Virginia Polytechnic Institute and State University Working on Covid 19 SIR Model Paper

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Vcubarncx

Mathematics

Virginia Polytechnic Institute and State University

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I dont understand this project . need help. I need the work to be detailed and the Matlab code so that I can compute ,too . thanks

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The Covid-19 pandemic This Project is due by 9:30 a.m. on Thursday, May 6 (our last class meeting). In this project, you will be creating a system of differential equations to model the Covid-19 pandemic, and then analyzing your model to make predictions about the course of the pandemic. Overview: You will be creating an S-I-R system of differential equations that models the spread of the Coronavirus through the population. As you hopefully recall from the worksheet we had on this type of model (it is still on BB!), this divides the population into subgroups who are (S)usceptiable, (I)nfected, or (R)ecovered. A system of differential equations models the movement of people into and out of each of these groups. Once you have set up your model, you will analyze it and use it to make predictions about the spread of the virus, and the impact that different measures can have on its spread. While the end of the pandemic is hopefully near, the latest news suggests that we might be in for another wave, so it is still important to understand the dynamics of its spread. The system will be coupled and non-linear, and so you will likely not be able to solve for explicit formulas for the different group sizes. As we’ve seen all semester, however, this doesn’t mean you won’t be able to analyze the behavior of the solutions. Qualitative methods, including graphical approximations (like slopefields) and numerical (like Euler’s Method or more sophisticated numerical approximation algorithms such as MATLAB’s ODE45), are perfectly suited for situations when we cannot explicitly solve the problem. The project description will provide guidance on how to build your model, but this is intended an outline to guide you and not step-by-step instructions. You will need to supplement what I provide with some research on your own – there will be some helpful mathematics in our book, useful data online, etc. And you may well think of improvement to the model beyond what I suggest – this outline will produce “good” results, but not “great”– if you can make it better, you should do so! You may seek help, and you may collaborate with others in this class, but ultimately what you submit must be your own work, not just a copied version of someone else’s. And I definitely am not interested in seeing someone else’s model (published or not) for how the virus is spreading – it is the act of creating the model that is important here. Step 1: The first thing you will need to do is select a time and place to model. Choose a focus that will not be chosen by others in the class – that way you can discuss your work with others with no worry that you will be getting the same answers (remember, working together is allowed, but multiple people submitting the same work is not!). As an example of what I mean, you may choose to look at how the virus first spread in Maryland last spring, but you could also concentrate on a different state, or different country – if you are from someplace not around here, or have family in a different part of the country or world, pick a place that is meaningful to you. Then decide whether you are going to work with data from when the pandemic first struck your region, or maybe a later wave. You have a lot of flexibility here, but please remember that the emphasis will be on the mathematical modeling, not public policy – so while it is great that Australia has pretty much eliminated the virus, and Israel has succeeded in vaccinating the majority of its population, these countries’ efforts should be admired (and possibly emulated), but would not make for a very interesting mathematical model at this point! As you build your model, you will need to make some simplifying assumptions, but want this to be at least somewhat realistic. Record the assumptions you make, and why you chose to make them, and be sure to discuss these in your analysis. Start your model simple, and then build it up to make it more realistic – don’t try to do everything at once! I suggest you start by modeling the spread in the absence of the vaccine (if you are modeling a time period before the past few weeks it is unlikely that many people were vaccinated anyway), and then go back and see what the impact would have been if different levels of the population were immune. Questions to think about: 1) What is the population size of the region you are modeling? Over a few week time period during the wave that you are studying, what was the size of I? Based on these values, estimate values of the derivative of I during that time period. 