Asian Journal of Mathematics & Statistics

Year: 2022 | Volume: 15 | Issue: 1 | Page No.: 1-8
DOI: 10.3923/ajms.2022.1.8

Estimation of Spatial Effects of COVID-19 in Africa: Spatial Panel Data Model (SPDM) Approach

Oyamakin S. Oluwafemi and Yusuf O. Olufemi

Abstract: Background and Objective: Data containing time-series observations of several spatial units are treated best using spatial panels this is because panel data offers extended modelling possibilities to researchers as compared to the single equation cross-sectional procedures, which was the primary focus of the spatial statistics as contained in the literature for a long time. This study estimated the spatial effect of COVID-19 in Africa by exploring the factors influencing the rate of confirmed cases and examining the spatial spillover effects of COVID-19 within the African continent and interpreting the most efficient and consistent model with direct and indirect spatial effects. Materials and Methods: The study considered the spatial effect of COVID-19 in Africa using the Spatial Panel Data Models (SPDM) approach. The COVID-19 data on 54 countries in Africa with confirmed cases of COVID-19 as of 12th May, 2020 were extracted from the COVID-19 dashboard of the Center for Systems Science and Engineering at the John Hopkins University (CSSE, JHU). Results: The study revealed a daily increase in the rate of confirmed cases and that an increase of 0.1527 per 100,000 people is expected in the coming weeks in Africa if the pattern of spread remains constant. Conclusion: Conclusively, we have been able to provide information about the effect of the spread of COVID-19 across the African continent. We also gathered from the results that the rate of death and recovery from COVID-19 in Africa has a significant positive effect on the spread of the virus within the continent.

Keywords: COVID-19, spatial effects, spatial panel data models, public health, modeling, health infrastructures and panel data

INTRODUCTION

Coronavirus disease (COVID-19) caused by a newly discovered novel coronavirus is an infectious disease¹. Mild to moderate symptoms and recovery without special treatment is what most people who fall sick with COVID-19 experience¹. The COVID-19 is mainly transmitted through droplets generated when an infected person coughs, sneezes, or exhales. These droplets are too heavy to hang in the air and quickly fall on floors or surfaces². It has been discussed at different levels how one can be infected, which is majorly by breathing in the virus if you are within proximity of someone who has COVID-19 or by touching a contaminated surface and then your eyes, nose or mouth. Various strategies have since been put in place by affected countries. These included the regular use of face masks, social distancing, washing of hands regularly and staying indoors among others. Most of these strategies are difficult to implement in Africa due to our limitations in terms of health infrastructures, the culture of buying and selling, modes of transportation and bad data management policies among others.

The spatial statistics literature has maintained a growing interest in the specification and estimation of relationships based on spatial panels in recent times. Spatial panels typically refer to data containing time-series observations of several spatial units². This property explained why the study adopted the use of the Spatial Panel Data Model (SPDM) since it offered an extended modelling possibility as compared to the single equation cross-sectional setting, which was the primary focus of the spatial statistics as used in econometric literature for a long time. Panel data are generally more informative and they contain more variation and less collinearity among the variables. The use of panel data results in greater availability of degrees of freedom and hence increases efficiency in the estimation. Panel data also allow for the specification of more complicated behavioural hypotheses, including effects that cannot be addressed using pure cross-sectional data³. With these in mind, this paper applied the spatial panel data models to determine the rate of spread of COVID-19 across the continent of Africa.

MATERIALS AND METHODS

Study area: The study was carried out at the Biostatistics Unit, Department of Statistics, University of Ibadan, Nigeria. The COVID-19 data was extracted from the COVID-19 dashboard of the Center for Systems Science and Engineering at the John Hopkins University (CSSE, JHU), sampling from 29 February to 12 May, 2020 for the 54 countries of Africa with confirmed cases of the Novel Coronavirus (COVID-19).

