The dynamics of heroin and illicit opioid use disorder, casual use, treatment, and recovery: A mathematical modeling analysis.

A leading crisis in the United States is the opioid use disorder (OUD) epidemic. Opioid overdose deaths have been increasing, with over 100,000 deaths due to overdose from April 2020 to April 2021. This paper presents a mathematical model to address illicit OUD (IOUD), initiation, casual use, treatment, relapse, recovery, and opioid overdose deaths within an epidemiological framework. Within this model, individuals remain in the recovery class unless they relapse back to use and due to the limited availability of specialty treatment facilities for individuals with OUD, a saturation treatment function was incorporated. Additionally, a casual user class and its corresponding specialty treatment class were incorporated. We use both heroin and all-illicit opioids datasets to find parameter estimates for our models. Bistability of equilibrium solutions was found for realistic parameter values for the heroin-only dataset. This result implies that it would be beneficial to increase the availability of treatment. An alarming effect was discovered about the high overdose death rate: by 2046, the disorder-free equilibrium would be the only stable equilibrium. This consequence is concerning because it means the epidemic would end due to high overdose death rates. The IOUD model with a casual user class, its sensitivity results, and the comparison of parameters for both datasets, showed the importance of not overlooking the influence that casual users have in driving the all-illicit opioid epidemic. Casual users stay in the casual user class longer and are not going to treatment as quickly as the users of the heroin epidemic. Another result was that the users of the all-illicit opioids were going to the recovered class by means other than specialty treatment. However, the change in the relapse rate has more of an influence for those individuals than in the heroin-only epidemic. The results above from analyzing this model may inform health and policy officials, leading to more effective treatment options and prevention efforts.

Optimal Control with RdCVFL for Degenerating Photoreceptors.

Both the rod and cone photoreceptors, along with the retinal pigment epithelium have been experimentally and mathematically shown to work interdependently to maintain vision. Further, the theoredoxin-like rod-derived cone viability factor (RdCVF) and its long form (RdCVFL) have proven to increase photoreceptor survival in experimental results. Aerobic glycolysis is the primary source of energy production for photoreceptors and RdCVF accelerates the intake of glucose into the cones. RdCVFL helps mitigate the negative effects of reactive oxidative species and has shown promise in slowing the death of cones in mouse studies. However, this potential treatment and its effects have never been studied in mathematical models. In this work, we examine an optimal control with the treatment of RdCVFL. We mathematically illustrate the potential this treatment might have for treating degenerative retinal diseases such as retinitis pigmentosa, as well as compare this to the results of an updated control model with RdCVF.

A mathematical model of GLUT1 modulation in rods and RPE and its differential impact in cell metabolism.

We present a mathematical model of key glucose metabolic pathways in two cells of the human retina: the rods and the retinal pigmented epithelium (RPE). Computational simulations of glucose transporter 1 (GLUT1) inhibition in the model accurately reproduce experimental data from conditional knockout mice and reveal that modification of GLUT1 expression levels of both cells differentially impacts their metabolism. We hypothesize that, under glucose scarcity, the RPE's energy producing pathways are altered in order to preserve its functionality, impacting the photoreceptors' outer segment renewal. On the other hand, when glucose is limited in the rods, aerobic glycolysis is preserved, which maintains the lactate contribution to the RPE.

Modeling the Dynamics of Heroin and Illicit Opioid Use Disorder, Treatment, and Recovery.

