Roughly 3. Stanford University researchers have developed a mathematical model that could help public health officials and policymakers curb an opioid epidemic that took the lives of an estimated 49, Americans last year. The model includes data about prescriptions, addictions and overdoses in the United States. Fung Professor in the School of Engineering. Brandeau is the senior author of a paper that describes how the model analyzes opioid use and addiction among Americans whose initial exposure to these drugs came either from being prescribed pills or gaining illicit access to pills.

The lead author is graduate student Allison Pitt.

## Modeling Illicit Drug Use Dynamics and Its Optimal Control Analysis

The paper cites the daunting facts of the opioid crisis: Between and , there was a percent spike in prescriptions for opioid painkillers. Today, roughly 3. Yet, as doctors have begun responding to the crisis by reducing prescriptions, overdose deaths have increased because former pill addicts, unable to get the quality-controlled drug, have started buying and overdosing on heroin. Against this bleak backdrop, the Stanford model projects that if doctors can cut prescriptions by 25 percent over a year period, while policymakers expand drug treatment programs, addictions and deaths will start falling by Margaret Brandeau , the Coleman F.

Humphreys said the model puts multiple parts of the opioid-crisis puzzle together so that government and private-sector officials can see how addressing any given aspect of the crisis will affect the big picture. Pitt, the lead author, spent the last two years collecting data on opioid use and addiction and other variables. The researchers based their model on data from Thereafter the model made projections for such variables as what fraction of American adults experience moderate to severe pain each year; how many opioid painkillers are prescribed; how many people have become addicted to opioids by starting with pills; and how many Americans who initially became addicted through exposure to pills die each year from overdoses of pharmaceutical opioids or heroin.

## How a mathematical model could help us curb the opioid epidemic | Stanford School of Engineering

In , when million opioid prescriptions were issued, 3. Of these, the researchers estimated that roughly 16, died as a result of a pill overdoses. Another 22, who were initially hooked on pills died that year from heroin overdoses. This crossover from pills to street drugs, which often occurs when pill addicts can no longer get prescription medications, is one tragic aspect of the opioid crisis. In addition to these 38, deaths related to pill exposure, an estimated 26, more Americans died of overdoses from other drugs, mainly from cocaine and methamphetamine in But the Stanford study does not factor these deaths into the model.

Instead, the researchers focused on overdoses that could be traced to pill exposure and projected forward 10 years to make educated guesses about how different scenarios would be likely to decrease addiction and death rates stemming from pharmaceutical opioids. The first, or baseline, scenario was to project the status quo. If current trends prevail, the model projects that million prescriptions would still be written in , and that by then an estimated 4. Pill-connected overdose deaths would likely soar to an estimated 50,, compared with 38, in After establishing this baseline, the researchers considered various scenarios by which the epidemic might be slowed: What if physicians reduced new prescriptions across the board?

What if there were controls on refills? This may suggest that other combinations of drugs may target complimentary pathways more efficiently. Sensitivity analysis on the model parameters can support the robustness of the simulation results. But it can also suggest what drugs do not work and what drugs are more likely to work. The results are shown in Fig. All the p -values were less than 0. A positive PRCC i. A negative PRCC for N means that an increase in the parameter will decrease the number of live neurons.

The other PRCC values can also be seen to be consistent with the model dynamics. This is also seen from Fig. Currently there is no drug that can cure, stop, or even slow the progression of the disease. NFTs destroy microtubles in neurons, which results in neurons death. Additionally, peripheral macrophages, responding to cue from MCP-1 produced by astrocytes, are attracted to the brain and increase the inflammatory environment, which is harmful to neurons. In the present paper, we developed a mathematical model of AD based on Fig.

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The model can be used to explore the efficacy of drugs that may slow the progress of the disease. Based on these findings we propose that clinical trials should use a combination therapy with etanercept f and aducanumab h. In Fig. This suggests that in an optimal regimen with fixed total amount, A, of the drugs, f should be significantly larger than h. We did not consider here, however, adverse side effects that are likely to limit the amount of drugs that can be given to a patient. When these limits become better known, one could then proceed to determine the optimal combination of etanercept and aducanumab for slowing the progression of AD.

The mathematical model developed in this paper depends on some assumptions regarding the mechanism of interactions involving amyloid, tau and neunofilaments in AD. There are currently not enough data to sort out competing assumptions. Hence the conclusion of the paper regarding combination therapy should be taken with caution. But dendritic pathologies also play an important role in the disease. Dendritic abnormalities in AD include dystrophic neuritides, reduction in dendritic complexity and loss in dendritic spines [ 36 , 37 ].

Recent studies also begin to address white matter degeneracy that could help identify high risk of AD [ 72 ]. If X does not tend to a steady state then the parameter K X will be taken to be the estimated average of X over a period of 10 years, the average survival time of AD patients [ 73 ]. Accordingly, we have the following relation [ 75 ]:. Hence, from the steady state of Eq. From the steady state of Eq. The human brain has billion neurons, and its weight is approximately g, so its volume is approximately ml. Peripheral macrophages arrive later, and their immune response may perhaps exceed that of microglia, but this is currently not known [ 12 , 84 ].

Motivated by the inflammatory immune attack in AD [ 85 ], we assume that, in steady state, the proinflammatory macrophages exceed the anti-inflammatory macrophages, and that proinflammatory peripheral macrophages exceed the proinflammatory microglias. Accordingly we take. Half-life of tau proteins is 60 hours [ 88 ]. Assuming that at steady state of Eq. In [ 89 ] the concentration of tau protein was taken uniformly in the tissue of patients.

We take the half-life of astrocytes to be the same as the half-life of ganglionic glial cells, that is, days [ 91 ]. By the steady state of Eq. Actually, in a mouse model of AD, the number of activated astrocytes is increasing [ 58 ]. In mice experiments [ 92 ], macrophages phagocytosed apoptotic cells at rates that varied in the range 0. Concentration of HMGB-1 in neurons is 1. We assume that N d stabilizes somewhere below 2. Then, our assumption under Eq. In the case of infection in the lung by M.

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Peripheral macrophages immigrate into the brain of AD [ 96 , 97 ]. We assume that, because of the BBB, the concentration of monocytes in the brain capillaries must be significantly higher than the concentration of peripheral macrophages already in the tissue. When microglia cells are activated, they become either of M 1 or M 2 phenotype. Using the steady state equation. Since A is increasing in time, also P is increasing in time. Hence the steady state assumption needs to be revised. Role of genes and environments for explaining Alzheimer disease.

Arch Gen Psychiat. J Alzheimers Dis. Accessed 1 Sept Seeman P, Seeman N. Front Mol Neurosci. Bloom GS. Amyloid-beta and tau: the trigger and bullet in Alzheimer disease pathogenesis. JAMA Neurol. Int J Alzheimers Dis. Wray S, Noble W.