Understanding the Covid-19 epidemic and the hidden laws that govern how diseases spread

A whisper. That’s how it always begins. Then, the whisper becomes a murmur, the murmur a rumor, and the rumor spreads like wildfire among family, friends, close ones, until it expands outside our inner circle and no one can make sense of what is truly right or wrong… or where it all began in the first place. In our interconnected world today, misinformation and disease alike can spread more readily than ever before. Epidemics are just one instance of how widespread proliferation occurs in populations. So how can we make sense of why things spread? Or perhaps more importantly, how can we stop them? As it turns out, the spread of the Covid-19 virus and misinformation surrounding it aren’t so different once we comprehend the rules of contagion.

Two important concepts to differentiate when discussing epidemics are virulence and pathogenicity. The former describes the potential of an infectious agent to cause harm, and it’s best described by case fatality rate (CFR). The latter describes the ability to cause disease and is measured by R0 – the number of new infections generated from one patient. This variable is crucial for understanding the nature of infectious diseases. Many factors affect R0, but mathematically R0 can be best understood using the Susceptible-Infected-Recovered or SIR model. 

The SIR model is constructed upon a few base assumptions. First, using the categories of susceptibles (\(S(t)\)), infected (\(I(t)\)), and recovered (\(R(t)\)), the SIR model assumes that the three categories encapsulate all individuals with everyone beginning as a susceptible individual. Thus, \(S(t)+I(t)+R(t)=N\), the number of people in our designated population whether it be the world or a single country. Next, the model assumes that once a susceptible person becomes infected, that person becomes immune. Thus, the infected person is moved to the recovered category (assuming that the death rate is small in comparison to the recovery rate). Once a person recovers, the model assumes that they can no longer get the disease, nor can they give it to anyone else.

Now, we can mathematically represent this model through a system of differential equations, each representing how the three categories (SIR) change with respect to time. The idea is to note a rate of change with respect to time, also known as the derivative, for each of these categories to construct a model that adheres to all three equations. In order to define these derivatives, we need to think about what affects each of the three individual categories. 

Beginning with the rate of change for the susceptibles group \(\frac{\partial S}{\partial t}\), the way that a susceptible person may leave the group and become infected is if a susceptible person and an infected person come into contact. Then, there exists some probability that the susceptible person becomes infected because of that interaction. Thus, we can write \(\frac{\partial S}{\partial t}=-aSI\), where \(a\) is some positive proportionality constant based on the idea that as there exist more susceptible and more infected peoples (and thus more interactions between them), there is a higher chance of infection for susceptible people. 

Now looking at the change in our infected group \(\frac{\partial I}{\partial t}\), the number of infected people increases when susceptible people become infected. Hence, we want to add the quantity \(aSI\) that we lost from the susceptibles into the infected group. However, that’s not the only change that can occur. We also have to account for the loss of infected people as they slowly begin to recover. To account for that change, we subtract out the term \(bI\) where \(b\) is another positive constant affecting the rate of recovery based on the idea that as more infected people exist, more will recover as well. Therefore, we can write \(\frac{\partial I}{\partial t}= aSI-bI\). Finally, for the recovered group, if we are losing \(bI\) from the infected group, we are gaining that amount in the recovered group; thus, \(\frac{\partial R}{\partial t}=bI\).

This is the basic system of equations that can model the spread of an infectious disease. Much like any other system of equations, all three of these must be concurrently true in order to satisfy the model. While solving this system of nonlinear equations turns out to be quite challenging, these equations have a lot of nice features that we can use to analyze the spread of disease. Examining the graph of what \(I(t)\) may look like for a particular population, we can see that the graph resembles a sort of hump-like figure, where the number of infected goes up quickly in the beginning, begins to flatten out at its peak, and then slowly falls down again as more people recover and less susceptible people remain. Because a major goal for anyone who studies epidemics is to reduce the number of infected, our equation for the change in infected population, \(\frac{\partial I}{\partial t}\), is particularly interesting. 

One of the big questions regarding the \(\frac{\partial I}{\partial t}\) term is whether or not the number of infected people is increasing or decreasing. The way academics can tell whether or not we are having an increase or decrease in number of infected is by examining if \(\frac{\partial I}{\partial t}\) is less than 0 or greater than 0. By writing out the terms of \(\frac{\partial I}{\partial t}\) with the \(I\) factored out, we can see that \(\frac{\partial I}{\partial t}= I(aS-b)\). Setting this expression to be less than 0, we can divide out the \(I\) term, move the \(b\) to the other side of the equation, and divide both sides by \(b\) to find that the number of infected people is decreasing if \(\frac{aS}{b}<1\). As it turns out, the \(\frac{aS}{b}\) term is actually given a special name that may sound familiar: R0

So how exactly does this all relate to our current epidemic?

One of the things people around the world have been encouraged to do is wash their hands. What this actually means in regards to R0 is that people are encouraged to lower the transmission rate represented by our term \(a\) in \(\frac{aS}{b}\). A lot of the global health policies encouraging quarantine and other social distancing practices have all been an attempt to lower the transmission rate in order to increase the probability that the ratio R0 is less than 1. Another factor we can alter is the \(S\) term or the number of people who are susceptible. This is often done by creating a vaccine in order to create herd immunity. The \(b\) term is a little bit harder to change since we essentially have to keep the people who are currently infected alive so their immune systems can eventually help them recover.

All in all, R0 can be thought to mathematically represent whether an epidemic will continue or not. However, even if R0 is less than 1, outbreaks can still be surprisingly large. A classic calculation shows that for R0 less than 1, the average number of people in a chain of infections can be found by calculating \(\frac{1}{1-\text{R}_0}\). So with R0 = 0.8, there will be an average of 5 people in a chain. Even so, a small percentage of chains will have sizes much larger than the mean. This often occurs because of so-called “superspreaders,” or individuals who by choice or necessity have interacted with many people. Merely 1 “superspreader” can cause a large outbreak with hundreds of people infected through a single chain. 

Therefore, in the context of the current Covid-19 epidemic, it can be misleading to look at our data and celebrate once R0 falls below 1. Epidemics differ by country, state, and county, so if we are not truly careful, we run the risk of propagating the epidemic among particular places or groups. Additionally, hotspots will occur. As we can see through superspreaders, it is inevitable that even with small spread on average, there will exist cases where the disease spreads to many people from a single source. The SIR model and its qualities can only tell us so much, and many, much more complex models exist to refine this rudimentary model. As a whole, if we continue to test and trace the spread of the Covid-19 virus, we will prevail and get the virus under control. There will be flare-ups, and there will be cases of strain on local resources, but as long as we continue to offer our generosity for those in need, we will push through these trying times. Together, we will emerge anew.References