[Solved] FYS4150 Project5-Theoretical background and description of the system

(6)Δt=min{4aN,1bN,1cN}.“>t=min{4aN,1bN,1cN}.(6)(6)t=min{4aN,1bN,1cN}.

This construction allows us to reinterpret the values bIΔt“>bItbIt, and (7)P(S→I)=aSINΔt,P(I→R)=bIΔt,P(R→S)=cRΔt.“>P(SI)P(IR)P(RS)=aSINt,=bIt,=cRt.(7)(7)P(SI)=aSINt,P(IR)=bIt,P(RS)=cRt.

For each possible move, a random number between 0 and 1 is generated. If the number is less than the probability for the move, the move is taken.

Develop now a Monte Carlo algorithm which implements the last equations using the same parameters a“>aa, c“>cc for each population. Compare your results to those obtained with the deterministic differential equation solver and discuss your findings.

You will need to find the fraction of people when the system has been equilibrated. You will need to find the equlibrium expectation values and corresonding standard deviations.

Part c) Improvements, vital dynamics

The same principles used in this simple model can be extended to include more details about the population and disease.

For instance, vital dynamics can be easily added to the system so that the model can describe the spread of diseases which occur over longer stretches of time. If d“>dd is the death rate, and (8)S′=cR−aSIN−dS+eNI′=aSIN−bI−dI−dIIR′=bI−cR−dR“>SIR=cRaSINdS+eN=aSINbIdIdII=bIcRdR(8)(8)S=cRaSINdS+eNI=aSINbIdIdIIR=bIcRdR

Here, we have assumed that all the babies born into the population are initially susceptible.

Add now these additions to your ODE solver and Monte Carlo solver and discuss the results.

Part d) Seasonal Variation

For diseases such as influenza, the rate of transmission depends largely on the time of year. During the colder months, individuals are more likely to spend time in closer proximity to one another, resulting in a rate of transmission which oscillates. So we can let (9)a(t)=Acos(ωt)+a0,“>a(t)=Acos(t)+a0,(9)(9)a(t)=Acos(t)+a0,

where A“>AA is the maximum deviation from ω“> is the frequency of oscillation. Study the the effect of seasonal variations as well, with both the ODE and Monte Carlo solver. Try different values of the above new parameters and discuss your results.

Part e) Vaccination

Diseases with available vaccinations allow people to move directly from R“>RR, breaking the cyclic structure of the SIRS model. We assume that a susceptible individuals choice to become vaccinated does not depend on how many other susceptibles are vaccinated. We may, however, assume that the rate of vaccination (10)S′=cR−aSIN−fI′=aSIN−bIR′=bI−cR+f“>SIR=cRaSINf=aSINbI=bIcR+f(10)(10)S=cRaSINfI=aSINbIR=bIcR+f

Add now the effect of vaccination via the parameter f“>ff.

Introduction to numerical projects

Here follows a brief recipe and recommendation on how to write a report for each project.

• Give a short description of the nature of the problem and the eventual numerical methods you have used.
• Describe the algorithm you have used and/or developed. Here you may find it convenient to use pseudocoding. In many cases you can describe the algorithm in the program itself.
• Include the source code of your program. Comment your program properly.
• If possible, try to find analytic solutions, or known limits in order to test your program when developing the code.
• Include your results either in figure form or in a table. Remember to label your results. All tables and figures should have relevant captions and labels on the axes.
• Try to evaluate the reliabilty and numerical stability/precision of your results. If possible, include a qualitative and/or quantitative discussion of the numerical stability, eventual loss of precision etc.
• Try to give an interpretation of you results in your answers to the problems.
• Critique: if possible include your comments and reflections about the exercise, whether you felt you learnt something, ideas for improvements and other thoughts youve made when solving the exercise. We wish to keep this course at the interactive level and your comments can help us improve it.
• Try to establish a practice where you log your work at the computerlab. You may find such a logbook very handy at later stages in your work, especially when you dont properly remember what a previous test version of your program did. Here you could also record the time spent on solving the exercise, various algorithms you may have tested or other topics which you feel worthy of mentioning.

Format for electronic delivery of report and programs

The preferred format for the report is a PDF file. You can also use DOC or postscript formats or as an ipython notebook file. As programming language we prefer that you choose between C/C++, Fortran2008 or Python. The following prescription should be followed when preparing the report:

• Use Canvas to hand in your projects, log in at https://www.uio.no/english/services/it/education/canvas/ with your normal UiO username and password.
• Upload only the report file! For the source code file(s) you have developed please provide us with your link to your github domain. The report file should include all of your discussions and a list of the codes you have developed. Do not include library files which are available at the course homepage, unless you have made specific changes to them. Alternatively, you can just upload the address to your GitHub or GitLab repository.
• In your git repository, please include a folder which contains selected results. These can be in the form of output from your code for a selected set of runs and input parameters.
• In this and all later projects, you should include tests (for example unit tests) of your code(s).
• Comments from us on your projects, approval or not, corrections to be made etc can be found under your Canvas domain and are only visible to you and the teachers of the course.