The Final Size of an Epidemic

Module 1: Low-Dimensional Dynamics

An SIR epidemic always ends with some susceptibles left. How many people are infected in total can be computed without solving the ODE in time (Kermack and McKendrick 1927; Hethcote 2000).

The final size relation

Divide $\dot S = -\beta S I$ by $\dot R = \gamma I$:

\[ \frac{dS}{dR} = -R_0 S \quad \Longrightarrow \quad S(t) = S(0)\, e^{-R_0 (R(t) - R(0))} . \]

As $t \to \infty$, $I \to 0$. With $R(0) = 0$ and $R(\infty) = 1 - S(\infty)$ you get

\[ \ln\frac{S(0)}{S(\infty)} = R_0 \bigl(1 - S(\infty)\bigr) . \tag{1}\]

For a small initial seed, $S(0) \to 1$, and the fraction ever infected, $z = 1 - S(\infty)$, satisfies

\[ z = 1 - e^{-R_0 z} . \tag{2}\]

For $R_0 \le 1$ the only solution in $[0, 1]$ is $z = 0$. For $R_0 > 1$ there is a unique positive solution, and it is the final size (Hethcote 2000).

Solving it numerically

Equation 2 has no closed-form solution in elementary functions, but a root finder solves it instantly.

import numpy as np
from scipy.optimize import brentq

for r0 in [1.5, 2.0, 3.0, 5.0]:
    z = brentq(lambda z: z - 1 + np.exp(-r0 * z), 1e-9, 1.0)
    print(f"R0 = {r0}: final size z = {z:.3f}, herd immunity threshold = {1 - 1 / r0:.3f}")

R0 = 1.5: final size z = 0.583, herd immunity threshold = 0.333
R0 = 2.0: final size z = 0.797, herd immunity threshold = 0.500
R0 = 3.0: final size z = 0.940, herd immunity threshold = 0.667
R0 = 5.0: final size z = 0.993, herd immunity threshold = 0.800

The final size is larger than the herd immunity threshold $1 - 1/R_0$. The epidemic overshoots: when $S$ reaches $1/R_0$ the number of infectious people is at its maximum, and they still cause many infections on the way down.

Peak prevalence

The same trick gives the maximum of $I$. From $dI/dS = -1 + 1/(R_0 S)$,

\[ I_{\max} = I(0) + S(0) - \frac{1}{R_0} - \frac{1}{R_0}\ln\bigl(R_0 S(0)\bigr), \]

valid when $R_0 S(0) > 1$ (Hethcote 2000). amlab.epidemics.peak_prevalence implements this formula.

Exercise

Check the final size relation against a simulation for $R_0 = 3$ ($\beta = 0.3$, $\gamma = 0.1$) with $I(0) = 10^{-4}$.

Solution

from scipy.integrate import solve_ivp

from amlab.epidemics import final_size, sir_rhs

sol = solve_ivp(sir_rhs, (0, 400), [1 - 1e-4, 1e-4, 0.0], args=(0.3, 0.1), rtol=1e-9)
print(f"Simulated: {1 - sol.y[0, -1]:.4f}")
print(f"Relation:  {final_size(3.0, s0=1 - 1e-4) + 1e-4:.4f}")

Simulated: 0.9405
Relation:  0.9405

The simulated fraction $1 - S(\infty)$ matches Equation 1 including the initial seed.

Extensions

The SIR model is the starting point of a large family. Each extension relaxes one assumption from Compartmental Models.

SEIR: an exposed class $E$ for people infected but not yet infectious. It changes the timing of the epidemic but, in the simplest case, not $R_0$ (Hethcote 2000; Keeling and Rohani 2008).
Vital dynamics: births and deaths allow endemic equilibria and recurrent epidemics (Hethcote 2000).
Age structure: contact rates that depend on age (Hethcote 2000; Keeling and Rohani 2008).
Spatial structure and networks: contacts limited to neighbours or to a contact network. Module 5 studies epidemics on networks (Pastor-Satorras et al. 2015).
Stochastic models: needed when the number of cases is small, for example to compute the probability that an outbreak dies out by chance (Keeling and Rohani 2008).

Session 6 implements these models and fits them to data.

References

Hethcote, Herbert W. 2000. “The Mathematics of Infectious Diseases.” SIAM Review 42 (4): 599–653. https://doi.org/10.1137/S0036144500371907.

Keeling, Matt J., and Pejman Rohani. 2008. Modeling Infectious Diseases in Humans and Animals. Princeton University Press.

