Stochastic Kermack-McKendrick Model

• Stochastic Kermack-McKendrick model is a probabilistic epidemic framework that integrates susceptible depletion, infection age, and renewal structure with random effects.
• It employs multiple methodologies including finite-population Markov jump processes, stochastic networks, and individual-based models with random infectivity and waning immunity.
• Deterministic Kermack-McKendrick equations emerge as mean-field or dense-network limits, providing rigorous bounds and insights into epidemic spread and final size.

Searching arXiv for the cited topic and papers to ground the article. The stochastic Kermack–McKendrick model denotes, in the cited literature, a family of epidemic constructions that preserve the Kermack–McKendrick concern with susceptible depletion, infection age, and renewal structure, while introducing randomness at different levels of description. These levels include finite-population Markov jump epidemics, stochastic network processes, individual-based models with random infectivity and waning-immunity trajectories, and stochastic microscopic derivations whose deterministic limits are Kermack–McKendrick-type PDE or Volterra systems (Wilkinson et al., 2016, Wilkinson et al., 2016, Angstmann et al., 2015, Forien et al., 2022). The same literature also makes clear that some works adjacent to this theme are deterministic age-of-infection models supplemented by stochastic simulations or early-phase stochastic surrogates rather than literal stochastic analogues of the full Kermack–McKendrick infection-age theory (Demongeot et al., 2022, Kobayashi, 2020, Velasco-Hernandez, 2021).

1. Scope of the concept

At its narrowest, a stochastic Kermack–McKendrick model is a probabilistic epidemic process whose transmission mechanism depends on time since infection and whose deterministic closure or limit recovers a Kermack–McKendrick renewal or infection-age equation. This is the role played by the directed continuous-time random-walk constructions in which an infected individual contributes force of infection according to an infection-age kernel, and by individual-based reinfection models in which each infection episode draws a random infectivity path and a random susceptibility path (Angstmann et al., 2015, Forien et al., 2022).

At a broader level, the term also includes stochastic SIR models that retain the Kermack–McKendrick compartmental backbone but randomize infection and recovery events directly. The classical finite-population “general stochastic epidemic” on the complete graph is of this type: it is Markovian, homogeneous-mixing, and does not carry a general infection-age density, yet it is explicitly compared with the deterministic Kermack–McKendrick SIR system and shown to lie on one side of it in expectation (Wilkinson et al., 2016).

The literature also identifies a third layer, in which Kermack–McKendrick equations arise as limits or reductions of richer stochastic mechanisms. Message-passing and pairwise systems derived from stochastic network epidemics converge to deterministic Kermack–McKendrick equations under dense symmetric-network scaling; gathering-based jump Markov epidemics yield generalized-incidence SIR ODEs as mean-field limits; kinetic particle systems reduce to classical SIR under spatial homogenization and Maxwell-type closure (Wilkinson et al., 2016, Cortez, 2022, Pulvirenti et al., 2020).

A recurrent terminological issue is that “stochastic” is not used uniformly. Some papers are deterministic in their main analytical object and use stochasticity only in simulation-based robustness checks or as hidden probabilistic interpretation of residence-time assumptions. This is explicit in the age-of-infection identifiability paper, the SARS-CoV-2 modeling review, and the reinfection-age Volterra paper (Demongeot et al., 2022, Velasco-Hernandez, 2021, Li et al., 5 Dec 2025).

2. The classical finite-population stochastic SIR formulation

A central rigorous benchmark is the finite-population general stochastic SIR epidemic on a complete graph. The population is a finite set $\mathcal V$ of size $N$, with deterministic initial susceptible set $\mathcal S_0$ and deterministic initial infected set $\mathcal I_0=\mathcal V\setminus\mathcal S_0$. While infected, each individual makes infectious contacts to any specified other individual according to an independent Poisson process of rate $\beta$, infection occurs immediately upon infectious contact, and the infectious period is an independent exponential random variable with parameter $\gamma$ (Wilkinson et al., 2016).

