Finding a recursive for virus outbreak

Question

Suppose a contagious person infects 3 new people in one day. After a day, these newly infected people become contagious themselves. Every contagious person goes into quarantine after one day, meaning they do not infect any more people. To summarize:

• a person becomes infected and is not contagious
• after 1 day they become contagious and infect 3 people
• 1 day after that they get put in quarantine

I want to find a recursive formula that gives me the number of people, that are not in quarantine (Infected or Contagious)

My approach was the following:

I have found an recursive Formula:

$$\overline{Q}_n =3* \overline{Q}_{n-1}$$ under the conditions $$ n>1, \overline{Q}_{0}=1, \overline{Q}_{1}=4$$

Is there a way to make the recursive formula so, such that only one initial value is required? Thanks in advance

Answer 1

You have two sequences involved:

• $I_n =$ number of infected but not contagious people on day $n$.
• $C_n =$ number of contagious people on day $n$.

There is no particular reason to worry about the number of quarantined people for this problem. The givens are:

• $I_0 = 1$
• $C_0 = 0$
• $C_n = I_{n-1}$
• $I_n = 3C_n$, $n > 0$

The $n > 0$ clause is necessary because the given data for day $0$ does not match the information that $I_n = 3C_n$, since $1 \ne 3\times 0$. I.e., the initial infection arrived in a way other than in accordance to the rules that hold going forward.

And it is this exception that you are running into in your sequence. $\overline Q_n = \frac 43 I_n$, but only for $n > 0$. That additional $\frac 13$ contagious person required by the later rules just isn't there on day $0$.