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In this study, we apply optimal control theory to an immuno-epidemiological model of HIV and opioid epidemics. For the multi-scale model, we used four controls: treating the opioid use, reducing HIV risk behaviour among opioid users, entry inhibiting antiviral therapy, and antiviral therapy which blocks the viral production. Two population-level controls are combined with two within-host-level controls. We prove the existence and uniqueness of an optimal control quadruple. Comparing the two population-level controls, we find that reducing the HIV risk of opioid users has a stronger impact on the population who is both HIV-infected and opioid-dependent than treating the opioid disorder. The within-host-level antiviral treatment has an effect not only on the co-affected population but also on the HIV-only infected population. Our findings suggest that the most effective strategy for managing the HIV and opioid epidemics is combining all controls at both within-host and between-host scales. 

In this paper, we present a multi-scale co-affection model of HIV infection and opioid addiction. The population scale epidemiological model is linked to the within-host model which describes the HIV and opioid dynamics in a co-affected individual. CD4 cells and viral load data obtained from morphine addicted SIV-infected monkeys are used to validate the within-host model. AIDS diagnoses, HIV death and opioid mortality data are used to fit the between-host model. When the rates of viral clearance and morphine uptake are fixed, the within-host model is structurally identifiable. If in addition the morphine saturation and clearance rates are also fixed the model becomes practical identifiable. Analytical results of the multi-scale model suggest that in addition to the disease-addiction-free equilibrium, there is a unique HIV-only and opioid-only equilibrium. Each of the boundary equilibria is stable if the invasion number of the other epidemic is below one. Elasticity analysis suggests that the most sensitive number is the invasion number of opioid epidemic with respect to the parameter of enhancement of HIV infection of opioid-affected individual. We conclude that the most effective control strategy is to prevent opioid addicted individuals from getting HIV, and to treat the opioid addiction directly and independently from HIV. 

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.