Intervention Analysis of COVID-19 Vaccination in Nigeria: The Naive Solution Versus Interrupted Time Series

Desmond Chekwube Bartholomew; Chrysogonus Chinagorom Nwaigwe; Ukamaka Cynthia Orumie; Godwin Onyeka Nwafor

doi:10.1007/s40745-023-00462-8

. 2023 Jan 19:1–26. Online ahead of print. doi: 10.1007/s40745-023-00462-8

Intervention Analysis of COVID-19 Vaccination in Nigeria: The Naive Solution Versus Interrupted Time Series

Desmond Chekwube Bartholomew ^1,^✉, Chrysogonus Chinagorom Nwaigwe ¹, Ukamaka Cynthia Orumie ², Godwin Onyeka Nwafor ¹

PMCID: PMC9851118 PMID: 38625335

Abstract

In this paper, an intervention analysis approach was applied to daily cases of COVID-19 in Nigeria in order to evaluate the utilization and effect of the COVID-19 vaccine administered in the country. Data on the daily report of COVID-19 cases in Nigeria were collected and subjected to two models: the naïve solution and the interrupted time series (the intervention model). Based on the Alkaike Information Criterion (AIC), sigma², and log likelihood values, the interrupted time series model outperformed the Naïve solution model. ARIMA (4, 1, 4) with exogenous variables was identified as the best model. It was observed that the intervention (vaccination) was not significant at the 5% level of significance in reducing the number of daily COVID-19 cases in Nigeria since the start of the vaccination on March 5, 2021, until March 28, 2022. Also, the ARIMA (4, 1, 4) forecasts indicated that there will be surge in the number of daily COVID-19 cases in Nigeria between January and April 2023. As a result, we recommend strict adherence to COVID-19 protocols as well as further vaccination and sensitization programs to educate people on the importance of vaccine uptake and avoid Corona virus spread in the year 2023 and beyond.

Keywords: COVID-19, Vaccination, Effect, Intervention analysis, Interrupted time series, Naïve solution

Introduction

Corona virus disease 2019 (COVID-19) is a highly communicable disease caused by Severe Acute Respiratory Syndrome Corona Virus 2 (SARS-CoV-2), which was reported first in Wuhan, People's Republic of China (PRC), in December 2019 [1]. According to [2], there are over 512 million reported cases of COVID-19 worldwide, with over 6.23 million deaths reported globally as of April 29, 2022. These deaths have led to enormous psychological, sociological, demographic, and economic crises around the world. The outbreak of COVID-19 seriously challenges every government for its capacity and management of public health systems facing the catastrophic emergency [3]. On February 27, 2020, Nigeria reported her first case of COVID-19 [4], and the cases continued to fluctuate. The first case in Nigeria was an Italian citizen who entered Nigeria on February 24, 2020, from Milan via Istanbul on a Turkish Airlines flight. Reference [5] stated that more than 290 candidate vaccines have been tested within a short interval because of the urgent global need in the face of the pandemic. People with the highest risk of contracting the disease are health workers, elderly people, and people with underlying health issues. People from these categories were prioritized in the vaccination rollout [6]. Vaccines are one of the active interventions that can reduce the dominance of diseases (such as COVID-19) globally [7]. Nigeria was the third African country to receive its first batch of COVID-19 vaccines. Nigeria got the first COVID-19 vaccine (Oxford-AstraZeneca) from the Japanese government in March 2021 and administered it all in July 2021 [8]. Statistics indicate that as of June 28, 2021, about 3,441,146 (88% of the total Oxford-AstraZeneca) doses had so far been utilized for the first and second dose vaccinations [9]. The Nigerian government received additional doses of COVID-19 vaccines before the end of September 2021. On February 6, 2022, the Federal Government received from the European Union another batch of COVID-19 vaccines (Johnson, Johnson Vaccine). Unfortunately, Nigeria could not achieve the World Health Organization’s (WHO’s) set target for all countries to vaccinate 10% of their populations by the end of September 2021 [10]. It is also stated in [10] that, as of April 11, 2022, about 13,588,718 persons have been fully vaccinated.

Some existing literatures have been published on COVID-19 symptoms, mortality rate, awareness, related diseases, vaccination, and side effects, as well as modeling of reported cases. Reference [11] showed that the significant factors affecting the willingness to receive the COVID-19 vaccine among Nigerians were occupation, education, religion, and a prior diagnosis of COVID-19. In 2020, Ref. [12] reported that the Libyan hospitals were unprepared for detecting and treating patients with COVID-19. Reference [13] highlighted the need for sensitization campaigns and programs to tackle factors affecting the willingness to receive the COVID-19 vaccine among nurses and midwives in Cyprus. Reference [14] recommended that developing strategies to increase trust is essential for rolling out COVID-19 vaccination programs in Libya. Reference [7] revealed that the acceptability of COVID-19 vaccines was low in Jordan and that age, belief, trust issues, and employment status were significant factors that affected people’s acceptance of the COVID-19 vaccines. Reference [15] listed the source of vaccine knowledge, the history of chronic disease, and education level as factors that affected the willingness to accept the vaccine. Reference [16] reported a 15% increase in the number of deaths due to COVID-19. Reference [17] argued that most of the students had trust issues about the effectiveness of the COVID-19 vaccine, which inhibited them from taking the vaccine willingly. Reference [18] revealed that the side effects on the day of vaccination in the Federal Medical Center, Yenagoa, Bayelsa State, Nigeria, were pain at the injection site, fever, headache, tiredness, and chills. Reference [19] observed that the pregnant population of the Federal Medical Center, Yenagoa, has good knowledge and attitude towards the appropriate COVID-19 practices. Reference [20] maintained that all socio-demographic factors considered in their research were significant predictors of willingness to receive the COVID-19 vaccine.

All the papers reviewed showed that many authors have done justice to determining the factors that influenced acceptance of the COVID-19 vaccine in different countries and also in Nigeria, but none has tried to identify the effect of the vaccine on the number of cases reported daily in Nigeria after the introduction of the vaccine (an intervention). This particular gap in knowledge is what this paper wants to fill. Thus, in this paper, the impact of vaccination on the number of cases reported by Statista.com since the vaccination started in Nigeria is determined using the best model between the Naïve Solution approach (Ordinary Least Squares Regression) and the Interrupted Time Series model. Predictions of daily cases of COVID-19 were made up to the end of the year, 2024 using the identified model.

