Spatio-temporal models are valuable tools for disease mapping and understanding the geographical distribution of diseases and temporal dynamics. Spatio-temporal models have been proven empirically to be very complex and this complexity has led many to oversimply and model the spatial and temporal dependencies independently. Unlike common practice, this study formulated a new spatio-temporal model in a Bayesian hierarchical framework that accounts for spatial and temporal dependencies jointly. The spatial and temporal dependencies were dynamically modelled via the matern exponential covariance function. The temporal aspect was captured by the parameters of the exponential with a first-order autoregressive structure. Inferences abo ut the parameters were obtained via Markov Chain Monte Carlo (MCMC) techniques and the spatio-temporal maps were obtained by mapping stable posterior means from the specific location and time from the best model that includes the significant risk factors. The model formulated was fitted to both simulation data and Kenya meningitis incidence data from 2013 to 2019 along with two covariates; Gross County Product (GCP) and average rainfall. The study found that both average rainfall and GCP had a significant positive association with meningitis occurrence. Also, regarding geographical distribution, the spatio-temporal maps showed that meningitis is not evenly distributed across the country as some counties reported a high number of cases compared with other counties.
Disease mapping has a long history and is becoming an indispensable tool for epidemiologists in analyzing disease incidence in a geographical location and over time. Often, disease counts are characterized by space and time structures. In most cases, spatial and temporal dependencies are modelled independently, implying that the spatial variation and temporal variation are modelled separately without considering their joint interactions [
Over the years, studies on spatio-temporal disease mapping have incorporated the Besag York Mollie (BYM) model [
In this paper, the main focus is to develop a Bayesian Spatio-temporal model that accounts for the spatial and temporal dependencies jointly. This will be facilitated by using the non-separable, stationary matern covariance function, specifically, the matern exponential covariance function for space-time process to model the spatial and temporal dependencies jointly. This will incorporate the space-time interaction between the geographical locations and also eliminate the spatial and temporal autocorrelation problems. The model will then be applied to the simulated data set and Kenya meningitis incidence data from 2013 to 2019. The study also seeks to identify significant risk factors associated with meningitis and the construction of spatio-temporal occurrence maps for different years both on the simulation data and real data will be done.
The rest of the paper is structured as follows: section 2 presents materials and methods utilized in the paper. In section 3, results are discussed both for simulation study and real data application, and finally, in section 4, conclusions to this study are presented.
We develop a spatio-temporal model based on the Poisson regression model. For each county we assume y i t ~ P o i s s o n ( η i t ) , where, y i t denotes meningitis cases (or counts) in the ith county at time t and η i t is the mean for Poisson distribution or the estimated disease count for meningitis cases. We then, formulate a spatio-temporal model as follows;
η i t = α + x ′ i t β _ + w i t + ε i t (1)
where α is a common intercept for the entire study region, β _ is a vector of coefficients, x ′ i t is a vector of covariates for all the 47 counties, w i t is the spatial-temporal dependencies, and ε i t is the white-noise error term for space-time interaction.
The common intercept α , was assigned a conjugate prior, which was assumed to have a normal distribution, N ( μ α , σ α 2 ) . The vector of coefficients β _ was assumed to have a Multivariate Normal distribution, M V N ( μ _ β , Σ β ) . The white-noise error term for space-time interaction ε i t was assumed to have a normal distribution, ε i t ~ N ( 0 , τ ε 2 ) . The precision parameter τ ε 2 was assigned a conjugate prior of an inverse gamma distribution τ ε 2 ~ I n v - G a m m a ( a ε , b ε ) . The spatial and temporal dependencies w i t , followed a zero-mean Multivariate Normal distribution with the matern covariance function, specifically, the matern exponential covariance function. That is; w i t ~ M V N ( 0 , Σ ( t ) ) , where Σ ( t ) is the matern exponential covariance function that is dependent on time t. This can be expressed as
Σ ( t ) = [ exp ( − | S _ i − S _ j | θ t ) , i , j = 1 , ⋯ , n ; t = 1 , ⋯ , T ]
where | S _ i − S _ j | is the Euclidean distance between two locations defined by their longitude and latitude coordinates. θ t = ϕ θ θ t − 1 + ξ t , ξ t ~ N ( 0 , σ ξ 2 ) i.e. θ t is A R ( 1 ) process. We assumed ϕ θ to follow a Uniform distribution
ϕ θ ~ U n i f o r m ( − 1 , 1 ) and σ ξ 2 was assigned a priori with a conjugate prior of inverse gamma distribution σ ξ 2 ~ I n v - G a m m a ( a ξ , b ξ ) . The set of parameters of interest was λ = ( α , β _ , ϕ θ , τ ε 2 , σ ξ 2 ) and the vector of hyperparameters was ψ = ( μ α , σ α 2 , μ β , Σ β , a ε , b ε , a ξ , b ξ ) . The choice of hyperparameters was selected in a manner that the prior specification does not swamp the data and hence has minimal effect on the overall result.
