Abstract
We consider SIR epidemic model in which population growth is subject to logistic growth in absence of disease. We get the condition for Hopf bifurcation of a delayed epidemic model with information variable and limited medical resources. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium is discussed. If the basic reproduction ratioℛ0<1, we discuss the global asymptotical stability of the disease-free equilibrium by constructing a Lyapunov functional. Ifℛ0>1, we obtain sufficient conditions under which the endemic equilibriumE*of system is locally asymptotically stable. And we also have discussed the stability and direction of Hopf bifurcations. Numerical simulations are carried out to explain the mathematical conclusions.
Highlights
From an epidemiological viewpoint, it is important to investigate the global dynamics of the disease transmission
In the recent years, based on SIR epidemic model, in order to investigate the spread of an infectious disease transmitted by a vector, Wang et al [3] have considered the asymptotic behavior of the following delayed SIR epidemic model: dS (t) dt τ), dI (t) dt γ) I, (1)
We have found the existence oeqf udiilsiebarsieu-mfreEe∗e.qTuhileibbraisaicE0reapnrdoEdu1 catniodnhnavuemabuenriRqu0e positive changes the stability of the disease-free equilibrium
Summary
It is important to investigate the global dynamics of the disease transmission. We consider the information variable Z(t), nonlinear incidence rate of the form βSG(I(t−τ)), and limited. R is the intrinsic growth rate of susceptibles, k is the carrying capacity of susceptibles, α is the saturation factor that measures the inhibitory effect, β is the transmission or contact rate, μ1, μ2 are the natural death rate of the infective and recovered individuals, γ is the natural recovery rate, ε is the disease-related mortality, b ≥ 0 is the maximal medical resources supplied per unit time, and ω > 0 is half-saturation constant.
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