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Malaria is a major cause of morbidity and mortality in Apac district, Northern Uganda. Hence, the study aimed to model malaria incidences with respect to climate variables for the period 2007 to 2016 in Apac district. Data on monthly malaria incidence in Apac district for the period January 2007 to December 2016 was obtained from the Ministry of health, Uganda whereas climate data was obtained from Uganda National Meteorological Authority. Generalized linear models, Poisson and negative binomial regression models were employed to analyze the data. These models were used to fit monthly malaria incidences as a function of monthly rainfall and average temperature. Negative binomial model provided a better fit as compared to the Poisson regression model as indicated by the residual plots and residual deviances. The Pearson correlation test indicated a strong positive association between rainfall and malaria incidences. High malaria incidences were observed in the months of August, September and November. This study showed a significant association between monthly malaria incidence and climate variables that is rainfall and temperature. This study provided useful information for predicting malaria incidence and developing the future warning system. This is an important tool for policy makers to put in place effective control measures for malaria early enough.

Uganda is one of the Sub-Saharan African countries where malaria is still endemic in over 90% of the country’s regions [

Transmission of malaria is very complicated. It can be determined by climatic or non-climatic factors. The impact of climatic variables on malaria patterns still remains controversial. The aim of this study is to model malaria incidences in Apac district, Northern Uganda with respect to climate variables specifically rainfall and temperature.

The greatest burden of malaria, remains in the heart land of Africa, characterized by limited infrastructure to monitor disease trends, large contiguous areas of high transmission, and low coverage of control interventions. The epidemiology of malaria varies widely in Uganda, from highland regions with low prevalence and unstable disease to large regions with dense agricultural settlement and some of the highest recorded malaria intensities in the world [

The climate in Uganda allows stable, year round malaria transmission with relatively little seasonal variability in most areas. Malaria is highly endemic in Uganda with some of the highest recorded entomological inoculation rates (EIR, infective mosquito bites per person per year) in the world, including rates of 1586 in Apac district and 562 in Tororo district measured in 2001 to 2002 [

Malaria has historically been a very serious health problem and currently poses the most significant threat to the health of the people in malaria prone areas. Uganda show that more than 55 percent of pediatric cases are due to malaria [

Malaria remains one of the leading health problems of the developing world, and Uganda bears a particularly large burden from the disease. Our understanding is limited by a lack of reliable data, but it is clear that the prevalence of malaria infection, incidence of disease, and mortality from severe malaria all remain very high. Uganda has made progress in implementing key malaria control measures, in particular distribution of insecticide impregnated bednets, indoor residual spraying of insecticides, utilization of artemisinin-based combination therapy to treat uncomplicated malaria, and provision of intermittent preventive therapy for pregnant women. However, despite enthusiasm regarding the potential for the elimination of malaria in other areas, there is no convincing evidence that the burden of malaria has decreased in Uganda in recent years. Major challenges to malaria control in Uganda include very high malaria transmission intensity, inadequate health care resources, a weak health system, inadequate understanding of malaria epidemiology and the impact of control interventions, increasing resistance of parasites to drugs and of mosquitoes to insecticides, inappropriate case management, inadequate utilization of drugs to prevent malaria, and inadequate epidemic preparedness and response. Despite these challenges, prospects for the control of malaria have improved, and with attention to underlying challenges, progress toward the control of malaria in Uganda can be expected [

The relationship between climatic variables and malaria transmission has been reported in many countries [

Rainfall and temperature anomalies are widely considered to be a major driver of inter-annual variability of malaria incidence in the semi-arid areas of Africa [

Based on the background study of malaria above, the impact of weather and environmental factors on dynamics of malaria has attracted considerable attention in recent years, yet uncertainties around future disease trends under environment change remain. The role of climate as a driving force for malaria incidences is still a subject of considerable attention [

In this study, the association between malaria incidences and climate variables are modeled using Poisson and negative binomial Regression models respectively. The significance of rainfall and temperature on the malaria incidences are determined. This knowledge is important since it gives clear understanding of malaria incidences predictors. This is necessary for the development of malaria warning systems in Apac district, Northern Uganda and hence enable effective malaria control measures to be put in place in a timely manner.

Several studies have been carried out on malaria incidence. [

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In this study, data on monthly Malaria incidence in Apac district for the period January 2006 to December 2016 were obtained from the Ministry of health, Uganda. Climate data were obtained from Uganda National Meteorological Authority. The response variable is the malaria incidence where as the climate variables are the explanatory variables.

