loglikelihoodepiILM: 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
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.
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.
it has the same structure as the control.sus, but for the transmissibility function (>0).
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 parameter (>=0), representing random infections that are unexplained by other parts of the model. Default value is zero.
the notification effect parameter for SINR model. The default value is 1.
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).
Value
Returns the log likelihood value.
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.