# Probabilistic Study of the Effect of Anti-Epileptic Drugs Under Uncertainty: Cost-Effectiveness Analysis

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## Abstract

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## 1. Motivation and Preliminaries

**Theorem**

**1**

## 2. Solving the Randomized Model

#### 2.1. Computing the 1-PDF of ${X}_{n}$

#### 2.1.1. 1-PDF of ${X}_{n}^{1}$

#### 2.1.2. 1-PDF of ${X}_{n}^{2}$

#### 2.1.3. 1-PDF of ${X}_{n}^{3}$

#### 2.1.4. 1-PDF of ${X}_{n}^{4}$

#### 2.2. Computing the Pdf of the Time

#### 2.2.1. PDF of ${N}_{1}$

#### 2.2.2. PDF of ${N}_{4}$

#### 2.3. Computing the Approximated Pdf of the Incremental Cost-Effectiveness Ratio (Icer)

- ${c}_{D}$ is the daily AEDs cost.
- ${c}_{A}$ is the average cost of concomitant Monotherapy.
- ${c}_{T}$ is the Titration total cost.

## 3. Numerical Example

#### 3.1. 1-Pdf of the Solution Sp

#### 3.2. Pdf of the Time

- Step 1: To sample 500,000 values of the form $({p}_{12},{p}_{13},{p}_{14})$, according with the distributions assumed for RVs ${P}_{12}$, ${P}_{13}$ and ${P}_{14}$.
- Step 2: For each sample in Step 1 we solve the non-linear equation$${\rho}_{3}=\frac{{p}_{13}({(1-{p}_{14})}^{n}-{(1-{p}_{12}-{p}_{13}-{p}_{14})}^{n})}{{p}_{12}+{p}_{13}},$$
- Step 3: To plot the histogram with the 500,000 values of n. An approximation of the PDF of ${N}_{3}$ is obtained by normalizing this histogram.

#### 3.3. 1-PDF of ICER

- Probabilities of response, partial response and discontinuation [21]. Brivaracetam: ${p}_{12}=0.02520$, ${p}_{13}=0.13765$, ${p}_{14}=0.0572$. Eslicarbazepine: ${p}_{12}=0.0112$, ${p}_{13}=0.1160$, ${p}_{14}=0.0452$. Retigabine: ${p}_{12}=0.0122$, ${p}_{13}=0.1577$, ${p}_{14}=0.0763$. These quantities will be used to calculate the different expectation in the cost and the effectiveness formulas, $\mathbb{E}\left[{X}_{j}^{i}\right]$, where ${X}_{j}^{i}$ is the solution SP in cycle j of the state i given in Formula (6).
- Probabilities of adverse events, Ataxia, Dizziness, Fatigue, Nausea and Somnolence [23] As previously, these probabilities will be considered independent absolutely continuous RVs. We choose for each RV a Beta distribution with parameters a and b calculated as before considering the following mean $\mu $. Brivaracetam: ${\mu}_{A}=0.00747$, ${\mu}_{D}=0.03198$, ${\mu}_{F}=0.02717$, ${\mu}_{N}=0.01242$, ${\mu}_{S}=0.03439$. Eslicarbazepine: ${\mu}_{A}=0.01489$, ${\mu}_{D}=0.05588$, ${\mu}_{F}=0.01489$, ${\mu}_{N}=0.02955$, ${\mu}_{S}=0.02955$. Retigabine: ${\mu}_{A}=0.01242$, ${\mu}_{D}=0.06059$, ${\mu}_{F}=0.03681$, ${\mu}_{N}=0.01735$, ${\mu}_{S}=0.03681$.
- Titration cost (Total cost) [26]. Brivaracetam: ${c}_{T}=0\u20ac$. Eslicarbazepine: ${c}_{T}=31.36\u20ac$. Retigabine: ${C}_{T}=106.56\u20ac$.
- Utility of each state in all treatments: ${u}_{1}=0.683$, ${u}_{2}=0.869$, ${u}_{3}=0.805$, ${u}_{4}={u}_{1}$.
- Utilities of adverse events [23]: ${u}_{A}=-0.28$, ${u}_{D}=-0.22$, ${u}_{F}=-0.25$, ${u}_{N}=-0.23$ and ${u}_{S}=-0.22$.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Plot of the 1-PDFs in each of the four states described in Figure 1 with the Brivaracetam treatment. (

