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#### Use MATLAB to solve the below questions 1) On January 10 December 2021, 100 students attend a legal rave in Mumbai

###### Math

Use MATLAB to solve the below questions

1) On January 10 December 2021, 100 students attend a legal rave in Mumbai. While having fun, but unbeknownst to them, 10 of the participants had just been exposed (and cannot transmit yet) to COVID-19, while other 10 are already infectious (and can transmit further). The infection starts to spread in the City according to the following model

We here assume equal birth and death rates, and the parameter values N = 250, 000, a = 1/14, β = 8 and γ = TR/TC and m = TD/TC (in the notations from the below Table).

Total Cases (TC)                Total Deaths (TD)             Total Recovered (TR)      Active Cases (AC)             Serious Critical (SC)

3,316,019             87,295   1,503,654             1,725,070             3,722

Table: Coronavirus information

•             Solve the problem numerically (you may use a built-in routine) and discuss the dynamics of the population compartments (S, E, I, R) if the conditions stay the same.

•             Estimate in how many days will the 30 beds available in the (ICU) at the Mumbai  Hospital be filled. Discuss the emergency budget required to order extra beds at £200k/unit, to accommodate all ICU patients at the peak.

•             Discuss some actions for “flattening the curve” and model their impact.

Hint. Relate to the model parameters and discuss the role played by R0.

•             Formulate and discuss (briefly) two extensions of the model, which consider other realistic features (vaccination, mutations of the virus (delta, omicron), reinfection, etc). Hint. The evaluation checks the model description, equations, simulations, analysis.

Question 2:

VERHUSLT MODEL

A population of 300 coyotes is relocated to the park at time t = 0. In this environment, the evolution of the coyote population x(t) is described by the equation:

•             Derive the exact formula describing the evolution of the coyote population x(t).

•             Implement Euler’s method for solving this model. Implement the 4th order Runge Kutta method (RK) for this model. Compare the Exact solution, Euler and Runge Kutta  solutions, illustrating the accuracy of the methods.

Hint: Plot (t, x(t)) obtained from all methods on the same graph, for 0 ≤ t ≤ 60.

•             (You can solve this model by using ODE 45) Build a model with a hunting rate proportional to x(t), i.e., H(t) = k(t)x(t). Function k(t) has a period of 12 months (k(t + 12) = k(t)), while k_min = 0, k_max = 1 and k(0) = 0. Plot your solution over five years, using a time-unit of 1 month.