Analysis and contrast of Chadian-Senegalese Covid-19 outbreaks

Abstract

As COVID-19 is an emerging pandemic, analysing its evolution is necessary to understand it in order to find appropriate answers. In this paper, we aim to observe and analyse it at the Chadian-Senegalese level. Thus, we collect public data in order to present via curves, histograms and tables the main characteristics of this pandemic. In this way, we implement a python program to construct these. We focus only on extracting long-term data without predictive models. We observed that there are mainly two waves (outbreak) per year with stable or even decreasing infection and death rates. We also identified moments of growth and relaxation of the disease. These results can be used to identify times when treatment or prevention should be intensified.

Keywords:

• Outbreak
• Covid-19
• data compilation
• fitting curve
• Fourier analysis
• periodic signal

Maths:

• 33-00
• 62-00
• 65-04

1. Introduction

The COVID-19 pandemic that swept through Africa in early 2020 left no country untouched. Being a new disease, each state has had to adopt a response strategy. The common point of these strategies is the daily communication of data via ministries in charge of health. Despite they are not organized in a standard way, we note among other things, the number of samples tested, positive cases, cured, deaths, vaccinated, imported, etc. The number of infected people, the rates of infection, recovery, etc. can be deduced from this. These data require appropriate analysis and observations over a period of years in order to understand the main characteristics of this pandemic: the number of waves per year, the peak and lull times, the basic reproduction rate etc.

One can quote the site of the ministries, in particular that of Senegal [1] which has even undertaken the illustration of the spatial data. There are also interactive sites providing information on the global situation [2–4]. In addition to well-illustrated baseline data, there is a lack of thorough analysis of the underlying data.

In contrast, Mathematical models have required little attention due to lack of knowledge around the disease. However, after more than two years of pandemic, several studies suggesting mathematical models appeared. Most of them are variants of the SEIR model [5–10]. It is worth noting that these SEIR-type models are comparable to research conducted by A. Jajarmi and al. on outbreaks such as dengue fever or cholera [11,12]. Some of these approach uses fractional models [13,14], in this perspective, M. Aychluch and al. shows that the good way to limit virus transmission is by increasing test of population [15]. Other authors, such as Zenebe and al. have used the optimal control approach to address this topic, they pointed out that in comparison to the absence of control cases, the suggested controls have a considerable impact on disease burden [16,17].

In short, these approaches have in common the forecasting of reported and active cases, some also estimate the basic reproduction rate $R_0$. Although the quality of their mathematical results is well presented, there is little investigation into the validation of these models with real data.

To address the issues raised above, we propose to observe and analyse the cases of Chad and Senegal for at least two years [18].

First, we designed a Python program containing the modules ‘matplotlib.pyplot’, ‘numpy’, ‘datetime’. The aim of this algorithm is to collect, store, and display graphs of variables and parameters of the pandemic [19–21].

Our main contribution is the creation of graphics that depict basic and underlying data. As a result, the following conclusions have been reached: The Covid-19 pandemic is a seasonal disease with two waves per year, with peak periods in Senegal around February and August; the infection rate ranges between 4% and 8%, with a death rate of 2% to 3%.

2. Methods & materials

We collected data from health authorities' daily public reports and stored it in a database in ‘.csv’ format (the first and last lines of which are shown in Table 1). After that, we analysed them to capture outbreak features. In Senegal, these cases derive from samples of people tested (see Table 2): Infected, contact (who comes into contact with an infected), untraceable, imported, active (in medical treatment), recovered, severe, and deceased. The number of vaccinated is added one year later. For Chadian side, we have the same data as in Senegal except cases such as severe, untraceable or vaccinated are omitted. In addition, beginning in mid-October 2021, Chadian authorities will no longer report data on a daily basis. They switched to monthly and then weekly reporting. From there, we have chosen to present the monthly evolution of the situation by compiling weekly data.

Table 1. Representation of brute data from Senegalese case.

