loglikelihoodepiILM {EpiILMCT}

Calculates the log likelihood

Description

Calculates the log likelihood for the specific compartmental framework of the continuous-time ILMs.

Usage

loglikelihoodepiILM(object, distancekernel = NULL, control.sus = NULL,

control.trans = NULL, kernel.par = NULL, spark = NULL, gamma = NULL,

delta = NULL)

Arguments

object

an object of class “datagen” that can be the output of datagen or as.epidat functions.

distancekernel

the spatial kernel type when kerneltype is set to “distance” or “both”. Choices are “powerlaw” for a power-law distance kernel or “Cauchy” for a Cauchy distance kernel.

control.sus

a list of values of the susceptibility function (>0):

1st:

a vector of values of the susceptibility parameters,

2nd:

an n \times n_{s} matrix of the susceptibility covariates,

3rd:

a vector of values of the power parameters of the susceptibility function,

where, n and n_s are the number of individuals and number of susceptibility parameters, respectively. Default = NULL means the model does not include these parameters.

control.trans

it has the same structure as the control.sus, but for the transmissibility function (>0).

kernel.par

a scalar spatial parameter for the distance-based kernel (>0), or a vector of the spatial and network effect parameters of the network and distance-based kernel (both). It is not required when the kerneltype is set to “network”.

spark

spark parameter (>=0), representing random infections that are unexplained by other parts of the model. Default value is zero.

gamma

the notification effect parameter for SINR model. The default value is 1.

delta

a vector of the shape and rate parameters of the gamma-distributed infectious period (SIR) or a 2 \times 2 matrix of the shape and rate parameters of the gamma-distributed incubation and delay periods (SINR).

Details

We label the m infected individuals i = 1, 2, \dots, m corresponding to their infection (I_{i}) and removal (R_{i}) times; whereas the N-m individuals who remain uninfected are labeled i=m+1, m+2, \dots, N with I_{i}= R_{i} = \infty. We then denote infection and removal time vectors for the population as \boldsymbol{I} = \{I_{1}, \dots, I_{m}\} and \boldsymbol{R} = \{R_{1}, \dots, R_{m}\}, respectively. We assume that infectious periods follow a gamma distribution with shape and rate \delta. The likelihood of the general SIR continuous-time ILMs is then given as follows:

where \theta is the vector of unknown parameters; f(.;\delta) indicates the density of the infectious period distribution; and D_{i} is the infectious period of infected individual i defined as D_{i}= R_{i}-I_{i}. The likelihood of the general SINR continuous-time ILMs is given by:

where D^{inc}_i and D^{delay}_i are the incubation and delay periods such that D^{inc}_i = N_i - I_i and D^{delay}_i = R_i - N_i, and

\lambda_{ij}^{-} = \Omega_{S}(j) \Omega_{T}(i) \kappa(i,j),

for i \in I(t), j \in S(t), and

\lambda_{ij}^{+} = \gamma (\Omega_{S}(j) \Omega_{T}(i) \kappa(i,j)),

for i \in N(t), j \in S(t).

Note, \lambda_{ij}^{+} is used only under the SINR model.

Value

Returns the log likelihood value.

See Also

contactnet, datagen, epictmcmc.