## Abstract

Medication adherence is a well-known problem for pharmaceutical treatment of chronic diseases. Understanding how nonadherence affects treatment efficacy is made difficult by the ethics of clinical trials that force patients to skip doses of the medication being tested, the unpredictable timing of missed doses by actual patients, and the many competing variables that can either mitigate or magnify the deleterious effects of nonadherence, such as pharmacokinetic absorption and elimination rates, dosing intervals, dose sizes, and adherence rates. In this paper, we formulate and analyze a mathematical model of the drug concentration in an imperfectly adherent patient. Our model takes the form of the standard single compartment pharmacokinetic model with first-order absorption and elimination, except that the patient takes medication only at a given proportion of the prescribed dosing times. Doses are missed randomly, and we use stochastic analysis to study the resulting random drug level in the body. We then use our mathematical results to propose principles for designing drug regimens that are robust to nonadherence. In particular, we quantify the resilience of extended release drugs to nonadherence, which is quite significant in some circumstances, and we show the benefit of taking a double dose following a missed dose if the drug absorption or elimination rate is slow compared to the dosing interval. We further use our results to compare some antiepileptic and antipsychotic drug regimens.

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## Data Availability Statement

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

## References

Ackloo E, Haynes RB, McDonald HP, Sahota N, Yao X (2008) Interventions for enhancing medication adherence (Review). Cochrane Lib. https://doi.org/10.1002/14651858.CD000011.pub4

Albassam A, Hughes DA (2020) What should patients do if they miss a dose? A systematic review of patient information leaflets and summaries of product characteristics. Eur J Clin Pharmacol 1–10

Bialer Meir (2007) Extended-release formulations for the treatment of epilepsy. CNS Drugs 21(9):765–774

Boissel JP, Nony P (2002) Using pharmacokinetic-pharmacodynamic relationships to predict the effect of poor compliance. Clin Pharmacokinet 41(1):1–6

Breckenridge A, Aronson JK, Blaschke TF, Hartman D, Peck CC, Vrijens B (2017) Poor medication adherence in clinical trials: consequences and solutions. Nat Rev Drug Discov 16(3):149–150

Brittain ST, Wheless James W (2015) Pharmacokinetic simulations of topiramate plasma concentrations following dosing irregularities with extended-release vs. immediate-release formulations. Epilepsy Behav 52:31–36

Brown MT, Sinsky CA (2013) Medication adherence: we didn’t ask and they didn’t tell. Fam Pract Manag 20(2):25–30

Burnier M (2019) Is there a threshold for medication adherence? lessons learnt from electronic monitoring of drug adherence. Front Pharmacol 9:1540

Chen C, Wright J, Gidal B, Messenheimer J (2013) Assessing impact of real-world dosing irregularities with lamotrigine extended-release and immediate-release formulations by pharmacokinetic simulation. Ther Drug Monit 35(2):188–193

Counterman ED, Lawley SD (2021) What should patients do if they miss a dose of medication? A theoretical approach. J Pharmacokinet Pharmacodyn 48:873–892. https://doi.org/10.1007/s10928-021-09777-6

Crauel H (2001) Random point attractors versus random set attractors. J Lond Math Soc 63(2):413–427

Ding JJ, Zhang YJ, Jiao Z, Wang Y (2012) The effect of poor compliance on the pharmacokinetics of carbamazepine and its epoxide metabolite using monte carlo simulation. Acta Pharmacol Sinica 33(11):1431–1440

Durrett R (2019) Probability: theory and examples. Cambridge University Press, Cambridge

Dutta S, Reed RC (2006) Effect of delayed and/or missed enteric-coated divalproex doses on valproic acid concentrations: simulation and dose replacement recommendations for the clinician 1. J Clin Pharm Ther 31(4):321–329

Elkomy MH (2020) Changing the drug delivery system: does it add to non-compliance ramifications control? A simulation study on the pharmacokinetics and pharmacodynamics of atypical antipsychotic drug. Pharmaceutics 12(4):297

Erdős P (1939) On a family of symmetric Bernoulli convolutions. Am J Math 61(4):974–976

