Severe Acute Respiratory Syndrome: Time Series Modeling Coursework

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Introduction

Severe Acute Respiratory Syndrome (SARS) whose symptoms include high fever and one or more respiratory symptoms had become a major concern Hong Kong in 2003 reporting 16 death cases within the first two months through the several research that have been carried out concerning the disease it has shown that the disease has low mortality and morbidity, in addition the SARS epidemic can wide spread including to people who have not been affected.

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Its believed that one of the set backs in solving the serious Hong Kong problem was to ensure proper health care for those who were invested, and also solving the peoples fear of SARS which was mostly caused by its novel, rapid nosocomical transmission and the high risk of infection of the hospital and health care workers. SARS disease spread from the health workers, local community and also to the visitors. In this study we shall asses how the fears of the people influenced their utilization of Taiwan health services.

Sever Acute Respiratory Syndrome epidemic in Taiwan was first recorded on 14th March 2003 and lasted for four month hence ending on 21st April 2003. However, the outbreak did not just end in April but extended to May and June mainly affecting the health care workers, visitors and the locals this seemed to exaggerate the outbreak over the entire island. On July 5th the situation at Hong Kong had persisted, this was when Taiwan was removed from the World Health Organization’s list of those countries affected by SARS.

Below is accumulative number of Sever Acute Respiratory Syndrome reported to the department of Hong Kong from 14 March to April 2003.

Date 1993Healthcare workersPatients, families, and visitorsCumulative total
Hospital where outbreak startedOther healthcare workersTotal
14 March30303
15 March36036036
16 March36+13*049049
17 March44+16*12722395
18 March47+17*208439123
19 March54+17*219258150
20 March58+17*249974173
21 March66+17*2711093203
22 March68+17*29114108222
23 March73+17*32122125247
24 March78+17*34129136265
25 March82+17*35134156290
26 March88+18*37143176319
27 MarchNANA149221370
28 MarchNANA153272425
29 MarchNANA156314470
30 MarchNANA162368530
31 MarchNANA164446610
1 AprilNANA168517685

In addition to the above information, retrieving all the claims presented to the National Health Insurance program between a specified period that is from January 1st 2000 and August 31st, 2003 and this include the inpatient care, dental services, Chinese medicine services and Western medicine ambulatory care is necessary in carrying out our research to determine whether the SARS results were associated with the changes in the medical service utilization rates.

The research can be done using an interrupted time-series design; this is efficient since multiple types of services were analyzed separately which helped to determine if utilization changes were fixed to certain services. Autoregressive Moving average (ARIMA) analysis is a method of time analysis procedure that involves the identification, parameter estimation, and forecasting of autoregressive integrated moving average (Box- Jenkins) models.

ARIMA procedure has no limits involving the order of autoregressive or moving average processes, the procedure also allows estimation to be done by exact maximum likelihood, conditional or unconditional least squares and besides, it does you can model intervention models and regression models with the ARIMA errors, transfer function models with fully general rational transfer functions and seasonal ARIMA models.

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For those reasons ARIMA procedure becomes most efficient in determining weather the SARS epidemic was significantly associated with changes in the medical service utilization rates. During the research we analyze using SAS (SAS Institute Inc, Cary, NC) to attain the actual values observed and compare them with those attained using the ARIMA producer and present them as percentages, we assume a 95% confidence interval based on the daily new SARS cases. ARIMA procedure involves a number of models and by applying them we can learn which the best is.

Box-Jenkins Modeling is well established techniques for analyzing stationary linear time series.

An indicator variable can be used to recognize the non-stationary features of the observations of SARS in Hong Kong. We can also analyze this approach by way of growth curve fitting or we can assume that the trend was as a result of stochastic and use difference to eliminate the trend. We can know report the results from the plain model:

Yt – µ = ∑ αi (Yt _ i – µ) + Zt + ∑ βj Zt – j,

In this case Y represents the number of daily new SARS cases at day t, Zt is the white noise where mean is zero and variance σ2.

We can use the sample autocorrelation coefficients and partial autocorrelation coefficients which showed a strong autocorrelation of lag one and a significant partial autocorrelations of lag one. The results of the ACF and PACF were similar to the research that was carried out daily showing different lengths indicating an ARMA (1, 0).

Fitting the above equation using the maximal likely method through the S-plus and a sample of 30 observations, we find the results to be α1 =0.9584 with a estimate coefficient significantly differing from zero with p-value less than 0.0001, in addition the results standard deviation to be 0.0028 without using the mean. Applying the Dickey –Fuller test on unit root the results showed that the estimated coefficients were different from the actual results.