Kermack, William O., and Anderson G. McKendrick. 1927. “A Contribution to the Mathematical Theory of Epidemics.” Proceedings of the Royal Society A 115 (772): 700–721. https://doi.org/10.1098/rspa.1927.0118.

Pastor-Satorras, Romualdo, Claudio Castellano, Piet Van Mieghem, and Alessandro Vespignani. 2015. “Epidemic Processes in Complex Networks.” Reviews of Modern Physics 87 (3): 925–79. https://doi.org/10.1103/RevModPhys.87.925.

--- title: "The Final Size of an Epidemic" subtitle: "Module 1: Low-Dimensional Dynamics" format: html --- An SIR epidemic always ends with some susceptibles left. How many people are infected in total can be computed without solving the ODE in time [@kermack1927contribution; @hethcote2000mathematics]. ## The final size relation Divide $\dot S = -\beta S I$ by $\dot R = \gamma I$: $$ \frac{dS}{dR} = -R_0 S \quad \Longrightarrow \quad S(t) = S(0)\, e^{-R_0 (R(t) - R(0))} . $$ As $t \to \infty$, $I \to 0$. With $R(0) = 0$ and $R(\infty) = 1 - S(\infty)$ you get $$ \ln\frac{S(0)}{S(\infty)} = R_0 \bigl(1 - S(\infty)\bigr) . $$ {#eq-final-size} For a small initial seed, $S(0) \to 1$, and the fraction ever infected, $z = 1 - S(\infty)$, satisfies $$ z = 1 - e^{-R_0 z} . $$ {#eq-final-size-z} For $R_0 \le 1$ the only solution in $[0, 1]$ is $z = 0$. For $R_0 > 1$ there is a unique positive solution, and it is the final size [@hethcote2000mathematics]. ## Solving it numerically @eq-final-size-z has no closed-form solution in elementary functions, but a root finder solves it instantly. ```{python} import numpy as np from scipy.optimize import brentq for r0 in [1.5, 2.0, 3.0, 5.0]: z = brentq(lambda z: z - 1 + np.exp(-r0 * z), 1e-9, 1.0) print(f"R0 = {r0}: final size z = {z:.3f}, herd immunity threshold = {1 - 1 / r0:.3f}") ``` The final size is larger than the herd immunity threshold $1 - 1/R_0$. The epidemic overshoots: when $S$ reaches $1/R_0$ the number of infectious people is at its maximum, and they still cause many infections on the way down. ## Peak prevalence The same trick gives the maximum of $I$. From $dI/dS = -1 + 1/(R_0 S)$, $$ I_{\max} = I(0) + S(0) - \frac{1}{R_0} - \frac{1}{R_0}\ln\bigl(R_0 S(0)\bigr), $$ valid when $R_0 S(0) > 1$ [@hethcote2000mathematics]. `amlab.epidemics.peak_prevalence` implements this formula. ::: {.callout-note} ## Exercise Check the final size relation against a simulation for $R_0 = 3$ ($\beta = 0.3$, $\gamma = 0.1$) with $I(0) = 10^{-4}$. ::: ::: {.callout-tip collapse="true"} ## Solution ```{python} from scipy.integrate import solve_ivp from amlab.epidemics import final_size, sir_rhs sol = solve_ivp(sir_rhs, (0, 400), [1 - 1e-4, 1e-4, 0.0], args=(0.3, 0.1), rtol=1e-9) print(f"Simulated: {1 - sol.y[0, -1]:.4f}") print(f"Relation: {final_size(3.0, s0=1 - 1e-4) + 1e-4:.4f}") ``` The simulated fraction $1 - S(\infty)$ matches @eq-final-size including the initial seed. ::: ## Extensions The SIR model is the starting point of a large family. Each extension relaxes one assumption from [Compartmental Models](compartmental-models.qmd). - **SEIR:** an exposed class $E$ for people infected but not yet infectious. It changes the timing of the epidemic but, in the simplest case, not $R_0$ [@hethcote2000mathematics; @keeling2008modeling]. - **Vital dynamics:** births and deaths allow endemic equilibria and recurrent epidemics [@hethcote2000mathematics]. - **Age structure:** contact rates that depend on age [@hethcote2000mathematics; @keeling2008modeling]. - **Spatial structure and networks:** contacts limited to neighbours or to a contact network. Module 5 studies epidemics on networks [@pastor2015epidemic]. - **Stochastic models:** needed when the number of cases is small, for example to compute the probability that an outbreak dies out by chance [@keeling2008modeling]. ## Next Session 6 implements these models and fits them to data.