INTRODUCTION
Coronavirus disease (COVID19) caused by a newly discovered novel coronavirus is an infectious disease^{1}. Mild to moderate symptoms and recovery without special treatment is what most people who fall sick with COVID19 experience^{1}. The COVID19 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 surfaces^{2}. 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 COVID19 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 timeseries observations of several spatial units^{2}. 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 crosssectional 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 crosssectional data^{3}. With these in mind, this paper applied the spatial panel data models to determine the rate of spread of COVID19 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 COVID19 data was extracted from the COVID19 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 (COVID19).
Spatial panel models: Consider a simple pooled linear regression model with spatial specific effects but without spatial interaction effects^{4}:
y_{it} = x_{it}β+μ_{i}+ε_{it} 
(1) 
where, i is an index for the crosssectional 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 σ^{2}, while μ_{i} denotes a spatial specific effect. The standard reasoning behind spatial specific effects is that they control for all spacespecific timeinvariant variables whose omission could bias the estimates in a typical crosssectional 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 prespecified nonnegative 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:
y_{it} =x_{it}β+μ_{i}+ϕ_{it} 
(3) 
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 nonspherical 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 rownormalized 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 rownormalization, W might be normalized such that the elements of each column sum to one. This type of normalization is sometimes used in social economics literature^{5}. ^{}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 W_{0} denotes the spatial weights matrix before normalization, one may also divide the elements of W_{0} by its largest characteristic root, ω_{0,max} to get W = (1/ω_{0,max}) W_{0} or normalize W_{0} by:
W = D^{–}^{1/2}W_{0}D^{–}^{1/2} 
where, D is a diagonal matrix containing the row sums of the matrix W_{0}. The first operation may be labelled matrix normalization since it affects that the characteristic roots of W_{0} 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 Millo^{6}. proposed the second operation which affects the characteristic roots of W, which are also identical to the characteristic roots of a rownormalized W_{0}.
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) techniques^{7}. 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 rownormalized 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 decay^{8}.
RESULTS AND DISCUSSION
To achieve the stated objective of this study, Novel Coronavirus (COVID19) cases data was extracted from the COVID19 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 (COVID19). This study analyzed the relationship between the rate of confirmed cases (Reconfirmed), the death rate (Rdeath) and the recovery rate (Rrecovery) of COVID19 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.
Table 1: 
COVID19 descriptive statistics in Africa as at 12th of May, 2020 
Country/Region 
Population 
Total confirmed cases 
Deaths 
Recovered 
R_{confirmed} 
R_{death} 
R_{recovery} 
Algeria 
43851044 
6067 
515 
2998 
13.83547 
8488.545 
49414.87 
Angola 
32866272 
45 
2 
13 
0.136918 
4444.444 
28888.89 
Benin 
12123200 
327 
2 
76 
2.697308 
611.6208 
23241.59 
Burkina Faso 
20903273 
766 
51 
588 
3.664498 
6657.963 
76762.4 
Cabo Verde 
555987 
267 
2 
58 
48.02271 
749.0637 
21722.85 
Cameroon 
26545863 
2689 
125 
1524 
10.12964 
4648.568 
56675.34 
Central African Republic 
4829767 
143 
0 
10 
2.960805 
0 
6993.007 
Chad 
16425864 
357 
40 
76 
2.173402 
11204.48 
21288.52 
Congo (Brazzaville) 
89561403 
333 
11 
53 
0.371812 
3303.303 