Interrupted Time Series (ITS) is a statistical analytic method that involves following a period before and after an intervention (COVID-19 vaccination) at a certain moment in time in order to examine the intervention's impact within a single group or population (the Nigerian population). The interruption refers to a controlled external influence (vaccination) or combination of influences (such as season, COVID-19 protocols, e.g., a nose mask, social distancing, washing of hands, etc.). Changes in the level and slope of the time series, as well as the statistical significance of the intervention parameters, are used to assess the intervention's effects [21]. If you have more observations before and after the intervention (typically), the model will be more robust. The ITS design is free of difficulties due to between-group differences because it is based on watching a single population over time, but it is subject to time-varying confounders such as COVID-19 protocols (e.g., nose masks, social distancing, washing of hands, etc.) occurring around the time of the intervention that may also affect the outcome [22]. The ITS model has been used by numerous researchers in drug utilization research [23], evaluation of public health [24], behavioral data [25], and large-scale health intervention [26]. Some of the challenges and applications of interrupted time series are given by [27].

Data science is an interdisciplinary field that seeks useful insights from massive amounts of raw and structured data. Structured data is data in a standardized format that has a well-defined structure, follows a durable order, and is easily accessed by humans and programs. Names, dates, addresses, credit card numbers, stock information, geo-location, and other types of structured data are examples. Structured data is well-organized and easy for machines to understand. Data science professionals employ a variety of methodologies to gain answers, including computer science, predictive analytics, statistics, and machine learning to comb through huge information in an attempt to find solutions to issues that have not yet been imagined. It is an umbrella term for a variety of fields that use massive databases to mine information. Data mining is an important technique in data science because it allows data scientists to ask the relevant questions. Data mining is the pattern recognition study of enormous collections of information, or "big data," and has several business uses, including web mining and text mining [28]. Big data analytics is simply a subset of data science in which the data volumes are massive and require logistical solutions to cope with. The key concern is gathering, storing, extracting, processing, and interpreting information from these massive data sets in an efficient manner. Business, engineering, and healthcare can all benefit from big data [29]. Reference [30] discusses various areas of data science, such as real-time analytics, real-time decision-making, and deep learning. Because the dates and number of COVID-19 cases are well-defined, the COVID-19 case data can be considered as both raw and structured data. An interrupted time series model is a data science strategy utilized in this study to provide a solution to the problem of assessing the impact of COVID-19 immunization on Nigerian daily COVID-19 cases. Other data science tools, such as cluster analysis [31], a computational model, and a contact matrix [32], have been used to solve new challenges using COVID-19 cases data.

The remaining sections of this paper are divided into four, namely: Sect. 2 contains the materials and methods; Sect. 3 contains the results and discussion; Sect. 4 contains the summary; and Sect. 5 contains the conclusions.

Materials and Methods

In this section, the naïve solution and the interrupted time series model for modeling the Nigerian daily COVID-19 cases are described. The models are given in 2.1 and 2.2 respectively.

The Naïve Solution (Ordinary Least Square Regression Model)

As given in [33], the multiple linear regression model for this study can be written (in non-matrix form) as in (1):

Y = β_{o} + β_{1} X_{1} + β_{2} X_{2} + β_{3} X_{3} + β_{4} X_{4} + e

where

Y is the daily COVID-19 cases

X₁ is a count variable which indicates the days passed from February 28 2020

X₂ is a dummy variable indicating observation collected before $(X_{2} = 0)$ or after $(X_{2} = 1)$ the vaccination started

X₃ is a count variable indicating days passed since the vaccination started (where before vaccination started $X_{3} = 0$ and after vaccination has started $X_{3} = 1, 2, 3, . . .$ till the last observation)

X₄ is a dummy variable indicating the season of the observation $(X_{4} = 0)$ represents observation collected during dry season while $(X_{4} = 1)$ represents observations collected during wet season (tripreport.com)

$β_{i}, i = 0, 1, 2, 3, 4$ are the regression coefficients

e representing a normal random error.

The matrix form can simply be expressed as in (2)

\underline{Y} = \underline{X β} + \underline{ε}

where

\underset{752 \times 1}{Y} = [\begin{matrix} y_{1} \\ y_{2} \\ . \\ . \\ . \\ y_{752} \end{matrix}] ; \underset{752 \times 5}{X} = [\begin{matrix} 1 & x_{11} & x_{12} & x_{13} & x_{14} \\ 1 & x_{21} & x_{22} & x_{23} & x_{24} \\ . & . & . & . & . \\ . & . \\ . & . \\ 1 & x_{n 1} & x_{n 2} & x_{n 3} & x_{n 4} \end{matrix}] where n = 752 ;

\underset{4 \times 1}{β} = [\begin{matrix} β_{o} \\ β_{1} \\ β_{2} \\ β_{3} \\ β_{4} \end{matrix}] ; \underset{752 \times 1}{ε} = [\begin{matrix} e_{1} \\ e_{2} \\ . \\ . \\ . \\ e_{752} \end{matrix}]

The sample estimate of $β$ can be written as

\hat{β} = [\begin{matrix} β_{o} \\ β_{1} \\ β_{2} \\ β_{3} \\ β_{4} \end{matrix}] ; X^{'} X = [\begin{matrix} n & \sum x_{i 1} & \sum x_{i 2} & \sum x_{i 3} & \sum x_{i 4} \\ \sum x_{i 1} & \sum x_{i 1}^{2} & \sum x_{i 1} x_{i 2} & \sum x_{i 1} x_{i 3} & \sum x_{i 1} x_{i 4} \\ \sum x_{i 2} & \sum x_{i 2} x_{i 1} & \sum x_{i 2}^{2} & \sum x_{i 2} x_{i 3} & \sum x_{i 2} x_{i 4} \\ \sum x_{i 3} & \sum x_{i 3} x_{i 1} & \sum x_{i 3} x_{i 2} & \sum x_{i 3}^{2} & \sum x_{i 3} x_{i 4} \\ \sum x_{i 4} & \sum x_{i 4} x_{i 1} & \sum x_{i 4} x_{i 2} & \sum x_{i 4} x_{i 3} & \sum x_{i 4}^{2} \end{matrix}]

