Homework #2 – Math 377 Fall 2021
数学建模作业代写 A human can become infected when bitten by an infected mosquito, and a mosquito becomes infected when it bites an infected human.
Due Date: Sep 28th at 11:00pm. The system accepts late assignments with a penalty of 33% per day.
How to submit: We will submit homeworks and tests using software called Crowdmark. You will receive and email with the link to submit.
Academic Integrity: You are allowed to collaborate with your classmates while working on home- works, but your submission must be written in your own words and reflect your own understanding of the problems. Copying homework from another classmate is not permitted. While this will not generally be necessary for homeworks in this course, you are alsopermitted to use the internet provided you properly reference any external resource you use. Copying without referencing is a violation of academic integrity and will be strictly enforced.
Problem 1: Let’s adjust the basic SIR model we did in class to account for vaccinations. 数学建模作业代写
Part a) Suppose a town of 5000 people initially has 1 infected people, 0 recovered people, and of the remaining a constant percentage v of them have been vaccinated. We will assume that vaccinated people and recovered people are 100% immune, and in fact we will keep all other assumptions from the basic SIR model. If the coefficient representing transition from Susceptibles to Infectious is a = 0.0001 and the coefficient representing transition from Infectious to Recovered is b = 0.2, derive what the minimum percentage v such that the rate of change of Infectious people is negative (and thus I(0) will decline to zero). Note: the a and b were made up numbers consistent with the arbitrary choice made in class, they don’t represent COVID-19.
Part b) In this problem let us keep the same same numbers as before.
We will make the model more complex in two ways. Firstly, let’s assume that vaccinations in this town start at 0 people, but then increase at a rate of 50 people per day until 50% of the town is vaccinated (and we will assume we ONLY vaccinate Susceptible people). At this point, we assume nobody else is willing or able to be vaccinated. Secondly, let’s imagine that recovered people, but not vaccinated people, can in fact transition to Susceptible people (imagine waning immunity) and do so at a rate of d = 0.01.
i.Write out two systems of ODEs describing the rate of change of Susceptible, Infectious, Vac- cinated, and Recovered people for the two relevant time periods, namely when the Vaccinated rate is growing and when it stops growing. 数学建模作业代写
ii.Use computer software to provide a plot from t = 0 to t = 150 of these curves. You may use whatever software you are familiar with such as MATLAB, R, or Python. I will specifically support MATLAB in this course, which we won’t use extensively (except possible for your projects) but might from time to time. In a separate page on Brightspace I will sharemy MATLAB file that you can tweak, instructions on how to install MATLAB, and a video walkthrough of the file.
Problem 2: In this problem we are going to come up with a new system of differential equations again somewhat analogous to the SIR model, but modeling a different phenomenon. Malaria spreads through mosquito bites. 数学建模作业代写
A human can become infected when bitten by an infected mosquito, and a mosquito becomes infected when it bites an infected human. Humans don’t pass malaria to other humans and mosquitoes don’t pass malaria to other mosquitoes. Mosquitoes have a relatively short lifespan so you should include vital dynamics (births and deaths) of mosquitoes, but can ignore them for humans.
- Sketcha Mind Map demonstrating the relationships between the different components of this model
- State a system of differential equations that models this scenario, carefully defining any con- stants or variables.
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