A simple mathematical explanation on how to slow the spread and end the pandemic
Introduction
COVID-19 is highly contagious and it spreads from an infected individual to a healthy person easily. Let us keep in mind that COVID-19 is a deadly disease killing a significant percentage of people who get infected and is no way comparable to the seasonal flu. Therefore, if millions of people get infected, hundreds of thousands will die, a very sad reality. The death of so many individuals is an extremely sad situation by itself and should be prevented at all costs. For people who are worried about the economy, the key point is that significant loss of lives over an extended span of time will inevitably cause the economy to collapse. So, no matter from which angle we look at this situation, the main crux of the matter is that we need to prevent large number of deaths.
Therefore, in this article, I explain how to slow the spread and end the pandemic using simple mathematics so that every layperson can understand it. I rely only on high school mathematics so it is easy to understand.
Disclaimer to the mathematically inclined: This is a very simplified modeling approach, but it is sufficient to understand why social distancing and strict lockdowns for long durations are essential.
To understand how COVID-19 will affect our society we consider three cases (I am going to mention up front that Case 3 is the only bright case). I assume that people reading this article are already familiar with terms such as social distancing, lockdowns, and stay-at-home orders.
Case 1: What happens when we implement limited social distancing and quarantining measures.
Let us assume that there are 100 infected people in a community who are still asymptotic. Let us assume that an infected person infects 2 healthy people even before they realize that they have the disease and practice social distancing. For COVID-19, an infected person infecting a minimum of 2-3 healthy individuals when no social distancing measures are in place is a good assumption (just look at the number of cases in the United States). Therefore, 100 infected people will infect 2*100 = 200 healthy people. These 200 people will in turn infect 2*200 = 400 healthy people. In the next step, the number of newly infected people will be 2*400 = 800. Therefore, as one can see the numbers keep getting multiplied by 2 and the number of cases increases rapidly. This process leads to an exponential increase in the number of cases and that is why there have been calls to flatten the curve.
Case 2: What happens when we implement social distancing and minimal lockdown measures such as suggested stay-at-home orders?
When we practice social distancing and venture outside less often due to the lockdown, the chance of an infected person coming in close proximity of a healthy person decreases. With this kind of approach, let us assume a reduced rate of spread of every 10 people infecting 11 healthy individuals (i.e., each infected person infects 1.1 healthy people on average), a sharp reduction from an infected person infecting 2 healthy individuals. If we again consider 100 infected people in a community, then these 100 people infect an additional 1.1*100 = 110 healthy people. These newly infected 110 people infect 1.1*110 = 121 healthy people. Thus we can see that even though the rate of spread has been reduced drastically, the number of new cases still is increasing at an alarming rate. Sadly, as long as the rate is above 1 (i.e., each infected individual infects more than 1 healthy person), the number of new cases will increase quickly.
Case 3: What happens when we implement social distancing, strict lockdown (similar to the ones in Europe) and greater testing.
With these strict measures in place, the chance of infected people coming in contact with healthy individuals decreases further. While at first glance it might appear that this kind of extreme step should immediately drive the rate of spread from an infected person to 0, this is unlikely to happen. This is because people still have to venture out for groceries, outdoor exercise, and healthcare needs. Importantly, essential workers still have to keep working. Most essential workers (e.g., postal workers, sanitation workers) may not have adequate protection and therefore are at risk of contracting the disease and spreading it to their co-workers as well as the greater community. Additionally, there are always a group of individuals who flout these rules, thus increasing the overall risk of spread.
Let us assume that with these strict measures 10 infected people are only capable of infecting 7 healthy individuals (i.e., each infected person infects around 0.7 healthy people on average). Once again, if there are 100 infected people, they will infect 0.7*100 = 70 healthy people. But, the good news is that number of new cases have now decreased. These 70 people will then infect 0.7*70 = 49 healthy individuals. This number will slowly decrease and go to 0. Therefore, we see that making sure that the rate of spread is below 1 is key to decreasing the number of cases.
Now, the big question is - When to reopen?
Our above discussion focuses on the kinds of measures needed to contain the spread of the virus. The question of when to reopen is a difficult one, but if we focus on the economy and not mathematics, we will land in much greater trouble. To answer the question, let us break it down into simpler questions.
Question 1. How quickly can we drive the number of new infections to zero?
To understand this, let us again consider the case when the rate of spread is less than 1 (i.e., 0.7). Let us consider two scenarios:
Scenario 1: We have 100 infected individuals
Scenario 2: We have 1000 infected individuals
In scenario 1, we will take 10 cycles to bring down the number of new infections to 3. On the contrary, in scenario 2, we will take 100 cycles to bring down the number of new infections to less than 3. In reality, the value will be much less than 100 in scenario 2 (this is the result of our simple model), the main point of this exercise is that if we have more number of undetected cases in the community, it will take a significantly longer time to reduce the number of newly reported cases to 0.
Question 2. What happens if we reopen too early?
Let us assume that when we reopen, the number of undetected and asymptomatic cases in the community is 50. Let us assume that when we reopen we still practice social distancing and some lockdown (i.e., we land in Case 2). The rate of spread to healthy individuals increases to more than 1 per infected person (say 1.1). The number of cases thus increases to 1.1*50 = 55. These 55 people then infect 1.1*55 = 61 healthy individuals. As we can see the number of new cases again increases quickly.
Question 3. So what should we do?
We should first enforce strict lockdowns similar to the ones implemented in China and other countries to drive the number of new cases to zero. After the number of cases have been driven to zero, we should still enforce the strict lockdown for some duration (2 weeks) because the virus will still be around in the environment and infect healthy individuals.
The ease of the lockdown has to be gradual and we should be very careful as the virus will still infect healthy individuals because it is now part of our eco-system. Because the virus has spread this drastically, it will be foolish to assume that it will disappear completely.
Rapid testing and contact tracing will be very powerful tools in this situation because we can quickly identify new infections and quarantine individuals who have been in contact with the infected persons. This will drive the rate of spread per infected person below 1 (remember Case 3 is the only optimistic scenario) and we can push the number of cases close to zero again.
Conclusion
The above analysis shows that strict lockdown, rapid testing, and contact tracing are the key tools we have at our disposal here. We as a society need to be patient and acknowledge the power of the enemy. The enemy has nothing to lose and only way to defeat it is by limiting its spread. The road ahead is bumpy, but if we do not unite as a society, our chances of winning this battle will be limited and a lot of innocent lives will be lost. If we take a strong dose of the medicine early, then we can win; otherwise, we will have to keep taking it for an extended period of time. I understand that this requires a lot of sacrifice and also restricts our free movement, but what use is freedom to a dead person?
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