### A FOUR DIMENSIONAL FUNCTIONAL DIFFERENTIAL EQUATION FOR THE DYNAMICS OF CD4+ T-CELLS AND HIV-1

#### Abstract

A four compartment delay model for HIV-1 and T-cell concentrations is given so that the time delay $\tau \ge 0$ corresponds directly (both by definition and location in the equations) with the fact that conversion processes require positive time. Hence, at $\tau = 0$, the system reduces to the model in \cite{1}. Since the characteristic equation is \pagebreaktranscendental, the classical Routh-Hurwitz criteria, of course, cannot be used directly. The approach in the present paper uses the \textit{derived polynomial} $h_\tau (r)$. Part of what is new includes applying the Routh-Hurwitz criteria to $h_\tau (r)$. Results do not depend on obtaining roots as such, and so can be used in the stability analysis of delay systems of any order. We obtain new explicit sufficient criteria for sustained stability of steady states across real intervals of delay $\tau $. Critical delays are calculated that produce bifurcation in the steady states. For the non-infected steady states, a delay dependent upper bound function $N_{crit} (\tau )$ for the lysing rate is explicitly calculated. This upper bound is connected with the PDK stability parameter $N_{crit}^{PDK} $ by the equality $N_{crit}^{PDK} = N_{crit} (\tau = 0)$. As would be required on medical grounds, the function $N_{crit} (\tau )$ increases in $\tau $. Throughout the paper, qualitative results are illustrated with numerical simulations.

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