Monthly malaria incidences for the period 2007-2016 was used. The data was obtained from the Ministry of health, Uganda.

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The generalized linear models were applied to fit the malaria incidence data as a function of rainfall and average temperature. Count data regression models

Malaria incidences | Mean | Standard deviation | range |
---|---|---|---|

9746 | 3017.864 | 5034 - 19,589 |

can be represented and understood using generalized linear models (GLM) framework [

f ( Y i ) = λ y e − λ y ! , y = 0 , 1 , 2 , ⋯ (1)

f ( Y ) is the probability that the discrete random variable Y takes non-negative integer values, λ is the parameter of the Poisson distribution. It can be proved that;

E ( Y ) = V a r ( Y ) = λ (2)

A unique feature of Poisson distribution is that the mean is equal to the variance [

λ i = E ( y i / x i ) = exp ( x i β ) (3)

log ( λ i ) = x i β (4)

This is the model for analyzing count data. Under this model as λ i increases, the conditional variance of y increases. The Poisson regression model can be thought of as a non-linear model [

The Negative binomial regression model allows the conditional variance of y to exceed the conditional mean. The mean λ is replaced with the random variable λ ˜ :

λ ˜ i = exp ( x i β + ε i ) (5)

where ε is a random error that is assumed to be uncorrelated with x.

λ ˜ i = exp ( x i β ) exp ( ε i ) = λ i exp ( ε i ) = λ i δ i (6)

The assumption is that δ has a gamma distribution with parameters: E ( δ ) = 1 and V a r ( 1 v ) .

The expected value of y for the Negative binomial distribution is the same as for Poisson distribution but the conditional variance differs:

V a r ( y i / x i ) = λ i ( 1 + λ i v i ) = exp ( x i β ) ( 1 + exp ( x i β ) v i ) (7)

since λ and v are positive, the conditional variance of y must exceed the conditional mean, v is the same for all individuals:

v i = α − 1 (8)

for α > 0 , α is the dispersion parameter since increasing α increases the conditional variance of y.

V a r ( y i / x i ) = λ i ( 1 + λ i α − 1 ) = exp ( x i β ) ( 1 + exp ( x i β ) α − 1 ) = λ i ( 1 + α λ i ) = λ i + α λ i 2 (9)

If α = 0 , the mean and variance are equal [

Estimation of parameters in Poisson regression relies on maximum likelihood estimation (MLE) method. Maximum likelihood estimation gives an understanding of the values of the regression coefficients that are more likely to have given rise to the data. The maximum likelihood estimation for Poisson regression is discussed in detail below; let Y i be the mean for the i^{th} response, for i = 1 , 2 , ⋯ , p . The mean response is assumed to be a function of a set of explanatory variables, X 1 , X 2 , ⋯ , X p , the notation λ ( X i , β ) is used to denote the function that relates the mean response λ i and X i (the values of the explanatory variables for case i) and β (the values of the regression coefficients). Let’s consider the Poisson regression model in the form below;

λ i = λ ( X i , β ) = e X i , β (10)

Then, from the Poisson distribution;

P ( Y ; β ) = [ λ ( X i β ) ] Y e − λ ( X i β ) Y ! (11)

The likelihood function is given as,

L ( Y ; β ) = ∏ i = 1 N P ( Y ; β ) (12)

= ∏ i = 1 N [ λ ( X i β ) ] Y e − λ ( X i β ) Y ! (13)

= ∏ i = 1 N [ μ ( X i β ) ] Y e − ∑ i = 1 N λ ( X i β ) Y ! (14)

The next thing to do is taking natural log of the above likelihood function. Then, differentiate the equation with respect to β and equate the equation to zero. The log likelihood function is given as,

log L ( Y i , β ) = ∑ i = 1 N [ Y i log λ ( X i , β ) ] − λ ( X i , β ) − log ( Y i ! ) (15)

∂ ∂ β [ log L ( Y ; β ) ] = 0 (16)

The solution to the set of maximum likelihood given above must generally be obtained by iteration procedure. One of the procedure is known as iteratively re weighted least squares. This procedure will estimate the values of β . Maximum likelihood estimation produces Poisson parameters that are consistent, asymptotically normal and asymptotically efficient [