15915.92 
Congo (Kinshasa) 
5518087 
1102 
44 
146 
19.97069 
3992.74 
13248.64 
Cote d'Ivoire 
26378274 
1857 
21 
820 
7.039884 
1130.856 
44157.24 
Djibouti 
988000 
1256 
3 
886 
127.1255 
238.8535 
70541.4 
Egypt 
102334404 
10093 
544 
2326 
9.862763 
5389.874 
23045.68 
Equatorial Guinea 
1402985 
439 
4 
13 
31.29043 
911.1617 
2961.276 
Eritrea 
3546421 
39 
0 
38 
1.0997 
0 
97435.9 
Eswatini 
1160164 
184 
2 
28 
15.85983 
1086.957 
15217.39 
Ethiopia 
114963588 
261 
5 
106 
0.227028 
1915.709 
40613.03 
Gabon 
2225734 
863 
9 
137 
38.77373 
1042.874 
15874.86 
Gambia 
2416668 
22 
1 
10 
0.910344 
4545.455 
45454.55 
Ghana 
31072940 
5127 
22 
494 
16.49989 
429.1008 
9635.264 
Guinea 
13132795 
2298 
11 
816 
17.49818 
478.6771 
35509.14 
Kenya 
53771296 
715 
36 
26 
1.329706 
5034.965 
3636.364 
Liberia 
5057681 
211 
20 
259 
4.171872 
9478.673 
122748.8 
Madagascar 
27691018 
186 
0 
85 
0.671698 
0 
45698.92 
Mauritania 
4649658 
9 
1 
28 
0.193563 
11111.11 
311111.1 
Mauritius 
1271768 
332 
10 
101 
26.10539 
3012.048 
30421.69 
Morocco 
36910560 
6418 
188 
398 
17.38798 
2929.261 
6201.309 
Namibia 
2540905 
16 
0 
6 
0.629697 
0 
37500 
Niger 
24206644 
854 
47 
322 
3.527957 
5503.513 
37704.92 
Nigeria 
206139589 
4787 
158 
2991 
2.322213 
3300.606 
62481.72 
Rwanda 
12952218 
286 
0 
34 
2.208116 
0 
11888.11 
Senegal 
16743927 
1995 
19 
11 
11.91477 
952.381 
551.3784 
Seychelles 
98347 
11 
0 
648 
11.18489 
0 
5890909 
Somalia 
15893222 
1170 
52 
959 
7.361629 
4444.444 
81965.81 
South Africa 
59308690 
11350 
206 
153 
19.13716 
1814.978 
1348.018 
Sudan 
43849260 
1661 
80 
742 
3.787977 
4816.376 
44671.88 
Tanzania 
59734218 
509 
21 
10 
0.852108 
4125.737 
1964.637 
Togo 
8278724 
199 
11 
126 
2.403752 
5527.638 
63316.58 
Tunisia 
11818619 
1032 
45 
4357 
8.731985 
4360.465 
422189.9 
Uganda 
45714007 
129 
0 
173 
0.282189 
0 
134108.5 
Zambia 
18383955 
441 
7 
183 
2.398831 
1587.302 
41496.6 
Zimbabwe 
14862924 
36 
4 
92 
0.242213 
11111.11 
255555.6 
Mozambique 
31255435 
104 
0 
740 
0.332742 
0 
711538.5 
Libya 
6871292 
64 
3 
55 
0.931411 
4687.5 
85937.5 
GuineaBissau 
1968001 
820 
3 
117 
41.66665 
365.8537 
14268.29 
Mali 
20250833 
730 
40 
9 
3.60479 
5479.452 
1232.877 
Botswana 
2351627 
24 
1 
17 
1.02057 
4166.667 
70833.33 
Burundi 
11890784 
15 
1 
7 
0.126148 
6666.667 
46666.67 
Sierra Leone 
7976983 
338 
19 
72 
4.237191 
5621.302 
21301.78 
Malawi 
19129952 
57 
3 
24 
0.297962 
5263.158 
42105.26 
South Sudan 
11193725 
194 
0 
2 
1.733114 
0 
1030.928 
Western Sahara 
597339 
6 
0 
6 
1.004455 
0 
100000 
Sao Tome and Principe 
219159 
208 
5 
4 
94.90826 
2403.846 
1923.077 
Comoros 
869601 
11 
1 
0 
1.264948 
9090.909 
0 
The rates are multiplied by 100,000 
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.
Fig. 1: 
Spread of COVID19 in Africa as of May 12th, 2020 
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 crosssectional 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 COVID19 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 COVID19 in Africa: The application of the Pesaran^{11} test for general crosssectional dependence, Croissant and Millo^{6 }is a versatile way of determining how dependence is linked spatially in the crosssection of a panel dataset.
The results from the analysis considering the standard linear model and the other sixpanel data models considered is as summarized in Table 2. The parameters of the spatial panel data models were estimated with the quasimaximum likelihood estimator according to Lee and Yu^{12} and the pvalues were calculated with the robust standard error. The models were estimated to include both the temporal time effects and the individual crosssectional 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 R^{2} (0.9834), Likelihood Ratio Statistic (76.881), as well as the LM 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 COVID19 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.
Table 2: 
Spatial Panel Models for COVID19 in Africa 

