## Abstract

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.

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.

Original language | English |
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Publication status | Published - Feb 2010 |

Event | First Workshop on Dynamical Systems Applied to Biology and Natural Sciences - Duration: 31 Mar 2011 → … |

### Other

Other | First Workshop on Dynamical Systems Applied to Biology and Natural Sciences |
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Period | 31/03/11 → … |

## Keywords

- HIV/AIDS
- basic reproduction number
- loss of infectivity
- equilibrium and stability analysis
- global stability
- modelling
- spread
- injecting drug users