Abstract
In many areas, researchers might think that a differential equation model is required, but one might be forced to use an approximate difference equation model if data is only available at discrete points in time. In this paper, a detailed comparison is given of the behavior of continuous and discrete models for two representative time-delay models, namely a model for HIV and an extended logistic growth model. For each model, there are seven different time-delay versions because there are seven different positions to include time delays. For the seven different time-delay versions of each model, proofs are given of necessary and sufficient conditions for the existence and stability of equilibrium points and for the existence of Andronov–Hopf bifurcations in the differential equations and Neimark–Sacker bifurcations in the difference equations. We show that only five of the seven time-delay versions have bifurcations and that all bifurcation versions have supercritical limit cycles with one having a repelling cycle and four having attracting cycles. Numerical simulations are used to illustrate the analytical results and to show that critical times for Neimark–Sacker bifurcations are less than critical times for Andronov–Hopf bifurcations but converge to them as the time step of the discretization tends to zero.
Highlights
In recent years, time-delay differential equation and difference equation models have been studied by many authors as they are useful tools for modeling a wide variety of systems in areas including traditional areas such as physics and engineering and newer areas such as disease transmission, medical research, optimal drug treatment, bioeconomics, agriculture, finance, insurance, and environmental protection
9 Conclusions In this paper, the effects of time delays and the associated Andronov–Hopf and Neimark– Sacker bifurcation properties have been studied for two one-dimensional models
We have studied the dynamical behavior of a differential equation model and an equivalent difference equation model obtained from a forward Euler approximation
Summary
Time-delay differential equation and difference equation models have been studied by many authors (see, e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]) as they are useful tools for modeling a wide variety of systems in areas including traditional areas such as physics and engineering and newer areas such as disease transmission, medical research, optimal drug treatment, bioeconomics, agriculture, finance, insurance, and environmental protection. We first investigate the properties of the seven different versions of differential equation models and prove conditions for the existence and stability of equilibrium points and for the existence of Andronov–Hopf bifurcations at critical values of the time delays. We investigate the properties of discretized versions of the models and prove conditions for the existence and stability of equilibrium points and for the existence of Neimark–Sacker bifurcations at critical values of the time delays We show that both differential equation and difference equation models may have bifurcations from both disease-free and endemic equilibrium points. It can be seen that each of these Euler approximation equations are difference equations of order m + 1
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