Escribano C, Sastre MA, Torrano E (2003) Moments of infinite convolutions of symmetric bernoulli distributions. J Comput Appl Math 153(1–2):191–199

Fermín JL, Lévy-Véhel J (2017) Variability and singularity arising from poor compliance in a pharmacokinetic model ii: the multi-oral case. J Math Biol 74(4):809–841

Garnett WR, McLean AM, Zhang Y, Clausen S, Tulloch SJ (2003) Simulation of the effect of patient nonadherence on plasma concentrations of carbamazepine from twice-daily extended-release capsules. Curr Med Res Opin 19(6):519–525

Gibaldi M, Perrier D (1982). Pharmacokinetics Marcelly Dekker, 2 edition

Gidal BE, Majid O, Ferry J, Hussein Z, Yang H, Zhu J, Fain R, Laurenza A (2014) The practical impact of altered dosing on perampanel plasma concentrations: pharmacokinetic modeling from clinical studies. Epilepsy Behav 35:6–12

Gidal BE, Ferry J, Reyderman L, Piña-Garza JE (2021) Use of extended-release and immediate-release anti-seizure medications with a long half-life to improve adherence in epilepsy: a guide for clinicians. Epilepsy Behav 120:107993

Gilbert A, Roughead L, Sansom L et al (2002) I’ve missed a dose; what should I do? Aust Prescr 25(1):16–17

Gu JQ, Guo YP, Jiao Z, Ding JJ, Li GF (2020) How to handle delayed or missed doses: a population pharmacokinetic perspective. Eur J Drug Metabol Pharmacokinet 45(2):163–172

Hard ML, Wehr AY, Sadler BM, Mills RJ, von Moltke L (2018) Population pharmacokinetic analysis and model-based simulations of aripiprazole for a 1-day initiation regimen for the long-acting antipsychotic aripiprazole lauroxil. Eur J Drug Metab Pharmacokinet 43(4):461–469

Haynes RB (1976) A critical review of “determinants” of patient compliance with therapeutic regimens. Compliance Therapeut Regimens, pp 26–39

Haynes RB, McDonald HP, Garg A, Montague P (2002) Interventions for helping patients to follow prescriptions for medications. Cochrane Database Syst Rev. https://doi.org/10.1002/14651858.CD000011

Howard J, Wildman K, Blain J, Wills S, Brown D (1999) The importance of drug information from a patient perspective. J Soc Adm Pharm 16(3/4):115–126

Jessen B, Wintner A (1935) Distribution functions and the riemann zeta function. Trans Am Math Soc 38(1):48–88

Jonklaas J, Bianco AC, Bauer AJ, Burman KD, Cappola AR, Celi FS, Cooper DS, Kim BW, Peeters RP, Rosenthal MS, Sawka AM et al (2014) Guidelines for the treatment of hypothyroidism: prepared by the american thyroid association task force on thyroid hormone replacement. Thyroid 24(12):1670–1751

Kershner R, Wintner A (1935) On symmetric bernoulli convolutions. Am J Math 57(3):541–548

Kim J, Combs K, Downs J, Tillman F (2018) Medication adherence: the elephant in the room. US Pharm 43(1):30–34

Larry A (2015) Bauer. Clinical pharmacokinetic equations and calculations. McGraw-Hill Medical, New York, NY

Lawley SD, Keener JP (2019) Electrodiffusive flux through a stochastically gated ion channel. SIAM J Appl Math 79(2):551–571

Lawley SD, Mattingly JC, Reed MC (2015) Stochastic switching in infinite dimensions with applications to random parabolic PDE. SIAM J Math Anal 47(4):3035–3063

Levy Gerhard (1993) A pharmacokinetic perspective on medicament noncompliance. Clin Pharmacol Ther 54(3):242–244

Li J, Nekka F (2007) A pharmacokinetic formalism explicitly integrating the patient drug compliance. J Pharmacokinet Pharmacodyn 34(1):115–139

Lowy A, Munk VC, Ong SH, Burnier M, Vrijens B, Tousset EP, Urquhart J (2011) Effects on blood pressure and cardiovascular risk of variations in patients? adherence to prescribed antihypertensive drugs: role of duration of drug action. Int J Clin Pract 65(1):41–53