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Using the ARIMA method we can predict the number of SARS cases, in other words we can forecast SARS cases in k days ahead. Taking an example of a day a head forecasting we attained a mean of 21, standard deviation of 22 and therefore a 95%confidence interval would be ( 0, 64). Computation was not possible at the lower bound since it was negative therefore replacing it with zero.

Below are the observations showing the different lengths.

Dayαµσ( a)( b)( c )
220.822315223.889-42.13555
230.835114923.291-46.13752
240.892114523.765-18.11139
250.906914123.560-15.10628
260.93231372346-1.9128
270.949913322.833-0.7812
280.949212922.333-0.7717
290.966212522.216-0.5912
300.958412121.8210.6412
310.961611821.4160.5826
320.94911521.131-0.7220
330.952511221.124-0.6534
340.943411020.838-0.7916
350.95210720.520-0.618
360.956210420.1120.529
370.955210220.313-0.524
380.957499208-0.473
390.95759719.87-0.457
400.95549419.2110.491
410.95769219.15-0.422

Random walk is another model that can be efficiently used in this research since it will help forecasting the SARS cases, we shall later relate the daily format of the SARS cases to the utilization of the medical centers. In Random walk modeling we shall assume that the observed SARS cases were stationary and used the ARMA model to demean the time series. The mean value of one day ahead was adequate but the standard deviation estimated from the process was relatively large hence resulting to a negative lower bound for the 95% confidence intervals for many one day ahead projections. (Table 2).

At the start of the time series SARS cases there too many variations and this resulted to the major variations of the standard deviations, to stabilize the standard deviations log transformation of the time series is needed. Differencing the time series is an effective way to remove a stochastic trend. That is we let Xt = log Yt – log Yt_1. The first order difference could now remove the linear trend plotting the ACF and PACF showed that Xt was not serially correlated for this reason a simple ARIMA (0, 1, 0) random walk model can be used for log Yt:

log Yt = log Yt-1 + µ + Zt,

We can also predict Yt and 95% confidence intervals taking an example of a single day forecasting for log Y t+1 at day t + 1, the 95% confidence interval would therefore be

Log Yt + µt – 1.96σt * log Yt + µt + 1.96σt

Assuming σt is the sample standard deviation of Zt up to day t and a sample of 30 days µ30 = -0.0840, σ30 = 0.2374 considering previous forecasting of 95% confidence intervals we can find the future actual SARS observations.

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Below is a table showing the estimates of the parameters in the log- transformed random walk model and the forecasting of the daily new SARS cases:µ the estimate of the mean in the log scale, σ the estimate of the standard deviation of the innovation in log scale,

  1. the day’s forecast,
  2. 95% confidence interval,
  3. future actual observation.

Table showing estimation of the parameters in the log transformed random walk model and forecasting of the daily new SARS cases.

Dayαµ( a)( b)( c )
22-0.04530.167225.9980
23-0.04030.167756.10655
24-0.05480.175237.07352
25-0.05480.174935.6939
26-0.06420.173726.5128
27-0.07440.172618.3728
28-0.07170.17267.1712
29-0.09930.221110.2517
30-0.6280.23167.1712
31-0.0710.231114.4226
32-0.0520.282411.3220
330.0710.271918.5734
34-0.0520.29328.2716
35-0.07230.31154.148
36-0.910.3274.169
37-0.0850.3282.74
38-0.1040.3441.53
39-0.1090.3333.127
40-0.0850.3863.131
41-0.1310.4610.22

Monitoring the health of a population is an important activity and in this research we use the quantitative methods to come up with analyzes to carry out a forecasting on SARS in Hong Kong by applying two different methods (Random Walk and Box- Jenkins). These two methods involved the time series analyzes. ARIMA techniques were also applied, ARIMA techniques were named after Box-Jenkins’ seminar work in the 1960. From the research the tables could show the forecasting attained from the methods as well as the actual observations.

Box Jenkins model assumed the daily SARS cases were constant but this was not true because the government in China had had put so much interventions to the disease after April 21. In this light the results of ARMA model were greater than the observed case results, on the other hand Random walk model that used log was more accurate since it used the present daily SARS cases to predict the future results. The transformation of the daily cases by Random Walk reduced the variability significantly.

The two models in this article are used for different purposes; ARMA model assisted in showing the difference between the observed and the forecasted cases and could also be used as a measure of the effectiveness of the interventions. Random walk was used to provide more accurate results on short term forecasting task since it made most use of the present actual cases.

Figure 1 which compares the observed trends in expenditures for the ambulatory and inpatients care the predicted trends estimated by ARIMA model that assumes absence of the SARS epidemic the shows a reduction in the expenditure compared to the expected. The models above compared the present and the future cases of the SARS disease the number of cases related to the both ambulatory and inpatients services which were quite similar and corresponded to each transition period of the SARS epidemic. There were significant records of SARS cases before April but the number increased significantly in June and gradually in July and August before the epidemic was over.