The Bayesian hierarchical model consists of the data model, process model, and parameter model.
The data model is the probability distribution of the observed cases given the parameters of the model. In this study, the observed cases ( y i t ) followed a Poisson distribution with mean η i t i.e., y i t ~ P o i s s o n ( η i t ) , i = 1 , ⋯ , n , t = 1 , ⋯ , T which was written conditionally as y i t | η i t , λ ~ P o i s s o n ( η i t ) and the likelihood function was given as:
P ( y 1 _ , ⋯ , y T _ | η 1 _ , ⋯ , η T _ , λ ) = P ( y { 11 } , ⋯ , y { n , T } | η { 11 } , ⋯ , η { n , T } , λ ) = ∏ i = 1 n ∏ t = 1 T e − η i t η i t y i t y i t ! (2)
The process model was generated based on the spatio-temporal model specified in Equation (1) and further expressed in vector form as:
η t _ = X t β _ + w t _ + ε t _ (3)
where X t in Equation (3) is a design matrix at time t (non-stochastic) with n × ( p + 1 ) dimension and β _ = [ α β 1 ⋮ β p ] has a ( p + 1 ) × 1 dimension. We assumed ε t _ to be independent, thus; η t _ − X t β _ − w t _ = ε t _ ~ N ( 0 _ , τ ε 2 I ) was expressed conditionally as; η t _ | X t β _ , θ t , τ ε 2 ~ N ( 0 _ , τ ε 2 I ) . Written as:
f ( η t _ | X t , β _ , θ t , τ ε 2 ) = ∏ t = 1 T | τ ε 2 I | − 1 2 exp { − 1 2 τ ε 2 ∑ t = 1 T ( η t _ − x t β _ − w t _ ) ′ I ( η t _ − x t β _ − w t _ ) } (4)
The spatio-temporal dependency, w t _ was also expressed conditionally as:
f ( w t _ | θ t ) = | Σ ( t ) | − 1 2 exp { − 1 2 w t _ ′ ( Σ ( t ) ) − 1 w t _ } . (5)
The matern exponential covariance function, Σ ( t ) was given as:
Σ ( t ) = { σ i j t } = [ exp ( − | S _ i − S _ j | θ t ) , i , j = 1 , ⋯ , n ; t = 1 , ⋯ , T ] ,
where, we assumed θ t in the matern exponential covariance function to follow an AR1 process written as; θ t = ϕ θ θ t − 1 + ξ t and, ξ t ~ N ( 0 , σ ξ 2 ) . We generated the conditional distribution of θ t from t = 1 , ⋯ , t as:
f ( θ 1 , ⋯ , θ t ) = f ( θ 1 ) f ( θ 2 | θ 1 ) f ( θ 3 | θ 2 ) ⋯ f ( θ t | θ t − 1 ) = 1 2 π ( σ ξ 2 1 − ϕ θ 2 ) exp { − 1 − ϕ θ 2 2 σ ξ 2 ( θ 1 ) 2 } ∏ t = 2 T 1 2 π σ ξ 2 exp { − 1 2 σ ξ 2 ( θ t − ϕ θ θ t − 1 ) 2 } (6)
Now, the process model was given as the product of Equation (4), Equation (5), and Equation (6) denoted as P ( η 1 _ , ⋯ , η T _ , λ ) , written as:
P ( η 1 _ , ⋯ , η T _ , λ ) ∝ { ∏ t = 1 T | τ ε 2 I | − 1 2 exp { − 1 2 τ ε 2 ∑ t = 1 T ( η t _ − x t β _ − w t _ ) ′ I ( η t _ − x t β _ − w t _ ) } } × { | Σ ( t ) | − 1 2 exp { − 1 2 w t _ ′ ( Σ ( t ) ) − 1 w t _ } } × { 1 2 π ( σ ξ 2 1 − ϕ θ 2 ) exp { − 1 − ϕ θ 2 2 σ ξ 2 ( θ 1 ) 2 } ∏ t = 2 T 1 2 π σ ξ 2 exp { − 1 2 σ ξ 2 ( θ t − ϕ θ θ t − 1 ) 2 } } (7)
