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Proper understanding of global distribution of infectious diseases is an important part of disease management and policy making. However, data are subject to complexities caused by heterogeneities across host classes and space-time epidemic processes. This paper seeks to suggest or propose Bayesian spatio-temporal model for modeling and mapping tuberculosis relative risks in space and time as well identify risks factors associated with the tuberculosis and counties in Kenya with high tuberculosis relative risks. In this paper, we used spatio-temporal Bayesian hierarchical models to study the pattern of tuberculosis relative risks in Kenya. The Markov Chain Monte Carlo method via WinBUGS and R packages were used for simulations and estimation of the parameter estimates. The best fitting model is selected using the Deviance Information Criterion proposed by Spiegelhalter and colleagues. Among the spatio-temporal models used, the Knorr-Held model with space-time interaction type III and IV fit the data well but type IV appears better than type III. Variation in tuberculosis risk is observed among Kenya counties and clustering among counties with high tuberculosis relative risks. The prevalence of HIV is identified as the determinant of TB. We found clustering and heterogeneity of TB risk among high rate counties and the overall tuberculosis risk is slightly decreasing from 2002-2009. We proposed that the Knorr-Held model with interaction type IV should be used to model and map Kenyan tuberculosis relative risks. Interaction of TB relative risk in space and time increases among rural counties that share boundaries with urban counties with high tuberculosis risk. This is due to the ability of models to borrow strength from neighboring counties, such that nearby counties have similar risk. Although the approaches are less than ideal, we hope that our study provide a useful stepping stone in the development of spatial and spatio-temporal methodology for the statistical analysis of risk from tuberculosis in Kenya.

Studying the spatio-temporal pattern of infectious diseases is an important health problem in biomedical research. This is because infectious diseases adversely affect the health of the human population as well as of domestic and wild animals. This calls for the need to employ appropriate approaches to explain the spatio-temporal pattern of such diseases. This will enable us to have a proper understanding of the sources of infectious diseases as well as how such diseases spread nationally or globally. The data used to study the spread of infectious diseases are complicated due to heterogeneities that could be of spatial or non-spatial in nature [

Various authors have proposed methods to study the spread of infectious diseases [

There are various approaches to handle the MAUP [

Application of Bayesian methods in disease modeling and mapping has received great attention in biomedical research. For instance, Clayton and colleagues [

Several spatio-temporal models have been developed and applied to count data. Bernardineli and colleagues [

Iddrisu and Amoako [

We introduce the data used in this paper in Section 2. We discuss the methods used in Section 3. Section 3.1 discusses spatial distribution of diseases with much focus on the conditional autoregressive (CAR) model formulation for modeling spatial distribution of diseases [

The data used in this paper are obtained from Kenya DHS. The map of Kenya showing the counties is displayed in

Variables | No. of counties | Mean | SD | Median | Min | Max | 95% CI |
---|---|---|---|---|---|---|---|

TB Cases | 47 | 17830 | 22348.07 | 12531 | 1348 | 149600 | (7200, 20560) |

HIV Prevalence (%) | 47 | 4.289 | 2.797 | 3.800 | 1.000 | 16.430 | (2.950, 4.700) |

Proportion of poor | 47 | 0.5196 | 0.184 | 0.5013 | 0.1157 | 0.9434 | (0.3778, 0.6369) |

Illiteracy (%) | 47 | 24.47 | 19.958 | 16.00 | 2.80 | 77.30 | (12.10, 29.80) |

House hold 5km away from Hospital (%) | 47 | 77.76 | 16.399 | 80.80 | 19.20 | 99.00 | (72.05, 86.72) |

Firewood (%) | 47 | 78.52 | 20.175 | 84.60 | 1.80 | 96.70 | (74.95, 90.65) |

Altitude (m) | 47 | 1361 | 602.2214 | 1432 | 151 | 2274 | (1138, 1813) |

Mean House Hold Size | 47 | 5.383 | 0.799 | 5.250 | 3.800 | 6.900 | (4.775, 6.050) |

equal values at the low and high scale. These variables are almost symmetric or normally distributed. Finally, the variables, percentage of people who are at 5km distance away from hospital, and the percentage of those who use firewood have most of their data values concentrated at the low scale.

