Deterministic, stochastic and fractional mathematical approaches applied to AMR

Sebastian Builes; Jhoana P. Romero-Leiton; Leon A. Valencia; Sebastian Builes; Jhoana P. Romero-Leiton; Leon A. Valencia

doi:10.3934/mbe.2025015

Mathematical Biosciences and Engineering

2025, Volume 22, Issue 2: 389-414. doi: 10.3934/mbe.2025015

Previous Article Next Article

Research article

Deterministic, stochastic and fractional mathematical approaches applied to AMR

1.
Institute of Mathematics, University of Antioquia, Medellin, Colombia
2.
Department of Mathematical Sciences, University of Puerto Rico at Mayagüez, Puerto Rico, USA

Received: 02 December 2024 Revised: 24 January 2025 Accepted: 05 February 2025 Published: 10 February 2025

In this work, we study the qualitative properties of a simple mathematical model that can be applied to the reversal of antimicrobial resistance. In particular, we analyze the model from three perspectives: ordinary differential equations (ODEs), stochastic differential equations (SDEs) driven by Brownian motion, and fractional differential equations (FDEs) with Caputo temporal derivatives. Finally, we address the case of Escherichia coli exposed to colistin using parameters from the literature in order to assess the validity of the qualitative properties of the model.
- ordinary differential equation,
- stochastic differential equation,
- fractional differential equation,
- Caputo derivative,
- Brownian motion,
- stability,
- qualitative properties
Citation: Sebastian Builes, Jhoana P. Romero-Leiton, Leon A. Valencia. Deterministic, stochastic and fractional mathematical approaches applied to AMR[J]. Mathematical Biosciences and Engineering, 2025, 22(2): 389-414. doi: 10.3934/mbe.2025015

Related Papers:

Abstract

In this work, we study the qualitative properties of a simple mathematical model that can be applied to the reversal of antimicrobial resistance. In particular, we analyze the model from three perspectives: ordinary differential equations (ODEs), stochastic differential equations (SDEs) driven by Brownian motion, and fractional differential equations (FDEs) with Caputo temporal derivatives. Finally, we address the case of Escherichia coli exposed to colistin using parameters from the literature in order to assess the validity of the qualitative properties of the model.

