Abstract
A delayed worm propagation model with birth and death rates is formulated. The stability of the positive equilibrium is studied. Through theoretical analysis, a critical valueτ0of Hopf bifurcation is derived. The worm propagation system is locally asymptotically stable when time delay is less thanτ0. However, Hopf bifurcation appears when time delayτpasses the thresholdτ0, which means that the worm propagation system is unstable and out of control. Consequently, time delay should be adjusted to be less thanτ0to ensure the stability of the system stable and better prediction of the scale and speed of Internet worm spreading. Finally, numerical and simulation experiments are presented to simulate the system, which fully support our analysis.
Highlights
In recent years, Internet is undoubtedly one of the fastest increasing scientific technologies, which brings about convenience in people’s daily work and changes people’s life in variety of aspects
Enlightened by the researches in epidemiology, plenty of models have been constructed to predict the spread of worms and some containment strategies have been taken into consideration
Birth and death rates are widely applied in epidemiology because individuals in the ecological system may die during the spread of diseases
Summary
Internet is undoubtedly one of the fastest increasing scientific technologies, which brings about convenience in people’s daily work and changes people’s life in variety of aspects. Birth and death rates are widely applied in epidemiology because individuals in the ecological system may die during the spread of diseases. In order to imitate the real world, birth and death rates should be introduced to worm propagation model. Time delay is introduced in the worm propagation model along with birth and death rates, and its stability is analyzed. In order to guarantee the simplification and stability of the worm propagation system, time delay should be decreased appropriately by a decrease in the window size. Related work on time delay and birth is death rates and introduced. Afterwards we present the delayed worm propagation model with birth and death rates and analyze the stability of the positive equilibrium.
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