Abstract In this article, we investigate global asymptotic stability of both equilibria, namely the virus-free and the virus-endemic equilibria, of a classical within-host HIV dynamical model. At first, we show global asymptotic stability of these equilibria for the time-continuous setting by introducing two suitable Lyapunov-functions. Afterwards, we introduce a time-discrete variant of this model by applying the implicit Eulerian time-stepping scheme and demonstrate that it can be recasted in an explicit manner. As our main contribution, we show that the same Lyapunov-functions of the time-continuous model transfer to the time-discrete case to establish global asymptotic stability of both equilibria where even non-equidistant time grids can be used. Finally, we compare our suggested non-standard finite-diffence-method with the classical explicit Eulerian time-stepping method, one second-order Runge-Kutta time-stepping scheme and an explicit-implicit non-standard finite-difference-method. We see that only both non-standard finite-difference-methods conserve non-negativity for arbitrary time-step sizes and that they behave similarly for small time-step sizes.

Abstract A quasisum is a function $$ F: I_1 \times \dots \times I_n \longrightarrow \mathbb {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>:</mml:mo> <mml:msub> <mml:mi>I</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>×</mml:mo> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>⟶</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> of the form $$\begin{aligned} F \left( x_1 \,, \dots , x_n \right) = g \bigl ( f_1(x_1) + \dots + f_n(x_n) \bigr ) \hspace{10mm} \left( x_1 \in I_1 \,, \dots , x_n \in I_n \right) \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mi>F</mml:mi> <mml:mfenced> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mspace/> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mfenced> <mml:mo>=</mml:mo> <mml:mi>g</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> </mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mspace/> <mml:mfenced> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>I</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mspace/> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mfenced> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> where $$ n \ge 2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> is an integer and $$ f_k: I_k \longrightarrow \mathbb {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>⟶</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> is a continuous, strictly monotone function defined on a nonempty open interval of $$ \mathbb {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> (for $$ k = 1 , \dots , n $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> ), moreover $$ g: f_1(I_1) + \dots + f_n(I_n) \longrightarrow \mathbb {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>:</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>I</mml:mi> <mml:mn>

Abstract In this paper a computer program, developed in the computer algebra system Maple is presented, which determines the type of functional equations.