Abstract
A nonstandard numerical scheme has been constructed and analyzed for a mathematical model that describes HIV infection of CD4+ T cells. This new discrete system has the same stability properties as the continuous model and, particularly, it preserves the same local asymptotic stability properties. Linearized Stability Theory and Schur-Cohn criteria are used for local asymptotic stability of this discrete time model. This proposed nonstandard numerical scheme is compared with the classical explicit Euler and fourth order Runge-Kutta methods. To show the efficiency of this numerical scheme, the simulated results are given in tables and figures.
1. Introduction
Mathematical models are used not only in the natural sciences and engineering disciplines, but also in the social sciences. The differential equations in these mathematical models are usually nonlinear autonomous differential equation systems which have only time-independent parameters. It is not always possible to find the exact solutions of the nonlinear models that have at least two ordinary differential equations. It is sometimes more useful to find numerical solutions of this type systems in order to programme easily and visualize the results. Numerous methods can be used to obtain the numerical solutions of differential equations. By applying a numerical method to a continuous differential equation system, it becomes a difference equation system, in other words discrete time system. While applying these numerical methods, it is necessary that the new difference equation system should provide the positivity conditions and exhibit the same quantitative behaviours of continuous system such as stability, bifurcation, and chaos. It is well known that some traditional and explicit schemes such as forward Euler and Runge-Kutta are unsuccessful at generating oscillation, bifurcations, chaos, and false steady states, despite using adaptative step size [1–6]. For forward Euler’s method, if the step size is chosen small enough and the positivity conditions are satisfied, it is seen that local asymptotic stability for a fixed point is saved while in some special cases Hopf bifurcation cannot be seen. Instead of classical methods, nonstandard finite difference scheme (NFDS) can be alternatively used to obtain more qualitative results and to remove numerical instabilities. These schemes are developed for compensating the weaknesses that may be caused by standard difference methods, for example, numerical instabilities. Also, the dynamic consistency could be presented well by NFDS [7]. The most important advantage of this scheme is that, choosing a convenient denominator function instead of the step size , better results can be obtained. If the step size is chosen small enough, the obtained results do not change significantly but if gets larger this advantage comes into focus.
The NFDS modeling procedures were given in 1989 by Mickens [8]. It removes the problems discussed above by using the suitable denominator function . The papers [2, 8–16] show how to choose the denominator function and apply this scheme to many models. Micken’s method can be summarized by using [13] as follows.
Let us consider the following ordinary differential equation: where is a parameter. The simplest nonstandard finite difference schemes are where depends on the step size and satisfies . It should be chosen where is calculated from a knowledge of the fixed points of (1), that is, Assume that the last equation has -real solutions and denote by Now define as and take as
In this paper, an NFDS scheme is applied to human immunodeficiency virus (HIV), which has spread rapidly around the world in recent years and thus it gains importance. In the last decade, the published papers about the epidemiology of HIV are less in number and they are not detailed enough. A few of these models were simulated using numerical methods such as Runge-Kutta or Euler methods. However, explicit methods are generally known to exhibit contrived chaos whenever the discretization parameters exceed certain values [17, 18].
A model about HIV infection of CD4+ T cells was presented by Perelson and Nelson [19, 20]. This model is given as follows: where , , denote the concentration of CD4+ T cells, the concentration of infected CD4+ T cells by the HIV viruses, and free HIV virus particles, respectively. is the infection rate. Each infected CD4+ T cell is assumed to produce virus particles during its life time [21]. is the maximum level of CD4+ T cell population density in the body. is logistic equation, where is the average specific T-cell growth rate [19]. , , , and are positive constants and is the source of CD4+ T cells from precursors, is the death rate of CD4+ T cells, is the death rate of infected cells, and finally is the viral clearance rate constant [22–24]. Nelson et al. focused on other models of HIV-1 infection in [25, 26]. These models deal with dynamics occurring after drug treatment. They analyzed the delay differential equation models of HIV-1 infection. Initially they give a standard model of HIV and then afterwards they give delay model of HIV. They analyze the model and give some lemma and proofs. Culshaw and Ruan consider a delay differential equation model of HIV infection of CD4+ T cells [27].
This paper is organized as follows: in Section 2, in order to obtain explicit solutions of (8), first the model is discretizated in a nonstandard form and this discrete model provides the positivity conditions. In Section 3, some lemmas and Linearized Stability Theorem are given for the local asymptotic stability of the discrete time systems. In Section 4, the theorical results obtained in former section are compared with the other numerical methods and the simulated results are given.
2. Discretization of the Model
The nonlinear differential equation system (1) will be discretizated as follows: If , , and are explicitly solve from (8), the following iterations will be obtained: where denominator functions are chosen as Detailed information about how to find different nonlocal terms to different denominator functions can be read in [9, 10, 13, 15]. Let and . In order to obtain positive iterations , , and we have to require or if then . If we take the numerical values and initial conditions in [21], for each nonnegative initial conditions , , and , the iterations , , and and consequently , , and are also nonnegative.
3. Stability Analysis of the Model
Some useful lemmas and a theorem should be given for local asymptotic stability of discrete systems. Especially, it is necessary to investigate Schur-Cohn criteria which deal with coefficient matrix of the linearized system as follows:(i), (ii), (iii), where and denote coefficient matrix of the linearized system and trace of the matrix, respectively. One can find information in [13, 28–31] about the usage of Schur-Cohn criteria which do not need many process as in continuous models.