The parameter model was given as the product of the prior distributions of the set parameters λ , which was denoted as P ( λ ) . In this case, α was contained in β _ and β _ was assumed to follow a Multivariate Normal distribution, M V N ( μ _ β , Σ β ) . Thus, P ( λ ) is given as:
P ( λ ) = { ( 2 π ) − p + 1 2 | Σ β | − 1 2 exp { − 1 2 ( β _ − μ _ β ) ′ ( Σ β ) − 1 ( β _ − μ _ β ) } } × { ( 1 τ ε 2 ) a ε + 1 exp ( − b ε τ ε 2 ) } × { ( 1 σ ξ 2 ) a ξ + 1 exp ( − b ξ σ ξ 2 ) } (8)
The joint posterior distribution was given as the proportional of the product of the data model, the process model, and the parameters model denoted as P ( η 1 _ , ⋯ , η T _ , λ | y 1 _ , ⋯ , y T _ ) . Thus, by applying Bayes’ rule, the joint posterior distribution was expressed as:
P ( η 1 _ , ⋯ , η T _ , λ | y 1 _ , ⋯ , y T _ ) ∝ P ( y 1 _ , ⋯ , y T _ | η 1 _ , ⋯ , η T _ , λ ) P ( η 1 _ , ⋯ , η T _ | λ ) P ( λ ) (9)
where P(.) in Equation (9) is the probability density function. The right-hand side of Equation (9) entails, P ( y 1 _ , ⋯ , y T _ | η 1 _ , ⋯ , η T _ , λ ) the data model, P ( η 1 _ , ⋯ , η T _ | λ ) the process model, and P ( λ ) the parameter model. The study assumed the hyperparameters are independent of the parameter model i.e., P ( λ ) = ∏ i dim ( λ ) P ( λ i ) .
In the Bayesian framework, the unknown parameters of the model are treated as random variables with assigned prior distributions. While the Bayesian method has proven to be effective for parameter estimation, it often presents challenges due to the intractability of the posterior distribution, making direct sampling impossible especially when the conditional distributions of the parameters are unknown. The remedy to this problem is to employ the Markov Chain Monte Carlo (MCMC) method called Metropolis-Hastings within Gibbs sampling algorithm to sample from the unknown distribution [
P ( β _ | ϕ θ , τ ε 2 , σ ξ 2 , η t , θ t , y t _ ) ∝ P ( η 1 _ , ⋯ , η T _ , λ | y 1 _ , ⋯ , y T _ ) ∫ P ( β _ | ϕ θ , τ ε 2 , σ ξ 2 , η t , θ t , y t _ )
P ( ϕ θ | β _ , τ ε 2 , σ ξ 2 , η t , θ t , y t _ ) ∝ P ( η 1 _ , ⋯ , η T _ , λ | y 1 _ , ⋯ , y T _ ) ∫ P ( ϕ θ | β _ , τ ε 2 , σ ξ 2 , η t , θ t , y t _ )
P ( τ ε 2 | β _ , ϕ θ , σ ξ 2 , η t , θ t , y t _ ) ∝ P ( η 1 _ , ⋯ , η T _ , λ | y 1 _ , ⋯ , y T _ ) ∫ P ( τ ε 2 | β _ , ϕ θ , σ ξ 2 , η t , θ t , y t _ )
P ( σ ξ 2 | β _ , τ ε 2 , ϕ θ , η t , θ t , y t _ ) ∝ P ( η 1 _ , ⋯ , η T _ , λ | y 1 _ , ⋯ , y T _ ) ∫ P ( σ ξ 2 | β _ , τ ε 2 , ϕ θ , η t , θ t , y t _ )
P ( η t _ | β _ , σ ξ 2 , τ ε 2 , ϕ θ , θ t , y t _ ) ∝ P ( η 1 _ , ⋯ , η T _ , λ | y 1 _ , ⋯ , y T _ ) ∫ P ( η t _ | β _ , σ ξ 2 , τ ε 2 , ϕ θ , θ t , y t _ )
The full conditional distribution for θ t was determined in three phases using the formula below. The first phase was when time t = 1 , the second phase was when time t = 2 to time t = T − 1 , and the last phase was when time t = T .