In this section we discuss the Bayesian spatial and spatio-temporal models used

Year | N | Mean | SD | Median | Min | Max | 95% CI |
---|---|---|---|---|---|---|---|

2002 | 47 | 685,586 | 435,819.5 | 604,298 | 83,985 | 3,034,397 | (407424, 869752) |

2003 | 47 | 705,776 | 450,910.0 | 625,506 | 86,443 | 2,600,859 | (420676, 890554) |

2004 | 47 | 727,220 | 466,775.6 | 647,886 | 89,058 | 2,710,706 | (434932, 912918) |

2005 | 47 | 747,631 | 481,824 | 669,403 | 91,549 | 2,815,838 | (449244, 934009) |

2006 | 47 | 768,909 | 497,822 | 691,637 | 94,150 | 2,924,309 | (462472, 956223) |

2007 | 47 | 791,147 | 514,520 | 714,590 | 96,851 | 3,034,397 | (475354, 979870) |

2008 | 47 | 814,422 | 531,548 | 738,321 | 99,662 | 3,146,303 | (490594, 1005057) |

2009 | 47 | 838,793 | 548,905 | 762,870 | 102,593 | 3,260,124 | (506907, 1031868) |

Year | N | Mean | SD | Median | Min | Max | 95% CI |
---|---|---|---|---|---|---|---|

2002 | 47 | 1747 | 2409.296 | 1,053 | 111.0 | 15979.0 | (679.5, 2006.5) |

2003 | 47 | 2028 | 2761.937 | 1,355 | 154.0 | 18360.0 | (746.5, 2386.0) |

2004 | 47 | 2249 | 2954.713 | 1,497 | 138.0 | 19871.0 | (928.5, 2620.0) |

2005 | 47 | 2302 | 2913.039 | 1,549 | 124.0 | 19487.0 | (938.5, 2737.5) |

2006 | 47 | 2452 | 2934.737 | 1,757 | 172 | 19,472 | (1044, 2844) |

2007 | 47 | 2416 | 2836.123 | 1,988 | 177.0 | 18901.0 | (966.5, 2786.5) |

2008 | 47 | 2291 | 2790.387 | 1,676 | 223.0 | 18589.0 | (850.5, 2623.0) |

2009 | 47 | 2346 | 2843.675 | 1,700 | 249 | 18,984 | (896, 2724) |

in disease modeling and mapping.

Spatial data are associated with a given location on the surface of the earth where such models allow for borrowing of strength between neighboring locations. In this way, neighboring locations/countries/regions/counties will have similar risks whiles distant counties are expected to show variation in risks. Waldo Tobler’s first law of spatial analysis states that “everything is related to everything else but near-by things are more related than distant things’’ [

The conditional autoregressive (CAR) models were introduced by Besag and Molié [

Application of the CAR model can be found in [

v i | v j ≠ i ~ N ( ∑ i ≠ j Φ i j v j , τ i 2 ) .

These are full conditionals where Φ i j is the weight of each observation on the mean of v i and also denotes the spatial dependence parameter. The Φ i j is non zero only if j ∈ S . Conventionally, we set Φ i j = 0 since we do not want to regress any observation on itself. Hence no region is a neighbor of itself. The v j denotes a vector of all observation except v i . Note that v i depends only on a set neighbours v j only if location j is a neighborhood set N i of v i . The τ i 2 is a potential unique variance for v i . For instance, if state i has M neighbours

and Φ i j = 1 M for every state that is a neighbour, and Φ i j = 0 otherwise, then

the conditional expectation of a state’s observation is the mean of all neighbours observations [

The CAR model is defined by mean and covariance function [

E [ v i | v j ≠ i ] = μ i + ∑ j ∈ N i Φ i j [ v j − μ i ]

and

v a r ( v i | v j ) = τ i 2

Detailed information on the conditional probability density function of a CAR random variable v i can be found in [

v i | v j ≠ i ~ N [ μ i + ρ ∑ j ∈ N i Φ ( i j ) ( v j − μ j ) , τ i 2 ] , i ∈ S

and the joint probability density is

v ~ N ( μ , B − 1 Σ D ) . (1)

The necessary and sufficient condition for (1) to be a valid joint probability density function is that its covariance matrix should be symmetric and positive definite (eigenvalues λ i > 0 , i , ⋯ , n ) [

The τ v 2 controls the overall variability of v i and ρ describes the overall effect of spatial dependence. The value of ρ is should be chosen carefully [

Choose ρ such that Σ D v − 1 is non singular and preferably with ρ ∈ ( 1 λ 1 , 1 λ n ) .

