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Many zombie movies are based on the premise that zombies

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QUESTION:

Many zombie movies are based on the premise that zombies do not stop infecting new victims until they are destroyed by a susceptible. In addition, the susceptibles destroy as many zombies as they can. We can model the spread of zombies in such a movie by assuming that infecteds (zombies) become recovereds (zombies who can not infect susceptibles) at a rate proportional to the size of the remaining susceptible population. We obtain the system

\(\frac{d S}{d t}=-\alpha S I \)

\(\frac{d I}{d t}=\alpha S I-\gamma S\)

(a) Calculate the equilibrium points of this model.

(b) Find the region of the phase plane where \(d I / d t>0\).

(c) Use \(\alpha=0.2\) and \(\gamma=0.1\) and sketch the phase portrait. What does the model predict for the spread of the zombies in this case?

The SIR model is particularly relevant to a homogenous population in an environment with little geographic distribution. A famous example of exactly this situation occurred in 1978 at a British boarding school. A single boy in the school of 763 students contracted the flu and the epidemic spread rapidly, as shown in Table 2.3. (We are assuming that the number of students confined to bed was the same as the number of infected students.)

Table 2.3

The daily count of the number of infected students.

\begin{array}{cc|cc|cc} \hline t & \text { Infected } & t & \text { Infected } & t & \text { Infected } \\ \hline 0 & 1 & 5 & 222 & 10 & 123 \\ 1 & 3 & 6 & 282 & 11 & 70 \\ 2 & 7 & 7 & 256 & 12 & 25 \\ 3 & 25 & 8 & 233 & 13 & 11 \\ 4 & 72 & 9 & 189 & 14 & 4 \\ \hline \end{array}

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QUESTION:

Many zombie movies are based on the premise that zombies do not stop infecting new victims until they are destroyed by a susceptible. In addition, the susceptibles destroy as many zombies as they can. We can model the spread of zombies in such a movie by assuming that infecteds (zombies) become recovereds (zombies who can not infect susceptibles) at a rate proportional to the size of the remaining susceptible population. We obtain the system

\(\frac{d S}{d t}=-\alpha S I \)

\(\frac{d I}{d t}=\alpha S I-\gamma S\)

(a) Calculate the equilibrium points of this model.

(b) Find the region of the phase plane where \(d I / d t>0\).

(c) Use \(\alpha=0.2\) and \(\gamma=0.1\) and sketch the phase portrait. What does the model predict for the spread of the zombies in this case?

The SIR model is particularly relevant to a homogenous population in an environment with little geographic distribution. A famous example of exactly this situation occurred in 1978 at a British boarding school. A single boy in the school of 763 students contracted the flu and the epidemic spread rapidly, as shown in Table 2.3. (We are assuming that the number of students confined to bed was the same as the number of infected students.)

Table 2.3

The daily count of the number of infected students.

\begin{array}{cc|cc|cc} \hline t & \text { Infected } & t & \text { Infected } & t & \text { Infected } \\ \hline 0 & 1 & 5 & 222 & 10 & 123 \\ 1 & 3 & 6 & 282 & 11 & 70 \\ 2 & 7 & 7 & 256 & 12 & 25 \\ 3 & 25 & 8 & 233 & 13 & 11 \\ 4 & 72 & 9 & 189 & 14 & 4 \\ \hline \end{array}

ANSWER:

Step 1 of 4

a) In order to find  the equilibrium points ,

Set \(\frac{d S}{d t}=0\)

\(\begin{array}{l} \frac{d S}{d t}=-\alpha S I \\ -\alpha S I=0 \end{array}\)

We see that \(S=0\) or \(I=0\)

Now we set \(\frac{d I}{d t}=0\)

\(\begin{array}{l} \frac{d I}{d t}=\alpha S I-\gamma S=S(\alpha I-\gamma) \\ =S(\alpha I-\gamma)=0 \end{array}\)

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