Class SISe

Class to handle the SISe model. This class inherits from SimInf_model, meaning that SISe objects are fully compatible with all generic functions defined for SimInf_model, such as run, plot, trajectory, and prevalence.

Details

The SISe model contains two compartments: Susceptible ($S$) and Infected ($I$). Additionally, it includes a continuous environmental compartment ($\varphi$) to model the shedding of a pathogen to the environment.

The model is defined by two state transitions:

$$S \stackrel{\upsilon \varphi S}{\longrightarrow} I$$ $$I \stackrel{\gamma I}{\longrightarrow} S$$

where the transition rate from susceptible to infected is proportional to the environmental contamination $\varphi$ and the transmission rate $\upsilon$. The recovery rate $\gamma$ moves individuals from infected back to susceptible.

The environmental infectious pressure $\varphi(t)$ in each node evolves according to:

$$\frac{d\varphi(t)}{dt} = \frac{\alpha I(t)}{N(t)} - \beta(t) \varphi(t) + \epsilon$$

where:

$\alpha$ is the shedding rate per infected individual.
$N(t) = S + I$ is the total population size in the node.
$\beta(t)$ is the seasonal decay/removal rate, which varies throughout the year.
$\epsilon$ is the background infectious pressure.

The environmental pressure is evolved using the Euler forward method and saved at time points in tspan.

Seasonal Decay ($\beta(t)$): The decay rate $\beta(t)$ is piecewise constant, defined by four intervals determined by the parameters end_t1, end_t2, end_t3, and end_t4 (days of the year, where 0 <= day < 365). The year is divided into four intervals based on the sorted order of these endpoints. The interval that wraps around the year boundary (from the last endpoint to day 365, then from day 0 to the first endpoint) receives the same rate as the interval preceding the first endpoint. Three orderings are supported:

Case 1: end_t1 < end_t2 < end_t3 < end_t4

Interval 1: [0, end_t1) with rate beta_t1
Interval 2: [end_t1, end_t2) with rate beta_t2
Interval 3: [end_t2, end_t3) with rate beta_t3
Interval 4: [end_t3, end_t4) with rate beta_t4
Interval 1 (wrap-around): [end_t4, 365) with rate beta_t1

Case 2: end_t3 < end_t4 < end_t1 < end_t2

Interval 3: [0, end_t3) with rate beta_t3
Interval 4: [end_t3, end_t4) with rate beta_t4
Interval 1: [end_t4, end_t1) with rate beta_t1
Interval 2: [end_t1, end_t2) with rate beta_t2
Interval 3 (wrap-around): [end_t2, 365) with rate beta_t3

Case 3: end_t4 < end_t1 < end_t2 < end_t3

Interval 4: [0, end_t4) with rate beta_t4
Interval 1: [end_t4, end_t1) with rate beta_t1
Interval 2: [end_t1, end_t2) with rate beta_t2
Interval 3: [end_t2, end_t3) with rate beta_t3
Interval 4 (wrap-around): [end_t3, 365) with rate beta_t4

These different orderings allow the model to handle seasonal patterns where, for example, a winter peak crosses the year boundary.

Details

See also