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Homework sets will be uploaded here, together with due dates (typically one week after uploading). Solutions will appear on Sakai . Exam dates will also be posted.
DOx denotes Chapter x of Prof. Ocone's notes. See Resources.
| Lecture | Date | Topics | Reading and homework assignments | |
|---|---|---|---|---|
| Week 1 | ||||
| 1 | Tue 01/22 | Introduction to course, comparison between stochastic and determinisitc framework, basic "death" process example, quick review of probability | ||
| 2 | Thu 01/24 | Special lecture from Professor Sanchez-Tapia on basic epidemic modeling | Lecture notes introducing epidemiological model (SIS model) Continue reading DO1 Homework 1 (due 02/07) |
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| Week 2 | ||||
| 3 | Tue 01/29 | Continue introductory basic "death" process model, review of probability, introduction to genetics and heredity | ||
| 4 | Thu 01/31 | Finish introduction to genetics, mathematical analysis of genetics in large population | Continue reading DO1 Section 3.1 of DO3 |
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| Week 3 | ||||
| 5 | Tue 02/05 | Finish genetics lecture, start mathematical analysis of genetics of large population models | Slides on genetics (based on DO1) Section 3.1 of DO3 |
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| 6 | Thu 02/07 | Genetic evolution models without selection | Section 3.2 of DO3 Homework 2 (due 02/14) |
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| Week 4 | ||||
| 7 | Tue 02/12 | Continue analysis of random mating and no selection, Hardy-Weinberg equilibrium, mutations, and overlapping generations | Section 3.2 of DO3 |
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| 8 | Thu 02/14 | More examples of modeling without selection, basic solutions of linear difference equations, overlapping generation model | Section 3.2 and 3.4 of DO3 Homework 3 (due 02/21) |
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| Week 5 | ||||
| 9 | Tue 02/19 | Solution of first-order difference equations, finish overlapping generations, introduction to selection models | Sections 3.2, 3.4, 3.6 (Appendix) of DO3 |
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| 10 | Thu 02/21 | Analysis of selection model, nonlinear difference equations, cobwebbing | 3.4 of DO3 Homework 4 (due 03/05) |
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| Week 6 | ||||
| 11 | Tue 02/26 | Finish analysis of selection models. | Section 3.4 DO3 |
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| 12 | Thu 02/28 | Exam 1 | Chapters 1, 2, 3 of DO3 (excluding Section 3.3) | |
| Week 7 | ||||
| 13 | Tue 03/05 | Introduction to finite population stochastic models. Markov chains. | Section 4.1 DO4 Homework 5 (due 03/14) |
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| 14 | Thu 03/07 | Basic properties of Markov chains. Random walks. Transition matrices and one-step probabilities. Examples from stem-cells and DNA. | Section 4.1 DO4 |
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| Week 8 | ||||
| 15 | Tue 03/12 | Moran and Wright-Fisher Markov chains. General birth-death processes. | Section 4.1 DO4 |
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| 16 | Thu 03/14 | Model of DNA. General properties of Markov chains. Asymptotic behavior of distributions. Matrix diagonalization. | Section 4.2 DO4 Homework 6 (due 03/28) |
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| Week 9 | ||||
| Spring Break! | ||||
| Week 10 | ||||
| 17 | Tue 03/26 | Finish matrix diagonalization example. Limiting distributions. Conditional and absolute expectations of Markov chains. | Section 4.2 DO4 |
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| 18 | Thu 03/14 | General Markov property. Classification of states. Stationary distributions. Examples from Wright-Fisher and Moran models. | Sections 4.4, 4.5 DO4 Homework 7 (due 04/09) |
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| Week 11 | ||||
| 19 | Tue 04/02 | Limiting distribution of Markov chains. Classification of states. Recurrence and transience. | Section 4.5 DO4 |
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| 20 | Thu 04/04 | Stationary distribution of Markov chains. Basic Ergodic theory. Examples from Wright-Fisher and Moran models. | Section 4.5 DO4 Exam 2 next Thursday (4/11) |
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| Week 12 | ||||
| 21 | Tue 04/09 | Stationary distributions and limit behavior for aperiodic irreducible Markov chains. Applications to Wright-Fisher and Moran models (with mutations). | Section 4.5 DO4 |
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| 22 | Thu 04/11 | Exam 2 | Chapters 3 (roughly) | |
| Week 13 | ||||
| 23 | Tue 04/16 | Introduction to Luria-Delbrück experiment related to spontaneous vs. directed drug resistance. Basic stochastic analysis and result of experiment. | ||
| 24 | Thu 04/18 | Rigorous analysis of Luria-Delbrück distribution. Review of moment-generating functions. | Mathematical details of Luria-Delbrück distribution Homework 8 (due 04/25) |
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| Week 14 | ||||
| 25 | Tue 04/23 | Finish analysis of Luria-Delbrück experiment. Introduce generalization to understand more statisitical properties. Moment-generating functions. | Mathematical details of Luria-Delbrück distribution | |
| 26 | Thu 04/25 | Analyze Luria-Delbrück distribution via generating functions. Approximation via integrals. Generalizations to different mutant growth rates and sotchastic growth of mutants. | Mathematical details of Luria-Delbrück distribution Homework 9 (due 05/06) |
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| Week 15 | ||||
| 27 | Tue 04/30 | Introduction to stochastic modeling of chemical reactions. Basic examples. Chemical master equation (CME) and mass-action kinetics. | Notes on stochastic chemical kinetics. Sections 4.1-4.3.2. | |
| 28 | Thu 05/02 | Further analysis of chemical reaction networks. Continous-time Markov chains and exponential waiting times. Transition probabilities. | Notes on stochastic chemical kinetics. Sections 4.4 and 4.5. Extra Credit: Please send me your groups and topics by 05/03 if you intend to complete a project. |
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