MATLAB Assignment 4 Solution

NJIT Math 222

Fourth MATLAB Assignment

Computational project part 1

I ran the code with $N = 10000$ , $γ = 1$ and $β$ varying. The assignment says to expect different behavior depending on whether or not $R_{0} > 1$ , where $R_{0} = \frac{β N}{γ}$ .

For these values of $N$ and $γ$ , this an epidemic when $β > 10^{- 4}$ . Therefore I tried three values of $β$ : $2 \times 10^{- 4}$ , $1.25 \times 10^{- 4}$ , and $9 \times 10^{- 5}$ . For the first two values, I saw growth, with a larger epidemic in the first case $β = 2 \times 10^{- 4}$ .

My results are in a

MATLAB-generated webpage . This was generated from

this MATLAB code .

Computational project part 2

Here you were asked to use phase-plane drawing software to examine solutions graphically. It was pointed out that $\frac{d I}{d t} = 0$ when $S^{*} = \frac{γ}{β}$ . From the following three images, we see that $S_{0} > S^{*}$ when $β > 10^{- 4}$ , which is when we see epidemics.

Phase plane with $\beta=2\times10^{-4}$, $S^*=5000$ — Phase plane with $β = 2 \times 10^{- 4}$ , $S^{*} = 5000$

Phase plane with $\beta=1.25\times10^{-4}$, $S^*=8000$ — Phase plane with $β = 1.25 \times 10^{- 4}$ , $S^{*} = 8000$

In the third case, $β = 9 \times 10^{- 5}$ $S_{0} = 11111 > N = 10000$ , so the infection number is decreasing at $t = 0$ .

Phase plane with $\beta=9\times10^{-5}$ — Phase plane with $β = 9 \times 10^{- 5}$

There is also a question asked “Can this model support a sustained epidemic.” The answer is that it cannot. In a sustained epidemic, the infection rate $I (t)$ would have to reach a nonzero steady state but the phase plane shows that $lim_{t \to \infty} I (t) = 0$ for all solutions, so it can’t sustain an epidemic. To support a sustained epidemic, a model must refresh its supply of susceptible individuals. Fortunately for the germs, we do that anyway by having babies! A more complete models will include births.