether the disease dies out in the population. The model always has the trivial equilibrium and the disease-free equilibrium whereas it admits the endemic equilibrium if σ 0 exceeds one. The disease-free equilibrium is globally asymptotically stable if σ 0 is less than one, while it is unstable if σ 0 is greater than one.By applying the time delay as a bifurcation parameter, the local stability of the endemic equilibrium is studied and the condition which is absolutely stable or conditionally stable is established. Furthermore, a Hopf bifurcation occurs under certain conditions. Numerical simulations are carried out to illustrate the main results.

Mobility-based SIR model for complex networks: with case study Of COVID-19

“…Different variations of the SIR model have also been proposed to capture various real-world scenarios. For example, introducing a delay in the model to capture the incubation period during the spreading (Zhang et al 2010;Xia et al 2012;Liu 2015;Arquam et al 2018) or the introduction of interventions such as antiviral drugs (Towers et al 2011). In a different work to represent the nonlinear nature of epidemic spread, a SIR rumor spreading model was proposed in which tie strengths were dependent on nodes' degree (Singh and Singh 2012).…”

Bifurcation of a Delayed SEIS Epidemic Model with a Changing Delitescence and Nonlinear Incidence Rate

“…Thus, it has been an increasingly urgent issue to understand how to prevent or slow down the transmission of infectious diseases. To this end, many mathematical models have been proposed for describing the spread process of infectious diseases [2][3][4][5][6][7][8][9][10]. However, all the epidemic models above do not consider the change of delitescence of the infectious diseases.…”

Stability analysis for a delayed SIR model with a nonlinear incidence rate

“…Liu [14] discussed the dynamic behaviors of system (1.3). The trivial equilibrium is always unstable, and the disease-free equilibrium is stable if the basic reproduction number is less than one.…”

“…Furthermore, the endemic equilibrium may lose its stability under certain conditions if the basic reproduction number is greater than one, which admits periodic behavior. Inspired by the works of Wang et al [24], Enatsu et al [4], Zhang et al [28], and Liu [14], it is reasonable to construct a more realistic disease model with the general nonlinear incidence rate of the…”

“…Furthermore, the model (1.2) exhibits bifurcation properties as the length of the delay increases past a critical value provided that 1 < R 0 < R 0 . Liu [14] also considered another extended epidemic model based on the consideration of some biological meaning:…”