The negative binomial regression model is derived by re writing Poisson regression model such that,

log λ = β 0 + β i X i + ε i (17)

where e ε i is a Gamma distributed error-term with mean 1 and variance α 2 . This addition allows the variance to differ from the mean as,

V a r ( Y ) = λ ( 1 + α λ ) = λ + α λ 2 (18)

α also acts as a dispersion parameter. Poisson regression model is regarded as a limiting model of the negative binomial regression model as α approaches zero, which means that the selection between these two models is dependent upon the value of α . The negative binomial distribution has the form,

P ( Y = y ) = Γ ( 1 α + y ) Γ ( 1 α ) y ! [ 1 α ( 1 α ) + λ ] 1 α [ λ ( 1 α ) + λ ] y (19)

where Γ ( . ) is a gamma function. This results in the likelihood function,

L ( Y i ) = ∏ i Γ ( 1 α + y i ) Γ ( 1 α ) y i ! [ 1 α ( 1 α ) + λ i ] 1 α [ λ i ( 1 α ) + λ i ] y i (20)

Maximum likelihood estimation is used to estimate parameters in negative binomial. In addition, the interpretation of regression coefficients for negative binomial regression is the same as for Poisson regression.

Deviance was used to test the goodness of fit of the model. Deviance is a measure of discrepancy between observed and fitted values. The deviance for Poisson responses takes the form

D = 2 ∑ { y i log ( y i μ ^ i ) − ( y i − μ ^ i ) }

The first term represents ‘twice a sum of observed times log of observed over fitted’. The second term, a sum of differences between observed and fitted values, is usually zero, because maximum likelihood estimations in Poisson models have the property of reproducing marginal totals. For large samples of the distribution, the deviance is approximately a chi-square with n − p degrees of freedom, where n is the number of observations and p the number of parameters. Therefore, the deviance can be used directly to test the goodness of fit of the model.

The Poisson regression is a non-normal regression, that is residuals are far from being normally distributed and the variances are non constant. Therefore we assess the model based on quantile residuals which removes the pattern in discrete data by adding the smallest amount of randomization necessary on cumulative probability scale. The quantile residuals are obtained by inverting the distribution function for each response.

Mathematically, let a i = lim y ↑ y i F ( y ; μ ^ , Θ ^ ) and b i = F ( y i ; μ ^ , Θ ^ ) where F is the cumulative function of the probability density function f ( y ; μ , Θ ) then the randomized quantile residuals for y i is r q , r = Φ − 1 ( u i ) with u i the uniform random variable on ( a i , b i ] . The randomized quantile residuals are distributed normally barring the variability in μ ^ and Θ ^ [

The expected malaria incidences was modeled using Poisson regression and the results are presented in

The residual deviance for the fitted Poisson regression was given as 89489 on 117 degrees of freedom.

The fitted Poisson model is given as from

log A = 9.8881 + ( 7.877 e − 04 ) R − ( 3.265 e − 02 ) T = 9.888 + 0.1443 R − 0.4419 T (21)

where A is the expected malaria incidences, R stands for rainfall and T stands for average temperature.

To check the goodness of fit of the fitted Poisson model, the value of the residual deviance 89,489 on 117 degrees of freedom was considered which is far greater than the number of degrees of freedom. Therefore it can be concluded that the model has lack of fit. Because the ratio between the residual deviance and the degrees of freedom is far greater than one which implies over dispersion exists. The fitted Poisson model had an AIC value of 90,813 and a null deviance of 100,505 on 119 degrees of freedom. The assumption of mean equal to variance was violated since the dispersion parameter was not approximately equal to the 1, an indication of over dispersion in the data. This meant that the parameters of the model had been over estimated and the standard errors had been under estimated which did not give a true reflection of the model that could provide appropriate expected malaria incidences from 2007 to 2016.

To address this error, Negative Binomial Regression was used to modify the model so that the case of over dispersion in the data was taken care of and the results were presented in

Figures 2-5, show plots of the deviance residuals against the normal quantiles based on Poisson model and Negative binomial models respectively.

