meanResponse(theta, designRes)make.designlist with elementsperiod=52 for weekly data,
for a univariate time series.
Per default, the number of cases at time point $t-1$, i.e. $lag=1$, enter
as autoregressive covariates into the model. Other lags can also be considered.
The seasonal terms in the predictor can also be expressed as
$\gamma_{s} \sin(\omega_s t) + \delta_{s} \cos(\omega_s t) = A_s \sin(\omega_s t + \epsilon_s)$
with amplitude $A_s=\sqrt{\gamma_s^2 +\delta_s^2}$
and phase difference $\tan(\epsilon_s) = \delta_s / \gamma_s$. The amplitude and
phase shift can be obtained from a fitted model by specifying amplitudeShift=TRUE
in the coef method.
For multivariate time series the mean structure is
$$\mu_{it} = \lambda_i y_{i,t-lag} + \phi_i \sum_{j \sim i} w_{ji} y_{j,t-lag} + n_{it} \nu_{it}$$
where
$$\log(\nu_{it}) = \alpha_i + \beta_i t + \sum_{j=1}^{S_i} (\gamma_{i,2j-1} \sin(\omega_j t) + \gamma_{i,2j} \cos(\omega_j t) )$$
and $n_{it}$ are standardized population counts. The weights $w_{ji}$ are specified in the columns of
the neighbourhood matrix disProgObj$neighbourhood.
Alternatively, the mean can be specified as
$$\mu_{it} = \lambda_i \pi_i y_{i,t-1} + \sum_{j \sim i} \lambda_j (1-\pi_j)/ |k \sim j| y_{j,t-1} + n_{it} \nu_{it}$$
if proportion="single" ("multiple") in designRes$control. Note that this model specification is still experimental.