Spatial panel models: Consider a simple pooled linear regression model with spatial specific effects but without spatial interaction effects⁴:

where, i is an index for the cross-sectional dimension (spatial units), with i = 1,..., N and t is an index for the time dimension (periods), with t = 1,..., T. y_it is an observation on the dependent variable at i and t, x_it an (1, K) row vector of observations on the independent variables and β a matching (K, 1) vector of fixed but unknown parameters. The ε_it is an independently and identically distributed error term for i and t with zero mean and variance σ², while μ_i denotes a spatial specific effect. The standard reasoning behind spatial specific effects is that they control for all space-specific time-invariant variables whose omission could bias the estimates in a typical cross-sectional study. When specifying interaction between spatial units, the model may contain a spatially lagged dependent variable or a spatial autoregressive process in the error term, known as the spatial lag and the spatial error model, respectively. The spatial lag model posits that the dependent variable depends on the dependent variable observed in neighbouring units and on a set of observed local characteristics:

where, δ is called the spatial autoregressive coefficient and w_ij is an element of a spatial weights matrix W describing the spatial arrangement of the units in the sample. It is assumed that W is a pre-specified non-negative matrix of order N2. Baltagi et al.⁵ studied the spatial lag model which was typically considered the formal specification for the equilibrium outcome of a spatial or social interaction process, in which the value of the dependent variable for one agent is jointly determined with that of the neighbouring agents.

The spatial error model, on the other hand, posits that the dependent variable depends on a set of observed local characteristics and that the error terms are correlated across space:

where, ϕ_it reflects the spatially autocorrelated error term and ρ is called the spatial autocorrelation coefficient. Baltagi et al.⁵ also noted that a spatial error specification does not require a theoretical model for a spatial or social interaction process, but, instead, it’s a special case of a non-spherical error covariance matrix. In the empirical literature on strategic interaction among the outcome variables such as confirmed cases, reported deaths and discharged/recovered persons, the spatial error model is consistent with a situation where independent variables omitted from the model are spatially autocorrelated and with a situation where unobserved shocks follow a spatial pattern.

In both the spatial lag and the spatial error model, stationarity requires that 1/ω_min<δ<1/ω_max and 1/ω_min<ρ<1/ω_max, where, ω_min and ω_max denote the smallest (i.e., most negative) and largest characteristic roots of the matrix W. While it is often suggested in the literature to constrain δ or ρ to the interval (-1, +1), this may be unnecessarily restrictive. For row-normalized spatial weights, the largest characteristic root is indeed +1, but no general result holds for the smallest characteristic root and the lower bound is typically less than -1.

As an alternative to row-normalization, W might be normalized such that the elements of each column sum to one. This type of normalization is sometimes used in social economics literature⁵. Note that the row elements of a spatial weights matrix display the impact on a particular unit by all other units, while the column elements of a spatial weights matrix display the impact of a particular unit on all other units. Consequently, row normalization affects that the impact on each unit by all other units is equalized, while column normalization effects that the impact of each unit on all other units are equalized.

If W₀ denotes the spatial weights matrix before normalization, one may also divide the elements of W₀ by its largest characteristic root, ω_0,max to get W = (1/ω_0,max) W₀ or normalize W₀ by:

where, D is a diagonal matrix containing the row sums of the matrix W₀. The first operation may be labelled matrix normalization since it affects that the characteristic roots of W₀ are also divided by ω_0,max, as a result of which ω_max = 1, just like the largest characteristic root of a row- or column- normalized matrix. Croissant and Millo⁶. proposed the second operation which affects the characteristic roots of W, which are also identical to the characteristic roots of a row-normalized W₀.

Two main approaches have been suggested in the literature to estimate models that include spatial interaction effects. One is based on the maximum likelihood (ML) principle and the other on instrumental variables or generalized method of moments (IV/GMM) techniques⁷. Although IV/GMM estimators are different from ML estimators in that they do not rely on the assumption of normality of the errors, both estimators assume ω_max denotes the smallest (i.e., most negative) and largest characteristic roots of the matrix W. While it is often suggested in the literature to constrain δ or ρ to the interval (-1, +1), this may be unnecessarily restrictive. For row-normalized spatial weights, the largest characteristic root is indeed +1, but no general result holds for the smallest characteristic root and the lower bound is typically less than -1. Importantly, the mutual proportions between the elements of W remain unchanged as a result of these two alternative normalizations. This is an important property when W represents an inverse distance matrix, since scaling the rows or columns of an inverse distance matrix so that the sum of the weights to one would cause this matrix to lose its interpretation for this decay⁸.

RESULTS AND DISCUSSION

To achieve the stated objective of this study, Novel Coronavirus (COVID-19) cases data was extracted from the COVID-19 dashboard of the Center for Systems Science and Engineering at the John Hopkins University (CSSE, JHU)^9,10. This covered between 29 February to 12 May, 2020 for the 54 countries of Africa with confirmed cases of the Novel Coronavirus (COVID-19). This study analyzed the relationship between the rate of confirmed cases (Reconfirmed), the death rate (R-death) and the recovery rate (R-recovery) of COVID-19 in Africa with the spatial and temporal effects of the disease. The study calculated the rates by creating categories for each variable of the population by country. The population statistics for each country were extracted from the website of World meter as projected by the Elaboration of data by the United Nations, Department of Economic and Social Affairs, Population Division. These statistics are presented in Table 1.

