Analysis of a Delayed SIR Model with Nonlinear Incidence Rate

Abstract

An SIR epidemic model with incubation time and saturated incidence rate is formulated, where the susceptibles are assumed to satisfy the logistic equation and the incidence term is of saturated form with the susceptible. The threshold valueℜ0determining whether the disease dies out is found. The results obtained show that the global dynamics are completely determined by the values of the threshold valueℜ0and time delay (i.e., incubation time length). Ifℜ0is less than one, the disease-free equilibrium is globally asymptotically stable and the disease always dies out, while if it exceeds one there will be an endemic. By using the time delay as a bifurcation parameter, the local stability for the endemic equilibrium is investigated, and the conditions with respect to the system to be absolutely stable and conditionally stable are derived. Numerical results demonstrate that the system with time delay exhibits rich complex dynamics, such as quasiperiodic and chaotic patterns.

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

“…Notice that if α 1 = α 2 = 0, system (1.4) becomes the model investigated by Wang et al [24]. If α 2 = 0, the incidence rate becomes the saturated one [25,28]. If α 1 = 0, system (1.4) reduces to an example given by Enatsu et al [4].…”

“…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…”

Global Stability Analysis of Logistically Grown SIR Model with Loss of Immunity, Inhibitory Effect, Crowding Effect and its Protection Measure

“…[15,[21][22][23][24]. Similarly some authors [25][26][27][28][29] used the rate of incidence in the form aSI 1 + bI . In this form of incidence the increase in the number of infected population increases the incidence term and it ultimately becomes proportional to the number of susceptible populations.…”

Stability analysis of delay seirepidemic model

“…Mathematical models that describe epidemiology are widely used for the analysis of global stability of infection free and endemic equilibrium. Many authors worked on the infectious diseases such as Berezovsky et al (2005), Esteva andMatias, (2001) Greenhalgh (1992), Hsu and Zee (2014), Yi et al (2009), Zhang et al (2008 and the references therein.…”