If $X(t)$ and $Y(t)$ are the numbers susceptible and infected at time $t$, then $(X(t),Y(t))$ is a continuous-time Markov chain with transitions

$N$0

$N$1

and $N$2 is the recovered count. The deterministic comparator is the classical Markovian Kermack–McKendrick system

$N$3

with $N$4, $N$5, and $N$6 (Wilkinson et al., 2016).

The principal result is one-sided and strict: for all $N$7,

$N$8

Thus the deterministic Kermack–McKendrick model is a strict lower bound on the expected number of susceptibles and a strict upper bound on the expected number of recoveries in the Markovian stochastic epidemic. At the end of the epidemic, if $N$9, then $\mathcal S_0$0, so the expected stochastic final size is strictly smaller than the deterministic final size (Wilkinson et al., 2016).

The proof inserts a message-passing approximation between the stochastic process and the deterministic ODE. On the complete graph,

$\mathcal S_0$1

Here $\mathcal S_0$2 is built from cavity-state non-transmission probabilities $\mathcal S_0$3, which are exact on trees and lower bounds on general graphs. In this setting the deterministic Kermack–McKendrick trajectory is not merely a heuristic large-$\mathcal S_0$4 average; it is an exact finite-$\mathcal S_0$5 lower bound for susceptibility and an exact finite-$\mathcal S_0$6 upper bound for recoveries under Poisson transmission and exponential recovery (Wilkinson et al., 2016).

3. Network epidemics, message passing, and dense-network limits

A more general stochastic Kermack–McKendrick framework is obtained on contact networks. In the network SIR model, each individual $\mathcal S_0$7 has an initial state $\mathcal S_0$8, an infectious period $\mathcal S_0$9, and neighbour-specific contact times $\mathcal I_0=\mathcal V\setminus\mathcal S_0$0; infectious periods and post-infection contact times may be correlated within an individual, while different individuals remain independent. For this model, the message-passing variable $\mathcal I_0=\mathcal V\setminus\mathcal S_0$1 denotes the probability that $\mathcal I_0=\mathcal V\setminus\mathcal S_0$2, placed in cavity state, receives no infectious contact from $\mathcal I_0=\mathcal V\setminus\mathcal S_0$3 by time $\mathcal I_0=\mathcal V\setminus\mathcal S_0$4, and the susceptible approximation is

$\mathcal I_0=\mathcal V\setminus\mathcal S_0$5

The message-passing system has a unique feasible solution, and it provides a rigorous upper bound for expected epidemic size at any fixed time $\mathcal I_0=\mathcal V\setminus\mathcal S_0$6 because it underestimates susceptibility and overestimates recovery; equality holds on trees and forests (Wilkinson et al., 2016).

In homogeneous $\mathcal I_0=\mathcal V\setminus\mathcal S_0$7-regular symmetric networks the system reduces to four equations. If $\mathcal I_0=\mathcal V\setminus\mathcal S_0$8 is the common message, then

$\mathcal I_0=\mathcal V\setminus\mathcal S_0$9

This reduction supports several structural conclusions. Cycles inhibit spread relative to the Bethe lattice, final size satisfies an explicit message-passing equation in terms of $\beta$0, and the threshold parameter is $\beta$1 in the small-initial-infection regime (Wilkinson et al., 2016).

For Poisson contact processes, the same message-passing system is equivalent to a non-Markovian pairwise model, with standard pairwise ODE and delay systems emerging as special cases for exponential and fixed infectious periods. The deterministic Kermack–McKendrick equations arise as the limit of a sequence of homogeneous message-passing systems when $\beta$2 and $\beta$3 uniformly on compact time intervals. In this sense, deterministic Kermack–McKendrick dynamics appear as a dense-network asymptotic description of an underlying stochastic epidemic rather than as an isolated deterministic postulate (Wilkinson et al., 2016).

The effect of infectious-period variability has also been analyzed in a broad stochastic SEIR network model that contains classic Kermack–McKendrick, pairwise, message passing, and spatial models as special cases. Under convexity conditions on contact-time survival functions, decreasing infectious-period variability while keeping its mean fixed increases the probability that infection reaches any given subset of the population by any given time; at fixed per-edge transmission probability, greater variability in posterior transmissibility likewise tends to dampen spread. For Poisson contact processes and arbitrarily distributed infectious periods, delay-differential and ordinary-differential systems can provide lower and upper bounds, respectively, for the probability that any given individual has been infected by any given time (Wilkinson et al., 2017).