References

[1]	M. A. Salam, M. Y. Al-Amin, M. T. Salam, J. S. Pawar, N. Akhter, A. A. Rabaan, et al., Antimicrobial resistance: a growing serious threat for global public health, Healthcare, 11 (2023), 1946. https://doi.org/10.3390/healthcare11131946 doi: 10.3390/healthcare11131946
[2]	A. H. Holmes, L. S. P. Moore, A. Sundsfjord, M. Steinbakk, S. Regmi, A. Karkey, et al., Understanding the mechanisms and drivers of antimicrobial resistance, Lancet, 387 (2016), 176–187. https://doi.org/10.1016/S0140-6736(15)00473-0 doi: 10.1016/S0140-6736(15)00473-0
[3]	D. I. Andersson, D. Hughes, Persistence of antibiotic resistance in bacterial populations, FEMS Microbiol. Rev., 35 (2011), 901–911. https://doi.org/10.1111/j.1574-6976.2011.00356.x doi: 10.1111/j.1574-6976.2011.00356.x
[4]	W. R. Morrow, E. A. Frazier, W. T. Mahle, T. O. Harville, S. E. Pye, K. R. Knecht, et al., Rapid reduction in donor-specific anti-human leukocyte antigen antibodies and reversal of antibody-mediated rejection with bortezomib in pediatric heart transplant patients, Transplantation, 93 (2012), 319–324. https://doi.org/10.1097/TP.0b013e318240cd4b doi: 10.1097/TP.0b013e318240cd4b
[5]	C. L. Moo, S. K. Yang, K. Yusoff, M. Ajat, W. Thomas, A. Abushelaibi, et al., Mechanisms of antimicrobial resistance (AMR) and alternative approaches to overcome AMR, Curr. Drug Discovery Technol., 17 (2020), 430–447. https://doi.org/10.2174/1389200221666200204124414 doi: 10.2174/1389200221666200204124414
[6]	A. J. Lopatkin, H. R. Meredith, J. K. Srimani, C. Pfeiffer, R. Durrett, L. You, Persistence and reversal of plasmid-mediated antibiotic resistance, Nat. Commun., 8 (2017), 1689. https://doi.org/10.1038/s41467-017-01763-6 doi: 10.1038/s41467-017-01763-6
[7]	D. I. Andersson, D. Hughes, Antibiotic resistance and its cost: Is it possible to reverse resistance?, Nat. Rev. Microbiol., 8 (2010), 260–271. https://doi.org/10.1038/nrmicro2319 doi: 10.1038/nrmicro2319
[8]	E. Peterson, P. Kaur, Antibiotic resistance mechanisms in bacteria: Relationships between resistance determinants of antibiotic producers, environmental bacteria, and clinical pathogens, Front. Microbiol., 9 (2018), 2928. https://doi.org/10.3389/fmicb.2018.02928 doi: 10.3389/fmicb.2018.02928
[9]	M. J. Bottery, J. W. Pitchford, V. P. Friman, Ecology and evolution of antimicrobial resistance in bacterial communities, ISME J., 15 (2021), 939–948. https://doi.org/10.1038/s41396-020-00820-4 doi: 10.1038/s41396-020-00820-4
[10]	W. O. Kermack, A. G. McKendrick, A contribution to the mathematical theory of epidemics, Pro. R. Soc. A, 115 (1927), 700–721. https://doi.org/10.1098/rspa.1927.0118 doi: 10.1098/rspa.1927.0118
[11]	Q. Lu, Stability of SIRS system with random perturbations, Phys. A Stat. Mech. Appl., 388 (2009), 3677–3686. https://doi.org/10.1016/j.physa.2009.05.021 doi: 10.1016/j.physa.2009.05.021
[12]	E. Tornatore, S. M. Buccellato, P. Vetro, Stability of a stochastic SIR system, Phys. A Stat. Mech. Appl., 354 (2005), 111–126. https://doi.org/10.1016/j.physa.2005.02.051 doi: 10.1016/j.physa.2005.02.051
[13]	A. Gray, D. Greenhalgh, L. Hu, X. Mao, J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 71 (2011), 876–902. https://doi.org/10.1137/10081856X doi: 10.1137/10081856X
[14]	R. Rifhat, Q. Ge, Z. Teng, The dynamical behaviors in a stochastic SIS epidemic model with nonlinear incidence, Math. Biosci., 325 (2020), 108362. https://doi.org/10.1016/j.mbs.2020.108362 doi: 10.1016/j.mbs.2020.108362
[15]	C. Xu, Global threshold dynamics of a stochastic differential equation SIS model, J. Math. Anal. Appl., 447 (2017), 736–757. https://doi.org/10.1016/j.jmaa.2016.10.077 doi: 10.1016/j.jmaa.2016.10.077
[16]	A. Mouaouine, A. Boukhouima, K. Hattaf, N. Yousfi, A fractional order SIR epidemic model with nonlinear incidence rate, Adv. Differ. Equations, 2018 (2018), 1–9. https://doi.org/10.1186/s13662-018-1835-2 doi: 10.1186/s13662-018-1835-2
[17]	Y. Guo, The stability of the positive solution for a fractional SIR model, Int. J. Biomath., 10 (2017), 1750014. https://doi.org/10.1142/S1793524517500141 doi: 10.1142/S1793524517500141
[18]	J. S. Builes, C. F. Coletti, L. A. Valencia, A stochastic differential equation approach for an SIS model with non-linear incidence rate, preprint, arXiv: 2404.13469. https://doi.org/10.48550/arXiv.2404.13469
[19]	R. Khasminskii, Stochastic stability of differential equations, Springer Sci. Business Media, 66 (2011).
[20]	M. Hassan, Nonlinear Systems, Prentice Hall, 3rd edition, 2002.
[21]	X. Mao, Stochastic Differential Equations and Applications, 2nd edition, Horwood Publishing, Chichester, UK, 2008.
[22]	C. Vargas-De-León, Volterra-type Lyapunov functions for fractional-order epidemic systems, Commun. Nonlinear Sci. Numer. Simul., 24 (2015), 75–85. https://doi.org/10.1016/j.cnsns.2014.12.013 doi: 10.1016/j.cnsns.2014.12.013
[23]	A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006.
[24]	D. Matignon, Stability results for fractional differential equations with applications to control processing, Comput. Eng. Syst. Appl., 2 (1996), 963.
[25]	J. Huo, H. Zhao, L. Zhu, The effect of vaccines on backward bifurcation in a fractional order HIV model, Nonlinear Anal. Real World Appl., 28 (2015), 38–53. https://doi.org/10.1016/j.nonrwa.2015.05.014 doi: 10.1016/j.nonrwa.2015.05.014
[26]	M. Mendelson, A. Brink, J. Gouws, N. Mbelle, V. Naidoo, T. Pople, et al., The One Health stewardship of colistin as an antibiotic of last resort for human health in South Africa, Lancet Infect. Dis., 18 (2018), e288–e294. https://doi.org/10.1016/S1473-3099(18)30349-6 doi: 10.1016/S1473-3099(18)30349-6
[27]	M. Rhouma, F. Beaudry, A. Letellier, Resistance to colistin: What is the fate for this antibiotic in pig production?, Int. J. Antimicrob. Agents, 48 (2016), 119–126. https://doi.org/10.1016/j.ijantimicag.2016.04.014 doi: 10.1016/j.ijantimicag.2016.04.014
[28]	M. J. Goyes-Baca, M. R. Sacon-Espinoza, F. X. Poveda-Paredes, Management of the health care system in Ecuador to confront the antimicrobial resistance, Revista Inf. Científica, 102 (2023), Epub 18-Ene-2023. https://doi.org/10.34257/RIC102.01
[29]	J. Wu, X. Dong, L. Zhang, Y. Lin, K. Yang, Reversing antibiotic resistance caused by mobile resistance genes of high fitness cost, Msphere, 6 (2021), e00356. https://doi.org/10.1128/msphere.01128-20 doi: 10.1128/msphere.01128-20
[30]	M. Viveiros, M. Dupont, L. Rodrigues, I. Couto, A. Davin-Regli, M. Martins, et al., Antibiotic stress, genetic response and altered permeability of E. coli, PloS One, 2 (2007), e365. https://doi.org/10.1371/journal.pone.0000365 doi: 10.1371/journal.pone.0000365
[31]	F. Baquero, J. L. Martinez, V. F. Lanza, J. Rodríguez-Beltrán, J. C. Galán, A. San Millán, et al., Evolutionary pathways and trajectories in antibiotic resistance, Clin. Microbiol. Rev., 34 (2021). https://doi.org/10.1128/CMR.00050-19

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)