Num	Dates	Samples	Infected	Imported	Contact	Untraceables	Recovred	Severe	Death	Vaccinated
0	2020-03-05	3	2	0	0	0	0	0	0	0
1	2020-03-06	3	1	2	0	0	0	0	0	0
2	2020-03-07	2	1	0	0	0	0	0	0	0
···	···	···	···	···	···	···	···	···	···	···
893	2022-08-15	850	25	1	0	24	22	1	0	330
894	2022-08-16	553	8	1	0	7	28	0	0	442
895	2022-08-17	835	22	2	0	20	16	0	0	418

Table 2. Description of daily variables with colour coding used on figures.

Name (color)	Notation	Value	Description
Samples (Blue)	$s\left(t\right)$	data	Number of tested people
Infected (Orange)	$i\left(t\right)$	···	Positif result of test
Recovred (Green)	$r_v\left(t\right)$	···	Healed after medical treatment
Contacts (Dark purple)	$c\left(t\right)$	···	Contact with a known infected person
Untraceables (Black)	$u\left(t\right)$	···	Unknown origin of infection
Imported (Olive)	$i_m\left(t\right)$	···	Infected outside the country
Severe (Red-Dashed)	$s_v\left(t\right)$	···	Patients in severe cases
Death (Magenta)	$d\left(t\right)$	···	Death from infection
Vaccinated (Black-Dashed)	$v\left(t\right)$	···	Received at least one dose of vaccine

Using these primary sources of data, we calculated the number of active cases per day, as well as the infection and cure rates. Table 3 gives notations and formulas for these variables. In general, for $x\left(t\right)$ representing some data (sample, infected, etc.), we denoted $X\left(t\right)=\sum _{\delta t=1}^{t}x(t-\delta t)$ it's cumulative value and for scaling purposes, we may normalize $x\left(t\right)$ in two manner : we called the first maximum normalization the value $\tilde{x}\left(t\right)=\frac{x\left(t\right)}{{\displaystyle \underset{0\le \delta t\le t}{max}\left\{x\right(t-\delta t\left)\right\}}}$ ; this gives time series varying between 0 and 1. The second named centered normalization, the value $\overline{x}\left(t\right)=\frac{x\left(t\right)-m_x}{{\sigma }_x}$ where $m_x$, ${\sigma }_x$ denote mean and standard deviation of $x(t-\delta t)$ for $0\le \delta t\le t$.

Table 3. Description of variables shown on Figure 1.

Name (color)	Notation	Value	Description
Active (Red)	$a\left(t\right)$	$I\left(t\right)-R_v\left(t\right)-D\left(t\right)$	Total number of patients
Infection Rate	${\tau }_i\left(t\right)$	${\tau }_i\left(t\right)={\displaystyle \frac{\tilde{i}\left(t\right)}{\tilde{s}\left(t\right)}}$	Number of infected in samples
Recovery Rate	${\tau }_{r_v}\left(t\right)$	${\tau }_{r_v}\left(t\right)={\displaystyle \frac{{\tilde{r}}_v\left(t\right)}{\tilde{i}\left(t\right)}}$	Number of recovred in infected
Susseptibles(Blue)	$s_p\left(t\right)$	${\displaystyle \frac{s\left(t\right)}{\tilde{s}\left(t_{\mathrm{\infty }}\right)}}$	Daily sample in total samples
Infectious ( Orange )	$i\left(t\right)$	Daily positif cases
Removed (Green)	$r\left(t\right)$	$r_v\left(t\right)+d\left(t\right)$	Daily number of recovered plus deceased

Having these brute facts, we built a Python framework to analyse them. It contains graphical and mathematical methods like interpolation and distribution. Then, to smooth curves representing primary data, we made interpolation via the fast Fourier transform (the ‘fft’ method of the ‘numpy’ module). Thus, for $x\left(t\right)$ given, we note $\hat{x}\left(\xi \right)$ its Fourier transform but before taking its inverse, we calculated the power spectral density (PSD) according to the formula: (1) $PSD=20\times {\log }_{10}\left(\frac{|\hat{x}\left(\xi \right)|}{〈\hat{x}\left(\xi \right)〉}\right),$(1) where $|\hat{x}\left(\xi \right)|$ is the modulus and $〈\hat{x}\left(\xi \right)〉$ is the average of modules taken as a reference module. The transform $\hat{x}\left(\xi \right)$ was then filtered by keeping only indices whose PSD was greater than an amplitude δ threshold.

