Codimension-Two Bifurcations of an SIR-Type Model for COVID-19 and Their Epidemiological Implications

TL;DR

This paper analyzes complex bifurcation phenomena in a COVID-19 SIR model, revealing intricate transitions between disease states and emphasizing the importance of parameter control for eradication strategies.

Contribution

It identifies and characterizes codimension-two bifurcations in a COVID-19 model, highlighting complex transitions involving equilibria and limit cycles near critical points.

Findings

Backward bifurcation leads to complex transitions in disease dynamics.

Parameter control is crucial for effective COVID-19 eradication.

Multiple bifurcation points influence the model's long-term behavior.

Abstract

We study the codimension-two bifurcations exhibited by a recently-developed SIR-type mathematical model for the spread of COVID-19, as its two main parameters -- the susceptible individuals' cautiousness level and the hospitals' bed-occupancy rate -- vary over their domains. We use AUTO to generate the model's bifurcation diagrams near the relevant bifurcation points: two Bogdanov-Takens points and two generalised Hopf points, as well as a number of phase portraits describing the model's orbital behaviours for various pairs of parameter values near each bifurcation point. The analysis shows that, when a backward bifurcation occurs at the basic reproduction threshold, the transition of the model's asymptotic behaviour from endemic to disease-free takes place via an unexpectedly complex sequence of topological changes, involving the births and disappearances of not only equilibria but also limit cycles and homoclinic orbits. Epidemiologically, the analysis confirms the importance of a proper control of the values of the aforementioned parameters for a successful eradication of COVID-19. We recommend a number of strategies by which such a control may be achieved.