Compartmental Models: SIR and SIS

Module 1: Low-Dimensional Dynamics

Reading

Compulsory: (britton2003essential?), chapter on epidemic models.
Recommended: (murray2002mathematical?), chapter on the dynamics of infectious diseases.

Compartmental models divide a population into groups according to infection status and describe the flows between them. The SIR model goes back to Kermack and McKendrick (Kermack and McKendrick 1927); (hethcote2000mathematics?) and Keeling and Rohani (2008) review the family.

The SIR model

Let $S$, $I$, $R$ be the fractions of the population that are susceptible, infectious and removed (recovered and immune, or dead). Then

\[ \begin{aligned} \dot S &= -\beta S I,\\ \dot I &= \beta S I - \gamma I,\\ \dot R &= \gamma I . \end{aligned} \tag{1}\]

$\beta$ is the transmission rate: a susceptible person meets infectious people at a rate proportional to $I$, and each contact transmits with a fixed probability.
$\gamma$ is the recovery rate. The infectious period is exponentially distributed with mean $1/\gamma$.
$S + I + R = 1$ is conserved, so only two equations are independent.

The SIR model describes a single outbreak of a disease that gives lasting immunity, such as measles in a population without vaccination (hethcote2000mathematics?).

The SIS model

If recovery gives no immunity, recovered individuals return to $S$:

\[ \dot S = -\beta S I + \gamma I, \qquad \dot I = \beta S I - \gamma I . \tag{2}\]

With $S = 1 - I$ this reduces to one equation, $\dot I = \beta I (1 - I) - \gamma I$, which is a logistic equation for $I$. Its long-run state is a disease-free equilibrium or an endemic equilibrium (hethcote2000mathematics?). SIS models are used for some bacterial infections and sexually transmitted diseases that do not confer immunity (Keeling and Rohani 2008).

Assumptions

Each assumption is a simplification you should be able to defend or replace.

Well mixed. Every individual is equally likely to meet every other. Real contacts are structured by households, schools, age and geography.
Constant rates. $\beta$ and $\gamma$ do not change in time. Seasons, behaviour change and interventions all change $\beta$.
Closed population. No births, deaths or migration during the outbreak. This is reasonable for a short epidemic, not for an endemic disease over decades.
Exponential infectious period. Recovery happens at a constant rate, so some people recover almost immediately and others stay infectious for a long time.
Deterministic and continuous. Fractions change smoothly. This fails at the start of an outbreak, when there are only a few cases and chance matters.

Exercise

In Equation 1, show that $S(t)$ is decreasing and $R(t)$ is increasing for all $t$. Can $I(t)$ increase?

Solution

$\dot S = -\beta S I \le 0$ and $\dot R = \gamma I \ge 0$ because $S, I \ge 0$. $\dot I = I(\beta S - \gamma)$ is positive when $S > \gamma/\beta$. So $I$ grows while the susceptible fraction is above $\gamma/\beta$, then decreases. This threshold is the subject of the next page.

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.

--- title: "Compartmental Models: SIR and SIS" subtitle: "Module 1: Low-Dimensional Dynamics" format: html --- ::: {.callout-note} ## Reading - **Compulsory:** @britton2003essential, chapter on epidemic models. - **Recommended:** @murray2002mathematical, chapter on the dynamics of infectious diseases. ::: Compartmental models divide a population into groups according to infection status and describe the flows between them. The SIR model goes back to Kermack and McKendrick [@kermack1927contribution]; @hethcote2000mathematics and @keeling2008modeling review the family. ## The SIR model Let $S$, $I$, $R$ be the fractions of the population that are susceptible, infectious and removed (recovered and immune, or dead). Then $$ \begin{aligned} \dot S &= -\beta S I,\\ \dot I &= \beta S I - \gamma I,\\ \dot R &= \gamma I . \end{aligned} $$ {#eq-sir} - $\beta$ is the transmission rate: a susceptible person meets infectious people at a rate proportional to $I$, and each contact transmits with a fixed probability. - $\gamma$ is the recovery rate. The infectious period is exponentially distributed with mean $1/\gamma$. - $S + I + R = 1$ is conserved, so only two equations are independent. The SIR model describes a single outbreak of a disease that gives lasting immunity, such as measles in a population without vaccination [@hethcote2000mathematics]. ## The SIS model If recovery gives no immunity, recovered individuals return to $S$: $$ \dot S = -\beta S I + \gamma I, \qquad \dot I = \beta S I - \gamma I . $$ {#eq-sis} With $S = 1 - I$ this reduces to one equation, $\dot I = \beta I (1 - I) - \gamma I$, which is a logistic equation for $I$. Its long-run state is a disease-free equilibrium or an endemic equilibrium [@hethcote2000mathematics]. SIS models are used for some bacterial infections and sexually transmitted diseases that do not confer immunity [@keeling2008modeling]. ## Assumptions Each assumption is a simplification you should be able to defend or replace. - **Well mixed.** Every individual is equally likely to meet every other. Real contacts are structured by households, schools, age and geography. - **Constant rates.** $\beta$ and $\gamma$ do not change in time. Seasons, behaviour change and interventions all change $\beta$. - **Closed population.** No births, deaths or migration during the outbreak. This is reasonable for a short epidemic, not for an endemic disease over decades. - **Exponential infectious period.** Recovery happens at a constant rate, so some people recover almost immediately and others stay infectious for a long time. - **Deterministic and continuous.** Fractions change smoothly. This fails at the start of an outbreak, when there are only a few cases and chance matters. ::: {.callout-note} ## Exercise In @eq-sir, show that $S(t)$ is decreasing and $R(t)$ is increasing for all $t$. Can $I(t)$ increase? ::: ::: {.callout-tip collapse="true"} ## Solution $\dot S = -\beta S I \le 0$ and $\dot R = \gamma I \ge 0$ because $S, I \ge 0$. $\dot I = I(\beta S - \gamma)$ is positive when $S > \gamma/\beta$. So $I$ grows while the susceptible fraction is above $\gamma/\beta$, then decreases. This threshold is the subject of the next page. ::: ## Next