Ma J (2017) Stochastic modeling of random drug taking processes and the use of singular perturbation methods in pharmacokinetics. The University of Utah, PhD thesis

Mattingly JC (1999) Ergodicity of \(2\)D Navier-Stokes equations with random forcing and large viscosity. Comm Math Phys 206(2):273–288

Millum J, Grady C (2013) The ethics of placebo-controlled trials: methodological justifications. Contemp Clin Trials 36(2):510–514

Norris JR (1998) Markov Chains. Cambridge University Press, Statistical & Probabilistic Mathematics

Osterberg L, Blaschke T (2005) Adherence to medication. New Engl J Med 353(5):487–497

Panayiotopoulos CP (2010) Atlas of epilepsies. Springer Science & Business Media, Berlin

Pellock JM, Brittain ST (2016) Use of computer simulations to test the concept of dose forgiveness in the era of extended-release (XR) drugs. Epilepsy Behav 55:21–23

Pellock JM, Smith MC, Cloyd JC, Uthman B, Wilder BJ (2004) Extended-release formulations: simplifying strategies in the management of antiepileptic drug therapy. Epilepsy Behav 5(3):301–307

Peres Y, Solomyak B (1998) Self-similar measures and intersections of Cantor sets. Trans Am Math Soc 350(10):4065–4087

Peres Y, Schlag W, Solomyak B (2000) Sixty years of Bernoulli convolutions. Fractal geometry and stochastics II. Springer, pp 39–65

Perucca Emilio (2009) Extended-release formulations of antiepileptic drugs: rationale and comparative value. Epilepsy Curr 9(6):153–157

Qiu Y, Zhou D (2011) Understanding design and development of modified release solid oral dosage forms. J Valid Technol 17(2):23

Reed RC, Dutta S (2004) Predicted serum valproic acid concentrations in patients missing and replacing a dose of extended-release divalproex sodium. Am J Heal Syst Pharm 61(21):2284–2289

Sabaté E, Sabaté E et al (2003) Adherence to long-term therapies: evidence for action. World Health Organization

Schmalfuß B (1996) A random fixed point theorem based on Lyapunov exponents. Random Comput Dynam 4(4):257–268

Solomyak B (1995) On the random series \(\sum \pm \lambda ^{n}\) (an Erdos problem). Ann Math 142:611–625

Spencer TJ, Mick E, Surman CB, Hammerness P, Doyle R, Aleardi M, Kotarski M, Williams CG, Biederman J (2011) A randomized, single-blind, substitution study of OROS methylphenidate (Concerta) in ADHD adults receiving immediate release methylphenidate. J Atten Disord 15(4):286–294

Stauffer ME, Hutson P, Kaufman AS, Morrison A (2017) The adherence rate threshold is drug specific. Drugs R&D 17(4):645–653

Sumarsono A, Sumarsono N, Das SR, Vaduganathan M, Agrawal D, Pandey A (2020) Economic burden associated with extended-release vs immediate-release drug formulations among medicare part d and medicaid beneficiaries. JAMA Netw Open 3(2):e200181–e200181

Sunkaraneni S, Blum D, Ludwig E, Chudasama V, Fiedler-Kelly J, Marvanova M, Bainbridge J, Phillips L (2018) Population pharmacokinetic evaluation and missed-dose simulations for eslicarbazepine acetate monotherapy in patients with partial-onset seizures. Clin Pharmacol Drug Dev 7(3):287–297

Tian-You H, Lau KS (2008) Spectral property of the Bernoulli convolutions. Adv Math 219(2):554–567

Udelson JE, Pressler SJ, Sackner-Bernstein J, Massaro J, Ordronneau P, Lukas MA, Hauptman PJ, Investigators T (2009) Adherence with once daily versus twice daily carvedilol in patients with heart failure: the compliance and quality of life study comparing once-daily controlled-release carvedilol CR and twice-daily immediate-release carvedilol IR in patients with heart failure (CASPER) trial. J Card Fail 15(5):385–393