\begin{matrix} (X^{'} X) \hat{β} = & [\begin{matrix} n {\hat{β}}_{o} + {\hat{β}}_{1} \sum x_{i 1} + {\hat{β}}_{2} \sum x_{i 2} + {\hat{β}}_{3} \sum x_{i 3} \\ {\hat{β}}_{o} \sum x_{i 1} + {\hat{β}}_{1} \sum x_{i 1}^{2} + {\hat{β}}_{2} \sum x_{i 1} x_{i 2} + {\hat{β}}_{3} \sum x_{i 1} x_{i 3} \\ {\hat{β}}_{o} \sum x_{i 2} + {\hat{β}}_{1} \sum x_{i 2} x_{i 1} + {\hat{β}}_{2} \sum x_{i 2}^{2} + {\hat{β}}_{3} \sum x_{i 2} x_{i 3} \\ {\hat{β}}_{o} \sum x_{i 3} + {\hat{β}}_{1} \sum x_{i 3} x_{i 1} + {\hat{β}}_{2} \sum x_{i 3} x_{i 2} + {\hat{β}}_{3} \sum x_{i 3}^{2} \\ {\hat{β}}_{o} \sum x_{i 4} + {\hat{β}}_{1} \sum x_{i 4} x_{i 1} + {\hat{β}}_{2} \sum x_{i 4} x_{i 2} + {\hat{β}}_{4} \sum x_{i 4}^{2} \end{matrix}] \\ X^{'} Y = & [\begin{matrix} \sum y_{i} \\ \sum x_{i 1} y_{i} \\ \sum x_{i 2} y_{i} \\ \sum x_{i 3} y_{i} \\ \sum x_{i 4} y_{i} \end{matrix}] \end{matrix}

The normal equation can be expressed as

X^{'} X \hat{β} = X^{'} Y

Hence,

\begin{matrix} [\begin{matrix} n {\hat{β}}_{o} + {\hat{β}}_{1} \sum x_{i 1} + {\hat{β}}_{2} \sum x_{i 2} + {\hat{β}}_{3} \sum x_{i 3} \\ {\hat{β}}_{o} \sum x_{i 1} + {\hat{β}}_{1} \sum x_{i 1}^{2} + {\hat{β}}_{2} \sum x_{i 1} x_{i 2} + {\hat{β}}_{3} \sum x_{i 1} x_{i 3} \\ {\hat{β}}_{o} \sum x_{i 2} + {\hat{β}}_{1} \sum x_{i 2} x_{i 1} + {\hat{β}}_{2} \sum x_{i 2}^{2} + {\hat{β}}_{3} \sum x_{i 2} x_{i 3} \\ {\hat{β}}_{o} \sum x_{i 3} + {\hat{β}}_{1} \sum x_{i 3} x_{i 1} + {\hat{β}}_{2} \sum x_{i 3} x_{i 2} + {\hat{β}}_{3} \sum x_{i 3}^{2} \\ {\hat{β}}_{o} \sum x_{i 4} + {\hat{β}}_{1} \sum x_{i 4} x_{i 1} + {\hat{β}}_{2} \sum x_{i 4} x_{i 2} + {\hat{β}}_{4} \sum x_{i 4}^{2} \end{matrix}] = [\begin{matrix} \sum y_{i} \\ \sum x_{i 1} y_{i} \\ \sum x_{i 2} y_{i} \\ \sum x_{i 3} y_{i} \\ \sum x_{i 4} y_{i} \end{matrix}] \end{matrix}

By differentiating the sum of squares of the residual of the normal equation with respect to $β$ , that is

\begin{matrix} e^{'} e = {(Y - X β)}^{'} (Y - X β) \\ = Y^{'} Y - 2 β^{'} X^{'} Y + β^{'} X^{'} X β \\ \frac{\partial (e^{'} e)}{\partial β} = - 2 X^{'} Y + 2 β (X^{'} X) = 0 \\ \Rightarrow \hat{β} = {(X^{'} X)}^{- 1} X^{'} Y provided that |X^{'} X| \neq 0 \end{matrix}

Intervention Analysis

The general stationary intervention model is given by (6), according to [34], who first introduced intervention modeling.

Φ (B) (y_{t} - μ - f (ω, δ, x, t)) = θ (B) ε_{t}

where B is the usual backshift operator defined by $B y_{t} = y_{t - 1}$ and by implication, using B before either a value of the series x_t or y_t an error term $ε_{t}$ means to move an element back one time.

The operator, which is the backshift, does not operate on coefficients because they are always fixed values that do not move in time. The white noise series that satisfies the independence and normality assumptions is given by (7).

Φ (B) = 1 - θ_{1} B - θ_{2} B^{2} - \dots - θ_{p} B^{p}

(7) and $ε_{t}$ in (6) imply the white noise series, with the usual mean zero and variance $σ^{2} > 0$ which satisfy the independence and normality assumptions. The characteristic roots of $Φ B$ and $θ B$ lies outside the unit circle.

$X = x_{t} = [x_{1 t}, x_{2 t}, \dots, x_{kt}]$ is a collection of intervention variables and

f (ω, δ, x, t) = \sum_{i = 1}^{k} \{\frac{ω_{i} (B)}{δ_{i} (B)}\} x_{kt}

where $ω_{i} (B)$ and $δ_{i} (B)$ are polynomials in B with roots lying outside the unit circle. The ratio $\frac{ω_{i} (B)}{δ_{i} (B)}$ is referred to as the transfer function corresponding to input variables $x_{kt}$ .

The intervention model may be expressed in the equivalent form as given in (9)

y_{t} = μ + f (ω, δ, x, t) + π (B) ε_{t}

where

$π (B) = \frac{θ (B)}{φ (B)} = 1 - π_{1} B - π_{2} B^{2}$ , $θ (B)$ and $φ (B)$ are the usual notations.

