Epidemic models

The following compartmental models are provided by epifx:

A deterministic SEIR ODE model;
A deterministic SEEIIR ODE model;
A stochastic SEEIIR ODE model; and
A NoEpidemic “null model”, which does not allow for epidemic activity.

Note

The behaviour of these models can be modified by providing specific lookup tables. See the documentation for each model, below, for further details.

Deterministic compartment models

class epifx.det.SEIR(*args: Any, **kwargs: Any)

An SEIR compartment model for a single circulating influenza strain, under the assumption that recovered individuals are completely protected against reinfection.

\[\begin{split}\frac{dS}{dt} &= - \beta S^\eta I \\[0.5em] \frac{dE}{dt} &= \beta S^\eta I - \sigma E \\[0.5em] \frac{dI}{dt} &= \sigma E - \gamma I \\[0.5em] \frac{dR}{dt} &= \gamma I \\[0.5em] \beta &= R_0 \cdot \gamma \\[0.5em] E(0) &= \frac{1}{N}\end{split}\]

Parameter	Meaning
\(R_0\)	Basic reproduction number
\(\sigma\)	Inverse of the incubation period (day ^-1)
\(\gamma\)	Inverse of the infectious period (day ^-1)
\(\eta\)	Inhomogeneous social mixing coefficient
\(\alpha\)	Temporal forcing coefficient
\(t_0\)	The time at which the epidemic begins

The force of infection can be subject to temporal forcing \(F(t)\), as mediated by \(\alpha\):

\[\beta(t) = \beta \cdot \left[1 + \alpha \cdot F(t)\right]\]

Note that this requires the forcing time-series to be stored in the lookup table 'R0_forcing'.

Note

The population size \(N\) must be defined for each scenario:

[model]
population_size = 1000

Initialise the model instance.

stat_Reff(hist)

Calculate the effective reproduction number \(R_\mathrm{eff}\) for every particle.

Parameters:: hist – The particle history matrix, or a subset thereof.

class epifx.det.SEEIIR(*args: Any, **kwargs: Any)

An SEEIIR compartment model for a single circulating influenza strain, under the assumption that recovered individuals are completely protected against reinfection.

\[\begin{split}\frac{dS}{dt} &= - \beta S^\eta (I_1 + I_2) \\[0.5em] \frac{dE_1}{dt} &= \beta S^\eta (I_1 + I_2) - 2 \sigma E_1 \\[0.5em] \frac{dE_2}{dt} &= 2 \sigma E_1 - 2 \sigma E_2 \\[0.5em] \frac{dI_1}{dt} &= 2 \sigma E_2 - 2 \gamma I_1 \\[0.5em] \frac{dI_2}{dt} &= 2 \gamma I_1 - 2 \gamma I_2 \\[0.5em] \frac{dR}{dt} &= 2 \gamma I_2 \\[0.5em] \beta &= R_0 \cdot \gamma \\[0.5em] E_1(0) &= \frac{1}{N}\end{split}\]

Parameter	Meaning
\(R_0\)	Basic reproduction number
\(\sigma\)	Inverse of the incubation period (day ^-1)
\(\gamma\)	Inverse of the infectious period (day ^-1)
\(\eta\)	Inhomogeneous social mixing coefficient
\(\alpha\)	Temporal forcing coefficient
\(t_0\)	The time at which the epidemic begins

The force of infection can be subject to temporal forcing \(F(t)\), as mediated by \(\alpha\):

\[\beta(t) = \beta \cdot \left[1 + \alpha \cdot F(t)\right]\]

Note that this requires the forcing time-series to be stored in the lookup table 'R0_forcing'.

Note

The population size \(N\) must be defined for each scenario:

[model]
population_size = 1000

stat_Reff(hist)

Calculate the effective reproduction number \(R_\mathrm{eff}\) for every particle.

Parameters:: hist – The particle history matrix, or a subset thereof.

class epifx.det.NoEpidemic(*args: Any, **kwargs: Any)

A model that assumes there will be no epidemic activity.

This may be a useful hypothesis against which to evaluate other models.

Stochastic compartment models

class epifx.stoch.SEEIIR(*args: Any, **kwargs: Any)

A stochastic SEEIIR compartment model.

The mean rates are defined by the following equations:

\[\begin{split}\frac{dS}{dt} &= - \beta \frac{S}{N} (I_1 + I_2) \\[0.5em] \frac{dE_1}{dt} &= \beta \frac{S}{N} (I_1 + I_2) - 2 \sigma E_1 \\[0.5em] \frac{dE_2}{dt} &= 2 \sigma E_1 - 2 \sigma E_2 \\[0.5em] \frac{dI_1}{dt} &= 2 \sigma E_2 - 2 \gamma I_1 \\[0.5em] \frac{dI_2}{dt} &= 2 \gamma I_1 - 2 \gamma I_2 \\[0.5em] \frac{dR}{dt} &= 2 \gamma I_2 \\[0.5em] \beta &= R_0 \cdot \gamma \\[0.5em] E_1(0) &= 10\end{split}\]

Parameter	Meaning
\(R_0\)	Basic reproduction number
\(\sigma\)	Inverse of the incubation period (day ^-1)
\(\gamma\)	Inverse of the infectious period (day ^-1)
\(t_0\)	The time at which the epidemic begins

The basic reproduction number \(R_0\) can be sampled from the lookup table 'R0', in which case each particle is associated with a specific \(R_0\) trajectory 'R0_ix'. In this case, there must be prior samples for 'R0_ix'. Note that this allows \(R_0\) to vary over time.

External exposures can be injected into the model from the lookup table 'external_exposures'. These exposures \(e_\mathrm{ext}(t)\) have the following effect on the model equations:

\[\begin{split}\frac{dS}{dt} &= - \beta \frac{S}{N} (I_1 + I_2) - e_\mathrm{ext}(t) \\[0.5em] \frac{dE_1}{dt} &= \beta \frac{S}{N} (I_1 + I_2) + e_\mathrm{ext}(t) - 2 \sigma E_1\end{split}\]

Note

The population size \(N\) must be defined for each scenario:

[model]
population_size = 1000

stat_generation_interval(hist)

Calculate the mean generation interval for each particle.

Parameters:: hist – The particle history matrix, or a subset thereof.