The research design adopted for this study is a descriptive and Box-Jenkins research design. Descriptive survey design is a research design in which data is collected consistently to explain and predict the given situation. For this purpose, non-seasonal Box Jenkins approach is used to find the best fitted, the best forecasting model and the accuracy of the forecasting values are checked by comparing residuals. The steps of the suggested model and its forecasting can be explained in the following steps. Determining whether the time series is stationary or not is a very important concept before making any inferences in time series analysis. Therefore, Augmented Dickey Fuller (ADF) and Phillips-Person (PP) tests will be used to check the stationarity of the data series. There are several methods that can be used to fit a time series model, among them, ARMA, ARIMA, and SARIMA model which will be used on the stationary data of this study.

The study was carried out based on monthly data on HIV prevalence as secondary data, which was collected from document based on January 2007 to December 2018 retrievable document from the Statistical data record on HIV prevalence from the record of Communicable diseases in Minna general hospital for both male and female.

Documentary evidence constitutes the instrument of data collection. The major sources of data are from Minna general hospital Statistical record on communicable diseases. The data for this study are secondary monthly HIV data sourced from the General hospital Minna in Niger state from January 2007 to December 2018.

The advances in Time Series enable researchers to use those techniques in their analysis to re-analyze the traditional rotation analysis applied in earlier studies [13]. The central idea behind model identification is a time series derived from ARIMA process which has some sort of theoretical autocorrelation properties. Fitting the empirical autocorrelation patterns with the theoretical ones helps to identify the potential tentative model for the given time series data. In this step, transformation of observed time series to stationary is inevitable.

The software that was used for the test is Eviews 4.0 version.

We can have combinations of the two processes to give a new series of models called ARMA(p, q) models. The Autoregressive model (AR) and moving average (MA).

Where

AR of order p is:

X n = m + e n + φ 1 X n − 1 + φ 2 X n − 2 + ⋯ + φ p X n − p (3.4)

for n ≥ 0, where {e_n} n ≥ 0 is a series of independent, identically distributed (iid) random variables, and m is a constant.

MA of order q is:

X n = m + e n + θ 1 e n − 1 + θ 2 e n − 2 + ⋯ + θ q e n − q , (3.5)

for n ≥ 1 where θ 1 , ⋯ , θ q are real numbers and m is a real number.

The general form of the ARMA(p, q) models where p is used for the number of autoregressive components, and q for the number of moving average components is written as:

X n = m 1 + ∑ k = 1 p φ k X n − k + ∑ j = 1 q θ j e n − j + e n , n ≥ 0 , (3.6)

where {X_n} n ≥ 1, is some constant, and the φ_k and θ_j are defined as for AR and MA models respectively.

Autoregressive (AR), Moving Average (MA) or Autoregressive Moving Average (ARMA) models in which differences have been taken are collectively called Autoregressive Integrated Moving Average or ARIMA models. A time series {Y_t} is said to follow an integrated autoregressive moving average model if the d^th difference W t = ∇ d Y t is a stationary ARMA process. If {W_t} follows an ARMA(p, q) model, we say that {Y_t} is an ARIMA(p, d, q) process. For example, for practical purposes, we can usually take d = 1 or at most 2.

Consider then an ARIMA(p, 1, q) process. With W t = Y t − Y t − 1 , we have

W t = ϕ 1 W t − 1 + ϕ 2 W t − 2 + ⋯ + ϕ p W t − p + ε t − θ 1 ε t − 1 − θ 2 ε t − 2 − ⋯ − θ q ε t − q (3.7)

Or, in terms of the observed series,

Y t − Y t − 1 = ϕ 1 ( Y t − 1 − Y t − 2 ) + ϕ 2 ( Y t − 2 − Y t − 3 ) + ⋯ + ϕ p ( Y t − p − Y t − p − 1 )     + ε t − θ 1 ε t − 1 − θ 2 ε t − 2 − ⋯ − θ q ε t − q . (3.8)

The ARIMA model (3.7) is for non-seasonal non-stationary data. A purely seasonal time series is the one that has only seasonal AR or MA parameters. Seasonal autoregressive models are built with parameter called seasonal autoregressive (SAR) parameters. The SAR parameters represent the autoregressive relationships that exist between time series data separated by multiples of the number of periods per season. Box and Jenkins have generalized this model to deal with seasonality. Their proposed model is known as the Seasonal ARIMA (SARIMA) model. In this model seasonal differencing of appropriate order is used to remove non-stationarity from the series. A first order seasonal difference is the difference between an observation and the corresponding observation from the previous year and is calculated as X t = Y t − Y t − s . For monthly time series S = 12 and for quarterly time series S = 4 This model is generally termed as the SARIMA(p, d, q) × (P, D, Q)_S.

