N1. Disease progression in a population

Recent high profile infectious disease outbreaks around the globe – including SARS-CoV-2, Ebola, and Zika – have reignited a general interest in understanding the spread of disease. The utility of mathematical and computational approaches to simulate and numerically analyze infections within populations has also benefited from major advances in computational power over the past several decades. As a result, public health officials and policymakers have increasingly relied on a quantitative understanding of infectious disease dynamics.

Mathematical models of infectious disease dynamics have a rich history that dates back more than 100 years. Mathematically simple formulations that describe the transition of individuals in a population between “compartments” that capture the infection status of individuals leads to surprisingly significant insight. Their elegance and simplicity allow the ease of expansion to more complexities through, for example, the addition of compartments. Expanding these models is often straightforward but the apparent simplicity can mask subtle, but important, model structure and parameterization choices.

Mathematical models that describe the population-level dynamics of infectious diseases are typically based on the classic Susceptible – Infectious – Recovered (SIR) framework, a structure inspired by the seminal work of Kermack and McKenrick in 1927 ^[1]. In SIR models, individuals are separated into compartments based on their infectious status. Thus, any individual can only be in a single compartment at a given time, although they can transition between compartments. In this section, we briefly introduce and provide an analysis of a simple SIR model. We then generalize the model analysis so that it can be applied to a large suite of compartmental disease models.

Constructing compartmental models: the SIR framework

Compartmental disease models divide a population into sub-populations (or compartments) based on each individual’s infectious status, and then they track the corresponding population sizes through time. Here, we first consider a model with straightforward sub-populations: susceptible individuals (S) become infected through contact with infectious individuals (I), and infectious individuals recover (R) at a fixed rate and confer lifelong immunity.

The susceptible compartment (S) consists of individuals who have never been infected with the pathogen and could become infected if they were to come into contact with an infected individual. The infected compartment (I) consists of the individuals who are currently infected and are able to infect susceptible individuals, were they to come into contact. The recovered compartment (R) consists of individuals who have recovered from the pathogen and are immune.  Even if a recovered individual comes into contact with an infected individual, they cannot become infected.

The fixed recovery rate ν (the Greek letter, “nu”) has the interpretation that 1/ν is the infectious period (L), the average amount of time individuals spend in the infectious compartment. The recovery rate is a constant that will be different for each disease, depending on how long individuals remain infectious.  Although there typically is some lag between acquiring infection and becoming infectious, here we do not distinguish between infected and infectious individuals; we instead assume that all individuals are infectious immediately upon infection. Almost all diseases have some delay between infection and infectiousness.  However, when that delay is short (hours or days) and the goal is understanding the long term dynamics (years in the future), the assumption that individuals are immediately infectious may be reasonable.

Examples of some infections that are frequently modeled within the SIR framework include many common childhood diseases such as chicken pox and measles. An important common feature among these pathogens is that infection is thought to confer life-long immunity so that, from a modeling perspective, individuals move directly into (and remain in) the recovered compartment upon recovery.  We will work with diseases where immunity can wane later in this module, but for now we will assume that recovered individuals can never become susceptible again.

The force of infection refers to the number of people who become infected.  This value depends on a transmission rate (β), which is more precisely, the product of 1) the contact rate and 2) the probability of transmission given contact.  The contacts we are concerned about for transmission are the contacts between infectious and susceptible individuals.  Nobody can become infected if there are no susceptible individuals in the population.  And likewise, nobody can become infected if there are no infected individuals in the population.  The number of people infected therefore depends on the transmission rate (β), the susceptible population (S), and the infected population (I).

Licenses and Attributions

“Disease progression in a population” by Michelle McCully is adapted from “An introduction to compartmental modeling for the budding infectious disease modeler” in Letters in Biomathematics 5(1):195-221, 2018 by Julie C. Blackwood and Lauren M. Childs used under CC BY 4.0.  “Disease progression in a population” is licensed under CC BY-NC 4.0.