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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.
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This content will become publicly available on December 31, 2025
Optimal control of a multi-scale HIV-opioid model
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.
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- PAR ID:
- 10550438
- Publisher / Repository:
- Taylor and Francis
- Date Published:
- Journal Name:
- Journal of Biological Dynamics
- Volume:
- 18
- Issue:
- 1
- ISSN:
- 1751-3758
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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