Statistical analysis of the incidence and mortality of African horse sickness in South Africa.

Predictions for incidence from model 2 with climate change following

The highest ﬁve predictions for incidence from model 1 with climate

The highest ﬁve predictions for incidence from model 2 with climate

Observed average incidence shown over a year, exhibiting the seasonal

Schematic diagram representing the disease dynamics. Our model de-

Locations of cases of AHS in the Johannesburg and surrounds area for

Locations of the weather stations for which data was acquired from the

Model 1 : log(µ i ) = 31.7656+3.4912sinyr − 2.9563.T max+1.4629T min −

Model 3: log(µ i ) = 18.4110+3.2204.sinyr − 1.6940.T max+0.9094.T min+

Ixopo Temperature

Ixopo Rainfall

Ladysmith Temperature

Ladysmith Rainfall

Pietermaritzburg Temperature

Pietermaritzburg Rainfall

Newcastle Temperature

Newcastle Rainfall

Vryheid Temperature

Vryheid Rainfall

African Horse Sickness

This form of the disease is usually the only form that will affect donkeys and zebras, which have a higher resistance to the clinical disease (Coetzer and Erasmus, 1994). It is unknown at this stage whether there is a link between the serotype and the presentation of the disease (Young, 2011Personal Communication).

History

Culicoides midges and African Horse Sickness

Disease Dynamics - as illustrated by deterministic models

Vector Seasonality
Vaccination

The rate at which they are infected is given by the interval between blood meals on an AHS host, α, the proportion of vectors that become infected after a blood meal on an infected host, β, times the number of susceptible vectors multiplied by the proportion infected. hosts. A proportion γ/(γ+µ) of these latently infected flies is expected to transition to the infectious class.

Figure 1.1: Compartmental model as given in papers by Lord et al., ([27] [28] [29]).

Methods for Prevention and Control

Vaccination as a Control Strategy
Stabling
Vector Control

In these cases, monovalent vaccines were used to protect against the specific serotype of the outbreak. The aim of these methods is to reduce vector abundance or reduce the bite burden and possible AHS infection to which the animals are exposed.

Figure 1.3: Map showing the surveillance, protection, and free zones for African Horse Sickness

Temperature and African Horse Sickness

AHS and Rainfall

Epidemiological Modeling

The climatic variables used were annual average daily maximum and minimum temperature, annual minimum temperature, number of days with temperatures below 0◦C, October-March rainfall, April-September rainfall and total annual rainfall.

Implications for AHS in South Africa

Objectives of the Study

Chi-Square tests of Association
Condensed Data

Plots of cases and mortality are included for both KwaZulu Natal and South Africa in Figure 2.6 and Figure 2.8 respectively. In Figure 2.6 it is clear that there is a very seasonal pattern in the occurrence of cases.

Table 2.1: List of provinces and their abbreviations used in the data, and the total number of cases reported for each province.

South African Weather Service Data

First, although the disease is notifiable by law, there is poor reporting of the disease and it is difficult to know what percentage of cases are actually recorded. It is also uncertain what percentage of recorded cases are truly AHS, as there can be some confusion with Equine Encephalitis (especially the pulmonary form) and only a small percentage of cases are conﬁrmed by laboratory tests. Those competing have a vested interest in disease reporting and control being improved as this facilitates the ability to move horses around the country.

The mean daily maximum temperature and the mean daily minimum temperature were the monthly averages of the maximum and minimum daily temperatures, respectively. This version of the model specification can be easily extended to non-normal data, leading to generalized linear models. The link function specifies the function that equates the mean of the response to the systematic component.

The vector component relatedg(µi) to the ithunite or observation-varied linear predictor.

Figure 2.9: Bargraph displaying percentage of cases where the ﬁelds Vaccination Status, Age, Presentation, Conﬁrmation of Case, and GPS co-ordinates were recorded

Exponential Family of Distributions

In case the response is binary, it also cannot be associated with a normal variable. Therefore, the generalized linear models (GLM) of Nelder and Wedderburn (1972) are used, where other assumptions about the distribution of the data can be made. However, for standard distributions such as binomial and Poisson, ϕ = 1 is usually assumed.

Many distributions including Normal, Exponential, Gamma, Chi-Square, Poisson and Binomial belong to the Exponential family of distributions. Ifϕ̸= 1 means that we cannot fully specify the probability models and hence the likelihood of the data, except perhaps to make assumptions about the first two moments.

Log-Likelihood Equation and Deviance

This is precisely the reason why Wedderburn (1974) developed the idea of quasi-probability. Assuming that a(ϕ) =ϕ/ωi, which is usually a reasonable assumption, we can write from equation (3.8): 3.9) The statistic D(y,µ)/ϕˆ is called the scaled deviation. This is a more general approach and is valid if the dispersion parameter is not equal to 1.

As with the deviation in equation (3.7), a smaller scale deviation will indicate a model with a better fit.

Mean and Variance of the GLM

Asymptotic Covariance Matrix for β

Fitting the GLM

The Newton-Raphson method is formed by taking the Taylor expansion of L(β) and taking the first-order derivative and then setting it equal to zero. The initial estimate for β must be substituted into the equation for β(0). An iterative cycle is calculated until the changes in ℓ(β(t)) are small. Then a local maximum of l(β) will be reached and the drain will be approximately zero.

The number of iterations required to compute the maximum likelihood estimate of β depends on the accuracy of the initial estimate. SAS proc GENMOD uses the Fisher scoring method up to a certain iteration (by default 1) that can be specified by the SCORING option on the MODEL statement, after which it uses the Newton-Raphson method until convergence. When using the canonical relation, however, it can be shown that the two methods are identical since H =−J.