Reference [35] in modeling and analyzing the impact of gross domestic product in Nigeria, an intervention approach stated that time series intervention in practice is adopted to evaluate the effect that an observed interrupted series has on the entire series. Reference [36] opined that for an equally spaced time points regression, an intervention model can be applied and also postulated that the regression would have an effect on a response variable observed at the same time points on one or more predictor variables.

In this paper, there are input series that can be dependent or independent observations. The input series can be a pulse or step function. The function is a pulse if it shows an intervention in a single time index, while it is a step if it indicates that the intervention continued to exist starting with the time index [37].

Mathematically, the two input functions are:

\begin{matrix} P_{t}^{(t_{o})} = \{\begin{matrix} 0 & if t \neq t_{0} \\ 1 & if t = t_{0} \end{matrix}) \end{matrix}

S_{t}^{(t_{o})} = \{\begin{matrix} 0 & if t < t_{0} \\ 1 & if t \geq t_{0} \end{matrix})

In a given input series, the impact of the intervention can be written as in (10)

f (t) = \frac{ω (B)}{δ (B)} I_{t} (t_{o})

where

I_{t} (t_{α}) = P_{t}^{(t_{o})} or S_{t}^{(t_{o})}

I_t is the indicator variable such that $I_{t} = 0, t < T = t_{0} and I_{t} = 1, t \geq T = t_{0}$

$P_{t}^{(t_{o})}$ is the pulse function

$S_{t}^{(t_{o})}$ is the step function

$ω (B) = ω_{o} + ω_{1} + ω_{2} B^{2} + \dots + ω_{s} B^{s}$ is the moving average operator of polynomial form and

$δ (B) = 1 - δ_{1} B - \dots - δ_{r} B^{r}$ is an autoregressive operator of polynomial form with r and s satisfied non-negative integers

(B) is the backshift operator in time series notation.

Assumptions

i.
The intervention occurred at the same time $t_{o}$
ii.
Zeros of $δ (B)$ are assumed to be on or outside the unit circle $ω (B)$ and $δ (B)$ have no common elements.

In order to implement the intervention model, one needs to identify the tentative model and estimate the parameters of the tentative model. In evaluating the intervention impact, two parts of the overall model have to be obtained: the basic model for the targeted series and the intervention impact or effect. The methods for evaluation entail these steps:

i.
Apply the series (data) pre-intervention point $t_{0}$ to model for the series.
ii.
Apply the model to forecast values for the post intervention time.
iii.
Obtain the differences between actual values after the intervention and the forecasted values.
iv.
Evaluate the step 3 to determine a model for the intervention effect.

Interrupted Time Series (ITS)

Unlike a 2-period before-and-after t-test, interrupted time series allows you to account for long-term trends in your data. It also has the ability to analyze outcomes using population-level data, show results in a simple graphical format, conduct stratified analyses with ease, and evaluate both planned and unintended implications of interventions. However, some of its drawbacks are: (i) a minimum of eight time periods before and after an intervention is required; (ii) it is difficult to analyze the impact of separate components of a program that are introduced simultaneously in time; and (iii) the lack of a suitable control population. Understanding the counterfactual is critical in an ITS. The counterfactual describes what would have happened to Y if the policy intervention had not taken place. Counterfactuals are basic ways of contrasting what happens as a result of a change with what would have happened if the change had never occurred.

The basic autoregressive integrated moving average (ARIMA) model and the intervention effect are understood in the overall estimation process. The ARIMA(p, d, q) for example [38] and the approach for determining the correct values for the p, d, and q orders with the usual conditions on autocorrelation function (ACF), partial autocorrelation function (PACF), and time plots are described in [39].

The autoregressive model of order p, $A R (p)$ can be written as in (11)

y_{t} = c + φ_{1} y_{t - 1} + φ_{2} y_{t - 2} + \dots + φ_{p} y_{t - p} + ε_{t}

where $ε_{t}$ is white noise, $y_{t - p}$ are lagged values of y_t up to order p, $φ_{1}, \dots, φ_{p}$ are the model parameters.

Equation (11) has the following constraints on the parameter:

. for an $A R (1)$ model: $- 1 < φ_{1} < 1$ .

. for an $A R (2)$ model: $- 1 < φ_{2} < 1, φ_{1} + φ_{2} < 1, φ_{1} - φ_{2} < 1 .$

In this paper, $p > 3$ , R program took care of the complicated restrictions.

The moving average model of order q, $M A (q)$ can be written as in(12)

y_{t} = c + ε_{t} + θ_{1} ε_{t - 1} + θ_{2} ε_{t - 2} + \dots + θ_{q} ε_{t - q}

where y_t is the weighted moving average of the past few forecast errors.

Equation (12) has the following constraints on the parameter:

. for an $M A (1)$ model: $- 1 < θ_{1} < 1$ .

. for an $M A (2)$ model: $- 1 < θ_{2} < 1, θ_{1} + θ_{2} < 1, θ_{1} - θ_{2} < 1 .$

Again, for $q > 3$ , R program took care of the complicated restrictions.

The combination of the Eqs. (11) and (12) gives the full model as written in (13)

y_{t}^{'} = c + φ_{1} y_{t - 1}^{'} + φ_{2} y_{t - 2}^{'} + \dots + φ_{p} y_{t - p}^{'} + θ_{1} ε_{t - 1} + θ_{2} ε_{t - 2} + \dots + θ_{q} ε_{t - q} + ε_{t}

where $y_{t}^{'}$ is the differenced series.

Given that $y_{t}^{'}$ is differenced, then we call Eq. (13); ARIMA $(p, d, q)$ .

Where p is the order of the autoregressive part; d is the number of times the series is differenced; q is the order of the moving average part.

The Partial Autocorrelation Plot (PACF) and Autocorrelation Plot (ACF) are used to determine the values of p and q respectively. The determination of the values of p and q depends on the number of significant spikes that crosses the significant region (the blue shaded region, this blue area depicts the 95% confidence interval and is an indicator of the significance threshold) in Fig. 11. The basic guideline for interpreting the ACF and PACF plots are as following: Look for tail off pattern in either ACF or PACF. If tail off at ACF → AR model → Cut off at PACF will provide order p for AR(p). If tail off at PACF → MA model → Cut off at ACF will provide order q for MA(q).

