Epidemics and the SIR Model: Threshold, Peak and Extinction

In January 1918, an unusual influenza begins circulating in American military barracks. Within months, it spreads to every continent and kills between 50 and 100 million people — more than World War I. In 2020, a new coronavirus paralyzes the global economy within weeks. These events raise a fundamental question: can we predict the trajectory of an epidemic?

The partial answer is yes — provided one understands the assumptions and limitations of the models. The SIR model, proposed by Kermack and McKendrick in 1927, is the starting point of modern mathematical epidemiology. Simple in its formulation, it captures the essential mechanisms of infectious disease spread and allows key quantities such as the basic reproduction number $R_{0}$ to be defined.

The SIR model divides the population into three mutually exclusive compartments. S denotes susceptible individuals: they have not yet been infected and can be. I denotes infectious individuals: they carry the pathogen and can transmit it. R denotes removed individuals: they have been infected and are now immune (or deceased, in simplified versions). The total population N = S + I + R is assumed constant in the basic version.

These compartments are abstractions. In reality, the boundary between susceptible and infectious is not sharp: there is an incubation period during which the individual is infected but not yet contagious. The boundary between infectious and recovered depends on the duration of the infectious period, which varies between individuals. These simplifications are deliberate and must be made explicit.

The SIR model is described by three coupled ordinary differential equations. The first describes the decrease in susceptibles: dS/dt = −βSI/N. The second describes the dynamics of infectious individuals: dI/dt = βSI/N − γI. The third describes the accumulation of recovered individuals: dR/dt = γI.

The parameter β is the transmission rate: it represents the average number of effective contacts per unit time between an infectious individual and susceptibles. The parameter γ is the removal rate: its inverse 1/γ is the average duration of the infectious period. These two parameters summarize all the disease biology in the basic model.

The basic reproduction number $R_{0}$ is defined as the average number of secondary cases produced by one infectious individual in a fully susceptible population. In the SIR model, $R_{0}$ = β/γ. If $R_{0}$ < 1, each infectious individual produces on average less than one secondary case: the epidemic dies out. If $R_{0}$ > 1, the epidemic can grow. This threshold is the central quantity of mathematical epidemiology.

The effective reproduction number $R_{t}$ is $R_{0}$ multiplied by the fraction of the population still susceptible: $R_{t}$ = $R_{0}$ × S/N. At the start of the epidemic, S ≈ N and $R_{t}$ ≈ $R_{0}$. As the epidemic progresses, S decreases and $R_{t}$ declines. When $R_{t}$ = 1, the number of infectious individuals is at its maximum. When $R_{t}$ < 1, the epidemic declines.

Herd immunity in the SIR model framework is reached when the fraction of the immunized population exceeds 1 − 1/$R_{0}$. For a disease with $R_{0}$ = 4, the threshold is 1 − 1/4 = 75%. For measles itself ($R_{0}$ between 12 and 18 in unvaccinated populations), the threshold is about 92–95%. These values assume homogeneous immunity in the population, which is rarely the case.

The time curves of the SIR model show three characteristic phases. In the growth phase, I increases exponentially as long as S is close to N. In the peak phase, I reaches its maximum when $R_{t}$ = 1, that is, when S = N/$R_{0}$. In the decline phase, I decreases because the fraction of susceptibles is too low to sustain transmission. The epidemic dies out before all susceptibles are infected: a fraction of the population escapes infection even without prior immunity.

The phase space of the SIR model is two-dimensional if we use S and I (R being determined by R = N − S − I). The trajectory starts near (N, 0) — almost the entire population is susceptible, very few infectious — and evolves toward a point on the I = 0 axis — the epidemic is over. This trajectory is a monotone curve in the (S, I) plane: I increases then decreases while S decreases continuously.

The trajectory of the classic SIR model in the (S, I) plane is not a spiral. It is a curve that rises, reaches a maximum and falls, without ever returning toward its starting point. There is no oscillation around an interior equilibrium. The final equilibrium is on the I = 0 axis, which is not an interior fixed point but a line of equilibria.

Spiral trajectories can appear in extensions of the SIR model that introduce additional mechanisms. The SIRS model (with waning immunity) can produce damped oscillations around an endemic equilibrium if parameters are in a certain range. The model with demographic turnover (births and deaths) can also produce oscillations. Seasonality of the transmission rate can force annual oscillations.

Models with delays — for example, a delay between infection and contagiousness — can produce sustained oscillations or chaotic behavior. Contact-network-structured models, where individuals do not mix homogeneously, can produce complex dynamics with successive waves. In all these cases, the spiral in phase space is a property of the extended model, not of the basic SIR model.

The SEIR model adds an E compartment for exposed individuals: infected but not yet contagious. This compartment captures the incubation period, which is important for diseases like COVID-19 (incubation 2–14 days) or Ebola (2–21 days). Adding E slows the initial growth of the epidemic and changes the shape of the time curves.