P ( θ t = 1 | β _ , σ ξ 2 , τ ε 2 , ϕ θ , η t _ , y t _ ) ∝ P ( η 1 _ , ⋯ , η T _ , λ | y 1 _ , ⋯ , y T _ ) ∫ P ( θ t = 1 | β _ , σ ξ 2 , τ ε 2 , ϕ θ , η t _ , y t _ )
P ( θ t = 2 to T − 1 | β _ , σ ξ 2 , τ ε 2 , ϕ θ , η t _ , y t _ ) ∝ P ( η 1 _ , ⋯ , η T _ , λ | y 1 _ , ⋯ , y T _ ) ∫ P ( θ t = 2 to T − 1 | β _ , σ ξ 2 , τ ε 2 , ϕ θ , η t _ , y t _ )
P ( θ t = T | β _ , σ ξ 2 , τ ε 2 , ϕ θ , η t _ , y t _ ) ∝ P ( η 1 _ , ⋯ , η T _ , λ | y 1 _ , ⋯ , y T _ ) ∫ P ( θ t = T | β _ , σ ξ 2 , τ ε 2 , ϕ θ , η t _ , y t _ )
The full conditional distribution for τ ε 2 and σ ξ 2 were found to be known and that of β _ , ϕ θ , η t _ , and θ t were unrecognizable, thus difficult to sample from them directly leading to the utilization of Metropolis-Hastings within Gibbs algorithm to sample from the conditional posterior distributions of the parameters in the spatio-temporal model formulated. The samples obtained were used to estimate the model parameters under the Squared Error Loss Function (SELF) and Mean Absolute Error Loss Function (MAELF). For each parameter, both the posterior mean and median as well as the Bayesian credible intervals were calculated. The convergence of the samples generated from the posterior marginals of the parameters was assessed using the trace plot as shown in the appendix.
A significant interest in understanding the evolution of meningitis lies in identifying associated risk factors that influence its occurrence. This study utilized the stepwise regression approach, in particular, the backward selection method in building regression models. Backward selection is a criterion for building regression models starting with a full model (model with all covariates) to a null model (model with the intercept only). The study builds regression models permutationally based on the number of covariates.
The study later computed a scale deviance for all the reduced models to test the significance and contribution of each covariate in the model. Scale deviance denoted by D * was defined as:
D * = − 2 ( log L R e d − log L F u l l )
where L R e d is the likelihood of the reduced model and L F u l l is the likelihood of the full model.
The log-likelihood of each reduced model in the study was calculated based on the number of regression coefficients present in each model. The scale deviance, D * followed a chi-square distribution with k-p degrees of freedom, D * ~ χ k − p 2 , where k is the total number of regression coefficients in the full model and p is the total number of regression coefficients in the reduced model. The scaled deviance, D*, was later used in performing the hypotheses tests on the sets of regression coefficients in the model. Statistically significant predictors were considered useful and cannot be dropped from the model, and thus, they were identified as risk factors that influence the occurrence of meningitis based on this study.
The model performance was examined via Deviance Information Criterion (DIC) which is due to [
The spatio-temporal maps were obtained by mapping stable posterior means for the specific location and time from the best model that includes the significant risk factors. The posterior means were evaluated from the posterior samples obtained in the MCMC sampling. This enabled the examination of the spatio-temporal variability (geographic patterns and temporal dependence) and also helped in evaluating risk areas. All the computation was performed in R programming language.
This section presents the results obtained for both the simulation study and real data application.
The spatio-temporal model formulated was fitted to the simulated data to test the methodological development of the study. The simulation is based on yearly disease counts along with three covariates based on 47 counties in Kenya for 10 years. The years were assumed to start from 2011 to 2020. A real case study will be examined in section 3.2. The dataset used for simulation was generated using the following algorithm.
1) Setting the value of the true parameters ( α , β _ , ϕ θ , τ ε 2 , σ ξ 2 ) = ( 0.5 , 0.5 , 0.5 , 0.01 , 0.01 ) respectively (these parameter values were chosen arbitrarily).
