16. Simplified SIR Model

Abstract

This chapter develops and analyzes a simplified compartmental SIR model, designed to isolate the fundamental transmission-recovery dynamics of an infectious disease outbreak. By neglecting demographic turnover—such as births and deaths—this model provides a high-fidelity representation of the rapid propagation phase typical of initial epidemic waves. We formulate the model as a system of ordinary differential equations (ODEs) and employ non-dimensionalization to identify the basic reproduction number ( $R_0$ ) as the primary determinant of outbreak scale. Through linear stability analysis of the continuum of disease-free equilibria (DFE), we derive the critical threshold conditions for epidemic invasion. We further extend this analysis to explore the effective reproduction number ( $R_e$ ), establishing a mathematical foundation for the concept of herd immunity.

Introduction¶

Outbreaks of infectious diseases can have major public health and socioeconomic impacts. It is therefore important to understand the dynamics of disease spread and characterize factors determining outbreak size and duration. The classic SIR compartmental model provides a useful framework for modeling infection transmission at the population level. It incorporates demographic fluxes due to births and deaths. However, in many outbreak settings, the timescales over which infections and recoveries occur are much faster than demographic turnover.

As such, during the initial spread and peak of an outbreak, the influence of births and deaths can be neglected as a first approximation. In this chapter, we develop and analyze a simplified version of the SIR model that ignores demographic fluxes. This allows us to focus specifically on the short-term infection dynamics over the course of an outbreak. Understanding how the basic infection and recovery processes drive temporal changes in disease prevalence at this timescale is highly relevant from a public health perspective.

Key determinants of outbreak size and duration need to be identified to effectively target intervention strategies. The simplified SIR model provides a tractable framework for characterizing the influence of transmission rate, recovery rate, and other epidemiological parameters on short-term infection dynamics. Analytical insights will be used to explore model predictions and make connections to real-world outbreak settings. Overall, this analysis aims to complement the full demographic SIR model by illuminating infection spread mechanisms operating over a distinct, epidemiologically important short-term timescale.

Model development¶

We develop a compartmental model to describe the transmission dynamics of an infectious disease within a closed population. The population is divided into three mutually exclusive health states: susceptible ( $S$ ); infected ( $I$ ); and recovered ( $R$ ). Individuals transition between states according to the following processes:

Susceptible individuals contract the infection upon contact with infectious individuals. The per capita rate of new infections is proportional to the product of the number of susceptible and infectious individuals, with proportionality constant $\beta$ known as the infection rate.
Infected individuals recover at a constant per capita recovery rate $\gamma$ .
In the absence of demographic factors like birth, death, immigration or emigration, the total population size $N=S+I+R$ remains fixed.

These assumptions allow formulation of the following system of ordinary differential equations:

\begin{align*} \frac{dS}{dt} &= -\beta SI, \\ \frac{dI}{dt} &= \beta SI - \gamma I, \\ \frac{dR}{dt} &= \gamma I. \end{align*}

(1)

The total population size is non-dimensionalized to 1 by defining normalized variables $s=S/N$ , $i=I/N$ , $r=R/N$ . A rescaled time $t'=\gamma t$ is also introduced. In terms of these, the equations become:

\begin{align} \frac{ds}{dt'} &= -R_0 s i, \\ \frac{di}{dt'} &= R_0 s i - i, \\ \frac{dr}{dt'} &= i, \end{align}

(2)

where $R_0=\beta N/\gamma$ denotes the basic reproductive number, defined as the average number of secondary infections produced by an infectious individual in a fully susceptible population. In subsequent sections, we analyze this system to characterize outbreak dynamics.

Analysis of Disease-Free Equilibria¶

To understand the conditions under which an outbreak occurs, we examine the steady-state solutions of the non-dimensionalized system in Eq. (2). An equilibrium is reached when all state derivatives vanish ( $\dot{s} = \dot{i} = \dot{r} = 0$ ). This occurs if and only if the infectious fraction $i = 0$ . Consequently, the system does not have a single equilibrium point, but rather a continuum of disease-free equilibria (DFE) defined by:

s^* = \eta, \quad i^* = 0, \quad r^* = 1 - \eta,

(3)

where $\eta \in [0, 1]$ represents the initial susceptible fraction. To determine the stability of these equilibria, we evaluate the Jacobian matrix $\mathbf{J}$ at $(\eta, 0, 1-\eta)$ :

\mathbf{J}(\eta,0,1-\eta) = \begin{bmatrix} 0 & -R_0\eta & 0 \\ 0 & R_0\eta-1 & 0 \\ 0 & 1 & 0 \end{bmatrix}.

(4)

The stability of the system is governed by the eigenvalues of the Jacobian matrix, $\mathbf{J}(\eta,0,1-\eta)$ :

$\lambda_1 = R_0\eta - 1$ .
$\lambda_2 = \lambda_3 = 0$ .

The corresponding eigenvectors provide insight into the system’s phase-space geometry. The vectors $\vec{v}_2 = [1,0,0]^T$ and $\vec{v}_3 = [0,0,1]^T$ correspond to the zero eigenvalues, representing the manifold of equilibria. This confirms that in the absence of disease ( $i=0$ ), any shift in the ratio of susceptible to recovered individuals results in a new, neutrally stable steady state.

The system’s epidemiological fate is dictated by the dominant eigenvalue, $\lambda_1$ :

Subcritical Case ( $R_0\eta < 1$ ): The DFE is locally stable. Any small introduction of infectious individuals will decay exponentially, as $\lambda_1 < 0$ . The disease fails to invade the population.
Supercritical Case ( $R_0\eta > 1$ ): The DFE is unstable. A single infection triggers a transient exponential increase in cases, manifesting as an epidemic outbreak.

