Abstract
An SIQR epidemic model with nonlinear incidence rate and two delays is studied under the assumption that a susceptible of the host population has a constant input. Local stability and existence of Hopf bifurcation are analyzed by regarding combination of the time delay due to the latent period of disease and the time delay due to the period that the infective and quarantined individuals need to be cured as the bifurcation parameter. Furthermore, the properties of the Hopf bifurcation are determined by using the normal form method and center manifold theory. Some numerical simulations are also carried out in order to verify our theoretical findings.
Highlights
For the last two decades, various epidemic models have been proposed and investigated in order to understand disease transmissions and behaviours of epidemics
It has been suggested by several authors that the disease transmission process may have a nonlinear incidence rate and the epidemic models with a nonlinear incidence rate have been studied by many researchers [ – ]
We can conclude that the Hopf bifurcation is supercritical, the bifurcating periodic solutions are stable and the period of the bifurcating periodic solutions decreases according to Theorem . 5 Conclusions In the present paper, a delayed SIQR epidemic model with constant input and nonlinear incidence rate is investigated based on the model studied in [ ]
Summary
For the last two decades, various epidemic models have been proposed and investigated in order to understand disease transmissions and behaviours of epidemics. The bilinear incidence rate is based on the law of mass action, which is more appropriate for communicable diseases, but not for sexually transmitted diseases [ ] It has been suggested by several authors that the disease transmission process may have a nonlinear incidence rate and the epidemic models with a nonlinear incidence rate have been studied by many researchers [ – ]. In [ ], Song and Pang proposed the following SIQR (susceptible-infective-quarantined-recovered) epidemic model with constant input and nonlinear incidence rate:. Song and Pang studied stability of system ( ) They neglected the time delay due to the latent period of the disease and the time delay due to the period that the infective and quarantined individuals need to be cured in system ( ). Τ is the time delay due to the period that the infective and quarantined individuals need to be cured.
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