Kermack–McKendrick Theory

Kermack–McKendrick theory is the foundational mathematical theory of infectious disease epidemiology that governs how transmissible pathogens spread through, peak within, and subside across host populations. Formulated by Scottish biochemist William Ogilvy Kermack and physician-lieutenant colonel Anderson Gray McKendrick in a series of three landmark papers titled "Contributions to the Mathematical Theory of Epidemics" (published in the Proceedings of the Royal Society of London in 1927, 1932, and 1933), the theory introduced the Susceptible–Infectious–Recovered (SIR) compartmental framework and its general renewal-equation integral formulation.

Prior to Kermack and McKendrick's 1927 analysis (which built upon Sir Ronald Ross's 1911 malaria transmission models), a central puzzle of public health was why historical epidemics—such as the 1906 Bombay plague outbreak or seasonal influenza waves—consistently terminated on their own while a large fraction of the population remained uninfected and without any decline in the intrinsic virulence or transmissibility of the pathogen. By formulating coupled nonlinear differential equations tracking Susceptible (S), Infectious (I), and Removed/Recovered (R) individuals in a closed population of size N, Kermack and McKendrick proved two fundamental theorems of mathematical biology:

1. The Epidemic Threshold Theorem: An introduced pathogen can generate a growing epidemic wave if and only if the initial density or fraction of susceptible hosts exceeds a threshold (S(0) / N > γ / β = 1 / R₀, where R₀ is the basic reproduction number).
2. The Final Size and Overshoot Theorem: Epidemics terminate not because susceptible hosts are exhausted (S(∞) = 0), but because the susceptible fraction drops below the threshold 1 / R₀, leaving a positive fraction of the population uninfected at the end of the outbreak (S(∞) > 0) governed by the transcendental equation R(∞) = 1 − exp(−R₀ R(∞)).

Contain the Contagion: Apply SIR compartmental dynamics, transmission thresholds, and vaccination strategies to halt global pandemics in Outbreak.

Mass-Action Incidence and Constant Recovery Rates

Although the general form of Kermack and McKendrick's 1927 paper modeled infectivity β(τ) and recovery γ(τ) as arbitrary functions of the age of infection τ (yielding a system of nonlinear Volterra integro-differential equations), the celebrated special case assumes constant transition rates in a closed, homogeneously mixing population of fixed total size N = S(t) + I(t) + R(t) over an acute outbreak timescale where demographic births and natural non-disease deaths are negligible.

The population is partitioned into three mutually exclusive state compartments:

• Susceptible (S(t)): Individuals who lack immunity and can contract the infection upon infectious contact.
• Infectious (I(t)): Infected individuals currently capable of transmitting the pathogen to susceptible contacts.
• Removed / Recovered (R(t)): Individuals who have cleared the infection and acquired permanent sterilizing immunity (or have died or been isolated), so they no longer participate in transmission.

Under frequency-dependent (standard) incidence, each infectious individual makes an average of c potentially infectious contacts per unit time regardless of total population size N, and a fraction S(t) / N of those contacts are with susceptible individuals. If each contact between an infectious and a susceptible person transmits the infection with probability p, the effective transmission rate is β = cp (in units of time⁻¹), and the total number of new infections per unit time (incidence) is βS(t)I(t) / N. (In density-dependent mass-action incidence, contact rates scale with population density, writing incidence as β_dSI where β = β_dN). Simultaneously, infectious individuals recover or are removed at a constant per-capita rate γ (time⁻¹), meaning the duration of infectiousness follows an exponential distribution exp(−γt) with mean infectious period D = 1 / γ. This yields the classic three-dimensional Kermack–McKendrick SIR system:

dS / dt = −β*SI* / N, dI / dt = (β*SI* / N) − γI, dR / dt = γI

Summing the three derivatives gives dS/dt + dI/dt + dR/dt = 0, confirming conservation of total population S(t) + I(t) + R(t) = N. Defining normalized population fractions s(t) = S(t) / N, i(t) = I(t) / N, and r(t) = R(t) / N (such that s + i + r = 1) simplifies the equations to ds/dt = −β*si*, di/dt = (βs − γ)i, and dr/dt = γi.

