# How Would We Know COVID-19 is in Decline?

In light of the current COVID-19 pandemic, I have written an article entitled How Does Disease Spread? in which I discuss how mathematicians attempt to understand the way that a virus is spreading in the population. In said article, my primary focus is building up the concepts and thought processes that are needed to understand how a mathematician would come up with equations to simulate a virus spreading. I now want to write a brief follow-up addressing a related question:

In light of the behavior of viruses, how can we determine whether things are improving or getting worse?

This question actually isn’t quite as easy as it seems. The initial, intuitive response would be to look at the current number of contagious people, and to call any increase in that number bad and any decrease good. Of course, it is a bad thing when more people are becoming ill and it is a great thing when the total number of infectious people is going down, but that is actually not a very good indicator. The reason, as I will try to explain in this article is that in reality, the improvement actually starts to happen before the number of active cases starts going down.

A Brief Review of How Viruses Spread

Because viruses spread through direct or indirect personal contact, it is absolutely fundamental that at the beginning of the spread of a virus, the virus grows at an exponential rate – which basically just means that the speed at which is spreads is directly related to the number of people that have the virus. This is why numbers have been changing so drastically. When more people are infected, there are more ‘opportunities’ for others to also get infected, and so the number of new infections keeps going up.

Mathematicians recognize exponential equations as one of the most important of all equations, not only because they appear in numerous real-life situations like virus spread and population growth, but also because they are interesting in their own right. Exponential equations are so important that there is a notation developed specifically to help us write down these – normally we just use the term exponents. As an example of this form of writing, the number $2^2$ just means $2*2$, the number $2^3$ means $2*2*2$, and more broadly, the number $2^n$ means 2 multiplied by itself $n$ times. There is nothing special about the 2 either – if $a$ is any number, writing $a^n$ just means multiplying $a$ by itself a total of $n$ times.

Exponential Growth is Highly Sensitive

The number of cases of a virus always begin with exponential growth. But eventually, there are factors that slow this down. This is the point of social distancing and all travel shutting down – these measures slow things down. And as it turns out, even tiny changes produce massive shifts in the outcome of exponential equations. To see why, we can use made-up viruses with made-up equations. Let’s say some evil genius created two new viruses – let’s call them Virus A and Virus B. Since he is a genius, after all, he is able to figure out the equations that tell him how the viruses will spread – which are exponential. He finds that if we begin at “Day Zero” with only one person affected by Virus A, then after $n$ days, there will be about $(1.1)^n$ people infected by Virus A. He does the same thing for Virus B, and the equation turns out as $(1.15)^n$.

At first, his reaction might be “Well, A and B are almost identical! So I can use either one to wreck havoc.” But if he look a little closer, these actually turn out to be massively different, even those 1.1 and 1.15 are nearly the same number. To see this, we will compare the value after one month – 30 days – of infection. The two equations give values of $(1.1)^{30} \approx 17$ and $(1.15)^{30} \approx 66$. If you went for 100 days, then Virus A has infected a little over 10,000 people, but Virus B has already infected more than 1,000,000 people!

This is one of the fundamentally important aspects of exponential growth that mathematicians have understood for centuries – changing the number ‘on the bottom’ by even the tiniest amount has immense consequences. As a side note, this is why everything each of us do during this crisis matters so much, and this is why we are taking such drastic measures. Because of what a virus is and how it spreads, mathematics tells us that even small improvements in sanitation can, over the course of a month or two – make a difference in the thousands or even the millions.

How Things Get Better

This is not exactly obvious from our discussion so far, but not all exponential equations become large rapidly – some of them actually become small very rapidly. The ideas we need are summarized in 3 bullet points:

• Multiplying a number by 1 doesn’t change anything.
• Multiplying a number by something between 0 and 1 makes it smaller.
• Multiplying a number by something larger than 1 makes it bigger.

Since exponential equations are all about multiplying numbers together, this matters quite a lot. The value of $(1.00000001)^n$ will eventually become astronomically large if you give it enough time, and the value of $(0.99999999)^n$ will, if you give it enough time, eventually become indistinguishable from zero. The moral of all of this is that for exponential equations, it is not increasing or decreasing that matters most, it is about whether the number we are multiplying by is bigger than 1 or less than 1. To interpret this in terms of the spread of the COVID-19 virus, what matters is not the current number of cases, but the number of new cases every day. If there are fewer new cases today than yesterday, that is what is really important. And more than that – when you compare two days, you don’t use subtraction, because exponential equations have nothing to do with addition or subtraction. You use division to compare days, not subtraction. What we really care about is the value

$\dfrac{\textrm{Number of new cases today}}{\textrm{Number of new cases yesterday}},$

and our goal is to make this number smaller than 1. Of course, on a day-to-day basis, this number – called the growth factor – may change quite a bit. But if we have several days in a row of a growth factor less than 1, that is a strong sign that the situation has entered the stage of getting better rather than worse. And once the growth factor becomes less than 1, it is only a matter of time until the peak number of infections will be in the past and we will be on our way to a recovery from the pandemic.

Conclusion and Clarification

Obviously, this is a simplification, I am not an epidemiologist and I have never professionally studied the models that professionals are using to model this virus. However, I can say as a mathematician that the growth rate absolutely does matter, and if you see things about the growth rate in the news or on websites that are tracking the statistics of the virus, there is a good reason those numbers are there. They help us a lot in understanding whether things are getting better or worse.

As I become more aware of the mathematics myself, I will continue to write more about what is going on and how professionals are understanding all the numbers. For now, I echo the advice that governments around the world have put forward – stay inside as much as possible, stay sanitary, and seek medical advice if you show any symptoms. Let’s beat this together.