2) Builds a data frame consisting of the C = 47 counties based on their respective longitude and latitude of the county headquarters (county center).
3) Create a Euclidean distance D.
4) Specify the time frame.
5) Create a function consisting of the 47 counties, time, and the true parameters of the Spatio-temporal model, set N to be the dimension of the counties.
6) For time t = 1 to T, do the following steps.
Step 6.1: Simulate the matern parameter θ t = ϕ θ θ t − 1 + ξ t , ξ t ~ N ( 0 , σ ξ 2 )
Step 6.2: Create matern exponential covariance matrix, Σ ( t ) , using a Euclidean distance
Step 6.3: Simulate w t = M V N ( 0 , Σ ( t ) )
Step 6.4: For iteration s = 1 to C (county), do the following sub-steps
Step 6.4.a: Simulate the covariate vector
x 1 , s , t = r u n i f o r m ( a 1 = 1 , b 1 = 2 ) , x 2 , s , t = r u n i f o r m ( a 2 = 1 , b 2 = 5 ) and x 3 , s , t = r u n i f o r m ( a 3 = 0 , b 3 = 2 ) , and equate it to x ′ s , t = ( x 1 , s , t , x 2 , s , t , x 3 , s , t )
Step 6.4.b: Simulate the white noise error term for space-time interaction (random error term) ε s , t ~ N ( 0 , τ ε 2 )
Step 6.4.c: Compute η s , t = α + x ′ s , t β _ + w s , t + ε s , t
Step 6.4.d: Simulate Y s , t = P o i s s o n ( η s , t )
Due to the complexity of the model, the study simulated only one set of data which was used to sample the parameters of interest from the joint posterior distribution via the Metropolis-Hastings within Gibbs sampling technique. For each parameter, trace plots were constructed to assess the convergence of the samples generated from the posterior marginals of the parameters, see Appendix A. Based on the plots, we can observe that the trace plot of β _ , ϕ θ , τ ε 2 , σ ξ 2 , θ t and η t were first in converging while that for α (intercept) was slow in converging but it eventually converged. The poor mixing of chain for α could as a result of high variation of the parameter α . The true value, estimated posterior mean value, and 95% credible intervals for each parameter were also included in their respective trace plot. We can observe that the true values for β _ , ϕ θ , σ ξ 2 , θ t and η t all fall within their corresponding 95% credible set and they were also close to their estimated mean value apart from τ ε 2 which was outside its 95% credible set. We believe the reason why the true value for τ ε 2 happens to be outside its credible set could have been attributed to the characteristics of the sample set generated for this study, which, in turn, led the credible set to be narrower. Point estimate and Bayesian credible set for each parameter were also computed and the results are shown from Tables 1-4.
| Parameters | True Value | Mean | Median | 95% Confidence Interval |
|---|---|---|---|---|
| α | 0.5 | 0.9009 | 0.8758 | (−0.5105, 2.6048) |
| ϕ θ | 0.5 | 0.7720 | 0.8221 | (0.2786, 0.9884) |
| τ ε 2 | 0.01 | 2.0475 | 2.0421 | (1.8565, 2.2544) |
| σ ξ 2 | 0.01 | 0.0176 | 0.0150 | (0.0063, 0.0437) |
| Parameter, ( β _ ) | True value | Mean | Median | 95% Credible Interval |
|---|---|---|---|---|
| β 1 | 0.5 | 0.5270 | 0.5220 | (−0.3316, 1.3880) |
| β 2 | 0.5 | 0.4806 | 0.4825 | (0.2232, 0.7257) |
| β 3 | 0.5 | 0.7456 | 0.7447 | (0.2758, 1.2113) |
credible set apart from θ 1 and θ 2 which falls out of the 95% credible set. Note that we were evolving the matern parameter θ t via the AR1 process, which means that the observations at the current time point are influenced by the recent observation. In this note, θ 1 lack prior information, which also affects θ 2 , thus causing the credible set to be wider. Despite the true value of θ 1 and θ 2 falling outside the credible set due to the aforementioned reasons, there is a positive indication that the matern exponential covariance function has captured the space-time interaction effect between locations and time.
