Class SISe3_sp

Class to handle the SISe3_sp model. This class inherits from SimInf_model, meaning that SISe3_sp objects are fully compatible with all generic functions defined for SimInf_model, such as run, plot, trajectory, and prevalence.

Details

The SISe3_sp model contains two compartments in three age categories: Susceptible ($S_1, S_2, S_3$) and Infected ($I_1, I_2, I_3$). Additionally, it includes a continuous environmental compartment ($\varphi$) to model shedding of a pathogen to the environment. Moreover, it includes a spatial coupling of the environmental contamination among proximal nodes to capture between-node spread unrelated to moving infected individuals.

The model is defined by six state transitions:

$$S_1 \stackrel{\upsilon_1 \varphi S_1}{\longrightarrow} I_1$$ $$I_1 \stackrel{\gamma_1 I_1}{\longrightarrow} S_1$$ $$S_2 \stackrel{\upsilon_2 \varphi S_2}{\longrightarrow} I_2$$ $$I_2 \stackrel{\gamma_2 I_2}{\longrightarrow} S_2$$ $$S_3 \stackrel{\upsilon_3 \varphi S_3}{\longrightarrow} I_3$$ $$I_3 \stackrel{\gamma_3 I_3}{\longrightarrow} S_3$$

where the transition rate from susceptible to infected in age category $k$ is proportional to the environmental contamination $\varphi$ and the transmission rate $\upsilon_k$. The recovery rate $\gamma_k$ moves individuals from infected back to susceptible.

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

$$\frac{d \varphi_i(t)}{dt} = \frac{\alpha \left(I_{i,1}(t) + I_{i,2}(t) + I_{i,3}(t)\right)}{N_i(t)} + \sum_k{\frac{\varphi_k(t) N_k(t) - \varphi_i(t) N_i(t)}{N_i(t)} \cdot \frac{D}{d_{ik}}} - \beta(t) \varphi_i(t)$$

where:

$\alpha$ is the shedding rate per infected individual.
$N(t) = S_1 + S_2 + S_3 + I_1 + I_2 + I_3$ is the total population size in the node.
$\beta(t)$ is the seasonal decay/removal rate, which varies throughout the year.
the last term is the spatial coupling among proximal nodes. $D$ is the rate of the local spread and $d_{ik}$ the distance between holdings $i$ and $k$.

The environmental infectious pressure $\varphi(t)$ 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