Fig. 11 — ACF and PACF plots of the first differenced series

Model Selection Metrics

The order of the model suggested by inspecting the ACF, PACF and series plots will be compared with other competing model orders. Some of the model selection metrics used in this paper include:

i.
$Akaike's Information Criteria (AIC) = - 2 * L o g (L) + 2 (p + q + k + 1)$ 14

where

L is the likelihood of the series

$k = 1,$ if $q \neq 0$ and $k = 0,$ if $q = 0$
ii.

$Bayesian Information Criteria (BIC) = A I C + [L o g (T) - 2] * (p + q + k + 1)$ 15
where

T is the maximum likelihood estimation limit.
iii.

$Hannan-Quinn information criterion (HQC) = 2 L_{max} + 2 k ln (ln (n))$ 16
where

L_max is the log likelihood estimation limit

n is the sample size

Note: smaller values of AIC, BIC and HQIC indicate a better model.
iv.
log likelihood of the data: This is the logarithm of the probability of the observed data coming from the estimated model. The larger the log likelihood, the better the model.

In this study, computer software (R programming language and Python) is used to obtain the estimates of the model. The intervention analysis is graphically represented in Fig. 1.

Fig. 1 — Typical intervention analysis chart.

Source: www.xboard.developers

Results and Discussion

Data Structure

The data used in this paper are described in Table 1 following the variables outlined in Eq. (1). The data are of the secondary type and obtained from www.statista.com. We need to investigate the presence of a permanent level shift that is expected to occur due to an intervention. We are going to assume the intervention point to be March 5, 2021, which was the day COVID-19 vaccination started in Nigeria, but this will be further verified by the structural break that will be identified in Table 2.

Table 1.

Description of data

Id	Date	Cases (Y)	Day (X₁)	Intervention (X₂)	DayAfter (X₃)	Season (X₄)
1	Feb, 28 2020	1	1	0	0	0
2	Feb, 29 2020	0	2	0	0	0
3	Mar, 01 2020	0	3	0	0	0
4	Mar, 02 2020	0	4	0	0	0
⁝	⁝	⁝	⁝	⁝	⁝	⁝
408	Apr, 10 2021	84	408	1	36	1
409	Apr, 11 2021	63	409	1	37	1
410	Apr, 12 2021	38	410	1	38	1
411	Apr, 13 2021	74	411	1	39	1
⁝	⁝	⁝	⁝	⁝	⁝	⁝
749	Mar, 17 2022	91	749	1	377	0
750	Mar, 18 2022	11	750	1	378	0
751	Mar, 19 2022	0	751	1	379	0
752	Mar, 20 2022	0	752	1	380	0

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Table 2.

Investigation of structural changes in the data

Call:
lm(formula = mydata_ts ~ 1)
Residuals:
Min	1Q	Median	3Q	Max
−332.9	−277.2	−163.4	141.6	3702.1
Coefficients:
Estimate	Std	Error	t value	Pr(>\|t\|)
(Intercept)	332.94	15.13	22.01	< 2e−16***

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Structural Changes and Break Point

Consider Table 2. The investigation of structural changes is now underway. First, we start by verifying if regression against a constant is significant at the 5% level of significance for the time series. The regression against a constant was done, and the result in Table 2 shows that there are structural changes in the time series since the p-value (2e−16) is less than 0.05. The presence of structural changes infers the possible presence of structural breaks in the data, which is checked in Table 3 and plotted in Fig. 1.

Table 3.

Investigation of structural breaks in the data

Optimal (m + 1)—segment partition:
Call:
Break points.formula(formula = mydatats ~ 1)
Breakpoints at observation number:
m = 1	372
m = 2	271	383
m = 3	271	383	510
m = 4	112	271	383	510
m = 5	112	271	383	510	622
Corresponding to break dates:
m = 1	372
m = 2	271	383
m = 3	271	383	510
m = 4	112	271	383	510
m = 5	112	271	383	510	622
Fit:
M	0	1	2	3	4	5
RSS	129,254,395	123,036,832	94,954,169	89,229,279	87,983,553	87,752,232
BIC	11,212	11,189	11,007	10,973	10,976	10,987

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The id variable shown in Table 1 is used to identify the Cases variable values and their corresponding dates. The id variable represents the observation number in Table 3, where m represents the structural changes. It is clear that, there were five structural changes $(m = 1, 2, 3, 4 and 5)$ corresponding to five different observation numbers and dates. The best structural change is the one with the smallest BIC (Bayesian Information Criterion) and RSS (Residual Sum of Squares) values. The graphical representation of the fits (BIC and RSS) is given in Fig. 2.

Fig. 2 — Plot of BIC and RSS against breakpoints

Figure 2 shows that though the BIC is smallest at $(m = 3)$ but the RSS is not smallest at that structural change rather it is smallest at $(m = 5)$ . This means that it is difficult to choose the best structural break date. We therefore construct a 95% confidence interval for the actual structural breakdate and choose a date within this interval. The confidence interval (lower bound = 326; breakdate = 372; and upper bound = 436) was constructed and graphed in Fig. 3.

Fig. 3 — Plot of the original time series against its structural break and its confidence interval

Following the confidence interval constructed and plotted in Figs. 3 and 4, we can confidently choose observation number 372 corresponding to March 5, 2021 as the structural break date (which is called the intervention point). This is because the breakpoints of optimal 2-segment partitioning in Fig. 2 identified 372 as the observation number corresponding to the break point. The red line in Fig. 3 represents the confidence interval for the structural break dates, while the black dotted line standing on the x-axis represents the structural break date (March 5, 2021), which was the day vaccination started in Nigeria. The blue line represents the fitted break point, which was cut at the break date. This fall indicated that COVID-19 cases dropped from 332 to 246; a t-test was conducted to verify further if the difference in mean was significant across the two time windows identified by the breakpoint date, March 5, 2021.

Fig. 4 — Confidence intervals for breakpoints of optimal 2-segment partition

Based on the reported p-value in Fig. 5, we reject the null hypothesis that the true difference is equal to zero. We therefore proceed with the naïve solution (OLS regression) to measure the impact of the intervention (COVID-19 vaccination) on daily COVID-19 cases in Nigeria, and the results are shown in Fig. 6.