This article shows that SARS epidemic had stronger effect on the inpatients than the ambulatory. Below is a figure that shows the observed and predicted expenditures for ambulatory and inpatients care before, during and after the epidemic. This was analyzed considering the number of SARS cases that were recorded daily using the models discussed above.

Comparing the observed trends in expenditures for the ambulatory and inpatients care

There were greater reductions in utilization in the medical services although the responses to expenditures were similar inpatients experienced the largest reductions while the ambulatory experienced the lowest.

In conclusion, the research showed significant utilization reductions at the point when SARS epidemic was wide spread. The random walk model also showed that in a short while the impact on utilization reductions was approximately $ 18.8 billion new dollar decrease in Hong Kong (6% of the annual National Health Insurance expenditure) in health care during April to August 2003.

An electronic copy of the data set, the SAS codes and the relevant output from the codes.

The Table gives a summary of the SARS cases recorded in the moth of January in reference to the cases recorded earlier in the previous moths showed above.

The AQ1 score is achieved using the Arima method. AQ1=$_$

$y_t$ represents the time spent. While $t$ the actual unit of time is months.

Month cases AQ
2/1/94 2275 84.60
2/1/94 2275 84.61
2/1/94 2275 84.62
2/1/94 2275 84.63
2/1/94 2275 84.64
2/1/94 2275 84.65
2/1/94 2275 84.66
2/1/94 2275 84.67
2/1/94 2275 84.68
2/1/94 2275 84.69
2/1/94 2275 84.70
2/1/94 2275 84.71
2/1/94 2275 84.72
2/1/94 2275 84.73
2/1/94 2275 84.74
2/1/94 2275 84.75
2/1/94 2275 84.76
2/1/94 2275 84.77
2/1/94 2275 84.78
2/1/94 2275 84.79
2/1/94 2275 84.80
2/1/94 2275 84.81
2/1/94 2275 84.82
2/1/94 2275 84.83
2/1/94 2275 84.84
2/1/94 2275 84.85
2/1/94 2275 84.86
2/1/94 2275 84.87

The following are some of the ARIMA codes used in the SARS analyses

Proc arima ; i var=y ; time series y(t) ; to detrend use (one of) the following ; instead ;

i var=y(1) ; simple difference w(t) = y(t) – y(t-1) ;

i var=y(12) ; seasonal difference w(t) = y(t) – y(t-12) ;

i var=y(1,12) ; simple difference giving w(t), then ;

seasonal difference giving w2(t): ;

data a ; infile in ;

input mon mmddyy8. count aqi ;

t = _N_ ;

format mon monyy5. ; retain t ;

w2(t) = w(t) – w(t-12) ; fitting an arima model ;

e p=(1) ; fit ar(1) to w(t) ;

e p=(1) plot ; fit ar(1) and plot the acf/pacf of ;

residuals ;

e p=(1) q=(1) ; fit arma(1,1) ;

e p=(1)(12) ; fit multiplicative ARIMA:

ar(1) x sar(1) (season lag=12) ;

forecasting from the fitted arima model ;

f out=r lead=1 back=0 id=t ;

only forecast 1 step ahead.

data=r output;

f out=r lead=12 id=t ; forecast 12 steps ahead.

include forecasts within series. output to data=r ;

The SAS program

options ls=78 formdlim=’ ‘

filename in ‘aqi.dat’ ;

sample line entry:

1/1/94 2339 86.63

proc print ;

proc plot;

plot y”t;

proc arima ;

i var=y(1) ;

e p=(1) plot ;

f out=r lead=12 id=t ;

detrend simple difference: w(t)=y(t)-y(t-1) ;

fit ar(1) to w(t). plot acf/pacf of residuals ;

forecast 12 steps ahead using arima(1,1,0) ;

proc print data=r;

look at all forecasts and 95% CI’s ;

proc gplot ;

endsas ; ending.

References

Donnelly CA, Ghani AC, Leung GM, et al. Epidemiological determinants of spread of causal agent of severe acute respiratory syndrome in Hong Kong. Lancet. 2003; 361:1761–1766.

Emanuel EJ. The lessons of SARS. Ann Intern Med. 2003; 139:589–591.

Lipsitch M, Cohen T, Cooper B, et al. Transmission dynamics and control of severe acute respiratory syndrome. Science. 2003; 300:1966–1970.

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IvyPanda. 2022. "Severe Acute Respiratory Syndrome: Time Series Modeling." July 26, 2022. https://ivypanda.com/essays/severe-acute-respiratory-syndrome-time-series-modeling/.

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