Weekly Updates
Week 1: I met with mentor to talk about possible projects. We eventually chose to model HIV and the evolution of another disease. I've done some background reading on modeling the evolution of HIV and on modeling the epidemiology of influenza. Dr. Fefferman asked me to develop a simple model that would describe how strains of influenza would evolve given a fixed mutation rate and the relative number of virions of each strain.
Week 2: I've worked on modeling the relative number of influenza strains that arise in a population. I've proposed four different models, two that consider one mutation and two that consider two mutation sites. I'm hoping to figure out how to model five mutation sites, as the literature suggest five antigenic clusters. However, the notation gets really messy even for the two mutation site model. I've coded the models in Matlab. Eventually, these will be used in a model that will describe the number of influenza virions in a person.
Week 3: After talking with Dr. Fefferman, I generalized the model for any N strain of viruses. So in the new model, any strain has the ability to mutate to any of the other N-1 strains. I also included exponential growth for all the strains, and now I'm trying to figure out how to include an immune response to the growth of virions and ultimately link all of this to a HIV model.
Week 4: This week wasn't as productive as I hoped it would be. I had to do a larger literature search in order to see what other people have done. I made an attempt at making a population model, but it doesn't seem like a tenable model. On a brighter note, my updated Matlab code works.
Week 5-6: I forgot what happened in each week, so I'm lumping them together. Many changes have been made to the individual model: virus growth is now described as logistic growth, an immune response with a time delay (not a delay DE) that reduces the growth rate, a clearance term that describes death of the virus, and the same mutation description. The results of this model are exciting! The model is able to describe situations where (1) the body can overcome the virus with a sufficient immune response, (2) the virus can overcome an immunocompromised body and (3) a dynamics in between (1) and (2). With this progress, the individual model is almost and just needs a description of cross-immunity.
Week 7: I made a few attempts to describe cross-immunity. Cross-immunity is the idea that if a person is exposed to a strain of a virus, then he/she should have an enhanced immune response to a closely related different strain of that virus. However, I could not get results that make much sense. This week also involved preparing for the presentation (which has been uploaded), so I'll continue with modeling cross-immunity next week.
Week 8: I attempted a bunch of different ways to describe cross-immunity, but none of them are giving satisfactory results. They have been showing either (1) complete suppression of all virus strains and (2) no effect at all. I have not found a way that will give a result in between the two extremes. Other than that, I worked on and finished the write-up for the summer.