Detailed description of how to choose ρ values and parameter estimation in the CAR model can be found in [

The BYM is also a spatial model for risk smoothing which have appeared in literature and has received much attention [

l o g ( μ i ) = l o g ( E i ) + e x p ( X ′ β + u i + v i ) ,

where β , E and ϑ are vectors of the covariate, the associated parameters, the expected number of cases, and the relative risks of TB prevalence respectively. The u i is the county level random effect capturing the residual log RR of disease in county i. The u i (UH) is sometime thought of as a latent variable which captures the effect of unknown or unmeasured area level covariates and v i has a CAR model structure. Detailed information on formulation of the BYM model, parameter estimation and application to the Kenya TB data can be found in [

Many disease mapping models are restricted to identification of spatial heterogeneity and clustering of diseases risk which are in fact constrained to a single time period. However, most data in public health are often in the form of time window for many years. Therefore, there is the need to consider models which account for spatial and temporal pattern of diseases risks. Several methods have been proposed to account for spatial and temporal pattern of diseases risks [

We now present the structure of data in space and time. Consider the case where a given region of interest is divided into N areas (regions, districts, counties or municipalities) indexed by i = 1 , 2 , ⋯ , n . Let the temporal dimension be indexed by t = 1 , 2 , ⋯ , T , representing each period of time under study. Let n i t be the number of persons-times at risk in region i at period t and y i t be the corresponding observed cases which are counts TB2. The observed data y i t depends on N i t , the number of people at risk in region i and period t in the study population observed. Let N i t s be the number of people at risk in the standard population, y i t s be the observed TB cases in the standard population and C i t s be the crude rate of TB cases in the standard population. Therefore, the

crude rate for region i and period t, C i t s is defined by C i t s = y i t s N i t s . It follows

that the number of TB cases expected in region i and period t, E i t is defined by

E i t = C i t s N i t = y i t s N i t s N i t . Therefore, the overall crude rate of TB cases is defined by C = ∑ i n ∑ t T y i t s N i t s and the overall number of expected TB cases is defined by E = ∑ i N ∑ t T C i t s N i t = ∑ i N ∑ t T y i t s N i t s N i t . We assumed that y i t follows the Poisson

distribution with expectation E ( y i t ) = μ i t = E i t ϑ i t , where ϑ i t denotes the disease risk in region i, at period t. The distribution of y i t is y i t ~ P o i s s o n ( E i t e x p ( η i t ) ) , where η i t = μ + Z i + A t + Z A i t + u i t is a linear predictor, μ denotes the grand mean, Z i the main effect of region i, A t the temporal trend effect in period t, Z A i t is interaction of risk in space and time and u i t is the unstructured random effect. The contribution of a given term may serve to increase or decrease the risk of disease. The intercept or μ gives a background amount of risk shared by all regions and periods. Most often, an unstructured extra variability term u i t is included in the model so as to capture the overall effect of the other unaccounted and unobserved effects. The random effect u i t is defined as

u e | τ u 2 ~ i i d N ( 0, τ u 2 ) , e ∈ { i , t , i t }

and A t is often modeled as a structured random effect.

In the following sections, we describe the spatio-temporal models used and parameter estimation. Markov Chain Monte Carlo via Gibbs sampling is used to obtain parameter estimates under each model [

This spatio-temporal model is based on the mode proposed by Bernardineli and colleagues [

Assume that the TB counts y i t ~ P o i s s o n ( E i t e x p ( η i t ) ) . According to Bernardineli and colleagues [

ϑ i t = exp ( η i t ) = exp ( μ + u i + v i + ( ϖ + δ i ) × ℑ t ) .

The log of the Poisson mean

μ i t = E i t exp ( μ + u i + v i + ( ϖ + δ i ) × ℑ t )

is therefore given by

log ( μ i t ) = log ( E i t ) + μ + u i + v i + ( ϖ + δ i ) × ℑ t .

