Compartmental Models in Epidemiology
Basic models of spread of infectious diseases and their limitations
Introduction
In this article I give a brief overview of compartmental models, discuss how they are applied in epidemiology, and what fundamental assumptions are necessary. We discuss the intrinsic property, R_0, the reproduction number of an infectious disease and an example reference of how to compute it. I also give brief descriptions of how these can be applied to the flu (common cold), COVID-19, and even Ebola. Lastly, I discuss the limitations of this model and try to point out the need for other mathematical techniques to account for this model’s limitations.
Epidemiological Models
The original epidemiological models assume constant density (homogeneity) and sufficiently large numbers of people in a given closed population (no births/deaths/immigration/emigration). The standard models of disease spreads are compartmental models. Below is an example of a Susceptible-Infected-Removed (SIR) model.
To model the spread of disease, we start with a population of people in the Susceptible state, S. If there is an Infected individual in the area (someone in state I), the infectious disease’s reproduction number (or R_0) would inform the rate of transition of individuals from susceptible into the infected state. Then, given enough time, infected individuals either Recover from the infection or die (and are Removed from the population). In either case, the infected individual has transitioned into the final state, R. You can see below a pictorial description of the flow of individuals through the 3 states of interest. The values alpha, beta, and gamma correspond to rates of transitions between states. Where alpha represents rate of spontaneous death (without infection), beta represents rate at which susceptible individuals become infected — R_0 is embedded in this value — and lastly, gamma represents a rate of death/recovery of individuals in the infected state.
Epidemiologists will describe the phases of the disease by creating more sophisticated compartmental models. It takes domain expertise to understand the phase of a disease. And this will be reflected in the compartment model used. For example, the common flu might be better portrayed by an SIS model, since individuals can become susceptible to flu again after being infected by it.
Now for more math. Every compartmental model can directly be translated into a system of ordinary differential equations (ODEs). Analytical solutions are attainable in very few cases. The solutions will be functions S(t), I(t), and R(t), corresponding to the number of individuals in that state at given time t. The more complicated the compartmental model is, the less likely an analytical solution of the ODEs exist. This is where modelers turn to numerical simulation to understand the number of people in each state as a function of time. Generally, Epidemiologists are interested in “flattening the infection curve”, I(t). Thus, modeling spread of disease as well as the effect of public intervention can inform public health policy and aid in flattening the curve. Below is an example of ODEs corresponding to the compartmental model given above.
Let us consider the total number of individuals in the system as a function of time and denote it by N(t) = S(t) + I(t) + R(t). Notice that when summing the above equations, we obtain the following interesting property.
Looking at the outer expressions, we see that the total number of individuals in the population does not change in time! This is the mathematical representation of the original assumption of a closed population.
Example of a COVID model
Below is the compartmental model to describe COVID-19 as described by Leontitis et al.
The rates of transitions from one state to the next can be attained by empirical observation. Solving the corresponding equations can help us understand how infectious disease spreads. Specifically, it is common to extract the reproductive number or R0 of the disease — the average number of people who get infected if an infected person is dropped into a pool of susceptible people. R0 is a measure of whether the disease is spreading (R0>1) or going extinct (R0<1).
An interesting Inverse Problem: How to compute the R_0?
Note that R_0 is an intrinsic property of the disease. So how do we measure it? In the real world, many more factors may affect this variable. One such method would be to estimate this single parameter, R_0, given many empirical observations about the number of individuals in each state of the disease at a given time point. Take for example the COVID 19 outbreak on the Diamond Princess cruise ship. Since the population on the cruise is closed, these compartmental models can be useful for estimating R_0. See this example: A major outbreak of the COVID-19 on the Diamond Princess cruise ship: Estimation of the basic reproduction number — PubMed (nih.gov)
Limitations
Realistically, spread of disease is influenced by many other factors like geography, environment, culture, and social factors (among others epidemiologists deems important). One can see the limitation of the assumption of constant density and large populations. For example, Ebola might spread a lot slower in less dense/low population numbers in rural areas of Africa (Sudan, Libera, Democratic Republic of Congo, amongst others).
As another example, COVID has been shown to not last long in sunlight. Thus, in certain geographies, this may be an important environmental factor that might influence the spread or collapse of the disease.
Final thoughts
I wanted to point out that compartmental models are not strictly confined to the study of Epidemiology. Note that such models are useful if you want to consider a population of individuals and the possible states each individual could be in. In general, each individual can only be in one state at a time.
So how do we account for heterogeneity in density, geography, environment, sociological, factors in modeling disease spread? A good mathematical modeler or domain expert will assess which factors are most influential in such a disease spread and incorporate them into a model.
I hope you’ve enjoyed this intro to studying disease spread. Stay tuned as I start to combine methods to create a holistic picture of spread of infectious diseases.