The Naïve Solution (Ordinary Least Square Regression)

The OLS regression model using variables $(X_{1}, X_{2}, X_{3} and X_{4})$ as independent variables to predict variable $Y$ in Fig. 6 estimates that the daily cases of COVID-19 in Nigeria decreased by 679 units (coefficient of Intervention variable) since the vaccination started until March, 20 2022 and this effect is statistically significant (p-value is virtually zero. It is also noteworthy that the model estimates a small (on average, 1.84, approximately 2 units) but statistically significant trend of a decrease (coefficient of the DayAfter variable) in the number of daily COVID-19 cases in Nigeria each day. The weather condition (season variable) was not significant at the 5% level of significance. The plot of the predictions using the OLS regression and the counterfactual is shown in Fig. 7.

Fig. 7 — Plot of the Naïve model prediction and counterfactual 95% confidence Interval

Figure 7 depicts how the naive Solution Model fits before and after vaccination began in Nigeria, as well as a projected counterfactual. The r-squared value shows that the model with all the independent variables only explained about 23.8% of the unexplained variations in the outcome variable (cases), whereas the counterfactuals show that the daily cases of COVID-19 would have increased if the vaccination had not been introduced, as indicated by the counterfactual. However, the blue line below the counterfactual black line indicates that the vaccination reduced the number of daily COVID-19 cases in Nigeria. The naïve model has a low fit (R-squared less than 75%), so another model needs to be tried and compared with the OLS model. The interrupted time series model is introduced and investigated in Sect. 3.4.

Interrupted Time Series Analysis

The study used two methods to check if the COVID-19 cases data is stationary (a series whose properties do not depend on the time at which the series is observed). A visual inspection of the data and the Dickey-Fuller test were used. We can easily see that the time series is not stationary as indicated in Fig. 8 since it shows a pattern. To further confirm this claim that the series is not stationary, the rolling mean and standard deviation are plotted in Fig. 9.

Fig. 8 — Plot of the Nigerian COVID-19 daily cases from Feb 28, 2020–March 20, 2022

Observe that the mean and standard deviation in Fig. 9 are not constant, and this confirms that the series is not stationary. The Dickey-Fuller test of the original series was also conducted, and the results are shown in Table 4.

Table 4.

Results of Dickey-Fuller Test before differencing

Test Statistic	−2.90292
p-value	0.05000
#Lags used	13
Number of observations used	738
Critical value (1%)	−3.43924
Critical value (5%)	−2.86546
Critical value (10%)	−2.56886
dtype: float64

Open in a new tab

Since the Dickey-Fuller p-value of 0.05 is not less than 0.05, we conclude that the original series is non-stationary. So now we need to transform the data to make it stationary. There are various transformations that can be done to stationarize the series, like deflation by CPI, logarithmic transformation, differencing, seasonal difference, adjustment, etc. The data were then differenced once and re-plotted in Fig. 10, and the Dickey-Fuller results for the first differenced series are shown in Table 5.

Fig. 10 — Plot of the rolling Mean and Standard deviation of the first differenced series

Table 5.

Results of Dickey-Fuller test after first differencing

Test statistic	−7.86E+00
p-value	5.28E-12
#Lags used	1.20E+01
Number of observations used	7.38E+02
Critical value (1%)	−3.44E+00
Critical value (5%)	−2.87E+00
Critical value (10%)	−2.57E+00
dtype: float64

Open in a new tab

The rolling mean and standard deviation plot in Fig. 10 suggests that the series is now stationary, and this is confirmed by the results of Dickey-Fuller in Table 5.

The series is stationary since the p-value of 5.28E−12 is virtually zero and less than 0.05 in Table 5. The ARIMA model identification with ACF and PACF plots of the first differenced series is shown in Fig. 11.

By examining the ACF plot in Fig. 11 to ascertain the order of the Moving Average, there are four significant spikes at lags 7, 10, 12 and 14 which suggests MA4 (q = 4) while the PACF has four significant spikes at lags 1, 2, 3 and 4, this suggests AR4 (p = 4). The data were differenced once to achieve stationarity and this suggests d = 1. This means that the potential model for the series is ARIMA (4, 1, 4). The ARIMA (4, 1, 4) with exogenous variables $X_{i}, i = 1, 2, 3, 4$ was compared with other competing models orders in Table 6.

Table 6.

Comparison of potential ARIMA models for the series

ARIMA	Log likelihood	AIC	BIC	HQIC	Par. estimates	p-value
(1,1,4)	−5,146.6	10,312	10,358	10,330	ar.L1 (0.0819)	0.413
					ma.L1 (−0.9403)	0.000
					ma.L2 (0.1145)	0.221
					ma.L3 (−0.0229)	0.385
					ma.L4 (0.1579)	0.000
					sigma² 52,380	0.000
(2,1,4)	−5,146.2	10,314	10,365	10,334	ar.L1 (0.0545)	0.684
					ar.L2 (0.0462)	0.687
					ma.L1 (−0.9131)	0.000
					ma.L2 (0.0482)	0.814
					ma.L3 (0.0162)	0.875
					ma.L4 (0.1541)	0.000
					sigma² (52,370)	0.000
(0,1,4)	−5,146.3	10,310	10,352	10,326	ma.L1 (−0.8601)	0.000
					ma.L2 (0.0448)	0.094
					ma.L3 (−0.0253)	0.310
					ma.L4 (0.1730)	0.000
					sigma² (52,380)	0.000
(4,1,4)	−5,141.5	10,308	10,369	10,332	ar.L1 (−0.5246)	0.000
					ar.L2 (−0.2155)	0.006
					ar.L3 (0.3563)	0.000
					ar.L4 (−0.0859)	0.028
					ma.L1 (−0.3236)	0.000
					ma.L2 (−0.1989)	0.005
					ma.L3 (−0.5424)	0.000
					ma.L4 (0.5687)	0.000
					sigma² (52,240)	0.000

Open in a new tab

Bolded p-value is significant at 5%

The model was compared with other competing model orders, as also suggested by the ACF and PACF plots in Fig. 11. This model comparison as given in Table 6 using the log likelihood, sigma2, AIC, and model parameters shows that ARIMA (4, 1, 4) is the best model because it has the highest log likelihood, smallest sigma2 and AIC, and all the model parameters are significant at the 5% level of significance. The summary of the ARIMA (4, 1, 4) is given in Fig. 12.

