Global dynamics of target-mediated drug disposition models and their solutions by nonstandard finite difference method

Abstract

The aim of this work is to study global dynamics of target-mediated drug disposition (TMDD) models and their solutions by nonstandard finite difference (NSFD) schemes. Firstly, we use comparison principles and the Lyapunov stability theory for ODEs to establish positivity, boundedness, local and global asymptotic stability of the TMDD models. Secondly, positivity-preserving NSFD schemes are proposed and their dynamical properties are analysed rigorously. Lastly, we perform a set of numerical simulations to support and illustrate the theoretical results and to show advantages of the NSFD schemes over standard ones. The results show that there is a good agreement between the numerical results and theoretical ones. In addition, the numerical simulations indicate that the constructed NSFD schemes are dynamically stable and efficient in replicating the complex dynamical properties of the continuous models.

Appendix A: Convergence of the NSFD schemes

The following results on the convergence of the NSFD schemes are proved similarly to Theorem 5.9 in [7].

Proposition 1

The NSFD scheme (11) for the model (1) is convergent of order 1.

Proof

We rewrite the model (1) in the vector form

$$\begin{aligned} \dfrac{dy}{dt} = f(y), \end{aligned}$$

where $y(t) = \big [L(t), R(t), P(t)\big ]^T$ and f denotes the right-hand side function of (1). It is easy to verify that (11) can be written in the form

$$\begin{aligned} y_{n + 1} = y_n + \varphi (h)F(y_n, h), \end{aligned}$$

where

$$\begin{aligned} F(y, 0) = f(y). \end{aligned}$$

(25)

Using the hypothesis $\varphi (h) = h + \mathcal {O}(h^2)$ as $h \rightarrow 0$ and (25) and repeating the proof of [7, Theorem 5.9], we obtain

$$\begin{aligned} \Vert y_{n} - y(t_n)\Vert = \mathcal {O}(h), \end{aligned}$$

(26)

for $i = 0, 1, 2, \ldots$. Consequently, the proof is complete. $\square$

Similarly to Proposition 1, we obtain the convergence of the NSFD (20) as follows.

Proposition 2

The NSFD scheme (20) for the model (6) is convergent of order 1.