Vadivelu N, Timchenko A, Huang Y, Sinatra R (2011) Tapentadol extended-release for treatment of chronic pain: a review. J Pain Res 4:211

Véhel PEL, Véhel JL (2013) Variability and singularity arising from poor compliance in a pharmacokinetic model i: the multi-iv case. J Pharmacokinet Pharmacodyn 40(1):15–39

Wheless JW, Phelps SJ (2018) A clinician’s guide to oral extended-release drug delivery systems in epilepsy. J Pediatr Pharmacol Ther 23(4):277–292

Wolfram Research (2019) Mathematica 12

## Acknowledgements

SDL was supported by the National Science Foundation (Grant Nos. DMS-1944574 and DMS-1814832). The authors gratefully acknowledge four anonymous reviewers whose comments significantly improved this paper.

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## Appendix

### Appendix

In this appendix, we prove the theorems of Sect. 3.

### Proof of Theorem 1

For any dosing protocol *f* and any integers \(N\ge M\), define

For \(t\in [0,\tau ]\), define the random variable

Further, for any \(N\in \mathbb {Z}\), define the almost sure limits,

and notice that \(A_{-\infty ,0}=A\), \(B_{-\infty ,0}=B\), and \(C_{-\infty ,0}(t)=C(t)\) where *A* and *B* are defined in (24) and *C*(*t*) is defined in (26). To see why the almost sure convergence in (57) is guaranteed, note first that the function *f* must be bounded since its domain \(\{0,1\}^{m+1}\) is finite. Hence, the terms in the sums in (55) are bounded by the terms in a geometric series, and so the Weierstrass M-test ensures the almost sure convergence in (57).

Notice that the drug concentration in (10) with \(f_{n}=f(X_{n})\) can be written as

Therefore,

Since \(\{X_{n}\}_{n\in \mathbb {Z}}\) is stationary, it follows that

where \(=_{d }\) denotes equality in distribution. Now, as in (57), we have that

Equations (58), (59), and (60) yield (26). We note that random variables akin to (60) are sometimes called random pullback attractors because they take an initial condition and pull it back to the infinite past (Crauel 2001; Mattingly 1999; Schmalfuß 1996; Lawley et al. 2015; Lawley and Keener 2019).

Since *f* is bounded, \(C_{0,N}(t)\) can be bounded by a deterministic constant independent of *N*, and thus (59), (60), and the bounded convergence theorem yield

Combining (61) with (58) and (59) yields the second equality in (27). Combining the second equality in (27) with the bounded convergence theorem yields the second equality in (28).

Finally, the same argument that gave the second equality in (27) gives the second equality in (29) upon noticing that (58), (59), and (60) all still hold when integrated from \(t=0\) to \(t=\tau \).

To obtain the first equalities in (27)-(29) for the time averages, we first note that Theorem 7.1.3 in Durrett (2019) ensures that \(\{C_{-\infty ,n}(t)\}_{n\in \mathbb {Z}}\) is ergodic for any fixed \(t\ge 0\) since \(\{X_{n}\}_{n\in \mathbb {Z}}\) is ergodic and stationary. We thus have that

by Birkhoff’s ergodic theorem (see, for example, Theorem 7.2.1 in Durrett (2019)). By (58), we have that for any \(t\in [0,\tau ]\),

Hence, in order to prove the first equality in (27), it remains to prove

To prove this, we first note the bound,

where \(f_{+}:=\sup _{x\in \{0,1\}^{m+1}}f(x)<\infty \) since \(\{0,1\}^{m+1}\) is finite.

If \(j\ge 1\) is an integer, then the binomial theorem implies the general identity for \(a,b\in \mathbb {R}\),

Therefore,

for a suitably chosen deterministic constant \(K_{0}\) independent of *n*. Since *f* is nonnegative, it follows that

which then yields (62) for the case that \(j\ge 1\) is an integer.

If \(j>0\) is not an integer, then notice that the function \(g(a)=a^{j-\lfloor j\rfloor }\) for \(a>0\) is concave and therefore subadditive, and thus

Therefore, (63) and (65) imply

for suitably chosen deterministic constants \(K_{1}>0\) and \(\gamma _{1}\in (0,1)\) which are independent of *n*. Hence, (64) holds with \(K_{0}\) replaced by \(K_{1}\) and \(\gamma \) replaced by \(\gamma _{1}\) and (62) follows.

