COVID-19, Mathematical Models, and Masks

on August 21, 2020

Four months, two weeks, and five days.

We are four months away from the one year anniversary of the first reported case of the novel coronavirus (SARS-CoV-2).

For many of us, this year has produced anything but stability and reassurance. Canceled graduations. Missed births. Last words via FaceTime. In a time of uncertainty, isolation, and grief, the frustration of the unknown and the realization of “we do not know” can be crippling. There is an unforgiving darkness, a heaviness hovering over humanity at the moment.

But what we do know is powerful.

Through the analysis of mathematical models, the spread of this virus can be mitigated. The act of social distancing. The proper use of masks. The practice of good hygiene, including frequent hand washing. The “math” behind the graphs you find on your preferred news source is often avoided in the explanation of such preventive measures. Yet, these recommendationsin addition to massive government shutdowns and quarantineshave resulted in the lowering of cases in many countries.

[SIR Model Example]

The importance of these measures are derived from the analysis of epidemiological models, in particular, the SIR model. The SIR model (Susceptible-Infectious-Removed) is the most widely used model for the spread of infections of COVID-19. This basic model and its variants (SEIR, etc.) allow for doctors, officials, and citizens to isolate peaks of the number of infections, the rate of effectiveness of preventive measures, etc. This formidable tool in stopping the spread of the virus is based on the analysis of a system of differential equations. 

The SIR model separates the total population (=N) that will be affected by the virus into three categoriesthe susceptible, the infectious, and the removed (S + I + R = N). “Susceptible” represents those who have not contracted the virus. “Infectious” represents those who have contracted the virus and are infectious to those in the susceptible group. “Removed” represents those who have recovered from the virus and cannot contract the virus again. However, in other models and in reality, it is possible to contract the virus multiple times. This model categorizes the process of infection as a susceptible person making prolonged contact (interaction) with a member of the infectious class. To go from the “infectious” group to the “removed” group, there is no interaction necessary between the two groups; this move in between groups is spontaneous.

[Diagram of SIR Model]

Analyzed over time, each group has a derived differential equation, which when analyzed together, becomes a system of differential equations. You may notice that in front of the ‘S’ variable and the ‘I’ variable, there are unknown quantities, or parameters, β and γ. These parameters, beta and gamma, respectively, represent the rate of virus transmission between S and I and the rate at which the infectious join the removed group. Therefore, a low β value and a high γ value are desired. The value of γ can increase through better treatment methods and environmental factors. This value mostly cannot be increased by the everyday practices of individuals of the population. However, the value of β is extremely affected by the use of masks in public spaces, hand washing, and maintenance of social distancing guidelines. This is something that we do know and can control.

[System of Differential Equations of SIR Model]

Forever changed, our typical processes and culture that preceded the worldwide spread of the novel coronavirus cannot continue in an un-modified capacity. We are not powerless; we have the power to control our actions. We have power every day to save the lives of others.