Opioid use disorder (OUD) has become a serious leading health issue in the USA leading to addiction, disability, or death by overdose. Research has shown that OUD can lead to a chronic lifelong disorder with greater risk for relapse and accidental overdose deaths. While the prescription opioid epidemic is a relatively new phenomenon, illicit opioid use via heroin has been around for decades. Recently, additional illicit opioids such as fentanyl have become increasingly available and problematic. We propose a mathematical model that focuses on illicit OUD and includes a class for recovered users but allows for individuals to either remain in or relapse back to the illicit OUD class. Therefore, in our model, individuals may cycle in and out of three different classes: illicit OUD, treatment, and recovered. We additionally include a treatment function with saturation, as it has been shown there is limited accessibility to specialty treatment facilities. We used 2002-2019 SAMHSA and CDC data for the US population, scaled to a medium-sized city, to obtain parameter estimates for the specific case of heroin. We found that the overdose death rate has been increasing linearly since around 2011, likely due to the increased presence of fentanyl in the heroin supply. Extrapolation of this overdose death rate, together with the obtained parameter estimates, predict that by 2038 no endemic equilibrium will exist and the only stable equilibrium will correspond to the absence of heroin use disorder in the population. There is a range of parameter values that will give rise to a backward bifurcation above a critical saturation of treatment availability. We show this for a range of overdose death rate values, thus illustrating the critical role played by the availability of specialty treatment facilities. Sensitivity analysis consistently shows the significant role of people entering treatment on their own accord, which suggests the importance of removing two of the most prevalent SAMHSA-determined reasons that individuals do not enter treatment: financial constraints and the stigma of seeking treatment for heroin use disorder.

The role of RdCVFL in a mathematical model of photoreceptor interactions.

Recent experimental and mathematical work has shown the interdependence of the rod and cone photoreceptors with the retinal pigment epithelium in maintaining sight. Accelerated intake of glucose into the cones via the theoredoxin-like rod-derived cone viability factor (RdCVF) is needed as aerobic glycolysis is the primary source of energy production. Reactive oxidative species (ROS) result from the rod and cone metabolism and recent experimental work has shown that the long form of RdCVF (RdCVFL) helps mitigate the negative effects of ROS. In this work we investigate the role of RdCVFL in maintaining the health of the photoreceptors. The results of our mathematical model show the necessity of RdCVFL and also demonstrate additional stable modes that are present in this system. The sensitivity analysis shows the importance of glucose uptake, nutrient levels, and ROS mitigation in maintaining rod and cone health in light-damaged mouse models. Together, these suggests areas on which to focus treatment in order to prolong the photoreceptors, especially in situations where ROS is a contributing factor to their death such as retinitis pigmentosa.

Optimal control with MANF treatment of photoreceptor degeneration.

People afflicted with diseases such as retinitis pigmentosa and age-related macular degeneration experience a decline in vision due to photoreceptor degeneration, which is currently unstoppable and irreversible. Currently there is no cure for diseases linked to photoreceptor degeneration. Recent experimental work showed that mesencephalic astrocyte-derived neurotrophic factor (MANF) can reduce neuron death and, in particular, photoreceptor death by reducing the number of cells that undergo apoptosis. In this work, we build on an existing system of ordinary differential equations that represent photoreceptor interactions and incorporate MANF treatment for three experimental mouse models having undergone varying degrees of photoreceptor degeneration. Using MANF treatment levels as controls, we investigate optimal control results in the three mouse models. In addition, our numerical solutions match the experimentally observed surviving percentage of photoreceptors and our uncertainty and sensitivity analysis identifies significant parameters in the math model both with and without MANF treatment.

The Vicodin abuse problem: A mathematical approach.

The prescription drug epidemic in the United States has gained attention in recent years. Vicodin, along with its generic version, is the country's mostly widely prescribed pain reliever, and it contains a narcotic component that can lead to physical and chemical dependency. The majority of Vicodin abusers were first introduced via prescription, unlike other drugs which are often experienced for the first time due to experimentation. Most abusers report obtaining their supply from a prescription, either their own or someone else's. Although the problem with prescription drug abuse is well known, there is no standard method of addressing the problem. To better understand how to do this, we develop and analyze a mathematical model of Vicodin use and abuse, considering only those patients who were initially prescribed the drug. Through global sensitivity analysis, we show that focusing efforts on abuse prevention rather than treatment has greater success at reducing the population of Vicodin abusers. Our results demonstrate that relying solely on rehabilitation and other treatment programs is not enough to combat the prescription drug problem in the United States. We anticipate that implementing preventative measures in both prescribers and patients will reduce the number of Vicodin abusers.

Quantifying the metabolic contribution to photoreceptor death in retinitis pigmentosa via a mathematical model.

Retinitis pigmentosa (RP) is a family of inherited retinal degenerative diseases that leads to blindness. In many cases the disease-causing allele encodes for a gene exclusively expressed in the night active rod photoreceptors. However, because rod death always leads to cone death affected individuals eventually lose their sight. Many theories have been proposed to explain the secondary loss of cones in RP; however, most fail to fully explain the different pathological transition stages seen in humans. Incorporating experimental data of rod and cone death kinetics from two mouse models of RP, we use a mathematical model to investigate the interplay and role of energy consumption and uptake of the photoreceptors as well as nutrient availability supplied through the retinal pigment epithelium (RPE) throughout the progression of RP. Our data driven mathematical model predicts that the system requires a total reduction of approximately 27-31% in nutrients available to result in the complete demise of all cones. Simulations utilizing retinal degeneration 1 (rd1) mouse cell count data in which cone death was delayed by altering cell metabolism in cones show that preventing a 1-2% decrease in nutrients available can permanently halt cone death even when 90% have already died. Our results also indicate that the ratio of energy consumption to uptake of cones, Dc, is mainly disrupted during the death wave of the rods with negligible changes thereafter and that the subsequent nutrient decrease is mainly responsible for the demise of the cones. The change in this ratio Dc highlights the compensation that the cones must undergo during rod death to meet the high metabolic demands of the entire photoreceptor population. Global sensitivity analysis confirms the results and suggests areas of focus for halting RP, even at later stages of the disease, through feasible therapeutic interventions.

Mathematical Model of the Role of RdCVF in the Coexistence of Rods and Cones in a Healthy Eye.

Understanding the essential components and processes for coexistence of rods and cones is at the forefront of retinal research. The recent discovery on RdCVF's mechanism and mode of action for enhancing cone survival brings us a step closer to unraveling key questions of coexistence and codependence of these neurons. In this work, we build from ecological and enzyme kinetic work on functional response kinetics and present a mathematical model that allows us to investigate the role of RdCVF and its contribution to glucose intake. Our model results and analysis predict a dual role of RdCVF for enhancing and repressing the healthy coexistence of the rods and cones. Our results show that maintaining RdCVF above a threshold value allows for coexistence. However, a significant increase above this value threatens the existence of rods as the cones become extremely efficient at uptaking glucose and begin to take most of it for themselves. We investigate the role of natural glucose intake and that due to RdCVF in both high and low nutrient levels. Our analysis reveals that under low nutrient levels coexistence is not possible regardless of the amount of RdCVF present. With high nutrient levels coexistence can be achieved with a relative small increase in glucose uptake. By understanding the contributions of rods to cones survival via RdCVF in a non-diseased retina, we hope to shed light on degenerative diseases such as retinitis pigmentosa.

Mathematical modeling of fungal infection in immune compromised individuals: The effect of back mutation on drug treatment.

We present a mathematical model that describes treatment of a fungal infection in an immune compromised patient in which both susceptible and resistant strains are present with a mutation allowing the susceptible strain to become resistant as well as a back mutation allowing resistant fungus to again become susceptible. The resulting nonlinear differential equations model the biological outcome, in terms of strain growth and cell number, when an individual is treated with a fungicidal or fungistatic drug. The model demonstrates that under any levels of the drug both strains will be in stable co-existence and high levels of treatment will never completely eradicate the susceptible strain. A modified model is then described in which the drug is changed to one in which both strains are susceptible, and subsequently, at the appropriate level of treatment, complete eradication of both fungal strains ensues. We discuss the model and implications for treatment options within the context of an immune compromised patient.

The mathematical and theoretical biology institute--a model of mentorship through research.

This article details the history, logistical operations, and design philosophy of the Mathematical and Theoretical Biology Institute (MTBI), a nationally recognized research program with an 18-year history of mentoring researchers at every level from high school through university faculty, increasing the number of researchers from historically underrepresented minorities, and motivating them to pursue research careers by allowing them to work on problems of interest to them and supporting them in this endeavor. This mosaic profile highlights how MTBI provides a replicable multi-level model for research mentorship.

