Spatial-temporal hurdle model vs. spatial zero-inflated GARCH model: analysis of weekly dengue fever cases

Abstract

Dengue fever is transmitted to humans through the bite of an infected mosquito and is prevalent in all tropical and subtropical climates worldwide. It is thus essential to model weekly dengue fever counts and other infectious diseases that exhibit spatial-temporal dynamics, overdispersion, spatial dependence, and a high number of zeros. To address these characteristics, this study introduces a spatial hurdle integer-valued GARCH (INGARCH) model and an improved version of the spatial zero-inflated generalized Poisson (ZIGP) INGARCH model with and without meteorological variables. Implementing two parameters in the distance function influences the spatial weight between two locations: one controls the decay rate, while the other shapes the decay curve. We employ these newly designed models to analyze time-series counts of infectious diseases - specifically, weekly cases of dengue hemorrhagic fever in four northeastern provinces of Thailand. Applying these models allow us to offer inferences, predictions, and model selections within a Bayesian framework through Markov chain Monte Carlo (MCMC) methods. We then compare models based on the Bayes factors and the mean squared error of fitting errors. The results for the spatial ZIGP INGARCH models are remarkably good, but the spatial INGARCH model incorporating meteorological variables outperforms the other two.

Appendix

We aim to obtain the conditional mean and variance for the spatial hurdle INGARCH model as presented in (1). Let $P(Y_{i,t}=y;{\lambda _{i,t},\eta _i})$ be a GP probability mass function with parameters $\lambda _{i,t}$ and $\eta _i$, where the mean and variance are:

$$\begin{aligned} E(Y_{i,t})=\frac{\lambda _{i,t}}{1-\eta _i} \end{aligned}$$

and

$$\begin{aligned} \hbox{Var}(Y_{i,t})=\frac{\lambda _{i,t}}{(1-\eta _i)^3} \end{aligned}$$

The conditional mean $E(Y_{i,t}|\mathcal {F}_{t-1})$ and variance $\hbox{Var}(Y_{i,t}|\mathcal {F}_{t-1})$ can then be derived as follows:

$$\begin{aligned} E(Y_{i,t}|\mathcal {F}_{t-1})= & {} \displaystyle \sum _{y\ge 0}yP(Y_{i,t}=y)\\= & {} \displaystyle \sum _{y>0}y\pi _{i,t}TGP(y;\lambda _{i,t},\eta _i)\\= & {} \displaystyle \sum _{y>0}y\pi _{i,t}\frac{P(y;{\lambda _{i,t},\eta _i})}{1-P(y=0;{\lambda _{i,t},\eta _i})}\\= & {} \frac{\pi _{i,t}}{1-\exp (-\lambda _{i,t})}\displaystyle \sum _{y>0}yP(y;{\lambda _{i,t},\eta _i})\\= & {} \frac{\pi _{i,t}}{1-\exp (-\lambda _{i,t})}\displaystyle \sum _{y\ge 0}yP(y;{\lambda _{i,t},\eta _i})\\= & {} \frac{\pi _{i,t}\lambda _{i,t}}{(1-\exp (-\lambda _{i,t}))(1-\eta _i)}\\ \hbox{Var}(Y_{i,t}|\mathcal {F}_{t-1})= & {} E(Y_{i,t}^2|\mathcal {F}_{t-1})-\left( E(Y_{i,t}|\mathcal {F}_{t-1})\right) ^2\\= & {} \displaystyle \sum _{y\ge 0}y^2P(Y_{i,t}=y)-\left( \displaystyle \sum _{y\ge 0}yP(Y_{i,t}=y)\right) ^2\\= & {} \displaystyle \sum _{y>0}y^2\pi _{i,t}TGP(y;\lambda _{i,t},\eta _i)-\left( \displaystyle \sum _{y>0}y\pi _{i,t}TGP(y;\lambda _{i,t},\eta _i)\right) ^2\\= & {} \pi _{i,t}\displaystyle \sum _{y\ge 0}y^2TGP(y;\lambda _{i,t},\eta _i)-\pi _{i,t}^2\left( \displaystyle \sum _{y\ge 0}yTGP(y;\lambda _{i,t},\eta _i)\right) ^2\\= & {} \pi _{i,t}\displaystyle \sum _{y\ge 0}y^2\frac{P(y;{\lambda _{i,t},\eta _i})}{1-P(y=0;{\lambda _{i,t},\eta _i})}-\pi _{i,t}^2\left( \displaystyle \sum _{y\ge 0}y\frac{P(y;{\lambda _{i,t},\eta _i})}{1-P(y=0;{\lambda _{i,t},\eta _i})}\right) ^2\\= & {} \frac{\pi _{i,t}}{1-\exp (-\lambda _{i,t})}\displaystyle \sum _{y\ge 0}y^2P(y;{\lambda _{i,t},\eta _i})-\left( \frac{\pi _{i,t}}{1-\exp (-\lambda _{i,t})}\right) ^2\left( \displaystyle \sum _{y\ge 0}yP(y;{\lambda _{i,t},\eta _i})\right) ^2\\= & {} \frac{\pi _{i,t}}{1-\exp (-\lambda _{i,t})}\left( \frac{\lambda _{i,t}}{(1-\eta _i)^3}+\left( \frac{\lambda _{i,t}}{1-\eta _i}\right) ^2\right) -\left( \frac{\pi _{i,t}}{1-\exp (-\lambda _{i,t})}\right) ^2\left( \frac{\lambda _{i,t}}{1-\eta _i}\right) ^2 \end{aligned}$$