Time varying methods to infer extremes in dengue transmission dynamics

Dengue is an arbovirus affecting global populations. Frequent outbreaks occur, especially in equatorial cities such as Singapore, where year-round tropical climate, large daily influx of travelers and population density provide the ideal conditions for dengue to transmit. Little work has, however, quantified the peaks of dengue outbreaks, when health systems are likely to be most stretched. Nor have methods been developed to infer differences in exogenous factors which lead to the rise and fall of dengue case counts across extreme and non-extreme periods. In this paper, we developed time varying extreme mixture (tvEM) methods to account for the temporal dependence of dengue case counts across extreme and non-extreme periods. This approach permits inference of differences in climatic forcing across non-extreme and extreme periods of dengue case counts, quantification of their temporal dependence as well as estimation of thresholds with associated uncertainties to determine dengue case count extremities. Using tvEM, we found no evidence that weather affects dengue case counts in the near term for non-extreme periods, but that it has non-linear and mixed signals in influencing dengue through tvEM parameters in the extreme periods. Using the most appropriate tvEM specification, we found that a threshold at the 70th (95% credible interval 41.1, 83.8) quantile is optimal, with extreme events of 526.6, 1052.2 and 1183.6 weekly case counts expected at return periods of 5, 50 and 75 years. Weather parameters at a 1% scaled increase was found to decrease the long-run expected case counts, but larger increases would lead to a drastic expected rise from the baseline correspondingly. The tvEM approach can provide valuable inference on the extremes of time series, which in the case of infectious disease notifications, allows public health officials to understand the likely scale of outbreaks in the long run.


Model 4: Constant bulk regression and generalized pareto distribution regression
We apply the same bulk distribution and extreme value distribution as Model 2 and 3, but additionally impose regression structure for the extreme parameters for Model 4. We set the following priors for β ξ and β σ : 3.1 Sampling β and σ 2 β is sampled from If x t < u (s) , then the parameter can be sampled by: lξ If x t < u (s) , then the parameter can be sampled by: lσ where β σ,t = p k=0 β ξ,t,k X t,k .

Sampling u
) . Therefore, u (s+1) is accepted with probability: Posterior distributions for V σ ,β σ also follow the same functional form.

Model 5: Constant bulk regression and time-varying generalized pareto regression
In Model 4, the coefficients are constant in the regression structure. We use linear dynamic model which allows time-varying coefficients in Model 5.
4.1 Sampling β and σ 2 β is sampled from If x t < u (s) , then the parameter can be sampled by: lξ If x t < u (s) , then the parameter can be sampled by: lσ where β σ,t = p k=0 β ξ,t,k X t,k .
where S t,k = lξ t − p i=0,i =k β ξ,t,i X t,i . Posterior distributions for V σ , W σ,k , β σ,t,k also follow the same functional form.

Model Assessment
We use the deviation information criterion (DIC) and log Bayes factor (logBF) to assess the appriopriateness of each model for the data generating process.
whereD (θ) is the average of D(θ) over all samples of θ, D(θ) is the value of D evaluated at the average of the samples of θ.
D(θ) = −2 × log(p(y|θ)) + C C is a constant that can be ignored during the calculation. BF = p(D|M 1 ) p(D|M 2 ) logBF = log(p(D|M 1 )) − log(p(D|M 2 )) where p(D|M ) denotes the likelihood that some data is produced under the assumption of model M.