How COVID was modeled in Italy
In our discussions of the role of networks in the past semester, we’ve been exploring how the structure and makeup of networks can influence the way that information spreads through them. These networks can also be used to model the spread of infectious diseases, whose behavior is not unlike semantic information, and our ability to understand how they are spread, how long they spread for, and how to mitigate their spread can mean the difference between hundreds and thousands of infected/dead. The compartmental models essentially divide human beings within a population between compartments to study how a disease diffuses through a network. In fact, we can think of the simplest model concerning the spread of contagions we discussed earlier in the class, the branching process model that begins with one person that enters a population and potentially infects k people with probability p. This is a basic form of a SIR model, which attempts to account for disease behaviors for arbitrary networks as well.
The SIR model divides the human population into three categories, Susceptible, Infectious, and Removed. The basic diffusion of the disease consists of infected persons only being able to pass the disease on to those other Susceptible nodes they have a connection to with probability a. After the wave of infections, the node that was infected has a probability of recovering from the disease with probability b. Note that removed can refer to people that have recovered from the disease as well as people that have died from it. Therefore the flow of the model goes from Susceptible -> Infectious -> Removed. Of course, there are assumptions that the SIR model makes, for example, the level of susceptibility is basically a binary value, you are either prone to getting the disease if exposed or you are not. The model also makes the assumption that people who have been removed from the disease are now immune to it and cannot be both susceptible or infected again if exposed. As you might have guessed, these kinds of assumptions are often inconsistent with the way that diseases spread in real life, people have different levels of immunity in regards to catching diseases, people also have different behaviors depending on if they are aware if they have the virus or not, etc.
For this reason, epidemiologists have made more complex variations of these compartmental models which are often based off of the basic SIR model. In the case of the COVID-19 pandemic, researchers in Italy have proposed using the SIDARTHE-model to better control the spread of the pandemic. COVID-19 displays rather peculiar characteristics when it comes to its spread, for example, it tends to spread quite quickly in individuals that have not shown significant systems. This is the reason that testing for COVID-19 is so important, and it also requires us to use additional parameters in our modeling of the disease. Modeling the spread of the disease requires us to account for differences in behavior in the cases of undetected vs. detected cases, as well as the differences in behavior depending on the relative severity of the cases, life-threatening vs. non-life-threatening. The researchers define the different compartments as follows “S, susceptible (uninfected); I, infected (asymptomatic or paucisymptomatic infected, undetected); D, diagnosed (asymptomatic infected, detected); A, ailing (symptomatic infected, undetected); R, recognized (symptomatic infected, detected); T, threatened (infected with life-threatening symptoms, detected); H, healed (recovered); E, extinct (dead)”. While our SIR model was linear and fairly simple, as you might have guessed, the SIDARTHE model flow is far more complicated.
Analyzing the different parameters and nodes of this network show how much more complicated both the virus and human behavior can make modeling its spread. For example, there are now four parameters that dictate whether a susceptible person exposed to the virus becomes infected or not, as well as new nodes to try and predict human behavior. For example, symptomatic people find out if they are actually sick or not through diagnosis and can possibly have it but never know, while people that display actual symptoms will find out they are sick through their condition. The “Critical” parameter mu and nu account for the possibility that the disease is especially dangerous and requires hospitalization. Unlike the SIR-model we discussed in class, this model makes a distinction between people that have healed and have died from the virus.
The purpose of developing this model was to accurately predict the rates of infection and mortality depending on certain restrictions put into place to mitigate the spread of the virus. When testing the model, the day 1 basic reproduction number (R0), or expected number of cases generated from an infected individual was 2.38. As the course of the pandemic went on and people began to take basic preventative measures, social distancing, masks, etc, along with government lockdowns, the R0 predictably went down. Similarly, the researchers also make the case that because diagnosis is so important in the behavior of asymptomatic/mild symptom carriers of the disease, a population-wide testing and tracing program across the country would drastically improve rates of infection and help us fight the disease more effectively. In the below example you can clearly see the effects of both a national lockdown (Figure a,b) and a milder lockdown combined with mass-testing and contact tracing (Figure c,d). As you can see, after the mass testing and lockdown orders after Day 50, there is a predictable dropoff in additional cases that improve mortality.
Therefore the model can be utilized as a tool for policymakers to consider the effects and methods involved in combating the virus, through methods such as social-distancing mandates and mass testing. Of course, there are other considerations in policy that must be considered that could not be factored into the calculation. Although we can effectively model the diffusion of COVID-19 through concepts like SIR-model which evolved to the SIDARTHE-model, we must also be cognizant of other facets that we should consider that graphs cannot explain. For example, both the quality and access of the medical resources available to those who are in the “Threatened” category must be taken into consideration for hospitals, restrictive measures do not improve the mortality of the infected either. In the United States, there has been plenty of pushback from these type of regulations on our behavior, and while it may be difficult to force individuals to comply with them, we can rest assured knowing that our knowledge of networks and diffusion allows us to make smart decisions that save lives.
Article Link: https://www.nature.com/articles/s41591-020-0883-7