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 $\sigma_{0}$ 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 $\sigma_{0}$ exceeds one. The disease-free equilibrium is globally asymptotically stable if $\sigma_{0}$ is less than one, while it is unstable if $\sigma_{0}$ 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.
Highlights
Mathematical models have been become the important tools in investigating transmission and control of infectious diseases
6 Conclusion In this paper, a delayed SIR vector disease model with incubation time delay is established, in which the growth of susceptible individuals follows the logistic function in the absence of disease and the more general form of the nonlinear incidence rate is considered
It is shown that the trivial equilibrium is always unstable
Summary
Mathematical models have been become the important tools in investigating transmission and control of infectious diseases. We prove that the solutions of system ( ) are uniformly bounded for all t ≥ It follows from the first equation of system ( ) that S (t) ≤ rS(t)( – S(t)), which implies lim supt→∞ S(t) ≤. Proof At the endemic equilibrium E∗, it follows from the second equation of system ( ) that F(S∗) = μ + γ. By using ( ), the characteristic equation at endemic equilibrium E∗ = (S∗, I∗, R∗) can be turned into (λ + μ ) λ + aλ + b – e–λτ (cλ + d) = , where a = μ + γ + F S∗ I∗ – r – S∗ , b = (μ + γ ) F S∗ I∗ – r – S∗ ,.
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