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

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.