Estimate | Standard Errors | z value | Pr (>|z|) | |
---|---|---|---|---|

Intercept | 9.888 | 3.087e^{−02} | 320.35 | <2e^{−16}*** |

Rainfall | 7.877e^{−04} | 1.493e^{−05} | 52.76 | <2e^{−16}*** |

Average Temperature | −3.265e^{−02} | 1.187e^{−03} | −27.51 | <2e^{−16}*** |

Estimate | Standard Errors | z value | Pr (>|z|) | |
---|---|---|---|---|

Intercept | 10.1940861 | 0.7788004 | 13.089 | <2e^{−16}*** |

Rainfall | 0.0008147 | 0.0003849 | 2.116 | 0.0343* |

Average Temperature | −0.0451456 | 0.0298939 | −1.510 | 0.1310 |

Estimate | Standard Errors | z value | Pr (>|z|) | |
---|---|---|---|---|

Intercept | 11.37160 | 0.54154 | 21 | <2e^{−16}*** |

Average Temperature | −0.08822 | 0.02178 | −4.05 | 5.12e^{−05}*** |

Estimate | Standard Errors | z value | Pr (>|z|) | |
---|---|---|---|---|

Intercept | 9.0252866 | 0.0436207 | 206.904 | <2e^{−16}*** |

Rainfall | 0.0011832 | 0.0002779 | 4.258 | 2.06e^{−05}*** |

for Poisson regression, the plot was not approximately linear just as for

From the

Three models for Negative Binomial regression were considered and compared using the Akaike information criterion (AIC). The results for the two Negative Binomial Regression models without collinearity respectively are given in

The Negative Binomial regression model whose results were presented in

The Negative Binomial regression model whose results were presented in

The Negative Binomial regression model whose results were presented in

The Model whose results were presented in

Pearson correlation test was performed and the results were presented in

From

Climate Variables | Correlation value |
---|---|

Rainfall and Average Temperature | −0.6985212 |

Malaria and Average temperature | Malaria and rainfall | |
---|---|---|

t-value | −3.0311 | 3.4594 |

P-value | 0.9985 | 0.0003771 |

confidence interval | [ − 0.4033168,1.0000000 ] | [ 0.1598574,1.0000000 ] |

degrees of freedom | 118 | 118 |

Malaria is transmitted by the female Anopheles mosquito. The female Anopheles mosquito go through four stages in their life cycle that is egg, larva pupa and adult [

Pearson correlation between rainfall and average temperature showed a strong negative correlation. This highlights the importance of removing one of the climate variables from the model to avoid invalid association due to collinearity. In this study, rainfall was the only climate variable considered in the Negative Binomial Regression model since it presented the best fit. Negative Binomial regression model relating expected malaria incidences, rainfall and temperature was not selected as the final model due to high correlation between rainfall and average temperature which affected the significance of individual climate variables to expected malaria incidences. The model results showed that average temperature was not significant in the model while rainfall was weakly significant. This result was seen to contradict the biology of mosquito development.The model relating malaria incidences and average temperature showed a significant positive relationship though it was not the model selected since it had a higher AIC value. Modeling has shown that optimal malaria transmission occurs at 25˚C and malaria transmission decreases at temperature above 28˚C [

In previous studies of climatic effects on malaria incidence, different results on the effect of rainfall on malaria incidence were found. [

In the present study, there was a positive significant effect between rainfall and malaria incidences, similar to previous findings [

The study had it’s own limitations such as short data length and not being able to include non-climatic variables in the models. The relationship between malaria incidences and climate variables a period of 10 years was not found to be sufficient enough to predict future occurrences. Malaria incidence is associated with socio-economic conditions of the people as well as malaria control measures. These factors were not incorporated in the models.

Malaria remains an important public health problem in Apac district, Northern Uganda. The objective of this study was to model the factors associated with malaria incidences in Apac District. The study used monthly data for the period January 2007 to December 2016 in Apac district. The Poisson regression did not accurately fit the data on malaria incidences due to over dispersion in the data. The Negative Binomial Model was a better fit. The result obtained suggested that rainfall was positively significant on monthly malaria incidences whereas average temperature was not a significant predictor for malaria incidences based on results from Pearson correlation test in Apac District. A positive relationship between rainfall and expected malaria incidences was observed based on the coefficient value of parameter estimates in

We are grateful to all the authors who contributed generously to this work.

The authors declare that they have no competing interests.

All the authors have read and approved the final manuscript.

Eunice, A., Wanjoya, A. and Luboobi, L. (2017) Statistical Modeling of Malaria Incidences in Apac District, Uganda. Open Journal of Statistics, 7, 901-919. https://doi.org/10.4236/ojs.2017.76063