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# Stability properties of a delayed HIV model with nonlinear functional response and absorption effect

#### Authors:

##### LK
Department of Mathematics, Faculty of Science, University of Ruhuna, Matara.

#### Wanbiao Ma,

##### CN
Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, P. R. China.

#### Songbai Guo

##### CN
Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, P. R. China.

## Abstract

This study investigated the impact of latent and maturation delay on the qualitative behaviour of a human immunodeficiency virus-1 (HIV-1) infection model with nonlinear functional response and absorption effect. Basic reproduction number (R0), which is defined as the average number of infected cells produced by one infected cell after inserting it into a fully susceptible cell population is calculated for the proposed model. As Ro (threshold) depends on the negatively exponential function of time delay, these parameters are responsible to predict the future propagation behaviour of the infection. Therefore, for smaller positive values of delay and larger positive values of infection rate, the infection becomes chronic. Besides, infection dies out with larger delays and lower infection rates. To make the model biologically more sensible, we used the functional form of response function that plays an important role rather than the bilinear response function. Existence of equilibria and stability behaviour of the proposed model totally depend on Ro. Local stability properties of both infection free and chronic infection equilibria are established by utilising the characteristic equation. As it is crucially important to study the global behaviour at equilibria rather than the local behaviour, we used the method of Liapunov functional. By constructing suitable Liapunov functionals and applying LaSalle’s invariance principle for delay differential equations, we established that infection free equilibrium is globally asymptotically stable if R0 ≤ 1, which biologically means that infection dies out. Moreover, sufficient condition is derived for global stability of chronic infection equilibrium if R0 > 1 , which biologically means that infection becomes chronic. Numerical simulations are given to illustrate the theoretical results.

J.Natn.Sci.Foundation Sri Lanka 2015 43 (3):235-245

##### Keywords: Basic reproduction number,HIV-1 infection,latent delay,Liapunov functional,local and global stability,mathematical dela
How to Cite: Pradeep, B.S.A., Ma, W. and Guo, S., 2015. Stability properties of a delayed HIV model with nonlinear functional response and absorption effect. Journal of the National Science Foundation of Sri Lanka, 43(3), pp.235–245. DOI: http://doi.org/10.4038/jnsfsr.v43i3.7953
Published on 24 Sep 2015.
Peer Reviewed