Math 183 Modeling Projects Descriptions

Math 183. Mathematical Modeling Fall 2016 Ideas for Modeling Projects 1. Microbial Competition in a Chemostat 

At the start of the semester we looked at a continuous, deterministic model for a singlespecies bacterial population growing in a chemostat. In this project, you will look at the case of more than one species of microbes growing in a chemostat. Start out by reading the survey article by Hansen and Hubbell in Science, 207:1491-1493, 1980. The general mathematic theory had been discussed previously by Hsu, Hubbell and Waltman in 1977 (SIAM J. Applied Mathematics, 32(2):366-383, 1977). Your task is to present the model and some of the analysis. The more recent paper by Hsu and Waltman (SIAM J. Applied Mathematics, 52(2):528-540, 1992) might also be helpful.    2. Dynamics of Infectious Diseases  

In class we have derived the SIR epidemiology model of Kermack and McKendrick developed in the 1930s. The goal of this project is to present an analysis of this model as well as more recent extensions of the model. A good reference for this project is the review article of Herbert W. Hethcote, The Mathematics of Infectious Diseases (SIAM Review, Vol. 42, No. 4. (Dec., 2000), pp. 599-653).   3. Probabilistic Models of Traffic Flow 

Traffic flow is, perhaps, more adequately modeled by a discrete system of moving cars and some random variables whose averages are those that come up in the continuous and deterministic models of Lighthill, Whitman and Ricarhds’. Jiabaru and Liu present one of those stochastic models of traffic flow in Transportation Research B, 46:156-174, 2012. In this project you’ll present the probabilistic model in Transportation Research B, 46:156-174, and discuss the various distributions used in the model. Pay close attention to the connections between the Jiabaru-Liu models and the Lighthill-Whitman-Richards one. 4. A Queueing Model for Road Traffic Flow 

Another probabilistic approach to traffic flow modeling is to use queueing theory. In 1961, Allan J. Miller proposed one of those models in Journal Royal Statistical Society B, 23(1):64-90, 1961. Your task is to explain the model and analysis proposed in that paper.

5. Age-Structured Models in Population Dynamics and Epidemiology We have seen in class an example of a population model in which there is an age structure; that is, the population density is a function of the age of the individuals in the population. These models can be analyzed using the method of characteristic curves. Another situation in which age is an important factor comes up when modeling the spread of an infectious disease; the disease may have different infection rates or mortality rates for different age groups in the population. In Mathematical Epidemiology by Brauer et al. (Lecture Notes in Mathematics, Volume 1946, Springer 2008), Li and Brauer discuss these types of models. In this project, you will i. ii.

explain how to incorporate an age structure in a population or epidemiology model; discuss how these models can be analyzed.

6. Modeling Mutations to Resistance 

In class we saw how to use a Poisson process to model the number of mutations in a bacterial colony as a function of time. In this project, you will extend the analysis to take into account the fact that the size of the bacterial colony is increasing in time. Part of the analysis was begun by Luria and Delbrück in their paper in Genetics, 28:491-511, 1943, in the context of mutations that led to strains of E. Coli bacteria that were resistant to certain virus. In their 1943 paper, Luria and Delbrück did not compute the distribution. This was done later by Lea and Coulson (J. Genetics, 49:264-285, 1949). Many years later, a very simple derivation of the distribution was obtained by Ma, Sandri and Sarkar (J. Applied Probability, 29(2): 255-267, 1992). The task for this project is to discuss the Luria-Delbrück distribution as a birth-andmutation random process and to present the Ma, Sandri and Sarkar solution.

7. Statistical Testing of Models 

The goal of this project is to test the two models presented in the Luria-Delbrück paper of 1943 (Genetics, 28:491-511, 1943). One of the models predicts a Poisson process, while the other predicts the Luria- Delbrück distribution. You will analyze the data from the experimental results presented in the paper. You will have to do some research on techniques for testing whether data from experiments fit the predictions of theoretical models. You will also have to delve into parameter estimation.

8. Modeling Algal Blooms An algal bloom, also known as a red tide, is an overgrowth of populations of aquatic microorganisms in certain coastal areas. Starting with the pioneering work of Kierkstead and Solbodkin in 1953 (J. Mar. Res., 12, 1953, 141-147) many mathematical models of harmful algal blooms have been proposed and analyzed since that time. The article by Peter Franks (Limnology and Oceanography, 42(5, Part II) 1997, 1273-1282) provides a very exhaustive survey of those models up to that year. Several species of algae form red patches when the population density increases, hence the name “red tide.” The microorganisms grow by cell division when the conditions are conducive to growth. The patches are such that the conditions in them (nutrient concentration, temperature, etc.) promote growth that is faster than normal. Thus, population growth is very rapid in the patches. Outside of the patches the growth is not as fast and the cells can be easily dispersed by the action of wind and waves. The dispersion of cells away from the patches can be modeled by diffusion. Problem: Construct a model that takes into account the effects of cell growth and diffusion in the formation of algal blooms and that can be used to estimate the size of patches that are conducive to growth. References:  Kierkstead, H. and Solbodkin, L., The size of water masses containing plankton blooms, Journal of Marine Research, 12, 1953, 141-147.  Franks, P. J. S., Models of harmful algal blooms, Limnology and Oceanography, 42(5, Part II) 1997, 1273-1282.

9. Problems provided through the Draper Center for Community Partnership at Pomona College  

Title: Effective and Transparent School Funding in Pomona USD Questions: i. Are district and school funding aligned with needs based on student data? ii. Creation of a report/info graphic that highlights district expenditures based on state of CA 8 priority areas. Contact Information: Organization: Gente Organizada Contact Person: Jesus Sanchez 626-419-3540 [email protected]

Notes:

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CA recently transformed the school funding model which "encourages" local funding and community involvement in budget process. Gente Organizada will meet with district leadership around their campaign in the coming week. It will be helpful for a student from the team to be open to attending the meeting.

 

10. Modeling the resection of DNA ends in mutant yeast strains Proposed by Professor Tina Negritto, Molecular Biology Program, Pomona College Objective: Develop a mathematical model of the resection of DNA ends. This model will compare the kinetics of DNA end resection in the different mutant strains and determine if they show the same or different rates of resection compared to the WT strain. Question: Can the kinetics of resection tell us something about the mechanisms of resection in these different mutants? Professor Negritto has provided some data from her laboratory showing the degradation of DNA ends for the various strains as a function of time. An MS Excel file containing data and charts may be found in the Sakai site for this class. In there you may find an MS Word document containing a longer description of the problem. Perhaps the theory of reaction kinetics might be helpful in modeling this situation.

11. Modeling Marital Interactions This project deals with an example of the use on mathematics in modeling social interactions. In the article, The Mathematics of Marital Conflict: Dynamic Mathematical Nonlinear Modeling of Newly wed Marital Interaction (Journal of Family Psychology, 1999 Vol. 13, No. 1, pp. 3-19), John Gottman, Catherine Swanson, and James Murray use difference equations to model the dynamics of marital interaction. The goal of this project is to expound on the theory of difference equations and to present the analysis of the Gottman-Swanson-Murray model. The goal of the analysis is to make predictions about the future of newly wed couples.