Table 1 revealed that Djibouti has the highest concentration rate of confirmed cases in Africa (127.126 cases per 100,000 populations). This is followed by Sao Tome (94.908 cases per 100,000 population) and then Cabo Verde (48.023 cases per 100,000 population). The least rate of confirmed cases was observed in Mauritania, Angola and Burundi. These figures are as observed by the 12th of May, 2020.

Estimation of spatial panel models: The standard weight matrix (W) was used to characterize the spatial relationship among the variables. The dimension of the W matrix in this study is 5454 which is the number of African countries under consideration. This study also standardized the rows of the W matrix with zero diagonal factors which were conceptualized with the spatial relationships within the polygon rook contiguity.

This is presented formally using the equation:

The standard weight matrix was converted into an appropriate format for processing in Stata 15 that uses the command "xsmle" for the spatial panel regression model. The spatial panel data model was used to monitor the influence of the dependent variable on spatial autocorrelation and to analyze specifically the controlling variables and their temporal spillover impacts. The traditional linear panel data model was contrasted to the spatial panel data model since the spatial panel data model takes spatial factors such as spillover effects and spatial dependency into account.

Before the estimation of the spatial panel data models, there is the need to test for cross-sectional dependence which is the primary issue when confronted with spatially referenced data and to determine the existence of the spatial dependence. This means finding out whether nearby cases exhibit a stronger correlation than distant cases.

Figure 1 presents the visual representation of the spatial dependence observed in the spread of COVID-19 in Africa as of May 12th, 2020. This might be an indication of a degree of spatial autocorrelation between the rate of spread of the virus within the African geographical space.

Estimation of spatial models for COVID-19 in Africa: The application of the Pesaran¹¹ test for general cross-sectional dependence, Croissant and Millo⁶ is a versatile way of determining how dependence is linked spatially in the cross-section of a panel dataset.

The results from the analysis considering the standard linear model and the other six-panel data models considered is as summarized in Table 2. The parameters of the spatial panel data models were estimated with the quasi-maximum likelihood estimator according to Lee and Yu¹² and the p-values were calculated with the robust standard error. The models were estimated to include both the temporal time effects and the individual cross-sectional effects.

Table 3 summarizes the temporal time effects for each of the estimated spatial panel data models. The initial step of the analysis was to remove the spatial Durbin model SDM (1), spatial durbin error model SDEM (2), Spatial lagged model SLX (3) and spatial error model SEM (5) because these models are observed to lack spatial effect and tested to be statistically insignificant. Therefore, the study selected the most parsimonious model from spatial autoregressive model SAR (4) and spatial autocorrelation model SAC (6). The coefficients estimated for the spatially lagged variables (LM_recovery and LM_death) in the spatial autocorrelation model are observed to be statistically significant at a 0.05 significance level. Besides, the R² (0.9834), Likelihood Ratio Statistic (76.881), as well as the L-M test of common spatial terms statistic (9.394), are higher for the spatial autocorrelation model than for spatial autoregressive model SAR, also, the corrected Akaike information criterion (30.542) and the bayesian information criterion (29.052) computed for the spatial autocorrelation model is observed to be lowest among every other candidate models. Note that these statistics are computed for small samples. The test of significance on LM_recovery and LM_death for the selected model are statistically significant at a 0.05 level of significance. Therefore, the spatial effects of the explanatory variables LM_recovery and LM_death are significantly different from zero. The Hausman test statistic (17.279) computed for the spatial autocorrelation model is observed to be more consistent for the fixed effect model than for the random effect model as p<0.01. Hence, the spatial autocorrelation model can be considered to be the most parsimonious spatial panel regression model for the spread of COVID-19 in Africa. Therefore, this study will interpret the influencing factors using the results obtained from the estimation of the spatial autocorrelation model in subsequent analysis.

This implies that the rate of confirmed COVID-19 cases for countries in Africa is spatially autocorrelated which is an indication that the spatial autocorrelation model provides an appropriate representation of COVID-19 spread in Africa and it will be employed to estimate the spatial effect of COVID-19 in Africa. Since the objective of this study is to explore the factors influencing the rate of confirmed cases and examine their spatial spillover effects.