4. Infection age, memory, and fractional stochastic derivations

A specifically Kermack–McKendrick notion of stochasticity appears when infection age is modeled explicitly and infectivity depends on time since infection. In the continuous-time random-walk construction, an individual infected at time $\beta$4 infects a particular susceptible in $\beta$5 with probability

$\beta$6

with the factorization

$\beta$7

If $\beta$8 is the probability that the individual remains infectious and alive from $\beta$9 to $\gamma$0, then the incidence flux satisfies the renewal equation

$\gamma$1

This is already a Kermack–McKendrick Volterra structure. The infected population can be written as

$\gamma$2

and the corresponding infection-age density is

$\gamma$3

The model is therefore equivalent to an infection-age-structured Kermack–McKendrick system with age-specific infectivity kernel $\gamma$4 (Angstmann et al., 2015).

The same construction identifies a memory kernel

$\gamma$5

so that the force of infection becomes a nonlocal Volterra operator acting on $\gamma$6. When

$\gamma$7

the Laplace multiplier of the memory kernel is $\gamma$8, and the convolution becomes a Riemann–Liouville fractional derivative of order $\gamma$9. The resulting fractional infectivity SIR system is not an ad hoc fractional replacement of $X(t)$0; the fractional operator is induced by an explicit stochastic infectivity-age law (Angstmann et al., 2015).

A complementary stochastic derivation places the heavy tail in recovery rather than transmission. In the directed CTRW model with births and deaths, infectious survival is

$X(t)$1

where $X(t)$2 is the probability of not yet recovering and $X(t)$3 is survival against death. The memory kernel

$X(t)$4

enters the generalized master equation for $X(t)$5. If the recovery waiting time is Mittag–Leffler with power-law tail, then the recovery term becomes fractional and the model is consistent with both the age-structured Kermack–McKendrick formulation and the Hethcote–Tudor integral-equation SIR model. Here the non-Markovianity is in recovery, whereas in the infectivity-fractional model it is in transmission (Angstmann et al., 2015).

These two constructions show that stochastic Kermack–McKendrick theory need not be confined to Markovian Poisson infection and exponential recovery. It can also arise from explicit stochastic processes with infection-age-dependent infectivity, heavy-tailed residence times, and memory kernels that remain exactly interpretable in renewal terms (Angstmann et al., 2015, Angstmann et al., 2015).

5. Individual-based stochastic reinfection and waning immunity

A more general individual-based extension introduces repeated infection, random infectivity profiles, and random waning-immunity trajectories. In a population of size $X(t)$6, individual $X(t)$7 and infection episode $X(t)$8 draw an i.i.d. pair

$X(t)$9

where $Y(t)$0 is infectivity $Y(t)$1 units after the $Y(t)$2-th infection and $Y(t)$3 is susceptibility at that same elapsed time. If $Y(t)$4 counts infections of individual $Y(t)$5 on $Y(t)$6 and $Y(t)$7 is the age since the last infection, then the current infection intensity of $Y(t)$8 is

$Y(t)$9

where

$t$0

is average infectivity. The counting processes are constructed from independent Poisson random measures via

$t$1

This is a genuinely stochastic homogeneous-mixing epidemic with reinfection and non-Markovian episode-specific trajectories (Forien et al., 2022).

The main asymptotic result is a functional law of large numbers for empirical average susceptibility and average infectivity. The deterministic limit $t$2 solves a coupled nonlinear Volterra system:

$t$3

$t$4

The proof uses i.i.d. auxiliary processes and a propagation-of-chaos argument adapted to age-since-infection dynamics (Forien et al., 2022).

A notable asymmetry emerges in the limit: it depends on the full law of the susceptibility random functions but only on the mean infectivity functions. This is because infectivity contributes additively to the population force of infection, while susceptibility enters through exponential no-reinfection factors. The framework includes SIS, SIR, and SIRS as special cases, and under deterministic infectivity and susceptibility functions it recovers a Kermack–McKendrick SIRS PDE of the type studied by Inaba (Forien et al., 2022).

