NJIT Math 222
Fourth MATLAB Assignment
Computational project part 1
I ran the code with $N=10000$, $\gamma=1$ and $\beta$ varying. The assignment says to expect different behavior depending on whether or not $R_0>1$, where $R_0 = \frac{\beta N}{\gamma}$.
For these values of $N$ and $\gamma$, this an epidemic when $\beta> 10^{-4}$. Therefore I tried three values of $\beta$: $2\times 10^{-4}$, $1.25\times 10^{-4}$, and $9\times10^{-5}$. For the first two values, I saw growth, with a larger epidemic in the first case $\beta= 2\times10^{-4}$.
My results are in a
MATLAB-generated webpage . This was generated from
Computational project part 2
Here you were asked to use phase-plane drawing software to examine solutions graphically. It was pointed out that $\frac{dI}{dt}=0$ when $S^* = \frac{\gamma}{\beta}$. From the following three images, we see that $S_0> S^*$ when $\beta>10^{-4}$, which is when we see epidemics.
![Phase plane with $\beta=2\times10^{-4}$, $S^*=5000$](/media/math222/beta_0p0002_hu74f78e4a0b51da5ebac6f39b91eda5ee_200334_d06cab5b48132a838fdce4141a80bbca.webp)
![Phase plane with $\beta=1.25\times10^{-4}$, $S^*=8000$](/media/math222/beta_0p000125_hu2af8b23f53d3b98ca9d149ebd3f3a352_186785_2c2ff3b48ebc4db86d37664353dfe229.webp)
In the third case, $\beta=9\times10^{-5}$ $S_0=11111>N=10000$, so the infection number is decreasing at $t=0$.
![Phase plane with $\beta=9\times10^{-5}$](/media/math222/beta_0p00009_hu1813a56d341fb3d41f535cc3c8b82b0e_161516_0c8ea4d3d72b3b023c10e6053b2064ba.webp)
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.