| Parameter, ( θ t ) | Time | True Value | Mean | Median | 95% Credible Interval |
|---|---|---|---|---|---|
| θ 1 | 1 | 0.4331 | 0.0289 | 0.0234 | (0.0013, 0.0871) |
| θ 2 | 2 | 0.2219 | 0.0295 | 0.0250 | (0.0019, 0.0824) |
| θ 3 | 3 | 0.0695 | 0.0295 | 0.0253 | (0.0025, 0.0786) |
| θ 4 | 4 | 0.0282 | 0.0294 | 0.0253 | (0.0025, 0.0798) |
| θ 5 | 5 | 0.0024 | 0.0293 | 0.0252 | (0.0020, 0.0756) |
| θ 6 | 6 | 0.0062 | 0.0297 | 0.0257 | (0.0023, 0.0812) |
| θ 7 | 7 | 0.0050 | 0.0295 | 0.0254 | (0.0021, 0.0815) |
| θ 8 | 8 | 0.0006 | 0.0291 | 0.0250 | (0.0018, 0.0786) |
| θ 9 | 9 | 0.0136 | 0.0287 | 0.0246 | (0.0013, 0.0800) |
| θ 10 | 10 | 0.0217 | 0.0270 | 0.0226 | (0.0012, 0.0807) |
| Parameter, ( η t ) | County, (c) | Time, (t) | True value | Mean | Median | 95% Credible Interval |
|---|---|---|---|---|---|---|
| η ( 1 , 1 ) | 1 | 1 | 2.3680 | 2.3843 | 2.1839 | (0.3721, 5.4697) |
| η ( 5 , 4 ) | 5 | 4 | 4.5877 | 4.7158 | 4.6408 | (2.1585, 7.7396) |
| η ( 10 , 7 ) | 10 | 7 | 2.4074 | 2.3869 | 2.2181 | (0.4003, 5.4320) |
| η ( 20 , 7 ) | 20 | 7 | 3.5069 | 3.5500 | 3.4151 | (1.0403, 6.7647) |
| η ( 30 , 3 ) | 30 | 3 | 3.6489 | 3.3447 | 3.2082 | (0.9404, 6.4711) |
| η ( 38 , 3 ) | 38 | 3 | 3.0421 | 4.4188 | 4.3412 | (1.7754, 7.6089) |
| η ( 40 , 7 ) | 40 | 7 | 3.6504 | 4.7483 | 4.6614 | (2.1355, 7.8196) |
| η ( 45 , 6 ) | 45 | 6 | 4.7043 | 4.3970 | 4.3090 | (1.8578, 7.4275) |
that were randomly chosen. It can be observed that all the credible intervals for η t contained the true value.
For model selection and identification of significant risk factors for the simulation study, the study built 8 spatio-temporal models (one full model and seven reduced models) based on the three simulated covariates, x 1 i t , x 2 i t , and x 3 i t using the stepwise regression approach explained in section 2.5. The 8 models are shown below and the results for each of the model is presented in
Model 1; y ^ = α ^ + β ^ 1 x 1 i t + β ^ 2 x 2 i t + β ^ 3 x 3 i t
Model 2; y ^ = α ^ + β ^ 1 x 1 i t + β ^ 2 x 2 i t
Model 3; y ^ = α ^ + β ^ 1 x 1 i t + β ^ 3 x 3 i t
Model 4; y ^ = α ^ + β ^ 2 x 2 i t + β ^ 3 x 3 i t
Model 5; y ^ = α ^ + β ^ 1 x 1 i t
Model 6; y ^ = α ^ + β ^ 2 x 2 i t
Model 7; y ^ = α ^ + β ^ 3 x 3 i t
Model 8; y ^ = α ^
Model selection is the process of selecting the best model from a set of candidate models. In this work, the approach utilized in model selection was based primarily on DIC. The DIC was computed for all eight models as −2loglikelihood + 2pD, where pD is the effective number of the parameters present in that particular model. The DIC computed was later compared to all the eight models and the model with the smallest DIC was considered the best. It is clear from
| Models | DIC | −2 loglikelihood | D* | χ 0.05 , k − p 2 | p-value |
|---|---|---|---|---|---|
| Model 1 | 213.2963 | 205.2963 | |||
| Model 2 | 215.6407 | 209.6407 | 4.3444 | 3.8410 | 0.0371 |
| Model 3 | 215.9094 | 209.9094 | 4.6131 | 3.8410 | 0.0317 |
| Model 4 | 216.5483 | 210.5483 | 5.2520 | 3.8410 | 0.0219 |
| Model 5 | 217.5214 | 213.5214 | 8.2251 | 5.9910 | 0.0165 |
| Model 6 | 216.1748 | 212.1748 | 6.8785 | 5.9910 | 0.0321 |
| Model 7 | 219.9835 | 215.9835 | 10.6872 | 5.9910 | 0.0048 |
| Model 8 | 249.8568 | 247.8568 | 42.5605 | 7.8150 | 0.0000 |
count (see
evolution of the estimated disease counts in Kenya from 2011-2020 using simulated data. The maps were obtained by mapping posterior means from the specific location and time from the best model as stated in section 2.6. There is a rising trend for estimated disease counts from the model over the years (2011-2020). Considering that the spread of the disease was linked to three simulated covariates that showed significant influence, we believe that these factors might have played a role in the observed increase in risk over time. Upon visual inspection, there is a close resemblance between the map for observed simulated counts and that of estimated counts.