For a seasonal time series of order s, [14] proposed that {X_t} be modelled by:

A ( L ) Φ ( L s ) ∇ s d X t = B ( L ) Θ ( L s ) ε t (3.9)

where the series must have been subjected to seasonal differencing D times and non-seasonal differencing d times, ∇ s = 1 − L s , being the seasonal differencing operator. Moreover, Φ(L) and Θ(L) are the seasonal autoregressive and moving average operators respectively. These seasonal operators are polynomials in L.

Suppose that Φ ( L ) = 1 + φ 1 L + φ 2 L 2 + ⋯ + φ P L P and Θ ( L ) = 1 + θ 1 L + θ 2 L 2 + ⋯ + θ Q L Q , then the time series {X_t} is said to follow a multiplicative seasonal autoregressive integrated moving average model of orders p, d, q, P, D, Q and s, designated (p, d, q) × (P, D, Q)_s SARIMA model.

In this section, we discuss empirical results beginning with preliminary analysis conducted with the aim to determine the normality of the data. Skewness, kurtosis and Jarque-Bera show the normality of the distribution. A distribution is said to be normal when skewness is approximately zero and kurtosis is three. Also, the probability of the Jarque-Bera statistics tells whether the series is normal or not. The null hypothesis of the Jarque-Bera test says that the distribution is a normal one. Therefore, if the probability is less than 0.05, we reject the null hypothesis and conclude that the distribution is not normal (Table 1).

Furthermore, from the Jarque-Bera test for normality of each of the variables, it was observed in the above table that the variables “HIV prevalence” p-value is less than 0.1 (10%) level of significance and not at 5% level. Thus, the enter variable is normally distributed at 10% level of significance. This is a strong factor of the fundamental assumptions of the application of ARMA, ARIMA and SARIMA models. Hence, data differencing transformation is considered in order to correct for the normality assumption violation (Table 2).

*MacKinnon (1996) one-sided p-values.

Table 3 shows the results of parameter estimation and model selection for the ARMA models, where results of the different estimation parameter of ARMA were estimated with most of the parameter significant at 1% and 5%. AIC was used to select the best model that will be used for ARIMA and SARIMA model because it is the combination of AR and MA model. From the AIC, ARMA(2, 1) was selected to be the best model since it has the smallest AIC. With this selection, our ARIMA model will be AR(2) and MA(1) while the integrated difference will be of one (1) and two (2).

Using the best model in Table 2, the result of Table 3 shows the P-value for ARMA(2, 1) indicates there is no evidence that the residuals are dependent. This further confirms that the ARMA(2, 1) model is adequate.

* at 1%, ** at 5%.

Figure 1 presents the trends analysis of the monthly data on HIV prevalence during the period of 2007 to 2018. The HIV prevalence started in January 2007 at a very slow prevalence rate. Until about September, 2008 when there was a sharp increase on the prevalence from 50 units to about 170 units. This clearly suggests an outbreak in the HIV virus. Although a relative decline in this trend was similarly observed as from July 2009 through to mid-year 2012. Another sharp increase in the trend is also observed in November 2012 but declined to almost zero in May 2015. With a steady gradual steady increase observed from march 2016 till date. This shows that if something is not done immediately the trend will go out of control.

Table 3 shows the results of parameter estimation and model selection for the AR, MA, ARMA & SARIMA models, where results of the different estimation parameter of the models were estimated with most of the parameter significant at 1% and 5%. AIC was used to select the best model. The models AR, MA, ARMA AND SARIMA were considered because the data set is in stationary at its original state and thus requires no differencing and transformation. Hence, the order and combination of the AR and MA component of the model is determined from the Correlogram plot below (Table 4).

These plots are used to choose the order parameters for candidates ARMA model. The simple moving average (MA) model is a parsimonious time series model used to account for very short-run autocorrelation. It does have a regression like form, but here each observation is regressed on the previous innovation, which is not actually observed. A weighted sum of previous and current noise is called Moving Average (MA) model.

Model identification started with autocorrelation analysis. Plots of autocorrelation function (ACF) and partial autocorrelation function (PACF) (Figure 2) showed only the first lag of the ACF was significant (i.e. laying outside the grey

95% CI band). It was also observed that the first few lags of ACF did not decay with time. Based on the autocorrelation structure, several potential models were Table 5. Candidate models proposed.

identified.

