SIR Model

The SIR model (Susceptible–Infectious–Recovered) is the canonical mathematical framework of quantitative epidemiology, partitioning a closed or open host population of total size N into mutually exclusive epidemiological compartments whose sizes evolve continuously over time through coupled nonlinear ordinary differential equations (ODEs). Derived from the seminal 1927 formulation of Kermack-McKendrick theory by William Ogilvy Kermack and Anderson Gray McKendrick, and building on the earlier mosquito-malaria differential equations of the Ross-Macdonald model, the SIR system captures the fundamental feedback loop of acute transmissible disease: infection incidence grows through contact between susceptible (S) and infectious (I) hosts, while simultaneously depleting the susceptible pool until transmission falls below the rate of recovery (R).

From the three governing rate equations of the deterministic SIR system emerge the core analytical laws of infectious disease dynamics: the basic reproduction number (R₀ = β / γ), the epidemic threshold theorem requiring an initial susceptible fraction S(0) / N > 1 / R₀ for an outbreak to expand, the exact algebraic formula for peak infectious prevalence (I_max), the transcendental final epidemic size equation proving that epidemics burn out before exhausting all susceptibles, and the herd immunity vaccination threshold (p_c = 1 − 1 / R₀). Extensions of the core architecture—including SEIR (adding an Exposed latent period), SIRS (incorporating waning immunity), SIS (recurrent infections without immunity), and age- or network-structured metapopulation matrices—form the computational engine of modern pandemic forecasting and public-health intervention design.

Mass-Action vs. Frequency-Dependent Transmission

In the classical closed SIR model without vital dynamics (assuming outbreak timescales are much shorter than host demographic birth and death rates, so N = S(t) + I(t) + R(t) is constant), individuals transition unidirectionally along the pathway S → I → R. Under the homogeneous frequency-dependent (standard incidence) mixing assumption, each infectious individual makes an average of c potentially infectious contacts per unit time across the population regardless of total population density, and a fraction S(t) / N of those contacts occur with susceptible hosts. If each contact between an infectious and a susceptible individual transmits the pathogen with probability p, the effective transmission rate is β = c · p (units: time^-1).

The per-capita rate at which susceptible individuals acquire infection is defined as the force of infection λ(t) = β · I(t) / N. Multiplying λ(t) by the total number of susceptibles S(t) yields the instantaneous incidence rate (new infections per unit time): βS(t)I(t) / N. Simultaneously, infectious individuals clear the pathogen or are removed from the transmission chain (via sterilizing recovery, isolation, or disease-induced mortality) at a constant per-capita recovery rate γ (units: time^-1), corresponding to a memoryless infectious period of mean duration τ_I = 1 / γ (where the probability of remaining infectious after time t follows the exponential decay survival distribution P(T > t) = e^−γt).

The Three Coupled Ordinary Differential Equations

Combining the incidence and recovery fluxes yields the three coupled nonlinear ordinary differential equations of the closed deterministic SIR model:

• Susceptible Compartment: dS / dt = −β · S(t) · I(t) / N
• Infectious Compartment: dI / dt = β · S(t) · I(t) / N − γ · I(t)
• Recovered (Removed) Compartment: dR / dt = γ · I(t)

Summing the three derivatives confirms conservation of total population size: dS/dt + dI/dt + dR/dt = 0, so S(t) + I(t) + R(t) = N for all t ≥ 0. Normalization by total population size N defines the dimensionless state fractions s(t) = S(t)/N, i(t) = I(t)/N, and r(t) = R(t)/N (where s + i + r = 1), simplifying the system to ds/dt = −βsi, di/dt = (βs − γ)i, and dr/dt = γi.

The Epidemic Threshold Theorem, R₀, and R_e

Factoring the infectious rate equation reveals the condition required for an epidemic wave to grow upon introduction of an index case into a population with initial susceptible fraction s₀ = S(0) / N:

dI / dt = I(t) · [β · (S(t) / N) − γ] = γ · I(t) · [(R₀ · S(t) / N) − 1]

Here, R₀ = β / γ is the basic reproduction number—the expected number of secondary infections generated by one infectious case over an average infectious duration 1/γ in a completely susceptible population (s₀ ≈ 1)—and R_e(t) = R₀ · S(t) / N is the time-varying effective reproduction number.

• Epidemic Growth Phase: During the initial invasion when S(t) ≈ N, linearized integration yields exponential growth of the infectious cohort according to I(t) = I(0) · exp((R₀ − 1)γt), with intrinsic epidemic growth rate r = (R₀ − 1)γ and epidemic doubling time T_d = ln(2) / [(R₀ − 1)γ].
• Kermack-McKendrick Threshold Theorem: An epidemic wave can grow (dI/dt > 0) if and only if R_e(0) = R₀ · s₀ > 1, or equivalently when the initial susceptible fraction exceeds the critical threshold s₀ > 1 / R₀. Conversely, if s₀ < 1 / R₀—either naturally or via pre-existing vaccination of a fraction p_c = 1 − 1 / R₀ (herd immunity threshold)—then dI/dt < 0 from t = 0 onward, and transmission decays monotonically without generating an epidemic wave.