In a perfectly “naive” population—where every individual is susceptible ( $\eta = 1$ )—the condition for not having an outbreak simplifies to the foundational requirement: $R_0 < 1$ .

However, if $R_0>1$ , an epidemic invasion can still be blocked in a population with pre-existing immunity. An outbreak is prevented if the initial susceptible fraction, $\eta$ , is below the critical susceptible threshold::

\eta < s^*_c = \frac{1}{R_0}.

(5)

This inequality defines the mathematical basis for herd immunity. By taking into consideration that the immune fraction is $r = 1 - \eta$ , we can derive the minimum proportion of the population that must be immune to ensure that any introduced infection decays rather than spreads. This value is known as the Herd Immunity Threshold ( $H$ ):

H = \frac{1}{R_0}.

(6)

When this threshold is surpassed, the “chain of transmission” is broken. Although susceptible individuals still exist within the population, the density of available hosts is too low for an infected person to encounter enough susceptible people to replace themselves.

The Effective Reproduction Number ( $R_e$ )¶

While the basic reproduction number ( $R_0$ ) represents the maximum transmission potential of a pathogen in a fully susceptible population, real-world dynamics often involve a partially immune population. To describe the real-time potential for spread, we introduce the effective reproduction number ( $R_e$ ):

R_e(t) = R_0 \cdot s(t).

(7)

By substituting the initial state $s(0) = \eta$ , we find the initial effective reproduction number $R_{e, 0} = R_0 \eta$ . As established in the stability analysis, the population is protected from an epidemic if $R_{e, 0} < 1$ , which defines the critical threshold for the susceptible fraction: $\eta < s^*_c = 1/R_0$ .

From Eq. (2), it is evident that the infectious fraction $i(t)$ grows as long as $R_e(t) > 1$ and declines when $R_e(t) < 1$ . During a typical outbreak, $s(t)$ —and consequently $R_e(t)$ —declines monotonically as individuals contract the disease and move into the recovered class. The moment $R_e(t) = 1$ —the herd immunity threshold—is the peak of the epidemic curve; however, it is a common misconception that the outbreak ends at this point. Because a significant portion of the population remains infectious ( $i(t) > 0$ ) when the threshold is crossed, the “momentum” of the disease carries the outbreak forward. This leads to epidemic overshoot, where the final number of susceptible individuals is depleted far below the herd immunity level before the infection finally dies out.

These dynamics can be conceptualized through a combustion analogy:

Susceptible Fraction ( $s$ ): This represents the fuel for the epidemic fire.
Reproduction Number ( $R_0$ ): This represents the burn rate or oxygen level.

If the fuel level is below the inverse of the burn rate ( $s < 1/R_0$ ), the fire cannot be sustained and will flicker out. If the fuel level is high, the fire will grow until it has consumed enough “timber” to bring the fuel level down to exactly $1/R_0$ . However, the existing heat and flames (the current infectious population) continue to char the remaining wood until the fire is physically exhausted, resulting in the overshoot.

Discussion¶

In this chapter, we developed and analyzed a simplified SIR model, intentionally neglecting demographic turnover to isolate the intrinsic infection dynamics of a short-term outbreak. By non-dimensionalizing the system, we identified the basic reproduction number ( $R_0$ ) as the primary bifurcating parameter. This dimensionless constant acts as the “tipping point” that determines whether a localized introduction of a pathogen results in a self-limiting event or a full-scale epidemic invasion.

Our stability analysis of the disease-free equilibria (DFE) demonstrated that the threshold for an outbreak is not a fixed property of the pathogen alone, but rather the product of its infectiousness and the population’s vulnerability ( $R_0 s$ ). This highlights a critical public health insight: even a highly contagious pathogen ( $R_0 \gg 1$ ) can be effectively suppressed if the initial susceptible fraction $s$ is sufficiently low. This mathematical condition provides the formal basis for herd immunity, defining the precise proportion of a population that must be protected—whether through prior infection or vaccination—to prevent sustained transmission.

While this model is highly idealized, it establishes the foundational logic for all compartment-based epidemiological modeling. It demonstrates that outbreaks are not chaotic or random occurrences, but predictable trajectories dictated by the competition between transmission $(\beta)$ and recovery $(\gamma)$ .

However, users should be aware of the model’s limitations:

Homogeneous Mixing: We assume every individual has an equal probability of meeting every other individual.
Closed Population: For long-term study, births, deaths, and migration must be re-integrated to capture endemic behaviors.
Constant Parameters: In real-world settings, $\beta$ often fluctuates due to seasonal changes or behavioral interventions.

Exercises¶

Numerically solve the simplified SIR model using various combinations of parameter values. Ensure to verify that all sets of parameters for which $R_0 > 1$ result in epidemic outbreaks.
Numerically solve the simplified SIR model using parameter values such that $R_0 > 1$ . Plot the number of recovered patients at the end of the outbreak vs. $R_0$ . Discuss the results.
Consider a viral disease in which individuals remain infected and contagious for a period of 2 weeks. The disease causes virtually no fatalities, but there is neither a cure nor a vaccine. The basic reproduction number of the disease is 6. Assume that an epidemic begins, and health authorities start to act when 0.1% of the population is infected. The only available measure is lockdowns, which incur economic costs proportional to their duration. Design a strategy that allows for achieving herd immunity while minimizing the costs of lockdown, without allowing more than 5% of the population to be infected simultaneously.