Unmasking Covid-19 with Math modeling

Mathematical modeling can help us understand the spread of a disease in a given population. Although a modeling approach may shed light on any contagious disease progression in general, I will mainly focus on Covid-19, the pandemic disease with a humongous impact on everything ranging from our personal life to the international economy. In this article, I am NOT trying to fit the models to real-world data; rather I’d like to give you a qualitative flavor about how these disease models, we frequently hear about in the news, work. If you don’t like math, don’t worry; I’ll try to explain everything in as common terms as possible. So, let’s get started!

The framework I’m going to discuss here is known as SIR (Susceptible-Infectious-Removed) model, where a given population is divided into three categories:

• Susceptible (S): Number of vulnerable people. Susceptibility varies depending on the disease; for Covid-19, almost the entire population is prone to infection, irrespective of age and gender. For conditions like chicken-pox or polio, younger kids are more vulnerable to infection.   
• Infectious (I): Number of people who carry the disease-causing agent (coronavirus for Covid-19).
• Removed: The infected person can either recover from the disease (Recovered, R) or die, unfortunately, due to disease-related complications (Dead, D).

​There are two major processes here: Infection and recovery. Once a susceptible person comes in contact with an infectious individual, there is a possibility (given by the rate α) of disease transmission. Once infected, the individual can recover or die with certain possibilities (given by βR and βD, respectively, where combined rate, β = βR + βD). So one individual would be shifted to the “I” pool with a rate α (infection rate), and removed from the pool with a rate β (removal rate). 

Assumptions:

1. Immunity: Usually, when someone recovers from an infectious disease, they would not be infected again due to acquired immunity. For most viruses, this is a reasonable assumption; however, if the virus mutates and comes up with a “fresh” strategy to attack us, this assumption might not work. There is some evidence that recovered patients of Covid-19 exhibit acquired immunity. Therefore this assumption may be fair game in our model. 
2. Population: We consider the total population to be constant throughout the simulation, which is a reasonable assumption for shorter time spans, like a few months.
3. Demographics: The infection and recovery rates for Covid-19, are correlated with gender and age. But for the sake of simplicity, we are not dividing the population into age groups or genders here. 

 Key concepts and results:

​1. Basic reproductive number or R0
​R0 is a very tricky measure, especially for an ongoing epidemic.  In our basic SIR model, R0 is simply the ratio of infection and removal rate multiplied by the initial susceptible population. When R0 is larger than 1, epidemic situation arises in a population. In other words, when disease transmission occurs more frequently than recovery or death, more and more people get infected, and community spread takes the form of an epidemic. Current estimates show that R0 for Covid-19 has a range of 2-5. 

​The graph below shows a typical population behavior under such situations (R0 = 2.5). Consider a population of 1000 individuals (blue curve), all susceptible initially. One infected person starts to spread the disease (Day 0); the number of infected people (Orange curve) spikes around 45 days or so. As time goes by, more people recover from the disease (green curve); a smaller fraction of people die as well (red curve). ​​

2. Flattening the curve
Larger R0 causes higher disease progression in a population, so what should we do to control it? Since R0 is a combination of infection rate, removal rate, and initial susceptible population (α, β, and S0), infection rate is a feasible parameter to change; in the absence of a vaccine or effective drugs, we can’t change the recovery (or mortality) rate. As we decrease R0 (by reducing α), the rate of infection slows down (see "Flattening the Curve" graph), and the total number of infected cases decreases (compare the green curve, R0 = 1.5 with the red, R0 = 4). Let’s say your local hospital can accommodate 200 individuals at max, R0 = 4 scenario would overwhelm the healthcare system, and patients would not get the necessary treatments due to scarcity of resources. That’s why public health officials implemented “social distancing” protocols; the primary purpose is to reduce α and prevent saturated healthcare systems. A lower R0 slows down infection, but it lingers in the population for longer, allowing for a second wave of infection if R0 is increased. For example, R0 = 1.5 curve reaches a peak after 120 days (4 months). So what’s the point of delaying the inevitable? Hold on to that question!​

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3. Lockdown Policies
To implement “Social distancing,” countries across the globe have adopted a “lockdown” approach. Since every job does not have the luxury to operate remotely, these lockdowns have devastated our economy. This raises the most important question: how long should we continue with the lockdown?  I have demonstrated the predictions for different periods of lockdowns (Lockdown Modeling graphs). One month from the onset of the disease, if we shut down everything for 1 - 3 months, we can delay the infection spike (orange curve) as well as the mortality rate (red curve). But when we lift those public interventions, there would be another wave of disease progression with more death. So we are back to our previous question: “why delay the inevitable?” The answer to this question lies in the ongoing research activities against COVID-19. As we delay the disease progression, we are buying scientists more time to come up with an effective drug and a vaccine ultimately. In this game of “economy vs. human lives,” if we look at a situation where there are “periodic one-month lockdowns” for a year, interestingly we can find a balance between minimizing the mortality and keeping the economy alive! 

Model calibrations
The results shown here are super sensitive to the parameters (α and β) of the model. The fun thing is that you can play with those parameters to reproduce the real-world data. In case you have a hard time matching your simulated data to the real data, you can always add layers to your model. For example, the “infectious (I)” group can further be divided into multiple categories with mild, medium, and severe symptoms, and these high-resolution models would give better predictions about the real situation.   

Remarks
I hope that I have convinced you how mathematical modeling is powerful enough to give us insights about epidemic progression. These models help our administrators decide on appropriate public policies for ensuring our safety.
If you are a math lover and want to explore this field in greater detail, check out the following videos:

1. The MATH of Epidemics | Intro to the SIR Model: https://www.youtube.com/watch?v=Qrp40ck3WpI
2. Simulating an epidemic: https://www.youtube.com/watch?v=gxAaO2rsdIs

Author

Aniruddha Chattaraj, M.S. @AniChattaraj
PhD candidate studying systems biology at UCONN Health. Interested in mathematical modeling of biochemical systems, protein phase separation, and actin dynamics. Enjoys dancing, cooking, and participating in science communication.