# Math Insight

### Quiz 1

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Total points: 8
1. The SIR model for the outbreak of a disease \begin{align*} S_{t+1} - S_t &= -b S_t I_t\\ I_{t+1} - I_t &= b S_tI_t - aI_t\\ R_{t+1} - R_t &= a I_t \end{align*} gives the evolution of the number of susceptible ($S$), infective ($I$), and removed ($R$) individuals.

Let's focus on the first time step, the evolution from $t=0$ to $t=1$. Setting $t=0$ in the above model, \begin{align*} S_{1} - S_0 &= -b S_0 I_0\\ I_{1} - I_0 &= b S_0I_0 - aI_0\\ R_{1} - R_0 &= a I_0, \end{align*} gives the formulas for going from the numbers of individuals $(S_0, I_0, R_0)$ at time step zero to the numbers of individuals $(S_1, I_1, R_1)$.

The model has two parameters $a$ and $b$, which we'll give the following values: \begin{align*} a &= 0.22\\ b &= 0.00085. \end{align*}

1. If there are $S_0 = 16000$ susceptibles, $I_0 = 730$ infectives, and $R_0=82$ removed individuals at time step 0, how many of each type of individual are at time step 1?

$S_1 =$

$I_1 =$

$R_1 =$

2. If the initial number of susceptibles is $S_0=311$ and the number of infectives is some (unknown) positive number, will the number of infectives increase or decrease during the first time step?

The number of infectives will
. (Write increase or decrease in the blank.)

Hint: the change in the number of infectives is \begin{align*} I_1-I_0 &= b S_0I_0 - aI_0\\ &= (b S_0 - a)I_0. \end{align*} You have numerical values for $b S_0-a$ and you know the sign of $I_0$. What is the sign of $I_1-I_0$? Does this mean the number of infectives is increasing or decreasing?

3. What value of the initial number of susceptibles $S_0$ will cause the number of infectives to remain constant in the first time step?
(It's OK if this number is not an integer.) Again, we assume the initial number of infectives $I_0$ is some (unknown) positive number.

Call this special number of suceptibles the “critical number” of suceptibles.

If $S_0$ is above this critical number, will the number of infectives increase or decrease in the first time step?

If $S_0$ is below this critical number, will the number of infectives increase or decrease in the first time step?

Hint: the number of infectives will remain constant in the first time step if $I_1-I_0 = 0$. Use fact that $I_1-I_0 = (b S_0 - a)I_0$, as above.

4. If the initial number of susceptibles is $S_0=649$, then the number of infectives should increase in the first time step. In this case, what's the minimum whole number of susceptibles that should be vaccinated so that the number of infectives decreases in the first time step? In other words, by how much do you need to reduce $S_0$ so that the number of infectives decreases (or at least doesn't increase) in the first time step?

Minimum number of susceptibles to vaccinated =

2. Compute the next four points of the solution to the discrete dynamical system $\left\{ \begin{array}{r c l} x_{n+1} & = & \frac{7}{2} x_n (1 - x_n) \\ x_0 & = & \frac{ 1 }{ 2 }\\ \end{array} \right.$ The solution must be accurate to 3 decimal places (the thousandths place.)

$x_1 =$

$x_2 =$

$x_3 =$

$x_4 =$