The SIRS model replaces permanent immunity with temporary immunity: recovered individuals return to the S compartment after some time. This model can produce recurrent epidemics and, under certain conditions, endemic oscillations. It is relevant for diseases like seasonal influenza, where immunity wanes and new variants emerge.

Age-structured models divide the population into age groups with different contact rates. Contact-network-structured models explicitly represent interactions between individuals. These approaches are more realistic but also more data- and computation-intensive. They are used for major pandemics where contact heterogeneities are important.

Fitting the SIR model to real data is difficult for several reasons. Reported cases underestimate actual infections: under-reporting is systematic and varies across surveillance systems. Delays between infection, symptoms, testing and reporting blur the time curves. Mortality data are more reliable than case data, but they reflect infection with an additional delay.

Behaviors change in response to the epidemic. When people become aware of the risk, they reduce contacts, decreasing β. When epidemic pressure decreases, behaviors return to normal. This behavioral feedback is not captured by the basic SIR model and can produce successive waves that the model does not predict.

The homogeneous population assumption is particularly problematic. Human contacts are highly heterogeneous: some individuals have far more contacts than others (super-spreaders). Family, occupational and geographic structures create transmission clusters. These heterogeneities can reduce the effective herd immunity threshold below the prediction of the homogeneous model.

Vaccination reduces the number of susceptibles. In the SIR model, vaccinating a fraction p of the population is equivalent to moving these individuals directly into the R compartment. The effective reproduction number becomes $R_{t}$ = $R_{0}$(1 − p). For the epidemic to be unable to start, we need $R_{t}$ < 1, i.e., p > 1 − 1/$R_{0}$. This is the herd immunity condition.

Non-pharmaceutical interventions — physical distancing, mask wearing, school closures, quarantine — act primarily by reducing β. Their effectiveness depends on implementation and population adherence. More complex models can estimate the impact of each measure separately, but interactions between measures and behavioral substitution effects complicate interpretation.

The SIR model illustrates a general principle: epidemics are collective phenomena governed by thresholds. Below the threshold $R_{0}$ = 1, an epidemic cannot establish itself. Above it, it can grow. Vaccination and control measures act by lowering $R_{t}$ below this threshold. This principle is robust even if the quantitative details depend on the chosen model.

The relation to the site’s thesis is precise: infection spread among susceptible, infected, and removed people may share geometry, dynamics, or representation with another family of phenomena, but these levels are not identical. Comparison becomes scientific when it names the proposed shared mechanism, the variables that differ, and the test that could distinguish them.

The relation to the site’s thesis is precise: infection spread among susceptible, infected, and removed people may share geometry, dynamics, or representation with another family of phenomena, but these levels are not identical. Comparison becomes scientific when it names the proposed shared mechanism, the variables that differ, and the test that could distinguish them.

The relation to the site’s thesis is precise: infection spread among susceptible, infected, and removed people may share geometry, dynamics, or representation with another family of phenomena, but these levels are not identical. Comparison becomes scientific when it names the proposed shared mechanism, the variables that differ, and the test that could distinguish them.

The relation to the site’s thesis is precise: infection spread among susceptible, infected, and removed people may share geometry, dynamics, or representation with another family of phenomena, but these levels are not identical. Comparison becomes scientific when it names the proposed shared mechanism, the variables that differ, and the test that could distinguish them.

The relation to the site’s thesis is precise: infection spread among susceptible, infected, and removed people may share geometry, dynamics, or representation with another family of phenomena, but these levels are not identical. Comparison becomes scientific when it names the proposed shared mechanism, the variables that differ, and the test that could distinguish them.

In this closed model, N is constant. Fractions s, i and r sum to one: s counts susceptible people, i infectious people and r those removed from transmission. “Removed” is a model state, not automatically a particular administrative observation. Transmission moves a person from S to I; leaving the infectious period moves them from I to R. The basic SIR model has no arrow returning R to S. ^[1]

The term βsi combines susceptible and infectious fractions and assumes homogeneous mixing, without an explicit contact network. β and γ have units day⁻¹ when time is measured in days. With γ = 0.1 day⁻¹, the model’s exponentially distributed infectious period has mean 1/γ = 10 days. This illustrative choice does not calibrate any real disease. Adding the derivatives gives zero: every departure from one compartment enters another.

Orange is i(t), the currently infectious fraction, not new infections that day. Model incidence is βsi, a flow with units per day. Reported cases would additionally depend on delays and observation procedures. In the simulation, s decreases, r increases and i rises then falls. The peak of i occurs when inflow βsi equals outflow γi; it does not mark the disappearance of transmission.

Change β while keeping γ and the initial state fixed. Faster transmission changes peak height and timing and also the final fraction in R. Compare the full trajectories with the same units, not just their maxima. Results use fourth-order Runge–Kutta integration with a 0.2-day step. Within numerical accuracy, they represent the displayed assumptions, without claiming to reproduce public-health observations.

Here $R_{0}$ = β/γ refers to an entirely susceptible population. During the episode, the effective number is Rₑ(t) = βs(t)/γ. For positive i, the sign of i′ is the sign of Rₑ − 1. With β = 0.3 and γ = 0.1, $R_{0}$ is three; i peaks when s crosses one-third. Susceptible people and new infections remain after this crossing, but the infectious compartment declines because outflow exceeds inflow.