Given that y i t ~ P o i s s o n ( E i t e x p ( η i t ) ) with likelihood function denoted by

P r ( y , E , ϑ | u , v , δ , ϖ , τ u 2 , τ v 2 , τ δ 2 ) . (2)

The prior distribution P ( u ) of u follows a normal distribution defined as

P r ( u ) = ( 1 2π ) n / 2 ( 1 τ u ) n e x p ( − ∑ i = 1 n u i 2 2 τ u 2 )

and prior distribution P ( v ) of v has CAR structure (Section 3.1.1). Also, δ i is modeled as a CAR structure with prior distribution denoted by P ( δ ) and ϖ ~ N ( 0,0.005 ) with prior distribution P ( ϖ ) . The overall mean is defined as μ ~ N ( 0,0.01 ) . Therefore, the posterior distribution is defined as

P r ( u , v , δ , ϖ , τ u 2 , τ v 2 , τ δ 2 | y , E , ϑ ) ∝ P r ( y , E , ϑ | u , v , δ , ϖ , τ u 2 , τ v 2 , τ δ 2 ) × P r ( u ) P r ( v ) P ( δ ) P ( ϖ ) . (3)

One limitation of the model is the assumption of a linear time trend in each region. This limitation is resolve by [

The spatio-temporal Approach used in this section was developed by Waller and colleagues [

Assume that y i t ~ P o i s s o n ( E i t e x p ( η i t ) ) , where η i t = u i ( t ) + v i ( t ) and μ i t = E i t e x p ( η i t ) is the Poisson mean. The log relative risk for area i and period t is log ( ϑ i t ) = η i t . Therefore relative risk of disease is given by

ϑ i t = exp ( η i t ) = exp ( u i ( t ) + v i ( t ) ) , (4)

where for each period t, the model term u i ( t ) + v i ( t ) follows the BYM specification (See Section 3.1.2) with different precision parameter τ u ( t ) and τ v ( t ) for each period of time. The log of the Poisson mean μ i t = E i t exp ( u i t + v i t ) is therefore given by

log ( μ i t ) = log ( E i t ) + u i t + v i t (5)

The u i ( t ) and v i ( t ) are respectively uncorrelated and correlated heterogeneity terms which may vary with time. This approach results in spatio-temporal model where the spatial dimension is nested within time; thus in effect a spatial model is fitted for each period.

Given that y i t ~ P o i s s o n ( E i t e x p ( η i t ) ) with likelihood function P r ( y , E , ϑ | u , v , τ u 2 , τ v 2 ) , the prior distribution P r ( u ) of u follows a normal distribution and prior distribution P r ( v ) of v has CAR structure (See Section 3.1.1). Therefore, the posterior distribution is defined as

P r ( u , v , τ u 2 , τ v 2 | y , E , ϑ ) ∝ P r ( y , E , ϑ | u , v , τ u 2 , τ v 2 ) P r ( u ) P r ( v ) . (6)

Estimation of parameters from Equation (6) was achieved through Bayesian MCMC via Gibbs sampling.

Knorr-Held and Rasser [

η i t = μ + u i + v i + ℑ t + ψ i t ,

where the term u i + v i follows the BYM specification. The parameter ℑ t represents an unstructured or structured temporal effect and the parameter ψ i t is the space-time interaction. The log relative risk for area i and period t is log ( ϑ i t ) = η i t . Therefore the relative risk of disease is given by

ϑ i t = exp ( μ + u i + v i + ℑ t + ψ i t ) .

The log of the Poisson mean

μ i t = E i t exp ( μ + u i + v i + ℑ t + ψ i t )

is therefore given by

log ( μ i t ) = log ( E i t ) + μ + u i + v i + ℑ t + ψ i t .

It should be noted that u , v and ℑ are the main effects whiles ψ is the space-time interaction term. This model is used to study smooth temporal evolution of the estimated relative risk of TB prevalence in Kenya in each region at given point in time.