Based on the Ljung-Box p-value (0.93) in Fig. 12, we conclude that the model ARIMA (4, 1, 4) verifies that the time series is not auto-correlated and the errors are constant (heteroskedasticity p = 0.00). The ARIMA model estimates show that the daily COVID-19 cases decreased by 17.5% (175 units) on average daily, but this effect is not statistically significant (p = 0.670). There was also a decrease in COVID-19 cases each day after intervention by 1.92 units, which is in line with expectations. Although, the estimates of the $X_{i}, i = 1, 2, 3, 4$ exogenous variables were not statistically significant but the ARIMA model had smaller BIC and AIC values with higher log likelihood value when compared with the Naïve solution. These are strong indications that the interrupted time series best models the Nigerian daily COVID-19 cases within the periods considered in this paper.

The model diagnostics is plotted in Fig. 13 while the estimates (with counterfactual projections) are shown in Fig. 14.

Figure 13 depicts that the residuals do not show any trend (standardized residual plot), and the normal Q-Q plot suggests that the error may have come from a normal distribution population and is not auto-correlated (no significant spike in the correlogram plot).

It is clearly seen in Fig. 14 by observing the blue line that the ARIMA (4, 1, 4) model fits the dataset better than the OLS model since the line for model prediction using ARIMA (4, 1, 4) follows closely with the upward and downward movements of the Nigerian COVID-19 data. The residual analysis using qqplot to check if they follow a normal distribution is shown in Fig. 15.

Fig. 15 — Comparison of residuals of Naïve solution and ARIMA model using QQ plot

It is observed in Fig. 14 that the ARIMA model residuals are not only, in general, closer to the red line (normally distributed), but they also fit better than the OLS model theoretical quantiles. The forecast from the chosen ARIMA(4, 1, 4) with exogenous variables from March 21st, 2022, to April 10th, 2024, is shown in Fig. 16.

The forecast values of the daily COVID-19 cases in Nigeria using ARIMA (4, 1, 4) with exogenous variables are plotted in Fig. 16. The forecasts suggest increase in the number of daily COVID-19 cases starting from January 2023 which will drop around April 2023 and rise again around August 2023. The forecast is showing that the Nigerian government should not relent in her efforts to contain the COVID-19 virus even though currently in Nigeria it appears that the number of cases recorded daily has reduced.

Summary

The Nigerian COVID-19 daily cases were fitted to two models to determine the effect of COVID-19 vaccination in Nigeria: the nave solution and the interrupted time series model. The results show that the nave solution had a poor fit to the data, while the interrupted time series model gave a good fit to the data after first differencing the data. This led to the identification of ARIMA (4, 1, 4) with exogenous variables as a better model after comparing it with other competing ARIMA models based on the log likelihood, AIC, and the model parameters. It was observed that the impact of the intervention (vaccination) exercise to contain COVID-19 in Nigeria since March 5, 2021 is not statistically significant at the 5% level of significance. As expected, not many Nigerians are willing to offer themselves to take the vaccine, as only 13.3 million (6.5%) persons, as reported by [10], had been vaccinated as of April 2022. This is significantly lower than the expected WHO vaccination rate. The research has shown that the intervention (vaccination) variable coefficient is negative. It is important to note that the post intervention period indicated an upward trend (Figs. 7 and 14), which showed that despite the intervention (vaccination), the number of daily cases of COVID-19 in Nigeria is still high. However, the forecast in Fig. 16 showed that there would be an increase and decrease in the daily number of COVID-19 cases in Nigeria. Carefully studying the forecast shows that there will be high spikes in the number of COVID-19 cases in Nigeria from January 2023 to around April 2023, after which the curve flattens until August 2023. This is indicative that Nigerians and all persons living in Nigeria should observe the COVID-19 protocols such as face masking, hand washing, social distancing, etc., especially during the 2022 Christmas festival. Nigerians are also encouraged to take the vaccine in order to help further contain the COVID-19 virus.

Conclusion

In this paper, we have shown that an ordinary least squares (OLS) linear regression is not a good modeling approach for the Nigerian daily COVID-19 cases between February 28, 2020, and March 20, 2022, which is the coverage area for this research, since it gave higher AIC and BIC values with a relatively smaller log likelihood when compared to the interrupted time series model. Therefore, the ARIMA (4, 1, 4) with exogenous variables model is recommended as the best model for Nigerian daily COVID-19 case data between February 28, 2020, and March 20, 2022. Again, all persons living in Nigeria are advised to go for vaccination and observe all the COVID-19 protocols during the 2023 Christmas festival and beyond since a high spike in COVID-19 cases is forecasted in Fig. 16. It is also seen in Fig. 6 that the COVID-19 data have a high condition index number of 193,000; this indicates the presence of multicollinearity in the data and, as a result, will affect the estimates of the Nave solution model. For further studies, regularization regression methods such as principal component regression, partial least squares, or ridge regression that have the power to handle multicollinearity should be fitted and compared with the interrupted time series model results. Also optimization based data mining such as different versions of Multiple Criteria Programming (MCP) can be extensively applied to the COVID-19 cases data for data separations (classification analysis) [40].

Author Contributions

Section one (Introduction): OUC and BDC, Section two (Materials and Methods): NGO and BDC, Section three (Results and Discussions): BDC, Section Four and Five (Summary and Conclusion): NCC. All authors read and corrected the final draft manuscript.

Funding

This research was not funded by any organization.

Code Availability

The code (R and python files) used for generating the results in this paper are available at https://github.com/BartholomewDesmond/Intervention-Analysis-code.

Data Availability

https://www.statista.com/statistics/1110871/coronavirus-cases-in-nigeria/.

Declarations

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Conflict of interest

The authors declare no conflict of interest.

Footnotes

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Data Availability Statement

The code (R and python files) used for generating the results in this paper are available at https://github.com/BartholomewDesmond/Intervention-Analysis-code.

https://www.statista.com/statistics/1110871/coronavirus-cases-in-nigeria/.