Tracing the progression of retinitis pigmentosa via photoreceptor interactions.

Retinitis pigmentosa (RP) is a group of inherited degenerative eye diseases characterized by mutations in the genetic structure of the photoreceptors that leads to the premature death of both rod and cone photoreceptors. Defects in particular genes encoding proteins that are involved in either the photoreceptor structure, phototransduction cascades, or visual cycle are expressed in the rods but ultimately affect both types of cells. RP is "typically" manifested by a steady death of rods followed by a period of stability in which cones survive initially and then inevitably die too. In some RP cases, rods and cones die off simultaneously or even cone death precedes rod death (reverse RP). The mechanisms and factors involved in the development of the different types of RP are not well understood nor have researchers been able to provide more than a limited number of short-term therapies. In this work we trace the progression of RP to complete blindness through each subtype via bifurcation theory. We show that the evolution of RP from one stage to another often requires the failure of multiple components. Our results indicate that a delicate balance between the availability of nutrients and the rates of shedding and renewal of photoreceptors is needed at every stage of RP to halt its progression. This work provides a framework for future physiological investigations potentially leading to long-term targeted multi-facet interventions and therapies dependent on the particular stage and subtype of RP under consideration. The results of this mathematical model may also give insight into the progression of many other degenerative eye diseases involving genetic mutations or secondary photoreceptor death and potential ways to circumvent these diseases.

Mathematical modeling of fungal infection in immune compromised individuals: implications for drug treatment.

We present a mathematical model that describes treatment of a fungal infection in an immune compromised patient in which both susceptible and resistant strains are present. The resulting nonlinear differential equations model the biological outcome, in terms of strain growth and cell number, when an individual, who has both a susceptible and a resistant population of fungus, is treated with a fungicidal or fungistatic drug. The model demonstrates that when the drug is only successful at treating the susceptible strain, low levels of the drug cause both strains to be in stable co-existence and high levels eradicate the susceptible strain while allowing the resistant strain to persist or to multiply unchecked. A modified model is then described in which the drug is changed to one in which both strains are susceptible, and subsequently, at the appropriate level of treatment, complete eradication of both fungal strains ensues. We discuss the model and implications for treatment options within the context of an immune compromised patient.

A mathematical model for photoreceptor interactions.

The interactions between rods and cones in the retina have been the focus of innumerable experimental and theoretical biological studies in previous decades yet the understanding of these interactions is still incomplete primarily due to the lack of a unified concept of cone photoreceptor organization and its role in retinal diseases. The low abundance of cones in many of the non-primate mammalian models that have been studied make conclusions about the human retina difficult. A more complete knowledge of the human retina is crucial for counteracting the events that lead to certain degenerative diseases, in particular those associated with photoreceptor cell death (e.g., retinitis pigmentosa). In an attempt to gain important insight into the role and interactions of the rods and the cones we develop and analyze a set of mathematical equations that model a system of photoreceptors and incorporate a direct rod-cone interaction. Our results show that the system can exhibit stable oscillations, which correspond to the rhythmic renewal and shedding of the photoreceptors. In addition, our results show the mathematical necessity of this rod-cone direct interaction for survival of both and gives insight into this mechanism.

"Traveling wave" solutions of FitzHugh model with cross-diffusion.

The FitzHugh-Nagumo equations have been used as a caricature of the Hodgkin-Huxley equations of neuron firing and to capture, qualitatively, the general properties of an excitable membrane. In this paper, we utilize a modified version of the FitzHugh-Nagumo equations to model the spatial propagation of neuron firing; we assume that this propagation is (at least, partially) caused by the cross-diffusion connection between the potential and recovery variables. We show that the cross-diffusion version of the model, be- sides giving rise to the typical fast traveling wave solution exhibited in the original "diffusion" FitzHugh-Nagumo equations, additionally gives rise to a slow traveling wave solution. We analyze all possible traveling wave solutions of the model and show that there exists a threshold of the cross-diffusion coefficient (for a given speed of propagation), which bounds the area where "normal" impulse propagation is possible.