Create an SISe3_sp model — SISe3

Create an SISe3_sp model to be used by the simulation framework.

Usage

SISe3_sp(
  u0,
  tspan,
  events = NULL,
  phi = NULL,
  upsilon_1 = NULL,
  upsilon_2 = NULL,
  upsilon_3 = NULL,
  gamma_1 = NULL,
  gamma_2 = NULL,
  gamma_3 = NULL,
  alpha = NULL,
  beta_t1 = NULL,
  beta_t2 = NULL,
  beta_t3 = NULL,
  beta_t4 = NULL,
  end_t1 = NULL,
  end_t2 = NULL,
  end_t3 = NULL,
  end_t4 = NULL,
  distance = NULL,
  coupling = NULL
)

Arguments

u0: A data.frame with the initial state in each node, i.e., the number of individuals in each compartment in each node when the simulation starts (see ‘Details’). The parameter u0 can also be an object that can be coerced to a data.frame, e.g., a named numeric vector will be coerced to a one row data.frame.
tspan: A vector (length >= 1) of increasing time points where the state of each node is to be returned. Can be either an integer or a Date vector. A Date vector is coerced to a numeric vector as days, where tspan[1] becomes the day of the year of the first year of tspan. The dates are added as names to the numeric vector.
events: a data.frame with the scheduled events, see SimInf_model.
phi: A numeric vector with the initial environmental infectious pressure in each node. Will be repeated to the length of nrow(u0). Default is NULL which gives 0 in each node.
upsilon_1: Indirect transmission rate of the environmental infectious pressure in age category 1
upsilon_2: Indirect transmission rate of the environmental infectious pressure in age category 2
upsilon_3: Indirect transmission rate of the environmental infectious pressure in age category 3
gamma_1: The recovery rate from infected to susceptible for age category 1
gamma_2: The recovery rate from infected to susceptible for age category 2
gamma_3: The recovery rate from infected to susceptible for age category 3
alpha: Shed rate from infected individuals
beta_t1: The decay of the environmental infectious pressure in interval 1.
beta_t2: The decay of the environmental infectious pressure in interval 2.
beta_t3: The decay of the environmental infectious pressure in interval 3.
beta_t4: The decay of the environmental infectious pressure in interval 4.
end_t1: vector with the non-inclusive day of the year that ends interval 1 in each node. Will be repeated to the length of nrow(u0).
end_t2: vector with the non-inclusive day of the year that ends interval 2 in each node. Will be repeated to the length of nrow(u0).
end_t3: vector with the non-inclusive day of the year that ends interval 3 in each node. Will be repeated to the length of nrow(u0).
end_t4: vector with the non-inclusive day of the year that ends interval 4 in each node. Will be repeated to the length of nrow(u0).
distance: The distance matrix between neighboring nodes
coupling: The coupling between neighboring nodes

Value

SISe3_sp

Details

The SISe3_sp model contains two compartments in three age categories; number of susceptible (S_1, S_2, S_3) and number of infectious (I_1, I_2, I_3). Additionally, it contains an environmental compartment to model shedding of a pathogen to the environment. Moreover, it also includes a spatial coupling of the environmental contamination among proximal nodes to capture between-node spread unrelated to moving infected individuals. Consequently, the model has 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 per unit of time from susceptible to infected is proportional to the concentration of the environmental contamination $\varphi$ in each node. Moreover, the transition rate from infected to susceptible is the recovery rate $\gamma_1, \gamma_2, \gamma_3$, measured per individual and per unit of time. Finally, the environmental infectious pressure in each node is evolved by,

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

where $\alpha$ is the average shedding rate of the pathogen to the environment per infected individual and $N = S_1 + S_2 + S_3 + I_1 + I_2 + I_3$ the size of the node. Next comes the spatial coupling among proximal nodes, where $D$ is the rate of the local spread and $d_{ik}$ the distance between holdings $i$ and $k$. The seasonal decay and removal of the pathogen is captured by $\beta(t)$. The environmental infectious pressure $\varphi(t)$ in each node is evolved each time unit by the Euler forward method. The value of $\varphi(t)$ is saved at the time-points specified in tspan.

The argument u0 must be a data.frame with one row for each node with the following columns:

S_1: The number of sucsceptible in age category 1
I_1: The number of infected in age category 1
S_2: The number of sucsceptible in age category 2
I_2: The number of infected in age category 2
S_3: The number of sucsceptible in age category 3
I_3: The number of infected in age category 3

Beta

The time dependent beta is divided into four intervals of the year


where 0 <= day < 365

Case 1: END_1 < END_2 < END_3 < END_4
INTERVAL_1 INTERVAL_2     INTERVAL_3     INTERVAL_4     INTERVAL_1
[0, END_1) [END_1, END_2) [END_2, END_3) [END_3, END_4) [END_4, 365)

Case 2: END_3 < END_4 < END_1 < END_2
INTERVAL_3 INTERVAL_4     INTERVAL_1     INTERVAL_2     INTERVAL_3
[0, END_3) [END_3, END_4) [END_4, END_1) [END_1, END_2) [END_2, 365)

Case 3: END_4 < END_1 < END_2 < END_3
INTERVAL_4 INTERVAL_1     INTERVAL_2     INTERVAL_3     INTERVAL_4
[0, END_4) [END_4, END_1) [END_1, END_2) [END_2, END_3) [END_3, 365)