[CR1] 1.Chen N, Zhou M, Dong X, Qu J, Gong F (2020) Han Y (2020) Epidemiological and clinical characteristics of 99 cases of 2019 novel coronavirus pneumonia in Wuhan, China: a descriptive study. Lancet 395:507–513 [DOI] [PMC free article] [PubMed] [Google Scholar]

[CR2] 2.JHU (2021) COVID-19 Dashboard by the Center for Systems Science and Engineering (CSSE) at Johns Hopkins University (JHU). https://coronavirus.jhu.edu/map.html. Accessed 16 Apr 2021

[CR3] 3.Li J, Guo K, Herrera Viedma E, Lee H, Liu J, Zhong Z, Gomes L, Filip FG, Fang SC, Özdemir MS, Liu XH, Lu G, Shi Y (2020) Culture vs policy: more global collaboration to effectively combat COVID-19. Innovation. 10.1016/j.xinn.2020.100023 [DOI] [PMC free article] [PubMed] [Google Scholar]

[CR4] 4.NCDC (2020) NCDCCoronavirusCOVID-19Microsite. https://COVID-19.ncdc.gov.ng/

[CR5] 5.Haque A, Pant AB (2020) Efforts at COVID-19 vaccine development: challenges and successes. Vaccines 8:739 [DOI] [PMC free article] [PubMed] [Google Scholar]

[CR6] 6.Kochhar S, Salmon DA (2020) Planning for COVID-19 vaccines safety surveillance. Vaccine 38(6):19–48 [DOI] [PMC free article] [PubMed] [Google Scholar]

[CR7] 7.El-Elimat T, AbuAlSamen MM, Almomani BA, Al-Sawalha NA, Alali FQ (2021) Acceptance and attitudes toward COVID-19 vaccines: a cross-sectional study from Jordan. PLoS ONE 16(4):25–55 [DOI] [PMC free article] [PubMed] [Google Scholar]

[CR8] 8.Al-Mustapha AI, Abubakar MI, Oyewo M, Esighetti RE, Ogundijo OA, Bolanle LD, Fakayode OE, Olugbon AS, Oguntoye M, Elelu N (2022) Socio-demographic characteristics of COVID-19 vaccine recipients in Kwara State, North Central Nigeria. J Front Public Health. 10.3389/fpubh.2021.773998 [DOI] [PMC free article] [PubMed] [Google Scholar]

[CR9] 9.COVID-19 Vaccine Tracker; 2021. https://vac-lshtm.shinyapps.io/ncov_vaccine_landscape/. Accessed 16 Apr 2021

[CR10] 10.IMMAP/DFS COVID-19 Situation Analysis (June 2021) Source iMMAPPosted 30 Jul 2021. https://reliefweb.int/organization/immap

[CR11] 11.Adedeji-Adenola H, Olugbake OA, Adeosun SA (2022) Factors influencing COVID-19 vaccine uptake among adults in Nigeria. PLoS ONE 17(2):264–371 [DOI] [PMC free article] [PubMed] [Google Scholar]

[CR12] 12.Elhadi M, Msherghi A, Alkeelani M, Alsuyihili A, Khaled A, Buzreg A, Boughididah T, Abukhashem M, Alhashimi A, Khel S, Gaffaz R, Ben SN, Bahroun S, Elharb A, Eisay M, Alnafati N, Almiqlash B, Biala M, Alghanai E (2020) Concerns for low-resource countries, with under-prepared intensive care units, facing the COVID-19 pandemic. Infect Dis Health 25(4):227–232 [DOI] [PMC free article] [PubMed] [Google Scholar]

[CR13] 13.Fakonti G, Kyprianidou M, Toumbis G, Giannakou K (2021) Attitudes and acceptance of COVID-19 vaccination among nurses and midwives in Cyprus: a cross-sectional survey. Front Public Health 9:656138. 10.3389/fpubh.2021.656138 [DOI] [PMC free article] [PubMed] [Google Scholar]

[CR14] 14.Mannan KA, Farhana KM (2020) Knowledge, attitude and acceptance of a COVID vaccine: a global cross-sectional study. Int Res J Bus Soc Sci 6(4):2020 [Google Scholar]

[CR15] 15.Al-MarshoudiS A-B, Al-WahaibimA A-K, Al-MaaniA A-F, Al-Jahwari A, Al-Habsi Z, Al-Shaibi M, Al-Msharfi M, Al-Ismaili A, Al-Buloshi H, Al-Rawahi B, Al-Barwani K (2021) Knowledge, Attitudes, and Practices (KAP) toward the COVID-19 Vaccine in Oman: a pre-campaign cross-sectional study. MDPI Open J 9(6):602. 10.3390/vaccines9060602 [DOI] [PMC free article] [PubMed] [Google Scholar]

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PERMALINK

Intervention Analysis of COVID-19 Vaccination in Nigeria: The Naive Solution Versus Interrupted Time Series

Desmond Chekwube Bartholomew

Chrysogonus Chinagorom Nwaigwe

Ukamaka Cynthia Orumie

Godwin Onyeka Nwafor

Abstract

Introduction

Materials and Methods

The Naïve Solution (Ordinary Least Square Regression Model)

Intervention Analysis

Interrupted Time Series (ITS)

Fig. 11.

Model Selection Metrics

Fig. 1.

Results and Discussion

Data Structure

Table 1.

Table 2.

Structural Changes and Break Point

Table 3.

Fig. 2.

Fig. 3.

Fig. 4.

Fig. 5.

Fig. 6.

The Naïve Solution (Ordinary Least Square Regression)

Fig. 7.

Interrupted Time Series Analysis

Fig. 8.

Fig. 9.

Table 4.

Fig. 10.

Table 5.

Table 6.

Fig. 12.

Fig. 13.

Fig. 14.

Fig. 15.

Fig. 16.

Summary

Conclusion

Author Contributions

Funding

Code Availability

Data Availability

Declarations

Institutional Review Board Statement

Informed Consent Statement

Conflict of interest

Footnotes

References

Associated Data

Data Availability Statement

ACTIONS

PERMALINK

RESOURCES

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