Based on the spatial panel data model estimated for the 54 countries in Africa with confirmed cases of COVID-19 as of 12th May, 2020, this study estimated the spatial effect of COVID-19 in Africa by exploring the factors influencing the rate of confirmed cases and examining the spatial spillover effects of COVID 19 in within the African continent. Before the estimation of the model, the cross-sectional dependence of the data was examined using the Pesaran test which revealed that there exists cross-sectional dependence within the units. The maximum pseudo-R², LR-test, LM-test statistics and minimum AICc and BICc values were used to determine the most parsimonious spatial panel data regression models and to select the most efficient and consistent model which spatial effects of COVID-19 in Africa and it was observed that the spatial autocorrelation model presents an appropriate representation of the data based on the criteria. The selected model was therefore, considered using the dependent and independent variables separately. From the Spatial Autocorrelation model, this study examined the variables separately by splitting the effects of the independent variables into the total, indirect (spatial spillover effects) and direct effects to enhance the identification of the actual impacts and spatial interactions of the factor components on COVID-19 in Africa. We can, therefore, conclude from the total effect that the death rate from COVID-19 in Africa has a significant positive effect on the spread of the virus within the continent and the recovery rate harms the spread of the virus.

As observed from the results, the average direct effect when compared with the average indirect effect can be said to have reflected the actual effects of the influencing factors more comprehensively. The indirect effect for the recovery rate was computed to be equal to 1.073 (p<0.001) which implies that every 1% increase in the death rate in any of the African countries with reported cases will bring about a 1.073% increase in the rate of confirmed cases in other neighbouring African countries. Also, the indirect effect of the rate of recovery was computed to be equal to -2.398 (p<0.001) which is significant at a 5% level of significance.

Table 4 summarizes the temporal effects of the spatial autocorrelation model and these depict that the rate of spread of COVID-19 in the early period of the pandemic (January) experienced a slight increase across Africa which is not statistically significant. However, the forecast from the Spatial Autocorrelation model depicts a surg from the last week in February from where significant increases were observed in the rate of confirmed cases. Therefore, we can conclude that an increase of 0.1527 per 100,000 people is expected in the coming weeks if the pattern of spread remains constant. Also, considering the direct effect, the rate of death and recovery from COVID-19 in Africa has a significant positive effect on the spread of the virus within the continent.

This implies that a 1% increase in the death rate in any of the African countries with reported cases will bring about a 3.3% increase in the rate of confirmed cases in other neighbouring African countries while the recovered cases have a significant negative effect on the spread of the virus within the continent. This implies that a 1% increase in the death rate in any of the African countries with reported cases will bring about a 14% decrease in the rate of confirmed cases. Lastly, from the indirect effects, the rate of death was observed to maintain significant positive effects on COVID-19 spread in the neighbouring African countries and the rate of recovery has significant negative effects.

As a result of the temporal effects as observed from the analysis, we observed a daily increase in the rate of confirmed cases, Examining the number of confirmed cases on the 29th of February, 2020 (Study start period) and the 12th of May, 2020 (Study end period), This study observed that the rate of confirmed cases has increased from 0.09 cases per 100,000 population to 94 cases per 100,000 population. There is a need to address some limitations while discussing the results of this study. It is impossible to generalize the model for the death rate from COVID-19 in Africa due to the presence of a large difference in the number of deaths across the countries. Also, the time frame under consideration appears to be short considering the pattern of the period it takes a patient to recover from the virus, therefore, future studies can consider using data with a longer period.

CONCLUSION

This study considered the spatial effect of COVID-19 in Africa using the spatial panel data model approach and it has been able to provide information about the effect of the spread of COVID-19 across the African continent. It can be observed from the results that an increase of 0.1527 per 100,000 people is expected in the coming weeks if the pattern of spread remains constant. And also, the rate of death and recovery from COVID-19 in Africa has a significant positive effect on the spread of the virus within the continent.

SIGNIFICANCE STATEMENT

This study addressed the temporal and spatial effects of the spread of COVID-19 in Africa and discovered that the rate of death and recovery from COVID-19 in Africa has a significant positive effect on the spread of the virus within the continent. These findings will help future researchers to uncover critical areas of spatial panel data models and their application to published data that many researchers were not able to explore. Thus, a new theory on the rate of spread and effect of COVID-19 in Africa can be established.

REFERENCES