ACF plots display correlation between a series and its lags. In addition to suggesting the order of differencing, ACF plots can help in determining the order of the MA(q) model. Thus, as observed from the ACF plots we have MA(1, 2, 3, 4, 5, 6).

Based on the ACF/PACF plots the following candidate models was proposed (Table 5).

The candidate model with the smallest value of the residual sums of squares is the model that best fit the data at hand. Also, using order selection strategy proposed in Hannan and Rissanan (1982) and used by [15] and [16], the model with the least Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) is the best among other models under consideration.

> library(forecast)

> library(“ggplot2”)

> library(“forecast”)

> library(“tseries”)

> data = ts(read.csv(“data.hiv.csv”, header = TRUE, stringsAsFactors = FALSE))

> ma1 <- arima(data, order = c(0, 0, 1))

> ma2 <- arima(data, order = c(0, 0, 2))

> ma3 <- arima(data, order = c(0, 0, 3))

> ma4 <- arima(data, order = c(0, 0, 4))

> ma5 <- arima(data, order = c(0, 0, 5))

> ma6 <- arima(data, order = c(0, 0, 6))

> summary(ma1)

Call:

arima(x = data, order = c(0, 0, 1))

Coefficients:

ma1 intercept

0.6401 85.2213

s.e. 0.0594 4.8635

sigma^2 estimated as 1273: log likelihood = −719.33, aic = 1444.66

> summary(ma2)

Call:

arima(x = data, order = c(0, 0, 2))

Coefficients:

ma1 ma2 intercept

0.6542 0.3323 85.0295

s.e. 0.0869 0.0709 5.5283

sigma^2 estimated as 1125: log likelihood = −710.43, aic = 1428.87

> summary(ma3)

Call:

arima(x = data, order = c(0, 0, 3))

Coefficients:

ma1 ma2 ma3 intercept

0.6543 0.4557 0.4208 85.0665

s.e. 0.0847 0.0761 0.0764 6.4169

sigma^2 estimated as 939.9: log likelihood = −697.68, aic = 1405.37

> summary(ma4)

Call:

arima(x = data, order = c(0, 0, 4))

Coefficients:

ma1 ma2 ma3 ma4 intercept

0.6724 0.5009 0.5015 0.1576 85.0292

s.e. 0.0812 0.0890 0.0862 0.0740 7.0642

sigma^2 estimated as 912.2: log likelihood = −695.54, aic = 1403.08

> summary(ma5)

Call:

arima(x = data, order = c(0, 0, 5))

Coefficients:

ma1 ma2 ma3 ma4 ma5 intercept

0.6746 0.5279 0.5277 0.2259 0.1656 84.9291

s.e. 0.0869 0.1014 0.0925 0.0848 0.0793 7.6515

sigma^2 estimated as 884.3: log likelihood = −693.31, aic = 1400.63

> summary(ma6)

Call:

arima(x = data, order = c(0, 0, 6))

Coefficients:

ma1 ma2 ma3 ma4 ma5 ma6 intercept

0.6370 0.493 0.5412 0.2747 0.2829 0.1507 84.9344

s.e. 0.0871 0.100 0.0939 0.0884 0.1044 0.0952 8.1966

sigma^2 estimated as 869.9: log likelihood = −692.15, aic = 1400.31

> ar1 <- arima(data, order = c(1,0,0))

> summary(ar1)

Call:

arima(x = data, order = c(1, 0, 0))

Coefficients:

ar1 intercept

0.7637 84.4499

s.e. 0.0532 10.3551

sigma^2 estimated as 900.1: log likelihood = −694.55, aic = 1395.1

> arma1<-arima(data, order = c(1, 0, 1))

> arma2<-arima(data, order = c(1, 0, 2))

> arma3<-arima(data, order = c(1, 0, 3))

> arma4<-arima(data, order = c(1, 0, 4))

> arma5<-arima(data, order = c(1, 0, 5))

> arma6<-arima(data, order = c(1, 0, 6))

> summary(arma1)

Call:

arima(x = data, order = c(1, 0, 1))

Coefficients:

ar1 ma1 intercept

0.8448 −0.1980 84.4697

s.e. 0.0555 0.0981 12.3268

sigma^2 estimated as 878.1: log likelihood = −692.79, aic = 1393.58

> summary(arma2)

Call:

arima(x = data, order = c(1, 0, 2))

