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:
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!
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.
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: