Kermack–McKendrick theory

In its initial form, Kermack–McKendrick theory is a compartmental differential-equation model that structures the infectioned population in terms of age-of-infection, while using simple compartments for people who are susceptible (S) and recovered/removed (R). Specified initial conditions would change over time according to

${\displaystyle {\frac {dS}{dt}}=-\lambda S,}$

${\displaystyle {\frac {\partial i}{\partial t}}+{\frac {\partial i}{\partial a}}=\delta (a)\lambda S-\gamma (a)i,}$

${\displaystyle {\frac {dR}{dt}}=\int _{0}^{\infty }\gamma (a)i(a,t)\,da,}$

where ${\displaystyle \delta (a)}$ is a Dirac delta-function and the infection pressure

${\displaystyle \lambda =\int _{0}^{\infty }\beta (a)i(a,t)\,da.}$

Only in the special case when the removal rate ${\displaystyle \gamma (a)}$ and the transmission rate ${\displaystyle \beta (a)}$ are constant for all ages does the substitution ${\displaystyle I(t)=\int _{0}^{\infty }i(a,t)\,da}$ transform their theory into the simple SIR model. This basic model only accounts for infection and removal events, which are sufficient to describe a simple epidemic, including the threshold condition necessary for an epidemic to start, but can not explain endemic disease transmission or recurring epidemics.

In their subsequent articles, Kermack and McKendrick extended their theory to allow for birth, migration, and death, as well as imperfect immunity. In modern notation, their model can be represented as

${\displaystyle {\frac {dS}{dt}}=b_{0}+b_{S}S+b_{I}I+b_{R}R-\lambda S-m_{S}S,}$

${\displaystyle {\frac {\partial i}{\partial t}}+{\frac {\partial i}{\partial a}}=\delta (a)\lambda (S+\sigma R)-\gamma (a)i-\mu (a)i-m_{i}(a)i,}$

${\displaystyle I(t)=\int _{0}^{\infty }i(a,t)\,da}$

${\displaystyle {\frac {dR}{dt}}=\int _{0}^{\infty }\gamma (a)i(a,t)\,da-\sigma \lambda R-m_{R}R,}$

where ${\displaystyle b_{0}}$ is the immigration rate of susceptibles, b_j is the per-capita birth rate for state j, m_j is the per-capita mortality rate of individuals in state j, ${\displaystyle \sigma }$ is the relative-risk of infection to recovered individuals who are partially immune, and the infection pressure

${\displaystyle \lambda =\int _{0}^{\infty }\beta (a)i(a,t)\,da.}$

Kermack and McKendrick were able to show that it admits a stationary solution where disease is endemic, as long as the supply of susceptible individuals is sufficiently large. This model is difficult to analyze in its full generality, and a number of open questions remain regarding its dynamics.