It can be seen that since this does not depend on the observation yi, the value of.

Testing Goodness of Fit

If you find this difference close to zero, it means that model 2 fits almost as well as model 1. If χ2 calculated > χ2α;p−q, we will therefore conclude that model 2 is significantly worse in ﬁt than model 1 .

Binomial GLM

Poisson GLM for Counts Data

Poisson Generalized Linear Model for African Horse Sickness disease in-

African Horse Sickness Trust Data
South African Weather Service Data
Model Checking
Interaction and Quadratic Eﬀects
Estimating the Scale Parameter
Summary

Again, we are cautioned that the negative of the hessian is not positive definite, so we should not use this model. We require that the spread of the points be completely random and centered around zero. We have not considered the possibility that the effect of the variables may not be strictly linear on incidence.

There is slight clustering in one section of the graph, but it is not so bad that we believe the model assumptions are seriously violated. It can be seen that the occurrence of extremely low minimum temperatures is minimal regardless of rainfall. Therefore, we will stop iterations and use model 3, where the value of the scale parameter is 1.9332.

Although there is some evidence of overdispersion, estimating the scale parameter does not change the optimal model.

Table 3.1: SAS results for the model expressing cases of AHS in KZN as a function of year and month.

Binomial Generalized Linear Model for African Horse Sickness Mortality 59

Summary

This is not surprising, as an owner is much more likely to send a sample to the lab if their horse has died. However, the odds of mortality were not significantly different between these two levels, as the confidence interval includes one. This shows that vaccination protects the horse from mortality even if it is infected with the disease.

This is a biologically sound finding, as the horse would have circulating antibodies that could help the immune system fight disease. This is a very interesting discovery, as it can be assumed that timely vaccination would give the horse greater protection from disease and therefore mortality. Although we cannot test (with these data) whether the horse will have greater protection from contracting the disease when vaccinated in time, one might.

It is known (and verified by these data) that vaccination will protect the horse from mortality and will therefore affect these estimates.

Table 3.11: Stepwise Regression for Binomial Generalized Linear Model for Probability of Mortality

Generalized Estimating Equations

Speciﬁcation of Working Correlation Structure
Model Selection
Applications of Generalized Estimating Equations in Modeling

The working correlation matrix is chosen based on the assumed realistic correlation structure of the data. This is an advantage of the GEE method and the reason why the "working" covariance structure is sufficient for estimation. In this case, the REPEATED clause is used to specify the cluster variable along with the hypothesized correlation structure.

4.8) Another consideration in choosing the correlation structure is that in the case of GEEs, wrong choice will not affect the consistency of the estimates it provides. A horse vaccinated on time had odds of mortality of about 0.5 compared to an unvaccinated horse, while the odds for a horse vaccinated late were about 0.07. A horse with cardiac symptoms had odds of mortality 9 times higher than a horse with a mild case.

Once again, alternative treatments were observed to perform best, reducing the odds of mortality by 0.045 times that of an untreated horse.

Generalized Linear Mixed Models

Estimation in GLMM’s
Conditional Likelihood Method
Maximum Likelihood Estimation
Penalized Quasi-Likelihood
Adaptive Gauss-Hermite Quadrature
Applications of Generalized Linear Mixed Models to Modeling of

Other types of distribution forbi can be assumed, but they are not the focus of the present analysis. The conditional model is used rather than conditional means to produce an approximation of the conditional distribution of bgeee that looks like a normal distribution, which is preferable to work with. However, using this usual Gauss-Hermite quadrature, the function to be integrated is sampled at fixed points, regardless of the range of the actual function.

The best model was selected as the one with the minimum AIC, which is shown in Table 4.5 as the model in step 4. The final model solutions are shown in Table 4.6, where it can be seen that the significant variables are Classification, Vaccinated, Presentation, Treatment and Isolation. Although the “Place” variable accounts for some heterogeneity across cases, we hypothesize that outbreak could be another random variable.

The final results for this model are shown in Table 4.9, and the odds ratios in Table 4.10.

Table 4.2: Analysis of GEE parameter estimates for mortality data

Comparison of Techniques

The standard errors in the latter model are marginally larger than for the first, which is to be expected as it accounts for an extra source of variation. We therefore have confidence in the accuracy of all models, but probably most in the GLMMs and GEE, as they account for more of the variation by location. The choice to use either GLMM or the GEE lies only in the particular interest of the model - the GEE should be used if one is interested in population mean or marginal effects.

The GLMM should be used if one wishes to use a random intercept model and may be interested in the particular effect of a particular site. The A1 scenario family develops into three groups that describe alternative directions for technological change in the energy system. The predictions are given in terms of increase in degrees Celsius for temperature, and percentage increase for precipitation between the 1980 and 1999 period and these predictions for 2080-2099.

This illustrates why climate change predictions from these models should be taken with caution.

Table 4.9: Solutions for Fixed Eﬀects for Binomial GLMM for the Probability of Mor- Mor-tality with Place and Outbreak(Place) as random eﬀects.

Refractory Periods

We wish to add a term to the models that reflects the severity of previous outbreaks. Although it certainly explains some of the variation, it lacks similar features of fit as those found in the model for KZN. For this purpose, both the incidence and the mortality of the disease are modeled in different ways.

However, before proceeding with the modeling, it is important to understand the reliability of the data to be used. However, in plots of this model (Figure 3.2) it appeared that there may be other explanatory variables that could explain more of the variation in the data. We also investigated, as reported in Section 5.1, what role climate change may have in the prevalence of the disease in South Africa.

However, all four models are useful; depending on what the requirements of the model are.