Model Behaviour Part 2: Assumption and the SIR Model

Computer programs can model and extrapolate data about the world surprisingly accurately, some can even out-diagnose doctors [1] and function as digital birth control [2]. Despite these capabilities, models, much like humans, cannot account for every possible outcome. I can write a model that simulates rolling a six-sided die, but it won’t account for my friend coloring in the one spot so it has seven dots. This model assumes that the die will always give a readable result from one to six. In most cases this is a reasonable assumption, so it is taken to simplify the model, increasing its ease of use and its efficiency.

This same principle of making assumptions can be applied to any kind of model; the more a model assumes, the simpler it becomes and the less accurate. Deciding what to assume in a model is a question of optimizing simplicity, computational strain and accuracy. Look at the figures provided to show the difference assumptions can make:

Both make the following assumptions [3]:

• Fixed population size (population does not change over time)

• Constant recovery rate (the rate at which people recover from the infection is constant)

• Well-mixed population (infected people will have sufficient contact with susceptible to spread the infection)

• No Vital Dynamics (There are no births and no deaths, either from the infection or otherwise)

The first model (left) assumes that infection rate is constant, while the second one (right) models that infection rate changes as a function of the number of infected based on this equation:

New Infection Probability = 1 – (1 – Infection Probability)^ (Initial Number of Infected)

Infection probability increases during each generation, so there is a huge immediate increase in the infected population, leading to the sharp peak on the right. Varicella (AKA Chicken Pox) is a non-lethal disease that spreads from person to person. Since the likelihood of infection varies with how much contact you have with infected individuals, the second model will likely give a more accurate prediction of a Varicella outbreak [4].

While Varicella can be studied with the SIR model, other infections cannot be constrained by the same assumptions. Some pathogens are lethal, some have effective vaccines, some can affect how well-mixed the population is. Human factors like conferred immunity, herd immunity and incubation periods can also affect the spread of a disease.

When designing models, one of the most important things to do is to consider the assumptions you can make, and how they reflect the reality of the disease.

Here is a link to the SIR model I wrote for this piece and used to generate the figures:  

https://github.com/DJSiddharthVader/CSEB/blob/master/SIR%20Model%20Siddharth%20Reed.py