The following lemmas and theorem given in citejury, citejodar are relevant to the roots of characteristic polynomials.
Lemma 1. For the quadratic equation the roots satisfy , if and only if the following conditions are satisfied:(i), (ii), (iii).
Lemma 2 (Jury conditions, Schur-Cohn criteria, ). Suppose the characteristic polynomial is given by . The solutions , , of satisfy if the following three conditions are held:(i), (ii), (iii).
Theorem 3 (the linearized stability theorem). Let be an equilibrium point of the difference equation where the function is a continuously differentiable function defined on some open neighborhood of an equilibrium point . Then the following statements are true.(1)If all the roots of the characteristic polynomial have absolute value less then one, then the equilibrium point is locally asymptotically stable.(2)If at least one root of the characteristic polynomial has absolute value greater than one, then the equilibrium point is unstable.
Equilibrium points of (8) are found as follows: where Only fixed points and have real biological meaning: the uninfected steady state and the (positive) infected steady state [21, 22]. Firstly, let us examine the fixed point . Equation (10) is rewritten as follows: By using these equations, Jacobian matrix will be found: where Firstly, let us find Jacobian matrix of (10) around to analyze the stability of this fixed point. We obtain
where To analyze the stability of , we need to find eigenvalues [32] where The first eigenvalue is We can find the other eigenvalues from where By considering Lemma 1, when , , the following conditions are satisfied and then the fixed point is locally asymptotic stable(i), (ii), (iii). Finally, let us examine the fixed point . Jacobian matrix around the fixed point is obtained as follows:
where By considering Lemma 2, we write the characteristic polynomial of as follows: where where If Lemma 2 is satisfied, we can say that the fixed point is locally asymptotically stable. Finally, it is important to say that the stability depends on time step size as it can be seen in Jacobian.
4. Numerical Results
In this section, we will use the values and the initial conditions in [21]. These values are given as follows: is the basic reproduction number. Wang and Li [21] present that if the basic reproduction number , the HIV infection is cleared from the T-cell population; if , the HIV infection persists. In this section we will calculate and see whether is locally asymptotically stable or not for different values of . And by using the criterion given in Section 3, we will check the validity of the results. For the fixed point , we will use Lemma 2, and we will also conclude whether fixed point is asymptotically stable or not.
4.1. Analysis of the Fixed Point
For , firstly let us calculate the basic reproduction number The first eigenvalue is From (23), other eigenvalues are found as follows: From Lemma 1, we see that(i), (ii), (iii). Therefore, the fixed point is unstable for .
For , the basic reproduction number is; and the first eigenvalue is From (23), So according to Lemma 1,(i), (ii), (iii). As a result, the fixed point is unstable for .
For ; the basic reproduction number is The first eigenvalue is and other eigenvalues are found as follows: By Lemma 1,(i), (ii), (iii). Therefore, fixed point is unstable for . For , the basic reproduction number is We obtain the eigenvalues as , , and by Lemma 1, :(i), (ii), (iii). So, the fixed point is locally asymptotically stable for .
4.2. Analysis of the Fixed Point
For , let us find characteristic polynomial of : By using Lemma 2,(i), (ii), (iii). Therefore the fixed point is locally asymptotically stable for . For , let us find characteristic polynomial of : From Lemma 2,(i), (ii), (iii). We obtain that the fixed point is unstable for . For , let us find characteristic polynomial of : By using Lemma 2,(i), (ii), (iii). We have the fixed point locally asymptotically stable for .
5. Conclusions
In general, it is too hard to analyze the stability of nonlinear three-dimensional systems. In this paper, by using the proposed NFDS scheme, nonlinear ordinary differential equation system which describes HIV infection of CD4+ T cells, is discretizated and the behaviour of the model is investigated. It is seen that the local asymptotic stability results of the fixed points and of the discrete time system satisfying the positivity condition are the same as in [21]. In Tables 1 and 2, for different step size and for different values, the qualitative stability results, obtained by NFDS, of the fixed point and are respectively compared to classical methods such as forward Euler and Runge-Kutta. The fixed points and are locally asymptotically stable for the values given in these two tables. If step size is chosen small enough, the results of the proposed NFDS are similar with the results of the other two numerical methods. But if the step size is chosen larger, the efficiency of NFDS is clearly seen. In Table 3, stability results for fixed point are given for different values. It is shown in [21] that if , is unstable and HIV infection persist in T-cell population. If , then is unstable. So in case of , neither nor are stable (Figure 2). In Figures 1 and 3, the NFDS solutions of , and converges to fixed point as simulated for and , respectively. Also in Figure 3, Runge-Kutta and proposed NFDS scheme are compared graphically. All the numerical calculations and simulations are performed by using Maple programme. In conclusion, the efficiency of the proposed NFDS scheme is investigated and compared with other numerical methods.
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Acknowledgments
The authors would like to thank Professor Apostolos Hadjidimos and Professor Michael N. Vrahatis for their valuable suggestions during the preparation of this paper. Also, the authors are grateful to the reviewers for their constructive comments, which have enhanced the paper.