Having proven the first equality in (27), the first equality in (28) then follows from the bounded convergence theorem upon noting that

Finally, the first equality in (29) follows from applying the same argument that gave the first equality in (27) to the integrals \(\int _{0}^{\tau }C_{-\infty ,n}(t)\,d t\) and \(\int _{0}^{\tau }C_{0,n}(t)\,d t\). \(\square \)

### Proof of Theorem 2

Equations (32) and (35) follow immediately from the definition of *C*(*t*) in (60). It thus remains to prove (33) and (36), which generalizes the proof of Theorem 1 in [20]. Recalling the definitions in (55) and (57), notice that

where \(=_{d }\) denotes equality in distribution. Taking the expectation of (66) and rearranging yields (33).

To obtain \(\mathbb {E}[A^{2}]\), we first square (66), take expectation, and rearrange to obtain

By definition of expectation, we have that \(\mathbb {E}\big [(f(X_{1}))^{2}\big ] =\sum _{x}(f(x))^{2}\pi (x)\). To compute \(\mathbb {E}[{A}f(X_{1})]\), let \(1_{E}\in \{0,1\}\) denote the indicator function on an event *E*, which means \(1_{E}=1\) if *E* occurs and \(1_{E}=0\) otherwise. Thus,

Multiplying (66) by \(1_{X_{1}=x}\), taking expectation, and using that \(({A},X_{0})\) is equal in distribution to \((A_{-\infty ,1},X_{1})\) yields

The conditional expectation tower property (Theorem 5.1.6 in Durrett (2019)) implies

where *P* is in (18). Combining (69) and (70) yields the following system of linear algebraic equations for \(\mathbb {E}[{A}1_{X_{0}=x}]\),

If we define the vectors \({{u}}_{\alpha }\in \mathbb {R}^{2^{m+1}}\) and \(v\in \mathbb {R}^{2^{m+1}}\) by

then (34) solves (71). We note that the Perron–Frobenius theorem guarantees the invertibility of \(I-\alpha P^{\top }\) since \(I-\alpha P^{\top }=\alpha (\alpha ^{-1}I-P^{\top })\) and \(\alpha \in (0,1)\). Therefore, (69) implies

Combining (72) with (67) yields the formula for \(\mathbb {E}[A^{2}]\) given by (36) upon replacing \(\beta \) by \(\alpha \). The analogous argument yields \(\mathbb {E}[B^{2}]\) (given by (36) upon replacing \(\alpha \) by \(\beta \)). To obtain \(\mathbb {E}[AB]\), we first observe that

Taking the expectation of (73) and rearranging yields

Using (72) and the analogous equation for \(\mathbb {E}\big [{B}f(X_{1})\big ]\) yields the formula for \(\mathbb {E}[AB]\) in (36) and completes the proof. \(\square \)

### Proof of Theorem 5

Since missing doses can only decrease the concentration for the single dose protocol, we have the following pair of inequalities,

But, setting \(\xi _{n}=1\) for all *n* yields \(c^{single }(N\tau +t)=c^{perf }(N\tau +t)\) and \(C^{single }(t)=C^{perf }(t)\), and thus the inequalities in (74) and (75) can be replaced by equalities. Hence, we have obtained the first and third equalities in (40).

The second equality in (40) and the first equality in (43) follow from the convergence in distribution in (26) in Theorem 1. To see this, note first that for any dosing protocol, we have by (58) and (59) that