**Up**): Non-responders (

**left**) and Seizure free population (

**right**). (

**Bottom**): Partial responders (

**left**) and population with discontinuation symptoms (

**right**).

**Figure 3.**Plot of the expectation (Blue line) plus/minus 1.96 standard deviation (Black dashed line) of each of the four states described in Figure 1 with the Brivaracetam treatment, during two years. (

**Up**): Non-responders (

**left**) and Seizure free population (

**right**). (

**Bottom**): Partial responders (

**left**) and population with discontinuation symptoms (

**right**).

**Figure 4.**Plot of the expectation (Blue line) plus/minus 1.96 standard deviation (Black dashed line) of each of the four states described in Figure 1 with the Brivaracetam treatment for greater times. (

**Up**): Non-responders (

**left**) and Seizure free population (

**right**). (

**Bottom**): Partial responders (

**left**) and population with discontinuation symptoms (

**right**).

**Figure 5.**Plot of the PDFs of the times, ${N}_{1}$ and ${N}_{4}$, until given proportions, ${\rho}_{1}\in \{0.1,0.2,\dots ,0.9\}$ and ${\rho}_{4}\in \{0.05,0.1,\dots ,0.35\}$, remain in the corresponding state, respectively. (

**Left**): Non-responders. (

**Right**): Discontinuation.

**Figure 6.**Plot of the PDFs of the times, ${N}_{2}$ and ${N}_{3}$, until given proportions, ${\rho}_{2}\in \{0.01,0.02,\dots ,0.09\}$ and ${\rho}_{3}\in \{0.05,0.1,\dots ,0.45\}$, remain in the corresponding state, respectively. (

**Left**): Seizure Free. (

**Right**): 50% Responder.

**Figure 7.**Plot of expectation plus/minus 1.96 standard deviation functions of ICER for $n\in [1,8]$. (

**Left**): Brivaracetam VS Eslicarbazepine. (

**Right**): Brivaracetam VS Retigabine. The red straight line represents the threshold value 20,000€/QALY.

**Figure 8.**Comparison between Brivaracetam and Eslicarbazepine AEDs for $n\in [1,8]$. (

**Left**): Incremental cost. (

**Right**): Expectation incremental QALY. In both increments Eslicabazepine is considered the first treatment.

**Figure 9.**Comparison between Brivaracetam and Regitabine AEDs for $n\in [1,8]$. (

**Left**): Incremental cost. (

**Right**): Expectation incremental QALY. In both increments Regitabine is considered the first treatment.

**Table 1.**Expectation of time until a given proportion of population, ${\rho}_{1}$, remains non-responder, ${N}_{1}$.

${\mathit{\rho}}_{1}$ | $0.1$ | $0.2$ | $0.3$ | $0.4$ | $0.5$ | $0.6$ | $0.7$ | $0.8$ | $0.9$ |
---|---|---|---|---|---|---|---|---|---|

$\mathbb{E}\left[{N}_{1}\right]$ | $9.33167$ | $6.52255$ | $4.87933$ | $3.71344$ | $2.80911$ | $2.07022$ | $1.44549$ | $0.904332$ | $0.426994$ |

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## Share and Cite

**MDPI and ACS Style**

Barrachina-Martínez, I.; Navarro-Quiles, A.; Ramos, M.; Romero, J.-V.; Roselló, M.-D.; Vivas-Consuelo, D.
Probabilistic Study of the Effect of Anti-Epileptic Drugs Under Uncertainty: Cost-Effectiveness Analysis. *Mathematics* **2020**, *8*, 1120.
https://doi.org/10.3390/math8071120

**AMA Style**

Barrachina-Martínez I, Navarro-Quiles A, Ramos M, Romero J-V, Roselló M-D, Vivas-Consuelo D.
Probabilistic Study of the Effect of Anti-Epileptic Drugs Under Uncertainty: Cost-Effectiveness Analysis. *Mathematics*. 2020; 8(7):1120.
https://doi.org/10.3390/math8071120

**Chicago/Turabian Style**

Barrachina-Martínez, Isabel, Ana Navarro-Quiles, Marta Ramos, José-Vicente Romero, María-Dolores Roselló, and David Vivas-Consuelo.
2020. "Probabilistic Study of the Effect of Anti-Epileptic Drugs Under Uncertainty: Cost-Effectiveness Analysis" *Mathematics* 8, no. 7: 1120.
https://doi.org/10.3390/math8071120