About this Journal Submit a Manuscript Table of Contents
International Journal of Differential Equations

Volume 2014 (2014), Article ID 138094, 6 pages

http://dx.doi.org/10.1155/2014/138094
Research Article

A Discrete Model for HIV Infection with Distributed Delay

1Department of Mathematics and Computer Science, Faculty of Sciences Ben M’sik, Hassan II University, P.O. Box 7955 Sidi Othman, Casablanca, Morocco

2Centre Régional des Métiers de l’Education et de la Formation (CRMEF), 20340 Derb Ghallef, Casablanca, Morocco

Received 12 January 2014; Revised 9 April 2014; Accepted 9 April 2014; Published 27 April 2014

Academic Editor: Toka Diagana

Copyright © 2014 Brahim EL Boukari et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We give a consistent discretization of a continuous model of HIV infection, with distributed time delays to express the lag between the times when the virus enters a cell and when the cell becomes infected. The global stability of the steady states of the model is determined and numerical simulations are presented to illustrate our theoretical results.

1. Introduction

Nowadays, human immunodeficiency virus (HIV) that causes acquired immunodeficiency syndrome (AIDS) is a major health problem worldwide. From the World Health Organization (WHO), more than 35 million people are living with HIV/AIDS, and 1.6 million people died of this disease in 2012 [1]. Recent studies have been developed to know the dynamics of HIV infection, such as [29]. All these studies are based on continuous mathematical models. In reality, the statistical data are collected in discrete time, and the numerical simulations of continuous-time models are obtained by discretizing the models.

In this paper, we consider the model presented in [9] and we ignore the effect of the adaptive immune response. This model becomes as follows: where , , and denote the concentration of uninfected cells, infected cells, and free virus particles at time , respectively. The uninfected cells are produced at a constant , die at a rate , and become infected by free virus at a rate . Infected cells are lost at a rate . Free viruses are produced by infected cells at a rate and cleared at a rate . The authors [9] assumed that the uninfected cells are contacted by the virus particles at time and become infected cells at time , where is a random variable with a probability distribution over the interval and is limit superior of this delay. This probability distribution is assumed, for simplicity, to be a positive and integrable function on , satisfying . The term is the probability of surviving from time to time , where is the death rate of infected but not yet virus-producing cells.

Using the result presented in [9], it is not hard to see that the basic reproduction number of system (1) is given by .

We recall that the number is defined as the average number of secondary infections produced by one infected cell over its average life time, when all cells are uninfected.

In addition, the system (1) always has a disease-free equilibrium which is globally asymptotically stable if and a unique endemic equilibrium is globally asymptotically stable when .

Motivated by the works [1015] and that the statistical data are collected in discrete time, we propose the following discrete model obtained from (1) by using the rectangle method to approximate the integral and by applying the backward Euler discretization: where if is an integer, and if not, is the integer part of ( ). The sequences , , and denote the concentration of uninfected cells, infected cells, and free virus particles at time , respectively. The parameters in the system (2) are the same as those in (1). For simplicity, we may assume that . Similar to the continuous system (1), system (2) always has a disease-free equilibrium and an endemic equilibrium point , where is the basic reproduction number of (2) which is defined by

The aim of this work is to show that the discretization scheme used in system (2) preserves the positivity and boundedness of solutions and the global stability of both equilibria for the continuous model (1). Therefore, this discretization is dynamically consistent which means that all of the critical, qualitative properties of the solutions to the system of differential equations should also be satisfied by the solutions of the discrete scheme.

The paper is organized as follows. Section 2 deals with positivity and boundedness of solutions. In Section 3, we discuss the global stability of the equilibria. The numerical simulations are presented in Section 4 and the paper ends with a conclusion in Section 5.

2. Positivity and Boundedness of Solutions

Model (2) describes the evolution of a cell population that the cell number is nonnegative and bounded. For these biological reasons, we assume that the initial data for system (2) satisfy

Proposition 1. All solution of system (2) subject to condition (4) remains nonnegative and bounded.

Proof. From (2), we have Hence, by recurrence and (4), we have nonnegative and, thereafter, and are nonnegative.

For the boundedness, we put .

We have with . Then, Then, and are bounded.

By the third equation of (2), we have . Since is bounded, then there is such that , for all . Then, ; hence, , and then is bounded.

3. Global Stability

In this section, we will give the following main result that characterizes the global behavior of our model.

Theorem 2. (i)If , then is globally asymptotically stable.(ii) ; then is globally asymptotically stable.

Proof. For (i), we consider the following sequence defined by with , , and .

It is clear that for any and has the global minimum .

Consider Since , we have that We consider the set .

We have and, by (2), we have . By LaSalle’s invariance principle (see [16, Theorem 4.24]), we have that is globally asymptotically stable.

For (ii), we consider the following sequence defined by with , , and .

