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1. A network immuno-epidemiological model of HIV and opioid epidemics

   https://doi.org/10.3934/mbe.2023189  

   Gupta, Churni ; Tuncer, Necibe ; Martcheva, Maia ( January 2022 , Mathematical Biosciences and Engineering)

   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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2. Modeling the Effects of Morphine-Altered Virus Specific Antibody Responses on HIV/SIV Dynamics

   https://doi.org/10.1038/s41598-019-41751-8  

   Mutua, Jones M. ; Perelson, Alan S. ; Kumar, Anil ; Vaidya, Naveen K. ( April 2019 , Scientific Reports)

   Drugs of abuse, such as opiates, have been widely associated with enhancing HIV replication, accelerating disease progression and diminishing host-immune responses, thereby making it harder to effectively manage HIV infection. It is thus important to study the effects of drugs of abuse on HIV-infection and immune responses. Here, we develop mathematical models that incorporate the effects of morphine-altered antibody responses on HIV/SIV dynamics. Based on fitting the model to experimental data from simian immunodeficiency virus (SIV) infections in control and morphine-addicted macaques, we found that two of the most significant effects of virus specific antibodies are neutralizing viral particles and enhancing viral clearance. Using our model, we quantified how morphine alters virus-specific antibody responses, and how this alteration affects the key components of virus dynamics such as infection rate, virus clearance, viral load, CD4⁺T cell count, and CD4⁺T cell loss in SIV-infected macaques under conditioning with morphine. We found that in a subpopulation of SIV-infected morphine addicted macaques, the presence of drugs of abuse may cause significantly diminished antibody responses, resulting in more severe infection with increased SIV infectivity, a decreased viral clearance rate, increased viral load, and higher CD4⁺T cell loss.

    

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3. Calculating prescription rates and addiction probabilities for the four most commonly prescribed opioids and evaluating their impact on addiction using compartment modelling

   https://doi.org/10.1093/imammb/dqab001  

   Rivas, Samantha R ; Tessner, Alex C ; Goldwyn, Eli E ( February 2021 , Mathematical Medicine and Biology: A Journal of the IMA)

   null (Ed.)

   Abstract In 2016, more than 11 million Americans abused prescription opioids. The National Institute on Drug Abuse considers the opioid crisis a national addiction epidemic, as an increasing number of people are affected each year. Using the framework developed in mathematical modelling of infectious diseases, we create and analyse a compartmental opioid-abuse model consisting of a system of ordinary differential equations. Since $40\%$ of opioid overdoses are caused by prescription opioids, our model includes prescription compartments for the four most commonly prescribed opioids, as well as for the susceptible, addicted and recovered populations. While existing research has focused on drug abuse models in general and opioid models with one prescription compartment, no previous work has been done comparing the roles that the most commonly prescribed opioids have had on the crisis. By combining data from the Substance Abuse and Mental Health Services Administration (which tracked the proportion of people who used or misused one of the four individual opioids) with data from the Centers of Disease Control and Prevention (which counted the total number of prescriptions), we estimate prescription rates and probabilities of addiction for the four most commonly prescribed opioids. Additionally, we perform a sensitivity analysis and reallocate prescriptions to determine which opioid has the largest impact on the epidemic. Our results indicate that oxycodone prescriptions are both the most likely to lead to addiction and have the largest impact on the size of the epidemic, while hydrocodone prescriptions had the smallest impact. 

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4. Extracellular dynamics of early HIV infection

   https://doi.org/10.1002/mma.5237  

   Capistrán, Marcos A. ; Núñez‐López, Mayra ; Rempala, Grzegorz A. ( August 2018 , Mathematical Methods in the Applied Sciences)

   In this paper, we explore the interplay of virus contact rate, virus production rates, and initial viral load during early HIV infection. First, we consider an early HIV infection model formulated as a bivariate branching process and provide conditions for its criticalityR₀ > 1. Using dimensionless rates, we show that the criticality conditionR₀ > 1 defines a threshold on the target cell infection rate in terms of the infected cell removal rate and virus production rate. This result has motivated us to introduce two additional models of early HIV infection under the assumption that the virus contact rate is proportional to the target cell infection probability (denoted by). Using the second model, we show that the length of the eclipse phase of a newly infected host depends on the target cell infection probability, and the corresponding deterministic equations exhibit bistability. Indeed, occurrence of viral invasion in the deterministic dynamics depends onR₀and the initial viral loadV₀. If the viral load is small enough, eg,V₀ ≪ θ, then there will be extinction regardless of the value ofR₀. On the other hand, if the viral load is large enough, eg,V₀ ≫ θandR₀ > 1, then there will be infection. Of note,V₀≈θcorresponds to a threshold regime above which virus can invade. Finally, we briefly discuss between‐cell competition of viral strains using a third model. Our findings may help explain the HIV population bottlenecks during within‐host progression and host‐to‐host transmission.

    

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5. Stochastic investigation of HIV infection and the emergence of drug resistance

   https://doi.org/10.3934/mbe.2022054  

   Olabode, Damilola ; Rong, Libin ; Wang, Xueying ( January 2021 , Mathematical Biosciences and Engineering)

   Drug-resistant HIV-1 has caused a growing concern in clinic and public health. Although combination antiretroviral therapy can contribute massively to the suppression of viral loads in patients with HIV-1, it cannot lead to viral eradication. Continuing viral replication during sub-optimal therapy (due to poor adherence or other reasons) may lead to the accumulation of drug resistance mutations, resulting in an increased risk of disease progression. Many studies also suggest that events occurring during the early stage of HIV-1 infection (i.e., the first few hours to days following HIV exposure) may determine whether the infection can be successfully established. However, the numbers of infected cells and viruses during the early stage are extremely low and stochasticity may play a critical role in dictating the fate of infection. In this paper, we use stochastic models to investigate viral infection and the emergence of drug resistance of HIV-1. The stochastic model is formulated by a continuous-time Markov chain (CTMC), which is derived based on an ordinary differential equation model proposed by Kitayimbwa et al. that includes both forward and backward mutations. An analytic estimate of the probability of the clearance of HIV infection of the CTMC model near the infection-free equilibrium is obtained by a multitype branching process approximation. The analytical predictions are validated by numerical simulations. Unlike the deterministic dynamics where the basic reproduction number $ \mathcal{R}_0 $ serves as a sharp threshold parameter (i.e., the disease dies out if $ \mathcal{R}_0 < 1 $ and persists if $ \mathcal{R}_0 > 1 $), the stochastic models indicate that there is always a positive probability for HIV infection to be eradicated in patients. In the presence of antiretroviral therapy, our results show that the chance of clearance of the infection tends to increase although drug resistance is likely to emerge.

    

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