Modelling the spread of HIV/AIDS amongst injecting drug users taking into account variable infectivity and loss of infectivity

We start off this paper with a brief introduction to modeling Human
Immunodeficiency Virus (HIV) and Acquired Immune Deficiency Syndrome
(AIDS) amongst sharing, injecting drug users (IDUs). Then we describe the
mathematical model which we shall use which extends an existing
model of the spread of HIV and AIDS amongst IDUs by
incorporating loss of HIV infectivity over time. This is followed by
the derivation of a key epidemiological parameter, the basic
reproduction number $R_0$. Next we give some analytical equilibrium,
local and global stability results. We show that if $R_0 \le 1$ then
the disease will always die out. For $R_0 > 1$ there is the
disease-free equilibrium (DFE) and a unique endemic equilibrium. The
DFE is unstable. An approximation argument
shows that we expect the endemic equilibrium to be locally stable. We next discuss a more
realistic version of the model, relaxing the assumption that the number
of addicts remains constant and obtain some results for this model.
The subsequent section gives simulations for both models confirming that if $R_0 \le 1$ then
the disease will die out and if $R_0 > 1$ then if it is initially present the disease will tend
to the unique endemic equilibrium. The simulation results are compared with the original model with no
loss of HIV infectivity. Next the implications of these results for control strategies are considered. A
brief summary concludes the paper.