2) Each interaction between a member of S and a member of I has x% chance of spreading the disease (and moving that person out of S and into I). Recall that interactions between populations are proportional to the product of the population sizes. Based on your data and calculations in #1, you should be able to estimate x. What value of x makes sense for your model? How does this appear in the differential equations modeling S and I ? 3) How long, on average, do people remain in I? Suggestion: Not everyone remains in I for the same amount of time, but you can simplify your model by assuming that if people on average remain I for n days, then on any given day 1/n of the people in I leave I. How does this appear in the differential equation modeling I? 4) When people leave I (in #3) they have to go somewhere. The survivors may go to S (if they are able to get sick again), or they may become immune (in P but not in S), or they may die and be removed from P. What is the mortality rate, the likelihood of which some in I will die? What does this mean for the rate of change of S? 5) How has the population size been changing during this time frame? As a percentage, the population of a state or country won’t change very much in any given day or week, but in absolute numbers the population change will still be noticeable over these time frames. This could cause new people moving into S just from population growth, and the model should reflect that. Starting with these questions and the data you fill in (S, I, x), you can create a system of non-linear differential equations that model the spread of the coronavirus through the population. ➢ What is this system? ➢ Does your system have any equilibrium point(s)? If so, where are they, and what do they represent in terms of the model? ➢ Use MATLAB to plot a numerical solution for the system given the ICs you estimated in the setup. In particular, graph I vs time. What does the model predict? You may want to pick different time frames, from under a month to a year to several years. ➢ How would social distancing and face-covering be included in your model? If more (or fewer) people social-distance, how would that be included in the model? Check your model under different assumptions for such protective measures – are there noticeable differences in the predicted outcomes? Some more questions to consider and address in your submission: What assumptions did you make in setting up your model? How did these make the mathematics simpler? How much realism is lost in making these assumptions? Can you suggest ways to improve realism without the mathematics getting too much complicated? Are there other assumptions you might consider including? Note: when I first did this myself, last spring, I came up with what I considered reasonable values for S, I, and x, for Maryland, and predicted at the time that the pandemic was about to get a lot worse (which, unfortunately, it did), but that with better preventative measures the pandemic would have fizzled out. A good self-check for your model is whether you can duplicate this sort of finding. Once you answer all of the above, address the following: ➢ Assuming the vaccines are successful, they will have the effect of making people no longer susceptible, effectively moving them from S directly to R, without having to first go through I. “Herd Immunity” is the term for when enough people are removed from S that the rate of infection drops instead of grows, causing the pandemic to die out. Demonstrate this through graphs showing the impact when different fractions of the population choose to get vaccinated, and determine how much of the population must be vaccinated, according to your model, for herd immunity to occur. Rubric: S-I-R system of differential equations: Equations set up appropriately; values for all variables and parameters appropriately researched and/or derived (25 pts) Qualitative analysis of S-I-R model: numerical analysis correctly done and interpreted; graphs clear and well labeled to support analysis. (30 pts) Demonstration of herd immunity through graphs and explanations (15 pts) Write-up: (25 pts) This includes: ➢ ➢ ➢ ➢ ➢ Clear explanation of what you were modeling; Explanations of the mathematics that you included in your equations; Explanations of the assumptions that you incorporated into the model; Interpretations of your findings; Explanation of how the model reflects mask-wearing and other preventative measures, and what would change in the model if these measures were practiced by more (or fewer) people. ➢ The writing is easy to understand, and free of grammatical and spelling errors. All of the above adds up to a total of 95 points possible. The remaining 5 points, plus up to 10 extra credit, can be earned by improving the model over what I have suggested. Find something that you can include in the model beyond my guidance above, and do so!
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Explanation & Answer

