How do you model growth of a virus?

Question

So since this pandemic is going on, I thought to model the growth of the virus in my area (not seriously, just as an exercise). I tried some things on my own and took the average growth factor over the course of a week and did: (current no. of cases)*(growth factor)^(number of weeks). This method was kind of accurate but it was not realistic because the number of cases grew exponentially and didn't stop.

Later on, I learned about differential equations from YouTube and other sources (as calculus hasn't been taught to me in school yet). But I couldn't understand things fully.

In the logistic differential equation $\frac{dN}{dt} = kN(1-\frac{N}{L})$. Its solution is $N(t)=\frac{N_0*L}{N_0+(L-N_0)e^{-kt}}$. Where $N$ = number of cases at time $t$, $N_0$ = initial number of cases, $k$ = constant of proportionality, $L$ = limiting/carrying capacity.

I tried putting k as the growth factor but I don't know what to put as L as I don't know what the limiting capacity is. I can't put the limiting capacity as the total population as the number of cases asymptotes towards a number much lower than the total population. So then I decided to take the percent of the total population of countries affected by COVID-19, but I don't think that would work because there are differences in healthcare facilities and other factors between these places. How do I correctly model this? I hope the question was clear.

Answer 1

I did a work exactly about what you are talking about, what I did was an approximation of the effectiveness of different health care measurements, search, for example, how many people are at quarantine, so you can reduce your maximum capacity L. My research is in spanish but I would love to share it. Maybe you would want to do the investigation only for a certain country, making the recollection of measures that redule L much easier.

Answer 2

Since you are using a logistic equation for the number of cases, you can try to parametrize the two parameters in your model using case data. For this, you will need at least all the data until the peak of the epidemic.

For the pandemic, since you are looking to model using logistic equations, one simple way is to use the logistic patch model (see this). For each country/region/city/county/etc. (similar type), you can increase the number of patches by one.

As Clarinetist points out, you should only do this as an exercise to gain some understanding of the dynamics of a pandemic. However, I am unaware of any successful attempt at modeling/forecasting the current pandemic - the previous forecasts are proven to be completely off the mark. There are many reasons for this, but it is not the place to discuss this topic.