A network immuno-epidemiological model of HIV and opioid epidemics

In this paper, we introduce a novel multi-scale network model of two epidemics: HIV infection and opioid addiction. The HIV infection dynamics is modeled on a complex network. We determine the basic reproduction number of HIV infection, $ \mathcal{R}_{v} $, and the basic reproduction number of opioid addiction, $ \mathcal{R}_{u} $. We show that the model has a unique disease-free equilibrium which is locally asymptotically stable when both $ \mathcal{R}_{u} $ and $ \mathcal{R}_{v} $ are less than one. If $ \mathcal{R}_{u} > 1 $ or $ \mathcal{R}_{v} > 1 $, then the disease-free equilibrium is unstable and there exists a unique semi-trivial equilibrium corresponding to each disease. The unique opioid only equilibrium exist when the basic reproduction number of opioid addiction is greater than one and it is locally asymptotically stable when the invasion number of HIV infection, $ \mathcal{R}^{1}_{v_i} $ is less than one. Similarly, the unique HIV only equilibrium exist when the basic reproduction number of HIV is greater than one and it is locally asymptotically stable when the invasion number of opioid addiction, $ \mathcal{R}^{2}_{u_i} $ is less than one. Existence and stability of co-existence equilibria remains an open problem. We performed numerical simulations to better understand the impact of three epidemiologically important parameters that are at the intersection of two epidemics: $ q_v $ the likelihood of an opioid user being infected with HIV, $ q_u $ the likelihood of an HIV-infected individual becoming addicted to opioids, and $ \delta $ recovery from opioid addiction. Simulations suggest that as the recovery from opioid use increases, the prevalence of co-affected individuals, those who are addicted to opioids and are infected with HIV, increase significantly. We demonstrate that the dependence of the co-affected population on $ q_u $ and $ q_v $ are not monotone.