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1

Orientability in Euclidian

Mathematics
Student’s Name
Due Date

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Chapter 1

1.1 Background Overview to Manifold (orientable and non-orientable)
A series of ‘patches’ sewn together in a ‘smooth’ manner is generally defined in a
manifold. The parametric equation is seen in each patch and the sewing’s smoothness makes no
cusps, corners, or crosses.
For instance, we look at a 1 sheet x 2+y 2−z 2=1 hyperboloid. The hyperboloid is a
revolutionary surface generated by rotating the hyperbola x 2 − z 2 = 1, which lies in the z-axis
on the (x, z)- stage. This can be configured with t — t (cosh t, 0,sinh t). The differentiable
parametry;
φ(t, θ) = (cosh t cos θ, cosh tsin θ,sinh t), −∞ < t < ∞, −∞ < θ < ∞.
can be configured to indicate the hyperboloid of the revolution(Chelnokov, Deryagina,
and Mednykh, 2017. pp.1558-1576).
We like each hyperboloid point (x, y, z) to be unique to its co-orders (t, altern) and, on
the other hand, to have each pair of co-ordinates (t, nom) allocated uniquely. The hyperbola x 2 −
z 2 = 1, y = 0, points equate to all the (t, ̄) with an æ whole integer of the 2′′s. This is not the case
when the parametrization is set to above. We can obtain one-to-one parametrizations by limiting
the mapping · to certain open subsets of R2:
φ1(t, θ) = (cosh t cos θ, cosh tsin θ,sinh t), −∞ < t < ∞, 0 < θ < 3π/2,
φ2(t, θ) = (cosh t cos θ, cosh tsin θ,sinh t), −∞ < t < ∞, π < θ < 5π/2.
The intersection of the hyperboloid with a set open in R3 should be observed as
representing each ći. The image of μ1 is the portion of the hyperboloid within the open region 0
< oral < 3 · 2 and the picture of ± 2 is the portion of the hyperboloid in the open region · < 5 μ/2

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in the cylindrical (r, ó, or z) cordon on R3. The ̈1 and ̈2 mappings are distinct, because the ̈1
and ̈2 images may be smooth surface patches.
At the heart of the general concept of a manifold are the following properties:
• Each ̈i shall be an injection map and ̈ −1 I shall be continuous, that is, ̈ −1 I shall be the
constraint on the constant map hyperboloid as specified in open set R 3. This state prevents the
surface from intersecting(Ferrández,Garay,and Lucas, 1991.pp. 48-54). • For each ̈i, vectors
“never-suspended,” “never-suspended” are linearly independent. ̈ This requirement guarantees
that the tangent plane spanned at each point on the surface is well established.
A subset S of R3 and a smooth parametrization set, the images of which cover S and
fulfill the above characteristics, are known as standard surfaces.
In two regions referring to 0 < all < μ/> and to μ < 3т/2, in this context, pictures from ·1
and ·2 are sewed together.
• The mapping ̈1 can be derived from the mapping ̈2 via a smooth coordaining exchange
in the regions where the representations of ̈1 and ̈2 overlap and ̈2 from ̈1 through smooth
coordinate exchanges. This implies that mappings φ1 and φ2 are available in the respective
accessible domains of R2, so that −θ12 and θ21 In addition, the opposite mapping is ~ twenty and •
twenty-one.
Indeed: φ2(t, θ) = φ1(t, θ) for all (t, θ) with t ∈ R and and φ2(t, θ) = φ1(t, θ − 2π) for all
(t, θ) with t ∈ R and 2π < θ < 5π/2. The corresponding smooth change of coordinates θ12 : R ×
[(π, 3π/2) ∪ (2π, 5π/2)] → R × [(π, 3π/2) ∪ (0, π/2)] is give in accordance to θ12(t, θ) = ½ (t, θ),
for t ∈ R and π < θ < 3π/2, (t, θ − 2π), for t ∈ R and 2π < θ < 5π/2.
Likewise,
θ21 : R × [(π, 3π/2) ∪ (0, π/2)] → R × [(π, 3π/2) ∪ (2π, 5π/2)]

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is given with respect to θ21(t, θ) = ½ (t, θ), for t ∈ R and π < θ < 3π/2, (t, θ + 2π), for t ∈
R and 0 < θ < π/2.
These variations in coordinates provide important surface detail. The above property is a
pillar of the concept of a mult...


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I was having a hard time with this subject, and this was a great help.

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