Phase-Plane Integration and Peak Infectious Prevalence (I_max)

Although the nonlinear SIR equations do not admit a closed-form elementary expression for I(t) as an explicit function of time t, dividing the infectious differential equation by the susceptible differential equation eliminates time t and yields an exact separable phase-plane ODE:

di / ds = (βsi − γi) / (−βsi) = −1 + 1 / (R₀ · s)

Integrating both sides from the initial state (s₀, i₀) to any subsequent state (s, i) along the phase trajectory produces the conserved invariant of the SIR system:

i(s) + s − (1 / R₀) · ln(s) = i₀ + s₀ − (1 / R₀) · ln(s₀)

The epidemic wave reaches its peak infectious prevalence (i_max = I_max / N) at the exact inflection point where dI/dt = 0, which occurs when the susceptible fraction has been depleted to the herd immunity threshold s* = 1 / R₀. Substituting s* = 1 / R₀ (and assuming a small initial seed i₀ → 0, s₀ ≈ 1) into the phase-plane invariant yields the exact analytical formula for the maximum fraction of the population simultaneously infected at the peak of the outbreak:

i_max = 1 − (1 / R₀) · [1 + ln(R₀)]

For example, a pathogen with R₀ = 3.0 reaches its epidemic peak when s* = 1/3 (33.3% susceptible remaining), at which point i_max = 1 − (1/3)(1 + ln 3) ≈ 0.300 (30.0% of the entire population simultaneously infectious).

The Final Epidemic Size Equation and Herd Immunity Overshoot

Even after s(t) drops below 1 / R₀ and the epidemic wave begins to decline (dI/dt < 0), substantial numbers of infectious individuals (i_max) remain in the population and continue infecting remaining susceptibles as the wave winds down—a phenomenon termed epidemic overshoot. As t → ∞, the infectious cohort decays to zero (i(∞) = 0), and the total fraction of the population ever infected approaches the final epidemic size r_∞ = R(∞) / N = 1 − s_∞. Evaluating the phase-plane invariant at t → ∞ (with i₀ → 0, s₀ = 1, and s_∞ = 1 − r_∞) yields the transcendental final size relation:

1 − r_∞ = exp(−R₀ · r_∞)

Because exp(−R₀r_∞) is strictly positive for all finite R₀, s_∞ = 1 − r_∞ > 0 always holds: an epidemic never infects 100% of the susceptible population, instead extinguishing because of susceptible depletion rather than complete exhaustion of hosts. For R₀ = 2.0, although the herd immunity threshold is reached at 50% infected (p_c = 0.50), epidemic overshoot carries the final cumulative attack rate to r_∞ ≈ 0.797 (79.7% infected).

The SEIR Model and Next-Generation Matrix

Most biological pathogens exhibit an extrinsic or intrinsic incubation/latent period during which a newly infected host harbors replicating pathogen (compartment E, Exposed) but is not yet capable of transmitting it to others. Adding an Exposed compartment that transitions to the Infectious state at rate σ (corresponding to a mean latent period τ_E = 1 / σ) produces the four-equation SEIR system:

• dS/dt = −βSI / N
• dE/dt = βSI / N − σE
• dI/dt = σE − γI
• dR/dt = γI

In a closed population without demographic mortality during the latent phase, the SEIR model has the same basic reproduction number (R₀ = β / γ) and final epidemic size (r_∞) as the corresponding SIR model, because every individual entering E eventually enters I. However, the latent delay 1/σ increases the mean serial interval / generation time (T_g = 1/σ + 1/γ) and reduces the early exponential growth phase rate r (where I(t) ∝ e^rt), which satisfies the coupled Euler-Lotka characteristic eigenvalue equation:

R₀ = (1 + r / σ) · (1 + r / γ)

Endemic Demographic Equilibrium and Stochastic Extinction

When host demographic turnover (per-capita birth and natural mortality rate μ) or waning immunity (rate ω in the SIRS model) continuously replenishes the susceptible pool, an infection with R₀ > 1 does not permanently extinguish after the first wave. Instead, trajectories spiral via damped oscillations (with characteristic inter-epidemic period T ≈ 2π / √[μγ(R₀ − 1)]) into a stable endemic equilibrium where the susceptible fraction is pinned at s* = 1 / R₀ and the endemic infectious fraction is maintained at i* = (μ / β)(R₀ − 1).

Finally, when absolute case numbers are small (such as during initial zoonotic spillover or post-wave troughs), the deterministic ODE continuum breaks down and must be replaced by a continuous-time Markov chain (CTMC) Gillespie stochastic simulation or branching process. Even when R₀ > 1, a single introduced index case faces a non-zero probability of stochastic fadeout before establishing a self-sustaining chain of transmission; under a continuous-time birth-death branching process with infection rate β and recovery rate γ, the probability of minor outbreak extinction from I(0) = i₀ initial cases is exactly P_extinct = (1 / R₀)^i₀.

Why doesn't an epidemic infect 100% of the population when no one is vaccinated?

An epidemic stops growing as soon as the susceptible fraction drops to the herd immunity threshold s* = 1 / R₀, and eventually burns out at a final attack rate r_∞ governed by 1 − r_∞ = exp(−R₀r_∞). Because the exponential function never reaches zero for finite R₀, infectious individuals recover faster than they can locate the remaining scattered susceptibles, leaving a permanent unexposed fraction s_∞ > 0.

What is the difference between the herd immunity threshold and the final epidemic size?

The herd immunity threshold (p_c = 1 − 1/R₀) is the immune fraction at which the epidemic wave peaks and starts declining (dI/dt = 0), or the preventive vaccination level needed to block an outbreak prior to introduction. In an uncontrolled epidemic, however, cases present at the peak continue transmitting during the downward phase (epidemic overshoot), pushing the final epidemic size (r_∞) substantially higher than p_c.

How does adding an Exposed (E) compartment in the SEIR model change the outbreak curve?

Adding a latent period (1/σ) in the SEIR model does not alter R₀ = β/γ or the total cumulative number of people infected (r_∞) in a closed population, provided no natural deaths occur during latency. However, the waiting time in compartment E stretches the generation interval, slowing the daily growth rate r, delaying the date of the epidemic peak, and lowering peak hospital prevalence (I_max).

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