Derivation of the Threshold and Early Invasion Dynamics

To determine whether introducing a small initial number of infectious cases I(0) = I₀ ≪ N into a susceptible population S(0) = S₀ will spark an epidemic wave or decay monotonically, factor I out of the infectious differential equation at t = 0:

(dI / dt)_t=0 = I₀ (βS₀ / N − γ) = γI₀ ([β / γ]s₀ − 1) = γI₀ (R₀s₀ − 1) = γI₀ (R_e(0) − 1)

where R₀ = β / γ is the basic reproduction number (the expected number of secondary infections generated by one infectious case in a naive susceptible population s₀ = 1), and R_e(t) = R₀ s(t) is the effective reproduction number at time t. This relation establishes Kermack and McKendrick's Epidemic Threshold Theorem:

• Subcritical Regime (s₀ < 1 / R₀, or R_e(0) < 1): Each infectious individual infects fewer than one replacement before recovering, so dI / dt < 0 for all t ≥ 0, and the infection dies out monotonically without causing an epidemic wave.
• Supercritical Regime (s₀ > 1 / R₀, or R_e(0) > 1): Each infectious individual produces more than one secondary case, so dI / dt > 0 initially. During the initial invasion phase where s(t) ≈ s₀ ≈ 1, integrating dI/dt ≈ (β − γ)I yields an early exponential growth phase I(t) = I₀ exp(r_growt):

I(t) ≈ I₀ exp(r_grow t) = I₀ exp[γ(R₀ − 1)t] where r_grow = β − γ = γ(R₀ − 1) is the intrinsic epidemic growth rate (Malthusian parameter), corresponding to an epidemic doubling time t₂ = (ln 2) / [γ(R₀ − 1)].

Consequently, to prevent an outbreak via vaccination before pathogen introduction, public health programs do not need to vaccinate 100% of the population; immunizing a fraction p_c = 1 − 1 / R₀ reduces the susceptible fraction below the threshold s_c = 1 / R₀, establishing herd immunity.

Phase-Plane Integration and Peak Prevalence

Although the nonlinear SIR equations do not admit a closed-form elementary expression for S(t) and I(t) as explicit functions of time t, Kermack and McKendrick obtained the analytical trajectory in the (s, i) phase plane by dividing the di/dt equation by the ds/dt equation to eliminate time t:

di / ds = [(βs − γ)i] / (−β*si*) = −1 + γ / (βs) = −1 + 1 / (R₀s)

Separating variables and integrating from the initial state (s₀, i₀) to any subsequent state (s, i) along the epidemic curve yields the conserved invariant of the SIR system:

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

Because the infectious fraction i(t) reaches its epidemic peak (i_max) when di/dt = 0 (which occurs when di/ds = 0), every SIR epidemic peaks at the moment when the susceptible fraction drops to the herd immunity threshold s_peak = 1 / R₀. Substituting s = 1 / R₀ into the phase-plane integral (with s₀ + i₀ = 1 and s₀ ≈ 1) gives the maximum fraction of the population simultaneously infected at the peak of the wave:

i_max = 1 − (1 / R₀)(1 + ln R₀)

For a pathogen with R₀ = 2.5 (comparable to ancestral SARS-CoV-2 or pandemic influenza), the epidemic peaks when s = 1 / 2.5 = 40% of the population remains susceptible (60% cumulative infected), with a peak simultaneous prevalence of i_max = 1 − 0.4(1 + ln 2.5) = 23.3%.

