SISe3_sp modelCreate an SISe3_sp model to be used by the simulation
framework.
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
)SISe3_sp
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.
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.
a data.frame with the scheduled events, see
SimInf_model.
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.
Indirect transmission rate of the environmental infectious pressure in age category 1
Indirect transmission rate of the environmental infectious pressure in age category 2
Indirect transmission rate of the environmental infectious pressure in age category 3
The recovery rate from infected to susceptible for age category 1
The recovery rate from infected to susceptible for age category 2
The recovery rate from infected to susceptible for age category 3
Shed rate from infected individuals
The decay of the environmental infectious pressure in interval 1.
The decay of the environmental infectious pressure in interval 2.
The decay of the environmental infectious pressure in interval 3.
The decay of the environmental infectious pressure in interval 4.
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).
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).
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).
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).
The distance matrix between neighboring nodes
The coupling between neighboring nodes
The time dependent beta is divided into four intervals of the year
where 0 <= day < 365Case 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)
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:
The number of sucsceptible in age category 1
The number of infected in age category 1
The number of sucsceptible in age category 2
The number of infected in age category 2
The number of sucsceptible in age category 3
The number of infected in age category 3