Coefficients:

ar1 ma1 ma2 intercept

0.8311 −0.2073 0.0587 84.5462

s.e. 0.0640 0.1039 0.1051 12.0457

sigma^2 estimated as 876.1: log likelihood = −692.63, aic = 1395.27

> summary(arma3)

Call:

arima(x = data, order = c(1, 0, 3))

Coefficients:

ar1 ma1 ma2 ma3 intercept

0.768 −0.1151 0.026 0.1704 84.6665

s.e. 0.096 0.1311 0.109 0.0967 11.0960

sigma^2 estimated as 858: log likelihood = −691.16, aic = 1394.33

> summary(arma4)

Call:

arima(x = data, order = c(1, 0, 4))

Coefficients:

ar1 ma1 ma2 ma3 ma4 intercept

0.8049 −0.1446 0.0054 0.1729 −0.0938 84.6406

s.e. 0.0928 0.1231 0.1021 0.0932 0.1069 11.4072

sigma^2 estimated as 853.3: log likelihood = −690.78, aic = 1395.57

> summary(arma5)

Call:

arima(x = data, order = c(1, 0, 5))

Coefficients:

ar1 ma1 ma2 ma3 ma4 ma5 intercept

0.7282 −0.0884 0.0604 0.2236 −0.0561 0.1425 84.8513

s.e. 0.1266 0.1438 0.1100 0.0963 0.1093 0.1021 11.1302

sigma^2 estimated as 841.4: log likelihood = −689.83, aic = 1395.66

> summary(arma6)

Call:

arima(x = data, order = c(1, 0, 6))

Coefficients:

ar1 ma1 ma2 ma3 ma4 ma5 ma6 intercept

0.6865 −0.0478 0.0853 0.2518 −0.0288 0.1512 0.0439 84.9150

s.e. 0.1731 0.1839 0.1286 0.1198 0.1288 0.1054 0.0970 10.9617

sigma^2 estimated as 840.2: log likelihood = −689.73, aic = 1397.46

> sarma1<-arima(data, order = c(1, 0, 1), seasonal = list(order = c(1, 0, 1), period = 12))

> sarma2<-arima(data, order = c(1, 0, 2), seasonal = list(order = c(1, 0, 2), period = 12))

> sarma3<-arima(data, order = c(1, 0, 3), seasonal = list(order = c(1, 0, 3), period = 12))

> sarma4<-arima(data, order = c(1, 0, 4), seasonal = list(order = c(1, 0, 4), period = 12))

> sarma5<-arima(data, order = c(1, 0, 5), seasonal = list(order = c(1, 0, 5), period = 12))

> sarma6<-arima(data, order = c(1, 0, 6), seasonal = list(order = c(1, 0, 6), period = 12))

> summary(sarma1)

Call:

arima(x = data, order = c(1, 0, 1), seasonal = list(order = c(1, 0, 1), period = 12))

Coefficients:

ar1 ma1 sar1 sma1 intercept

0.8399 −0.1750 −0.6316 0.7791 84.4652

s.e. 0.0552 0.0977 0.3538 0.3155 13.1139

sigma^2 estimated as 845.2: log likelihood = −690.59, aic = 1393.17

> summary(sarma2)

Call:

arima(x = data, order = c(1, 0, 2), seasonal = list(order = c(1, 0, 2), period = 12))

Coefficients:

ar1 ma1 ma2 sar1 sma1 sma2 intercept

0.8245 −0.1879 0.0717 −0.6498 0.8021 0.0059 84.5483

s.e. 0.0630 0.1054 0.1039 0.6473 0.6575 0.1714 12.8811

sigma^2 estimated as 841.7: log likelihood = −690.35, aic = 1396.7

> summary(sarma3)

Call:

arima(x = data, order = c(1, 0, 3), seasonal = list(order = c(1, 0, 3), period = 12))

Coefficients:

ar1 ma1 ma2 ma3 sar1 sma1 sma2 sma3

0.7548 −0.0856 0.0523 0.1840 −0.1493 0.3056 −0.0128 0.1610

s.e. 0.0966 0.1308 0.1067 0.0948 0.5515 0.5475 0.1286 0.1154

intercept

83.514

s.e. 13.396

sigma^2 estimated as 812.1: log likelihood = −688.04, aic = 1396.08

> summary(sarma4)

Call:

arima(x = data, order = c(1, 0, 4), seasonal = list(order = c(1, 0, 4), period = 12))