where \(C_{M,N}(t)\) is defined in (56) and \(C(t):=C_{-\infty ,0}(t)\) is defined in (57). The inequality in (76) holds because *f* is nonnegative. The \(=_{d }\) in (76) denotes equality in distribution, where the probability measure \(\mathbb {P}\) on the set of sequences \(\xi =\{\xi _{n}\}_{n\in \mathbb {Z}}\) can be any measure as described in Sect. 3.1. In particular, we can take \(\mathbb {P}\) to be the probability measure for the case that \(\{\xi _{n}\}_{n\in \mathbb {Z}}\) are iid with \(\mathbb {P}(\xi _{n}=1)=p\in (0,1)\) as in Sect. 3.3. The important point is that for this choice of \(\mathbb {P}\), a supremum over sequences \(\xi \) is the same as an essential supremum over sequences \(\xi \) (since for any finite sequence \(\{\zeta _{i}\}_{i=1}^{M}\in \{0,1\}^{M}\) and any \(n\in \mathbb {Z}\), we have \(\mathbb {P}(\xi _{n}=\zeta _{1},\dots ,\xi _{n+M}=\zeta _{M})>0\)). Therefore, (76) implies that

To obtain equality in (77), fix \(t\in [0,\tau ]\) and let \(\delta >0\). By definition of supremum,

where we again take \(\mathbb {P}\) to be as in Sect. 3.3 so that a supremum over \(\xi \) and an essential supremum over \(\xi \) are equivalent. Hence, the convergence in distribution in (26) in Theorem 1 implies that we can take *N* sufficiently large so that

Since \(\delta >0\) is arbitrary, we thus have that

Since \(t\in [0,\tau ]\) is arbitrary, combining (78) and (77) implies that the inequality in (77) can be replaced by equality. Since this holds for any dosing protocol, we have obtained the second equality in (40) and the first equality in (43).

The final equality in (40) and the maximizing time \(t^{*}\) in (41) follow from a simple calculus exercise. The formula in (42) follows from combining (40)-(41) with the definition of \(\theta \) in (16).

We now prove the inequality in (43). Fix \(t\in [0,\tau ]\) and recall that \(C^{double }(t)\) is

where \(K_{n}(t)\) is the coefficient,

and

By definition of the double dose protocol, we have that for all \(n\ge 0\),

We claim that there is a finite nonnegative integer \(n^{*}\ge 0\) such that

That is, the sequence \(\{K_{n}(t)\}_{n\ge 0}\) is strictly increasing in *n* for \(n<n^{*}\) and nonincreasing in *n* for \(n>n^{*}\). To prove this claim, we momentarily treat *n* as a continuous variable and differentiate \(K_{n}(t)\) with respect to *n*,

Rearranging (83) shows that \(\frac{\partial }{\partial n}K_{n}(t)=0\) if and only if

Note further that the second derivative is negative at \(n_{0}\),

Hence, \(K_{n}(t)\le K_{n_{0}}(t)\) for all \(n\in \mathbb {R}\). Therefore, if \(n_{0}\le 0\), then the claim is satisfied with \(n^{*}=0\). If \(n_{0}>0\), then the claim is satisfied by either \(n^{*}=\lfloor n_{0}\rfloor \ge 0\) or \(n^{*}=\lceil n_{0}\rceil \ge 1\), where \(\lfloor \cdot \rfloor \) and \(\lceil \cdot \rceil \) denote the floor and ceiling functions, respectively. To distinguish between the case \(n^{*}=\lfloor n_{0}\rfloor \ge 0\) or \(n^{*}=\lceil n_{0}\rceil \ge 1\) for \(n_{0}>0\), one merely checks if \(K_{\lfloor n_{0}\rfloor }(t)<K_{\lceil n_{0}\rceil }(t)\) or \(K_{\lfloor n_{0}\rfloor }(t)>K_{\lceil n_{0}\rceil }(t)\). We note that if \(K_{\lfloor n_{0}\rfloor }(t)=K_{\lceil n_{0}\rceil }(t)\), then we can simply take \(n^{*}=\lfloor n_{0}\rfloor \). Therefore, we have verified the claim.

