Delay differential model for tumour–immune dynamics with HIV infection of CD4⁺ T-cells

Abstract

In this paper, we introduce ordinary and delay differential equations to describe the interactions between a malignant tumour and the immune system in vivo in the presence of human immunodeficiency virus (HIV) infection of CD4⁺ T-cells. In the delay model, we take into account the time lags required by the healthy effector cell components to recognize the pathogens and tumour cells. The models consist of four populations: tumour cells, healthy effector cells (CD4⁺ T-cells), effector cells infected by HIVs and free viral particles. The presence of delay term in the model leads to a notable increase in the complexity of the observed behaviour. We investigate the qualitative behaviour of the models and find the conditions that guarantee the asymptotic stability of the steady states. Numerical simulations are provided to illustrate and extend the theoretical results. The obtained results are consistent with the real phenomena and give a better understanding of cancer immunity and viral oncogenesis.

Keywords:

• DDEs
• HIV
• Hopf bifurcation
• immune response
• ODEs
• stability
• tumour

2000 AMS Subject Classifications:

• 93A30
• 92C50
• 34K60
• 92D25
• 92D30

Acknowledgements

This research is generously supported by the UAE University. The authors thank the referees and Professor Abdul Q.M. Khaliq for their valuable and constructive comments on the paper.

Notes

AIDS is characterized by impairing the function of the immune system and by various clinical expressions. It was first recognized in 1981 19.

The concentration of the healthy effector cells reflects the normal physiological level of the effector cells which is approximated to lie between 800 and 1200 13.

Additional information

Notes on contributors

Fathalla A. Rihan

Permanent address: Faculty of Science, Helwan University, Cairo, Egypt.