Coefficients:

ar1 ma1 ma2 ma3 ma4 sar1 sma1 sma2 sma3

0.7795 −0.1115 0.034 0.1849 −0.0602 0.5665 −0.4278 −0.1129 0.1898

s.e. 0.0991 0.1308 0.106 0.0937 0.1185 1.4295 1.4192 0.2226 0.1184

sma4 intercept

−0.1457 83.3802

s.e. 0.2379 12.8059

sigma^2 estimated as 808.5: log likelihood = −687.79, aic = 1399.58

> summary(sarma5)

Call:

arima(x = data, order = c(1, 0, 5), seasonal = list(order = c(1, 0, 5), period = 12))

Coefficients:

ar1 ma1 ma2 ma3 ma4 ma5 sar1 sma1

0.7210 −0.0708 0.0679 0.2325 −0.0364 0.1106 0.2541 −0.1284

s.e. 0.1306 0.1528 0.1115 0.1026 0.1226 0.1020 1.3827 1.3780

sma2 sma3 sma4 sma5 intercept

−0.0660 0.1698 −0.0931 −0.0366 83.5674

s.e. 0.1874 0.1119 0.2567 0.1631 12.2876

sigma^2 estimated as 802.2: log likelihood = −687.19, aic = 1402.38

> summary(sarma6)

Call:

arima(x = data, order = c(1, 0, 6), seasonal = list(order = c(1, 0, 6), period = 12))

Coefficients:

ar1 ma1 ma2 ma3 ma4 ma5 ma6 sar1 sma1

0.6824 −0.0383 0.0986 0.2631 −0.0172 0.1171 0.0532 0.476 −0.3471

s.e. 0.1673 0.1804 0.1330 0.1244 0.1307 0.1025 0.1065 NaN NaN

sma2 sma3 sma4 sma5 sma6 intercept

−0.0877 0.1773 −0.1227 −0.0133 0.0171 83.6397

s.e. NaN 0.0892 NaN 0.1252 NaN 12.6138

sigma^2 estimated as 801.4: log likelihood = −687.08, aic = 1406.16

Estimated value of the parameter of the best model

> summary(sarma1)

Call:

arima(x = data, order = c(1, 0, 1), seasonal = list(order = c(1, 0, 1), period = 12))

Coefficients:

ar1 ma1 sar1 sma1 intercept

0.8399 −0.1750 −0.6316 0.7791 84.4652

s.e. 0.0552 0.0977 0.3538 0.3155 13.1139

sigma^2 estimated as 845.2: log likelihood = −690.59, aic = 1393.17.

The result shows the estimation of the best model and also identifies the significance of its parameter. Based on the computed value of the coefficient for each parameter and its standard error, the absolute quotient value of the AR1, MA1, SAR1, SMA1 respectively, is greater than 0.05, it means that there is statistical sufficient evidence to say that the parameters are significant (Table 6).

*The best performing model.

Figure 3 shows the residual plot of the best model created as part of residual diagnostics of the model. This shows that the variance of the error term are seems to be constant. It also shows that the average of the residual is approximately equal to zero.

Figure 3 further shows the residual analysis to identify the normality of error terms. Since the computed p-value of Jarque-Bera test with p-value is greater than 0.05 level of significance, there is statistical evidence not to reject or fail to reject the null hypothesis of the normality of error term. This means that the error term is normally distributed.

Table 7 shows the residual analysis in identifying the independency of error term for Autoregressive Conditional Heteroskedasticity (ARCH). Since the computed p-value Box-Ljung test is equal to 0.1846 which is greater than the assigned alpha 5%, there is a statistical sufficient evidence to say that the error term is independent.

Figure 4 shows the Independency of error term generalized autoregressive conditional heteroskedasticity (GARCH) (informal way). It is however noticed

that no spike hits the line at any lag, this strongly suggests that the model is free of white noise (Figure 5).

One of the objectives of fitting and selecting the best model from AR/MA/ ARMA/SARIMA model to data is to be able to forecast its future values. The model that best fits the data going by the various statistics given in Table 8 below is SARIMA(1, 0, 1) × (1, 0, 1)₁₂.

Figure 6 shows the point forecast (blue), it indicates that the forecasted value from the created model has an increasing and decreasing trend from 2019 January-2019 October with a semi-continuous increase in January, 2019 till October, 2019.

Total lags used: 10.

The fitted number of HIV infections was calculated by optimum SARIMA(1, 0, 1) model from 2019 January-2019 October. The fitted number or the inbound forecast was similar to the observed number of HIV cases.