Modelling HIV/AIDS Cases in Zambia: A Comparative Study of the Impact of Mandatory HIV Testing

In this study, a time series modeling approach is used to determine an ARIMA model and advance counterfactual forecasting at a point of policy intervention. We consider monthly data of HIV/AIDS cases from the Ministry of Health (Copperbelt province) of Zambia, for the period 2010 to 2019 and have a total of 120 observations. Results indicate that ARIMA (1, 0, 0) is an adequate model which best fits the HIV/AIDS time series data and is, therefore, suitable for forecasting cases. The model predicts a reduction from an average of 3500 to 3177 representing 14.29% in HIV/AIDS cases from 2017 (year of policy activation) to 2019, but the actual recorded cases dropped from 3500 to 1514 accounting for 57.4% in the same time frame.


Introduction
Human Immunodeficiency Virus (HIV) infection remains the prominent cause of morbidity and mortality throughout the world. Since the start of the epidemic, around 76.1 million people have been infected and almost 35.4 million people have died from Acquired Immunodeficiency Syndrome (AIDs) related illnesses.
Globally, in 2019 there were an estimated 38 million people living with HIV.
The vast majority of people living with HIV are located in low and middle-income 19.6 million are living in East and Southern Africa which saw increasing HIV infections due to HIV service disruptions during COVID-19 and the slowing public health response to HIV [1].
As one of the Sub-Saharan countries, the cases in Zambia are not different and HIV prevalence in Zambia continued to decline. The recent Zambia Population HIV Impact Assessment (ZAMPHIA) survey shows a reduction in the prevalence of about 1.7 percentage points from 13.3% in 2014 to 11.6% in 2016 [2].
The health sector has also recorded remarkable progress on antiretroviral treatment (ART) coverage, which stands at 72% of the eligible people against the United Nations AIDS global target of 90% according to Zambia National Health Strategic Plan (2017-2021) [2]. In 2018, around 48,000 adults and 5400 children became newly infected with HIV in Zambia. In the same year around 1.2 million people in Zambia were living with HIV and 17,000 people died from HIV-related illnesses. Also, the number of people newly diagnosed with HIV and with active tuberculosis (TB) entering care has fallen significantly from 66% in 2015 to 10% in 2017 with reference to [3].
Reference [3] articulates that in 2018, 78% of all people living with HIV were on treatment. As of 2019, 87% of people living with HIV were aware of their status, and 89% on treatment and 75% were virally suppressed. [3] also urges that in Zambia there has been progress in the number of AIDS-related deaths since 2010, with a 37% decrease, from 26,000 deaths to 17,000 deaths. The number of new HIV infections has also decreased, from 56,000 to 48,000 in the same period. The 90-90-90 targets envision that, by 2030, 90% of people living with HIV will know their HIV status, 90% of people who know their HIV-positive status will be accessing treatment and 90% of people on treatment will have suppressed viral loads [3].
The policy intervention on HIV/AIDS cases in Zambia presents a comparative (also known as counterfactual or causal impact analysis) time series problem.
Reference [4] explains that this technique is used to study the causal effect resulting from the difference between the observed series and the series that would have been observed had the intervention not taken place. In this study, the Autoregressive Integrated Moving Average (ARIMA) model also known as the "Box-Jenkins" methodology following the work of Box and Jenkins was used to determine the ideal model of best fit to be used in forecasting the HIV/AIDS data. Box-Jenkins forecasting is of greatest use when the primary factors causing demand for products, services, revenue, and, in this instance, disease burden is believed to behave in the future as well as it did in the past [5].
The study by [6]  [7] employed intervention (comparative study) time series analysis to evaluate the effectiveness of a collaborative intervention to improve quality in pre-hospital ambulance care for acute myocardial infarction (AMI) and stroke. Their findings were such that, based on the estimated change in intercept and slope from pre to post intervention using segmented regression, they found insufficient evidence of a statistically significant effect on quality of care for stroke, although potential clinically important effects for AMI cannot be ruled out.
[8] used controlled interrupted time series analysis to quantify the effect of the introduction of 20 mph (32 km an hour) traffic speed zones on road collisions, injuries, and fatalities in London. The results were, the introduction of 20 mph zones was associated with a 41.9% (95% confidence interval 36.0% to 47.8%) reduction in road casualties, after adjustment for underlying time trends. The percentage reduction was greatest in younger children and greater for the category of killed or seriously injured casualties than for minor injuries. There was no evidence of casualty migration to areas adjacent to 20 mph zones, where casualties also fell slightly by an average of 8.0% (4.4% to 11.5%). Thus, they concluded that 20 mph zones are effective measures for reducing road injuries and deaths.
The significance of this inquiry is to determine if the policy of mandatory HIV/AIDS testing for all, has had any impact on the disease trend/pattern. This was done through development of forecasting models for predicting number of expected cases under this policy intervention so as to guide policy intervention and control measures on time. The rest of the paper is organized as follows: we discuss the methodology considered in Section 2. The results and discussion are contained in Section 3 and lastly, we conclude the paper in Section 4.

Methods and Materials
HIV/AIDS data for the study is taken from the administrative data submitted to where t X represents monthly observations. MS EXCEL package was used as a primary data storage and EVIEWS version 9.0 software is used to implement stochastic models and graphical representations.

Stochastic Modelling
The Box-Jenkins methodology developed by [5] is a systematic process which is executed by using an iterative process until an adequate model is achieved. The where, ,θ ∅ and t e are autoregressive parameter, moving average parameter and residual respectively. The residuals are assumed to be independent and identically distributed (i.i.d) normal random variables [9] [10].

Measure of Forecast Accuracy
The statistics are used to compare how well models fit the time series. Reference [11] explains that the best fit model is one with a high number of smaller errors.
These errors are; the Root Mean Square Error (RMSE), the Mean Absolute Percentage Error (MAPE), the Mean Percentage Error (MPE) and the Mean Absolute Error (MAE). Forecast error is given by, where t FE is the forecast error, t O is the observed value and t F is the forecast value.
The AIC and SBC are given by Other measures of forecast accuracy are given by In these formulas, L is the value of the likelihood function evaluated at the parameter estimates, n is the number of observations, k is the number of esti-

Results and Discussion
This  shows that the ACF decays exponentially and the PACF has a single spike at lag 1 indicating that the series is generated by an ARIMA (1, 0, 0) process. Figure 3 shows that the Dickey-Fuller test statistic (4.270636) which is greater than (2.886074 at 5% level of significance). Hence, we reject the NULL hypothesis and conclude that the series has no Unit Root (the series is stationary).     In Figure 6, the histogram shows that the average of residuals is approximately 0.

Diagnostic Checks
And that the (Jarque-Bera) normality test of residues is statistically significant at 5% level of significance. Thus, we conclude that the residues are normally distributed.

Forecasting
Box-Jenkins approach to forecasting stationary time series is reasonably stress-free.

Discussion
Our future ability to control HIV is dependent on the skills and proficiency of  The potential implication of this study is that by developing forecasting models for predicting HIV/AIDS cases in advance on a regular basis is to support internal decisions and planning for programs such as test and treat.

Conclusion
In this paper, the Box-Jenkins modelling process is used to determine an ARIMA to be done such as behavioural change sensitizations and also implementing programmes aimed at reducing risk behaviours.