Spatial panel models 
Variables 
SLM 
SDM (1) 
SDEM (2) 
SLX (3) 
SAR (4) 
SEM (5) 
SAC (6) 
R_{recovery} 
0.7802 
0.7935*** 
0.7935*** 
0.7935*** 
0.7928*** 
0.7832*** 
0.7935*** 
R_{death} 
28.9284 
28.6132*** 
28.6080*** 
28.6618*** 
28.6223*** 
28.6241*** 
28.6301*** 
Cons 
0.0001 
0.034** 
0.044** 
0.044** 
0.041** 
0.037** 
0.044** 
ρ 

0.036 

0.006*** 

0.006*** 
π 

0.051 
0.158** 
lag.recovery 
0.143 
0.011 
0.361 
0.239** 

0.0935 
Lag.deaths 

1.1341 
1.2741* 
2.8341** 

1.3046* 
SLM: Standard linear regression model, SDM: Spatial durbin model, SDEM: Spatial durbin error model, SLX: Spatial lagged x model, SAR: Spatial autoregressive model, SEM: Spatial error model, SAC: Spatial autocorrelation model, *p<0.10, **p<0.05 and ***p<0.01 
Table 3: 
Model statistics 
Temporal effects 
SLM 
SDM (1) 
SDEM (2) 
SLX (3) 
SAR (4) 
SEM (5) 
SAC (6) 
Fstat/LR stat 
71.179** 
63.445** 
75.117** 
69.362** 
73.693** 
71.514** 
76.851** 
R^{2}/Pseudo R^{2} 
0.9663 
0.9347 
0.9182 
0.9505 
0.9786 
0.9744 
0.9834 
LM test of common spatial terms 
9.339 
0.446 
9.381 
9.39 
9.38 
9.382 
9.394 
AIC_{c} 
32.571 
27.652 
28.553 
29.656 
31.351 
32.459 
30.542 
BIC_{c} 
30.865 
30.879 
31.549 
30.157 
29.951 
30.755 
29.052 
PesaranCD test stat: 17.279 prob<0.01, SAC model Hausman Test chi (23): 24.795 (prob<0.001), SAC Model LMr test chi (1): 7.512 (prob<0.050), SAC Model LMd chi (1): 9.045 (prob<0.001) and **p<0.05 
This implies that the rate of confirmed COVID19 cases for countries in Africa is spatially autocorrelated which is an indication that the spatial autocorrelation model provides an appropriate representation of COVID19 spread in Africa and it will be employed to estimate the spatial effect of COVID19 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 COVID19 as of 12th May, 2020, this study estimated the spatial effect of COVID19 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 crosssectional dependence of the data was examined using the Pesaran test which revealed that there exists crosssectional dependence within the units. The maximum pseudoR^{2}, LRtest, LMtest 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 COVID19 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 COVID19 in Africa. We can, therefore, conclude from the total effect that the death rate from COVID19 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 COVID19 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 COVID19 in Africa has a significant positive effect on the spread of the virus within the continent.
Table 4: 
Spatial effect of the independent variables on the spatial autocorrelation model 

Confidence interval 
Variables 
dy/dx 
Deltamethod Std. Err 
Probability 
Lower 
Upper 
Direct spatial effect 
R_{recovery} 
14.017 
0.281 
<0.001 
13.736 
14.298 
R_{death} 
3.375 
0.093 
<0.001 
3.282 
3.468 
Indirect spatial effects 
R_{recovery} 
2.398 
0.437 
<0.001 
1.961 
2.835 
R_{death} 
1.073 
0.084 
<0.001 
0.989 
1.157 
Total spatial effects 
R_{recovery} 
16.415 
0.718 
<0.001 
15.697 
17.133 
R_{death} 
4.448 
0.177 
<0.001 
4.271 
4.625 
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 COVID19 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 COVID19 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 COVID19 in Africa using the spatial panel data model approach and it has been able to provide information about the effect of the spread of COVID19 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 COVID19 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 COVID19 in Africa and discovered that the rate of death and recovery from COVID19 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 COVID19 in Africa can be established.