An improved optimistic three-stage model for the spread of HIV amongst injecting intravenous drug users

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 R0. Next we give some analytical equilibrium, local and global stability results. We show that if R0 &gte 1 then the disease will always die out. For R0 > 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 R0 &gte 1 then the disease will die out and if R0 > 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

Mathematical models of the spread of Hepatitis C among injecting drug users, the effects of heterogeneity

The world faces an immense burden of hepatitis C virus (HCV) infection related morbidity and mortality. Transmission of HCV is ongoing, and the incidence of HCV infection has been increasing in recent years. Approximately 130 - 150 million people are estimated to be chronically infected with HCV and each year an estimated three to four million individuals are newly infected (WHO, 2013; Mohd Hanafiah et al., 2013). In developed countries, injecting drug users are considered as being at the highest risk of prevalence of HCV. Thus, this thesis describes the spread of HCV amongst injecting drug users. We use a mathematical model to study the effect of heterogeneity on the progress of the disease by dividing the population of addicts into p groups where they are sharing injecting needles in q shooting galleries and investigate the epidemic behavior of the virus. Moreover, we estimate the basic reproductive number R₀ and show analytically that HCV is controlled by this number R₀, if R₀ ≤ 1 then the disease dies out and if R₀ > 1 the disease takes off in both addicts and needles and there is a unique endemic equilibrium. We look at analytical results on the effect of heterogeneity on the spread of HCV and optimal control of the epidemic by needle exchange and needle cleaning. Simulations with realistic parameter values estimated from data and the literature confirm the theoretical results and we numerically investigate the effect of heterogeneity on the spread of HCV. Then we extend the basic model to more realistic assumptions where addicts move in and out of groups, and investigate the HCV dynamic behaviour. We obtain similar analytical results again validated by simulations with realistic parameter values estimated from data and the literature.The world faces an immense burden of hepatitis C virus (HCV) infection related morbidity and mortality. Transmission of HCV is ongoing, and the incidence of HCV infection has been increasing in recent years. Approximately 130 - 150 million people are estimated to be chronically infected with HCV and each year an estimated three to four million individuals are newly infected (WHO, 2013; Mohd Hanafiah et al., 2013). In developed countries, injecting drug users are considered as being at the highest risk of prevalence of HCV. Thus, this thesis describes the spread of HCV amongst injecting drug users. We use a mathematical model to study the effect of heterogeneity on the progress of the disease by dividing the population of addicts into p groups where they are sharing injecting needles in q shooting galleries and investigate the epidemic behavior of the virus. Moreover, we estimate the basic reproductive number R₀ and show analytically that HCV is controlled by this number R₀, if R₀ ≤ 1 then the disease dies out and if R₀ > 1 the disease takes off in both addicts and needles and there is a unique endemic equilibrium. We look at analytical results on the effect of heterogeneity on the spread of HCV and optimal control of the epidemic by needle exchange and needle cleaning. Simulations with realistic parameter values estimated from data and the literature confirm the theoretical results and we numerically investigate the effect of heterogeneity on the spread of HCV. Then we extend the basic model to more realistic assumptions where addicts move in and out of groups, and investigate the HCV dynamic behaviour. We obtain similar analytical results again validated by simulations with realistic parameter values estimated from data and the literature

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