2. Finding a Fixed Point: Consider the following simplified scenario of the Covid-19 pandemic. We divide the entire population into three classes: - Susceptible (S) - Infected (I) - Recovered/Immune (R) The susceptible class has either not been exposed to the virus or has not been given the vaccine. The infected class will recover eventually but can infect others in the mean time. The recovered class has either had either been exposed to the virus and has recovered or has been successfully vaccinated. Lets suppose in this scenario that we do not consider the otherwise grim reality of this pandemic. We observe that it is difficult to change human behavior/interaction but suppose we can produce vaccinations. So, we study the effect of vaccination behavior of the outbreak. Suppose today is day zero and we capture the proportion of the population in each of the classes S,I,R in a column vector x0=⎣⎡0.90.090.01⎦⎤ Suppose that each day 1/800 of the infected individuals recover. Each day, 1 out of every 150 susceptible individuals becomes infected. Because of mutations in the virus, 1/5,000 of the recovered individuals become susceptible again. Through vaccination, we are able to move some fraction of the individuals in S directly to R; call this fraction p where 0≤p≤1. Use this information to find a matrix M such that
x1=Mx0 describes the change in the population classes from day 0 to day 1. This is the same matrix that describes the transition of the population from day k to day k+1. Note that if some number of individuals move from one class to another, you have to remove them from the class they were in originally (meaning the columns of your matrix M should all sum to 1 ). Suppose that we are able to successfully vaccinate 1 out of every 1000 of the susceptible people everyday; i.e., p=1/1000. (a) For this value of p, save the matrix M as A3 (b) Which day is the first day where at least 50% of the population is infected? Call this day D0. (c) The system reaches a steady state as the number of days that have passed goes to infinity. Estimate the fraction of the population that is infected in the long term. Call this F0. In order to prevent tolerance errors, for F0, if you use a for loop, pick a large value (say 100000 ) to be your final iteration, and break out of the loop only if the difference between two subsequent (absolute) values of the infected variable is less than 10−8. (d) Save a row vector A4=[D0,F0].
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