Abstract
The rise of bacterial resistance to antibiotics is a major Public Health concern. It is the result of two interacting processes: the selection of resistant bacterial strains under exposure to antibiotics and the dissemination of bacterial strains throughout the population by contact between colonized and uncolonized individuals. To investigate the resulting time evolution of bacterial resistance, Temime et al. (Emerg Infect Dis 9:411–417, 2003) developed a stochastic SIS model, which was structured by the level of resistance of bacterial strains. Here we study the asymptotic properties of this model when the population size is large. To this end, we cast the model within the framework of measure valued processes, using point measures to represent the pattern of bacterial resistance in the compartments of colonized individuals. We first show that the suitably normalized model tends in probability to the solution of a deterministic differential system. Then we prove that the process of fluctuations around this limit tends in law to a Gaussian process in a space of distributions. These results, which generalize those of Kurtz (CBMS-NSF regional conference series in applied mathematics, vol 36. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1981, chap. 8) on SIR models, support the validity of the deterministic approximation and quantify the rate of convergence.
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References
Aldous D (1978) Stopping times and tightness. Ann Probab 6(2):335–340
Andersson H, Britton T (2000) Stochastic epidemic models and their statistical analysis. In: Lecture notes in statistics, vol 151. Springer, New York
Badrikian A (1996) Martingales hilbertiennes. Ann Math Blaise Pascal S3:115–171
Beran R, Ducharme GR (1991) Asymptotic theory for bootstrap methods in statistics. Université de Montréal, Centre de Recherches Mathématiques, Montreal
Cars O, Hogberg LD, Murray M, Nordberg O, Sivaraman S, Lundborg CS, So AD, Tomson G (2008) Meeting the challenge of antibiotic resistance. Br Med J 337(7672):726–728
Clémençon S, Tran VC, de Arazoza H (2008) A stochastic SIR model with contact-tracing: large population limits and statistical inference. J Biol Dyn 2(4):392–414
EARS-Network (2012) Antimicrobial resistance surveillance in Europe 2011. ECDC, Stockholm
Ethier SN, Kurtz TG (2005) Markov processes. Characterization and convergence. Wiley, Hoboken
Ferland R, Fernique X, Giroux G (1992) Compactness of the fluctuations associated with some generalized nonlinear Boltzmann equations. Can J Math 44(6):1192–1205
Fournier N, Méléard S (2004) A microscopic probabilistic description of a locally regulated population and macroscopic approximations. Ann Appl Probab 14(4):1880–1919
Jacod J, Shiryaev AN (2003) Limit theorems for stochastic processes. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol 288, 2nd edn. Springer, Berlin
Joffe A, Métivier M (1986) Weak convergence of sequences of semimartingales with applications to multitype branching processes. Adv Appl Probab 18(1):20–65
Kufner A, Opic B (1984) How to define reasonably weighted Sobolev spaces. Comment Math Univ Carol 25(3):537–554
Kurtz TG (1981) Approximation of population processes. In: CBMS-NSF regional conference series in applied mathematics, vol 36. Society for Industrial and Applied Mathematics (SIAM), Philadelphia
Maher M, Alemyehu W, Lakew T, Gaynor B, Hang S, Cevallos V, Keenan J, Lietman T, Porco T (2012) The fitness cost of antibiotic resistance in Streptococcus pneumoniae: insight from the field. PLoS One 7(1):e29,407
Martinez JL (2008) Antibiotics and antibiotic resistance genes in natural environments. Science 321(5887):365–367
Méléard S (1998) Convergence of the fluctuations for interacting diffusions with jumps associated with Boltzmann equations. Stoch Stoch Rep 63(3–4):195–225
Méléard S, Roelly S (1993) Sur les convergences étroite ou vague de processus à valeurs mesures. C R Acad Sci Paris Sér I Math 317(8):785–788
Méléard S, Tran VC (2012) Slow and fast scales for superprocess limits of age-structured populations. Stoch Process Appl 122(1):250–276
Métivier M (1982) Semimartingales, de Gruyter studies in mathematics, vol 2. Walter de Gruyter & Co., Berlin (a course on stochastic processes)
Métivier M (1987) Weak convergence of measure valued processes using Sobolev-imbedding techniques. In: Stochastic partial differential equations and applications (Trento, 1985). Lecture notes in math., vol 1236. Springer, Berlin, pp 172–183
Oelschläger K (1990) Limit theorems for age-structured populations. Ann Probab 18(1):290–318
Pollard D (1984) Convergence of stochastic processes. In: Springer series in statistics. Springer, New York
Prellner K, Hermansson A, White P, Melhus A, Briles D (1999) Immunization and protection in pneumococcal otitis media studied in a rat model. Microb Drug Resist 5(1):73–82
Rebolledo R (1980) Central limit theorems for local martingales. Z Wahrsch Verw Geb 51(3):269–286
Revuz D, Yor M (1991) Continuous martingales and Brownian motion. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol 293. Springer, Berlin
Roelly-Coppoletta S (1986) A criterion of convergence of measure-valued processes: application to measure branching processes. Stochastics 17(1–2):43–65
Temime L, Boelle P, Courvalin P, Guillemot D (2003) Bacterial resistance to penicillin G by decreased affinity of penicillin-binding proteins: a mathematical model. Emerg Infect Dis 9(4):411–417
Temime L, Boëlle PY, Thomas G (2005) Deterministic and stochastic modeling of pneumococcal resistance to penicillin. Math Popul Stud 12(1):1–16
Acknowledgments
We wish to thank Michèle Thieullen for helpful discussions, and Viet Chi Tran for clarification of several points.
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Boëlle, PY., Thomas, G. Resistance to antibiotics: limit theorems for a stochastic SIS model structured by level of resistance. J. Math. Biol. 73, 1353–1378 (2016). https://doi.org/10.1007/s00285-016-0996-2
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DOI: https://doi.org/10.1007/s00285-016-0996-2
Keywords
- Bacterial resistance to antibiotics
- Measure valued Markov jump process
- Fluctuation process
- Weighted Sobolev space