The equality explains a dynamic mechanism; it does not directly provide a real vaccination target. Heterogeneous contacts, protection effectiveness and duration, behavior changes and imported cases can alter the relevant threshold. The slider deliberately leaves these mechanisms out. Conclusions from this laboratory should remain consequences of the model and its parameters, rather than forecasts or medical recommendations. ^[1]

In the (s,i) plane, each point is the same population’s state at one time. The horizontal axis is not a geographical position. Since s′ = −βsi ≤ 0, the closed SIR trajectory always moves left while transmission occurs; it cannot circle an interior point. A rise and fall in i makes an arc, not a spiral. This clarifies the earlier discussion of oscillations: they require an additional mechanism.

The SIRS extension adds waning protection: R returns to S at rate ωr. Equations become s′ = −βsi + ωr and r′ = γi − ωr, with the same equation for i. For β=0.3, γ=0.1 and ω=0.01 day⁻¹, the endemic equilibrium is s*=1/3, i*=2/33 and r*=20/33. Linearization gives a stable focus for these parameters, allowing damped winding. This result concerns this SIRS parameter choice, not every epidemic model.

The graph overlays both mechanisms with the same initial state. SIRS runs for eight hundred days to show successive returns; reference SIR retains its 180-day window. Loops belong to aggregate variables, not people’s paths or a virus inside an organ. Observed waves can also arise from seasonality or interventions: repeated waves alone do not identify the return mechanism shown here.

The concrete starting point is infection spread among susceptible, infected, and removed people. The SIR model divides a population into compartments and describes flows among them. A bell-shaped infected curve is not a spiral: it results from the product of susceptible people and infectious contacts. Parameters represent transmission and removal rates, not a geometric shape. This first distinction matters because a winding form can be a trajectory, a structure, a projection, or simply a graphical convention.

The differential equations give a threshold related to the basic reproduction number. When susceptibles fall enough, new infections drop below removals and the epidemic declines. Extensions add seasonality, age structure, mobility, or immunity, but each extension must be tied to data or an explicit assumption. ^[1] The model is not the object: it is a description whose assumptions must remain visible.

Two waves can produce a loop or spiral in a phase plot depending on chosen variables. This does not mean disease follows a spiral in space. The shape reveals relations among compartments and delays, while the mechanism remains transmission, infectious duration, and interventions. The counter-example is as instructive as the example: it shows which variable is missing when moving from one domain to another.

An analysis compares incidence, hospitalizations, and deaths, accounts for delays, and reports uncertainty intervals. Fitting a model to a curve does not prove parameter uniqueness: multiple combinations can produce a similar peak. Visualization should show this non-identifiability. A quantitative figure must distinguish published data, theoretical calculation, simulation, and conceptual diagram.

To understand infection spread among susceptible, infected, and removed people, separate at least the variable that rotates, the one that changes scale, and the one that transports information or matter. The SIR model divides a population into compartments and describes flows among them. A bell-shaped infected curve is not a spiral: it results from the product of susceptible people and infectious contacts. Parameters represent transmission and removal rates, not a geometric shape. A global form can remain stable while its components are replaced; conversely, a trajectory can change shape without a new local mechanism.

The site’s convergence appears here as return toward low transmission, but it is neither automatic nor universal. An epidemic can rebound when conditions change. The valid analogy concerns flows and thresholds, never a supposed spiral cause. We therefore cannot conclude from an image alone. We must ask which observation would falsify the hypothesis, then keep negative results in the comparison. This discipline is especially important when several models produce a similar outline.

An analysis compares incidence, hospitalizations, and deaths, accounts for delays, and reports uncertainty intervals. Fitting a model to a curve does not prove parameter uniqueness: multiple combinations can produce a similar peak. Visualization should show this non-identifiability. To make the result checkable, preserve parameters, units, resolution, initial conditions, and calculation method. An interactive visualization should also provide a readable initial state and a reset control; otherwise readers see an output without understanding its dependencies.

The model explains the differential equations give a threshold related to the basic reproduction number. when susceptibles fall enough, new infections drop below removals and the epidemic declines. extensions add seasonality, age structure, mobility, or immunity, but each extension must be tied to data or an explicit assumption. ^[1] but does not answer questions at every scale. We still need to determine how initial conditions, perturbations, and system boundaries alter the pattern. This is where research begins: not by adding an analogy, but by formulating a prediction that could be measured.

The site’s convergence appears here as return toward low transmission, but it is neither automatic nor universal. An epidemic can rebound when conditions change. The valid analogy concerns flows and thresholds, never a supposed spiral cause. ^[2]

The relation to the site’s thesis is precise: infection spread among susceptible, infected, and removed people may share geometry, dynamics, or representation with another family of phenomena, but these levels are not identical. Comparison becomes scientific when it names the proposed shared mechanism, the variables that differ, and the test that could distinguish them.