The model formulated in Equation (1) was also fitted to a real dataset, in particular, the Kenya meningitis incidence data from 2013 to 2019. We chose meningitis data as a practical application to our model. The dataset was obtained from the Kenya Health Services for all the 47 counties. Due to data availability constraints, we incorporated only two covariates in the model, Gross County Product (GCP) and averaged rainfall for all 47 counties in Kenya. The GCP and averaged rainfall data were extracted from the Kenya National Bureau of Statistics (KNBS) and weather and climate websites respectively. The two covariates were in different units and therefore standardization technique, in particular, the Z-transformation technique was applied to normalize them, thus making them directly comparable. Posterior estimates for each parameter were obtained using the MCMC approach and the results are shown in Tables 6-9.
| Parameters | Mean | Median | 95% Credible Interval |
|---|---|---|---|
| α | 25.9428 | 26.0098 | (23.5370, 28.4434) |
| ϕ θ | 0.6965 | 0.7838 | (−0.0611, 0.9892) |
| τ ε 2 | 470.3430 | 467.9256 | (394.3319, 556.0817) |
| σ ξ 2 | 0.0010 | 0.0005 | (0.0000, 0.0053) |
| Covariates | Mean | Median | 95% Credible Interval |
|---|---|---|---|
| Average rainfall | 4.2376 | 4.2305 | (1.7631, 6.6843) |
| GCP | 31.4808 | 31.5454 | (29.0711, 33.8914) |
| Parameter, ( θ t ) | Time | Mean | Median | 95% Credible Interval |
|---|---|---|---|---|
| θ 1 | 1 | 0.0391 | 0.0335 | (0.0014, 0.1227) |
| θ 2 | 2 | 0.0384 | 0.0338 | (0.0031, 0.0997) |
| θ 3 | 3 | 0.0379 | 0.0332 | (0.0027, 0.0992) |
| θ 4 | 4 | 0.0378 | 0.0334 | (0.0021, 0.1001) |
| θ 5 | 5 | 0.0373 | 0.0318 | (0.0029, 0.0985) |
| θ 6 | 6 | 0.0386 | 0.0347 | (0.0020, 0.1038) |
| θ 7 | 7 | 0.0396 | 0.0351 | (0.0033, 0.1067) |
| Parameter, ( η t ) | County, (c) | Time, (t) | Mean | Median | 95% Credible Interval |
|---|---|---|---|---|---|
| η ( 1 , 1 ) | 1 | 1 | 48.6108 | 48.5645 | (36.2418, 61.6172) |
| η ( 5 , 4 ) | 5 | 4 | 1.0169 | 0.7110 | (0.0269, 3.7355) |
| η ( 10 , 7 ) | 10 | 7 | 3.0007 | 2.6799 | (0.6126, 7.1402) |
| η ( 20 , 7 ) | 20 | 7 | 2.0547 | 1.7250 | (0.2478, 5.7125) |
| η ( 30 , 3 ) | 30 | 3 | 2.0259 | 1.7007 | (0.2473, 5.6310) |
| η ( 38 , 3 ) | 38 | 3 | 12.0490 | 11.8244 | (6.3742, 19.1485) |
| η ( 40 , 7 ) | 40 | 7 | 3.1410 | 2.7922 | (0.6578, 7.6542) |
| η ( 45 , 6 ) | 45 | 6 | 50.0138 | 49.7484 | (37.1190, 64.0171) |
experienced a dry or wet season at a particular time. Meningitis incidence is high during the dry season and less during the rainy season [
Factors such as climatic, socioeconomic, and demographic have been found to have a quantitative impact on meningitis occurrence [
Model 1: y ^ = α ^ + β ^ 1 a v g r a i n + β ^ 2 g c p
Model 2: y ^ = α ^ + β ^ 1 a v g r a i n
Model 3: y ^ = α ^ + β ^ 2 g c p
Model 4: y ^ = α ^
From cases 1 to 3, see Appendix B, it is evident that average rainfall and GCP are jointly significant predictors in the model. This therefore suggests that climatic factors and socioeconomic factors are risk factors associated with meningitis
| Models | DIC | −2 loglikelihood | D* | χ 0.05 , k − p 2 | p-value |
|---|---|---|---|---|---|
| Model 1 | 724.2428 | 718.2428 | |||
| Model 2 | 762.8767 | 758.8767 | 40.6339 | 3.8410 | 0.0000 |
| Model 3 | 794.5123 | 790.5123 | 72.2695 | 3.8410 | 0.0000 |
| Model 4 | 949.8568 | 947.8568 | 229.6140 | 5.9910 | 0.0000 |
occurrence in Kenya.
