Like a lot of people around the globe, I'm a little worried about the Coronavirus spreading and I thought rather than reading statistics from the news and media, I would take matters into my own hands and work things out (approximately).
I am trying to express an exponential virus growth in a population. So every day, the number of people infected doubles and once people have recovered they cannot re catch it. However, when I try and formulate the problem, I feel that my functions start to get recursive and that I am doing something wrong.
This is what I have tried so far and I am feeling confused by it.
p is the population of the country and k is the duration of viral infection in days.
$$ p = 6.6\times10^{7}\, $$
$$ k = 14 $$
$$ Y(d) = (1-(Y(d-1)/p)) 2^{d} – Y(d-k)\, $$
I thought I would see something in this kind of shape:
Please help.
Best Answer
According to your assumptions your model should rather look like $$ Y(d) = 2Y(d-1) -Y(d-k)$$ What you describe doubles the healthy portion of the population "minus the recovered". Although the above does not take into account the population cap, namely that $Y\le p$. A more suitable model would be e.g. $$ Y(d)= 2\bigg(1-\frac{Y(d-1)}p\bigg)Y(d-1) - Y(d-k) $$ while $Y$ is small compared to $p$ this is basically doubling the infected, but as soon as Y assumes values comparable to p the rate of growth slows. The other option to the first equation would be $$ Y(d)=\begin{cases} 2Y(d-1) - Y(d-k) & \mbox{if }Y(d)< p \\ p &\mbox{if } 2Y(d-1)-Y(d-k)\ge p \\ Y(d-1)-Y(d-k) & \mbox{if }Y(d-1)=p, \mbox{and after reaching p} \end{cases} $$