Given that y i t ~ P o i s s o n ( E i t e x p ( η i t ) ) with likelihood function

P r ( y , E , ϑ | u , v , ψ , ℑ , τ u 2 , τ v 2 , τ ψ 2 , τ ℑ 2 ) ,

the prior distribution Pr ( u ) of u has a normal distribution and prior distribution P r ( v ) of v has CAR structure (See chapter 3.1.1). According to [

P r ( ℑ | τ ℑ 2 ) ∝ e x p [ − τ ℑ − 2 2 ∑ t = 2 T ( ℑ t − ℑ t − 1 ) 2 ] . (7)

According to [

P r ( u , v , ψ , ℑ , τ u 2 , τ v 2 , τ ψ 2 , τ ℑ 2 | y , E , ϑ ) ∝ P r ( y , E , ϑ | u , v , ψ , ℑ τ u 2 , τ v 2 , τ ψ 2 , τ ℑ 2 ) × P r ( u ) P ( v ) P r ( ψ ) P ( ℑ ) .

The interaction type depends on which of the two possible type of temporal effects (unstructured or structured) interacts with the two main effects ( u i and v i ). Knorr-Held and Rasser defined four interaction types. Each of the four type of interactions has different prior interrelationships involving the interaction term ψ i t TB1.

1) Interaction type I: If the unstructured main effects ( ℑ t and u i ) are expected to interact, then the distribution of the interaction parameter ψ i t is defined as

P r ( ψ | τ ψ ) ∝ e x p [ − τ ψ 2 ∑ i = 1 n ∑ t = 1 T ( ψ i t ) 2 ] .

This may be considered as an independent unobserved covariate for each combination of region and period ( i , t ) , thus without any structure TB1,TB2. On the other hand, if spatial and temporal main effects are present in the model, then the interaction effect only denote independence in the deviations from them. The main effects can cause contribution to risk in neighboring regions or in consecutive period of time to be highly correlated. This is a global space-time heterogeneity effect and it is often modeled as ψ i t ~ N ( 0, τ ψ 2 ) . This interaction type has independent prior with no structure in space-time interaction TB1, TB2.

2) Interaction type II: This interaction effect is distributed as a random walk independently of other counties if we modelled ℑ t as a random walk [

[ ψ | τ ψ ] ∝ exp [ − τ ψ 2 ∑ i = 1 n ∑ t = 2 T ( ψ i t − ψ i , t − 1 ) 2 ]

This type of interactions has no structure in space TB4. This implies that each region has a specific evolution structure that is independent of that in the neighbouring region TB2, TB1.

3) Interaction type III: If we assumed that the unstructured temporal main effect ( ℑ t ) and the spatially correlated or structured main effect ( v i ) interact, then the interaction effect parameter ψ t = ( ψ 1 t , ⋯ , ψ n T ) , t = 1 , ⋯ , T follows an independent Intrinsic autoregressive distribution defined as [

[ ψ | τ ψ ] ∝ exp [ − τ ψ 2 ∑ t = 1 T ∑ i ~ l ( ψ i t − δ l t ) 2 ]

This interaction is assumed to have a spatial structure for each period, independent of adjacent periods (its neighbors in time). This interaction type is analogous to the clustering effect, which is often modeled as a CAR distribution (section 3.1.1) for each period [

4) Interaction type IV: Type IV is completely dependent on space and time theoretically [

[ ψ | τ ψ ] ∝ exp [ − τ ψ 2 ∑ t = 2 T ∑ i ~ l ( ψ i t − ψ l t − ψ i , t − 1 + ψ l , t − 1 ) 2 ]

[

The hyper-prior distribution for τ ℑ 2 and τ ψ 2 are modeled as gamma distribution. Estimation of all parameters was achieved with Bayesian MCMC via Gibbs sampling.

Best fitting spatio-temporal model to Kenya TB data was selected from the above candidate models based on their respective model’s DICs and pDs presented in ^{th} element of the chain for each model shown in

From

Model indicators | Bernardinelli et al., 1995 | Waller et al.,1997 | Knorr-Held et al., 20000 |
---|---|---|---|