The study highlights the importance of incorporating and modelling spatial and temporal dependencies jointly in the spatio-temporal disease mapping. The study contributes to the field of disease mapping by developing a spatio-temporal model in the Bayesian hierarchical framework that accounts for the spatial and temporal dependencies jointly across both space and time. The spatial and temporal dependencies were dynamically modelled together via the matern exponential covariance function where the parameter of the matern exponential θ t was treated as a function of time with first-order autoregressive structure so that it captures the space-time interactions between geographical locations and over time. Parameter estimation was performed using the MCMC approach and the posterior estimates were computed from a posterior sample of size 20,000 which was used to explain the model results. Spatio-temporal maps were obtained by mapping stable posterior means for the specific location and time from the best model that includes the significant risk factors. Both simulation data and Kenya meningitis incidence data from 2013 to 2019 were employed in the model alongside two covariates GCP and average rainfall. The study found that both average rainfall and GCP are positively associated with meningitis occurrence in Kenya. Concerning the geographical distribution of meningitis cases in Kenya, the spatio-temporal maps showed that meningitis is not evenly distributed across the country as some counties reported a high number of cases compared to other counties. Counties situated in the Rift Valley, Western side, Coastal, and Nairobi regions exhibit a notable high risk of meningitis, thus requiring prioritized attention in intervention planning. We believe the model developed will aid the policymakers and public health authorities in understanding the joint dynamic of the evolution of disease across different locations and over time. In this paper, we did not do a comparative study with existing models in the literature. We propose a further study on the comparison with other models be undertaken. A more Bayesian approach to selecting relevant significant predictors such as Bayesian shrinkage approach could be explored in this model over stepwise regression. Additionally, alternative measures of model selection such as BIC or Log-marginal likelihood could be considered for this model.
The authors declare no conflicts of interest regarding the publication of this paper.
Otieno, F.A., Tamba, C.L., Okenye, J.O. and Orawo, L.A. (2023) Dynamic Spatio-Temporal Modeling in Disease Mapping. Open Journal of Statistics, 13, 893-916. https://doi.org/10.4236/ojs.2023.136045
The red horizontal line displayed in the plots is the true value of the parameter. The green horizontal line is the estimated posterior mean value while the black horizontal lines are the Bayesian credible interval values
Case 1: Model 1: y = α + β 1 a v g r a i n + β 2 g c p
In case 1, we tested the significance of average rainfall and GCP jointly in the model. The hypothesis was given as:
H 0 : β 1 = β 2 vs. H A : at least one of β i ( i = 1 , 2 ) is not equal to zero.
It can be observed from
Case 2: Model 2: y = α + β 1 a v g r a i n
In case 2 the significance of the regression coefficient β 2 was tested using the following hypothesis
H 0 : β 2 = 0 vs. H A : β 2 ≠ 0 .
It can be perceived on
Case 3: Model 3: y = α + β 2 g c p
In this case, we examine the significance of the regression coefficient β 1 in the model. The hypothesis was given as:
H 0 : β 1 = 0 vs. H A : β 1 ≠ 0 .
From