Derivation of R(∞) and Kermack–McKendrick's Approximate Bell Curve

To find the cumulative fraction of the population that has been infected by the end of the epidemic as t → ∞ (where active infections vanish, i(∞) = 0, and s_∞ + r_∞ = 1), divide ds/dt = −β*si* by dr/dt = γi to obtain the linear differential relation:

ds / dr = −(β / γ)s = −R₀s ⇒ s(t) = s₀ exp[−R₀(r(t) − r₀)]

Taking the limit t → ∞ (with initial recovered fraction r₀ = 0, s₀ ≈ 1, and s_∞ = 1 − r_∞) yields the Kermack–McKendrick final size equation:

r_∞ = 1 − exp(−R₀ r_∞) ⇔ s_∞ = exp[−R₀(1 − s_∞)]

Because exp(−R₀ r_∞) is positive for any finite R₀, s_∞ is always greater than zero (s_∞ > 0 and r_∞ < 1). This proves analytically why no natural epidemic ever infects 100% of a susceptible population. Furthermore, notice why epidemic overshoot occurs: when the susceptible fraction crosses the herd immunity threshold s = 1 / R₀, new infections stop growing (dI/dt = 0), but the prevalence i(t) is at its maximum (i_max). As that large pool of active cases recovers over the waning half of the curve, they continue to infect additional susceptibles until i → 0, driving final cumulative attack rates r_∞ substantially above the herd immunity threshold p_c = 1 − 1 / R₀. For example, at R₀ = 2.0, the herd immunity threshold is p_c = 50%, yet an unmitigated epidemic overshoots to infect r_∞ = 79.68% of the population; at R₀ = 3.0 (p_c = 66.7%), the final attack rate reaches r_∞ = 94.05%.

In their 1927 paper, Kermack and McKendrick also expanded exp(−R₀ r) in a third-order Taylor series for small epidemics (R₀ r ≪ 1) to integrate dr/dt = γ(1 − r − s₀ exp[−R₀ r]) explicitly in time. They showed that the rate of new removals dR/dt follows a symmetric hyperbolic sech² bell curve, which they fitted to weekly plague deaths in Bombay from December 1905 to July 1906:

dR / dt ≈ [γ³N² / (2β²S₀)] α² sech²(½αγt − φ)

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

The herd immunity threshold (p_c = 1 − 1 / R₀) is the fraction of the population that must be immune when active prevalence I is near zero (such as via prophylactic vaccination before an outbreak) so that R_e ≤ 1 and an epidemic cannot grow. In contrast, during an unmitigated natural epidemic, when the cumulative infected fraction reaches 1 − 1 / R₀, the number of actively infectious people I(t) is at its all-time peak (i_max). Even though each of those peak cases subsequently infects fewer than 1 person on average (R_e < 1), the sheer number of active cases at the peak generates a large wave of additional infections on the downward slope of the curve—a phenomenon called epidemic overshoot—governed by r_∞ = 1 − exp(−R₀ r_∞).

How is the basic Kermack–McKendrick SIR model extended for latent periods or waning immunity?

Real infectious diseases frequently exhibit a latent incubation period after exposure before an individual becomes infectious, or immunity that wanes over time:

• SEIR Model: Adds an Exposed (E) compartment between S and I, where newly infected individuals progress from E to I at rate σ (mean latent period 1 / σ). Adding a latent period slows the temporal growth rate r_grow of the epidemic wave, but in a closed population without demographic turnover, the threshold condition (R₀ = β / γ > 1) and the final epidemic size equation r_∞ = 1 − exp(−R₀ r_∞) remain identical to the SIR model.
• SIRS and SIS Models: If recovered individuals lose immunity and return to the Susceptible compartment at waning rate ω (R → S), or if the pathogen confers no lasting immunity at all (I → S directly, as in gonorrhea), the disease does not burn itself out to I(∞) = 0; instead, for R₀ > 1 it converges to a persistent endemic equilibrium where s^∗ = 1 / R₀.

1. Kermack, W. O., & McKendrick, A. G. (1927). "A Contribution to the Mathematical Theory of Epidemics." Proceedings of the Royal Society of London. Series A, 115(772), 700–721. doi:10.1098/rspa.1927.0118
2. Kermack, W. O., & McKendrick, A. G. (1932). "Contributions to the mathematical theory of epidemics. II. — The problem of endemicity." Proceedings of the Royal Society of London. Series A, 138(834), 55–83. doi:10.1098/rspa.1932.0171
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