We now claim that if \(C^{double }(t)\) is to be maximized, then we must have that

The claim in (84) is vacuously true if \(n^{*}=0\), so suppose \(n^{*}\ge 1\). To prove the claim in (84), we start with \(n=0\). It is immediate that the maximizing value must either be \(g_{0}=1\) or \(g_{0}=2\), since setting \(g_{0}=0\) only makes the first term in (79) smaller compared to if \(g_{0}=1\) or \(g_{0}=2\), and it does not allow any other term to be larger than if \(g_{0}=1\). If \(g_{0}=2\), then we must set \(g_{1}=0\) by (81). However, we claim that \(C^{double }(t)\) is certainly larger if \(g_{0}=g_{1}=1\) compared to if \(g_{0}=2\) and \(g_{1}=0\). To see this, note first that the values of \(g_{i}\) for \(i\ge 2\) are unconstrained by either choice. Further, since \(n^{*}\ge 1\), (82) implies that \(K_{0}(t)<K_{1}(t)\) and thus

Therefore, \(C^{double }(t)\) is larger if \(g_{0}=g_{1}=1\) rather than \(g_{0}=2\) and \(g_{1}=0\). At this point in the argument, it is still not determined if \(C^{double }(t)\) is larger by taking \(g_{0}=g_{1}=1\) or \(g_{0}=1\) and \(g_{1}=2\). Nevertheless, we conclude that \(C^{double }(t)\) is maximized by setting \(g_{0}=1\) in this case that \(n^{*}\ge 1\). Repeating this argument shows that we must take \(g_{n}=1\) for all \(n<n^{*}\) in order to maximize \(C^{double }(t)\), and thus, we have verified the claim in (84).

We further claim that if \(C^{double }(t)\) is to be maximized, then

To see why (85) holds, observe that if \(g_{n}=0\) and \(g_{n-1}\ne 2\) for some \(n\ge 1\), then one could change the value of \(g_{n}\) to be \(g_{n}=1\) without changing the value of \(g_{i}\) for any \(i\ne n\), and this would make the value of \(C^{double }(t)\) larger. Hence, (85) holds for any sequence \(\{g_{n}\}_{n\ge 0}\) which maximizes \(C^{double }(t)\).

Now, let \(\{g_{n}\}_{n\ge 0}\) be any sequence as in (80)-(81) that satisfies (84) and (85). We claim that the corresponding value of \(C^{double }(t)\) in (79) satisfies

where \(n'\ge n^{*}\) is the smallest integer such that \(g_{n}=2\),

where we set \(K_{n'}(t)=0\) if \(n'=\infty \) in the case that \(g_{n}=1\) for all \(n\ge 0\) (note that (86) is trivially satisfied in this case). In words, the claim in (86) means that the concentration for the double dose protocol is always less than the concentration for perfect adherence plus the concentration from a single dose taken \(n'\) dosing times in the past. Note that (85) and (87) imply that

Using the definition of \(C^{double }(t)\) in (79), (88), and the definition of \(n'\) in (87), the claim in (86) is equivalent to

Define the sequence of \(\{S_{n}\}_{n\ge n'+1}\) by

where \(S_{n'+1}=0\). Note that since \(g_{n}\in \{0,1,2\}\), we are assured that

Further, (81) implies that

In addition, since (87) implies that \(g_{n'}=2\), (81) implies

In words, (90) means that successive terms in the sequence \(\{S_{n}\}_{n\ge n'+1}\) can change by at most \(\pm 1\), and (91) means that if successive terms decrease by 1, then the next term increases by 1. Since \(S_{n'+1}=0\) and \(S_{n'+2}=1\) by (92), we conclude that

Using summation by parts, we have that

by (93) and the fact that \(n'\ge n^{*}\) and \(K_{n}(t)\) is nonincreasing in *n* for \(n\ge n^{*}\) as in (82). In (94), we also used that \(\lim _{n\rightarrow \infty }K_{n}(t)=0\) and \(S_{n'+1}=0\). Hence, we have verified (89) and thus (86).

Therefore, (86) implies that

We obtained \(\sup _{t\in [0,\tau ]}C^{perf }(t)\) in (40)-(41), and a straightforward calculus exercise yields

where \(s^{*}\) is given in (44). The bound in (45) follows from combining (43) and (44) with the definition of \(\theta \) in (16). \(\square \)

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Counterman, E.D., Lawley, S.D. Designing Drug Regimens that Mitigate Nonadherence.
*Bull Math Biol* **84**, 20 (2022). https://doi.org/10.1007/s11538-021-00976-3

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DOI: https://doi.org/10.1007/s11538-021-00976-3