The same model yields a nonclassical endemic threshold. With

$t$5

and $t$6, extinction occurs when

$t$7

while supercritical behavior is governed by the same harmonic-mean quantity rather than by $t$8. This result makes heterogeneity in long-term post-infection susceptibility structurally important for persistence (Forien et al., 2022).

6. Adjacent models, deterministic analogues, and recurring misconceptions

Several adjacent models are relevant to the topic but are not, in the strict sense, full stochastic Kermack–McKendrick systems. One example is the birth–death process with immigration for the infected count $t$9, with rates $(X(t),Y(t))$0 and $(X(t),Y(t))$1. Its transient law is exactly negative binomial, with

$(X(t),Y(t))$2

and it explains overdispersion, a mode at zero when $(X(t),Y(t))$3, and large discrepancies between deterministic means and typical realizations. However, it is best understood as a stochastic linearized early-phase SIR model with importation, not as a full stochastic replacement of the Kermack–McKendrick infection-age system (Kobayashi, 2020).

Another example is the kinetic particle formulation in which agents move in physical space, infection corresponds to the reactive collision $(X(t),Y(t))$4, and recovery corresponds to $(X(t),Y(t))$5. The microscopic model is stochastic, but the main equations are Boltzmann-type PDEs for phase-space densities $(X(t),Y(t))$6, $(X(t),Y(t))$7, and $(X(t),Y(t))$8. After spatial homogenization and, crucially, Maxwell-molecule closure, the macroscopic system reduces to

$(X(t),Y(t))$9

This gives a stochastic microscopic underpinning of classical SIR while remaining distinct from standard stochastic compartment processes (Pulvirenti et al., 2020).

The gathering-based jump Markov SIR model occupies an intermediate position. Gatherings occur at rate $N$00, the gathering size $N$01 is random, and each susceptible attendee is infected with probability $N$02 given $N$03 infected attendees. The finite-$N$04 process is a continuous-time jump Markov chain, and its mean-field limit yields

$N$05

with

$N$06

Its basic reproduction number is

$N$07

This is a stochastic microscopic foundation for generalized incidence in a Markovian SIR model, not a non-Markovian infection-age Kermack–McKendrick theory (Cortez, 2022).

The same boundary appears in deterministic age-of-infection inference work. The model

$N$08

is deterministic, and the corresponding Volterra equation for incidence is deterministic as well. The stochastic component is confined to an individual-based simulation model used to test robustness, not to the main analytical epidemic formulation (Demongeot et al., 2022). Likewise, the deterministic review of Kermack–McKendrick-type models emphasizes that standard ODE formulations hide exponential waiting-time assumptions and that more realistic passage times are Gamma or Beta distributed, but it does not itself formulate a fully stochastic Kermack–McKendrick process (Velasco-Hernandez, 2021). The recent reinfection-and-infection-age paper is also deterministic: its central object is a nonlinear Volterra equation for incidence

$N$09

with the age-specific reproductive power

$N$10

which is a renewal kernel rather than a stochastic intensity model (Li et al., 5 Dec 2025).

A persistent misconception is therefore to treat every model with random simulation output, every stochastic surrogate for $N$11, or every deterministic renewal system with hidden probabilistic ingredients as a stochastic Kermack–McKendrick model in the same sense. The literature instead supports a sharper distinction. Some models are exact finite-population stochastic epidemics compared against Kermack–McKendrick bounds (Wilkinson et al., 2016); some are stochastic network or individual-based systems whose law-of-large-numbers limits are Kermack–McKendrick-type equations (Wilkinson et al., 2016, Forien et al., 2022); some are stochastic derivations of non-Markovian infection-age or fractional models (Angstmann et al., 2015, Angstmann et al., 2015); and some are deterministic models or early-phase stochastic approximations that are relevant to Kermack–McKendrick theory without constituting full stochastic analogues (Demongeot et al., 2022, Kobayashi, 2020, Velasco-Hernandez, 2021, Li et al., 5 Dec 2025).