D ¯ | 135492 | 4039.320 | 3818.720 |

pD | 487.536 | 594.296 | 375.306 |

DIC | 135979 | 4633.62 | 4194.03 |

Model indicators | Type I | Type II | Type III | Type IV |
---|---|---|---|---|

D ¯ | 3820.060 | 3818.600 | 3827.920 | 3826.910 |

pD | 376.637 | 374.377 | 363.510 | 362.494 |

DIC | 4196.700 | 4192.970 | 4191.430 | 4189.410 |

Model indicators | estimates | 95% Credible Interval |
---|---|---|

μ | -0.22 | (-0.54,0.70) |

τ v 2 | 10.8 | (1.39,50.10) |

τ u 2 | 9.17 | (3.57,28.60) |

τ ψ 2 | 11.3 | (9.55,13.20) |

adjacent counties (Nairobi and Kiambu) in the central part of Kenya show opposite trend in disease risk. This may be due to the fact that type IV interaction borrows strength from neighboring counties, hence, the decreasing trend in Nairobi county causes the estimated increase in Kiambu which is less populated than Nairobi, to be less pronounced. Again, high risk of TB prevalence is observed in the North, West, North-West and the Central counties and low risk in the South-East counties for 2002-2009.

years Ohio respiratory cancer dataset and found that interaction type II was appropriate; offering lowest deviance [

This paper explores application of spatio-temporal models used in disease modeling and mapping of TB relative risk in Kenya. These models were fitted to Kenya’s TB prevalence data from 2002-2009. Markov Chain Monte Carlo via Gibbs sampling was used for simulation of parameters from posterior distributions. Rubin and Gelman convergence diagnostics test was used to confirm

convergence of the Markov Chain. Thinning the Markov Chain and the over-relax algorithm though slow the speed of the MCMC but significantly reduces autocorrelation and number of iterations. Long-run MCMC iterations and high thinning sample size k is require for spatio-temporal models used in fitting Kenya’s TB data. The DIC of each model were compared and best model selected from the set of candidate models used in fitting Kenya’s TB prevalence data. Among the spatio-temporal models considered, the model proposed by Knorr-Held and Leaonhard [

We recommend the Knorr-Held and colleagues model [

We are satisfied that the models selected in this paper are from an appropriate class that led to the analysis of the Kenya’s TB data for 2002-2009. Further research is required for a standard or acceptable distribution type for space-time interaction ψ i j to be identified since comparing posterior deviance from interaction type that assumed t j should be modeled as structured could lead to one or more deficiencies to a given interaction type.

The authors declare that he has no competing interests.

AI performed the literature review, statistical analyses, and wrote the article. AA and NA contributed in reviewing the paper contributed to the writing and reviewing the manuscript and also provided consultation regarding analysis and interpretation of findings. The final version has been approved by the authors.

The author would like to acknowledge Dr. Thomas Noel Achia, for mentoring me on application of Bayesian methods to disease modeling and mapping.

The data used are from Kenya Demographic and Health Survey and can be found on https://dhsprogram.com/What-We-Do/Protecting-the-Privacy-of-DHS-Survey-Respondents.cfm

The study receives no funding.

Ethics approval and consent to participate statements can be found on https://dhsprogram.com/What-We-Do/Protecting-the-Privacy-of-DHS-Survey-Respondents.cfm. Procedures and questionnaires for standard DHS surveys have been reviewed and approved by the ICF International Institutional Review Board (IRB). Additionally, country-specific DHS survey protocols are reviewed by the ICF IRB and typically by an IRB in the host country. The ICF International IRB ensures that the survey complies with the U.S. Department of Health and Human Services regulations for the protection of human subjects (45 CFR 46), while the host country IRB ensures that the survey complies with laws and norms of the nation. Before each interview or biomarker test is conducted, an informed consent statement is read to the respondent, who may accept or decline to participate. A parent or guardian must provide consent prior to participation by a child or adolescent. DHS informed consent statements provide details regarding: the purpose of the interview/test, the expected duration of the interview, interview/test procedures, potential risks to the respondent. Potential benefits to the respondent, contact information for a person who can provide the respondent with more information about the interview/test. Most importantly, the informed consent statement emphasizes that participation is voluntary; that the respondent may refuse to answer any question, decline any biomarker test, or terminate participation at any time; and that the respondent’s identity and information will be kept strictly confidential.

Iddrisu, A.-K., Alhassan, A. and Amidu, N. (2018) Investigating Spatio-Temporal Pattern of Relative Risk of Tuberculosis in Kenya Using Bayesian Hierarchical Approaches. Journal of Tuberculosis Research, 6, 175-197. https://doi.org/10.4236/jtr.2018.62017