Consider Using the fact that , we have

Using the relations + , , and , we obtain since for any ; then .

We consider the set .

We have and, by (2), we have and . From LaSalle’s invariance principle, we deduce that is globally asymptotically stable.

4. Numerical Simulations

In this section, we present the numerical simulations to illustrate our theoretical results. In this section, we choose . First, we use the following data set: , , , , , , , , and . In this case, the basic infection reproduction number is . By using Theorem 2 (i), we deduce that is globally asymptotically stable. Numerical simulation illustrates our result (see Figure 1).

fig1
Figure 1: Demonstration of the stability of .

In Figure 2, we choose and do not change the other parameter values. By calculation, we have which satisfies the condition (ii) of Theorem 2. Hence, is globally asymptotically stable. Numerical simulation illustrates our result (see Figure 2).

fig2
Figure 2: Demonstration of the stability of .

In Figure 3, the parameter values are the same as those in Figure 1. Figure 3 gives in function of and shows that the growth of decreases the value of below , making the disease-free equilibrium globally asymptotically stable.

138094.fig.003
Figure 3: Plot of the basic reproduction number as a function of the delay . Here, , , , , , , , and .

5. Conclusion

In this work, we have proposed a discrete mathematical model of HIV infection by applying the backward Euler discretization, with distributed time delay. We have proved that, when , the disease-free equilibrium is globally asymptotically stable. When , the endemic equilibrium is globally asymptotically stable. More precisely, it is proved that this discretization guarantees the correct dynamic behavior regardless of the size of the time step.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the editor and the anonymous referees for their valuable comments and suggestions that have led to improvement of the quality of this work.

References

  1. WHO, Global Health Observatory (GHO), http://www.who.int/gho/hiv/en/.
  2. P. W. Nelson and A. S. Perelson, “Mathematical analysis of delay differential equation models of HIV-1 infection,” Mathematical Biosciences, vol. 179, no. 1, pp. 73–94, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. R. Xu, “Global dynamics of an HIV-1 infection model with distributed intracellular delays,” Computers & Mathematics with Applications, vol. 61, no. 9, pp. 2799–2805, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. B. EL Boukari, K. Hattaf, and N. Yousfi, “Modeling the therapy of HIV infection with CTL response and cure rate,” International Journal of Ecological Economics and Statistics, vol. 28, no. 1, pp. 1–17, 2013.
  5. K. Hattaf and N. Yousfi, “Dynamics of HIV infection model with therapy and cure rate,” International Journal of Tomography and Statistics, vol. 16, no. 11, pp. 74–80, 2011. View at Scopus
  6. K. Hattaf and N. Yousfi, “Optimal control of a delayed HIV infection model with immune response using an efficient numerical method,” ISRN Biomathematics, vol. 2012, Article ID 215124, 7 pages, 2012. View at Publisher · View at Google Scholar
  7. K. Hattaf and N. Yousfi, “Two optimal treatments of HIV infection model,” World Journal of Modelling and Simulation, vol. 8, pp. 27–35, 2012.
  8. K. Hattaf and N. Yousfi, “A delay differential equation model of HIV with therapy and cure rate,” International Journal of Nonlinear Science, vol. 12, no. 4, pp. 503–512, 2011. View at MathSciNet
  9. B. EL Boukari, K. Hattaf, and N. Yousfi, “HIV infection model with immunity and distributed delay,” Applied Mathematical Sciences, vol. 8, no. 14, pp. 653–661, 2014.
  10. Y. Enatsu, Y. Nakata, and Y. Muroya, “Global stability for a class of discrete SIR epidemic models,” Mathematical Biosciences and Engineering, vol. 7, no. 2, pp. 347–361, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. Y. Enatsu, Y. Nakata, Y. Muroya, G. Izzo, and A. Vecchio, “Global dynamics of difference equations for SIR epidemic models with a class of nonlinear incidence rates,” Journal of Difference Equations and Applications, vol. 18, no. 7, pp. 1163–1181, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. G. Izzo and A. Vecchio, “A discrete time version for models of population dynamics in the presence of an infection,” Journal of Computational and Applied Mathematics, vol. 210, no. 1-2, pp. 210–221, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. R. E. Mickens, Advances in the Applications of Nonstandard Finite Difference Schemes, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, USA, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  14. S. Elaydi and R. J. Sacker, “Global stability of periodic orbits of non-autonomous difference equations and population biology,” Journal of Differential Equations, vol. 208, no. 1, pp. 258–273, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. S. Jang and S. Elaydi, “Difference equations from discretization of a continuous epidemic model with immigration of infectives,” Mathematics Faculty Research, paper 32, 2004. View at Zentralblatt MATH · View at MathSciNet
  16. S. Elaydi, An Introduction to Difference Equations, Springer, New York, NY, USA, 3rd edition, 2005. View at MathSciNet