Figure 1. Senegal's global situation (A) The daily evolution of maximum normalized data: active, infected, etc. (C) Total number with proportions. (D) Cumulated number in Y-logarithmic base 10 scale. (E) Histogram of infected cases and power-law fitting (dashed-green).

In addition, to observe the dynamics of the pandemic by season, we grouped averages of data specific to each month. We also calculated the duration of each outbreak as well as their lengths, which are the areas of the equivalent curves (see Table 4).

Table 4. The features of the Senegalese and Chadian outbreaks; lag time is the amount of time between waves.

Senegal
Outbreak	Duration (lag time)	Length (%)	Min${}_l$	Peak (Date)	Min${}_r$
$1\mathrm{s}\mathrm{t}$ outbreak from 20/03/05 to 21/11/11	8 months 6 days	28.87 (27.08)	0	4324 ( 20/08/19 )	21
$2\mathrm{n}\mathrm{d}$ outbreak from 20/11/11 to 21/04/27	6 months 2 weeks	21.72 (20.37)	21	4983 (21/02/21)	132
$3\mathrm{r}\mathrm{d}$ outbreak from 21/04/27 to 21/11/20	7 months	47.35 (44.41)	132	15562 ( 08/08/21)	5
$4\mathrm{t}\mathrm{h}$ outbreak from 21/11/20 to 2022-04-24	5 months	8.67 (8.13)	5	5187 ( 22/01/16)	23
Chad
$1\mathrm{s}\mathrm{t}$ outbreak from 20/03/19 to 20/07/12	4 months	12.79 (14.76%)	0	401 ( 20/05/25 )	10
$2\mathrm{n}\mathrm{d}$ ‘mini outbreak’ from 20/07/12 to 20/12/12	5 months	10.53 (12.15%)	10	138 (20/10/15)	47
$3\mathrm{r}\mathrm{d}$ outbreak from 20/12/12 to 21/06/13	6 months	63.26 (73%)	47	830 ( 21/01/29)	2

Note: The numbers of infected at the start and end of an outbreak are denoted by Min${}_l$ and Min${}_r$.

3. Results & discussion

Figure 3 displays the global scenario in Senegal. In part (A), we represented the instantaneous evolution of active cases' number (red), infected (dashed-orange), severe (dashed-red), infection, and recovery rate (orange and dashed-green) (see Table 3). The x-axis represents time in days, with the initial time $t_0=0,\left(2020/03/05\right)$ representing the beginning of the pandemic and the final time $t_{\mathrm{\infty }}$ representing the present day. The y-axis represents the number of people; it varies between 0 and 1 because of maximum normalization, but the preponderant numbers are explicitly marked (for example, 4324 is the first peak reached on 2020/08/19, which lasted 5 months and 2 weeks). Note that the evolution of active cases for the last 7 days is illustrated in part(B). After that,we pictured totals at time $t_{\mathrm{\infty }}$ with their respective percentages (part(C)); the proportion of infected cases is taken compared to samples, that of other cases compared to infected and that of samples and vaccinated¹ are compared to the population (estimated at 16,70 million people). We employed a logarithmic scale of base 10 and observed a logistic progression of samples (blue), infected (orange), recovered (green), untraceable (black), etc. The x−y axes are equally important as in part (A). The last part (E) represents the histogram of infected cases (orange). We used the function ‘hist’ of the Python library ‘maplotlib’ with 45 bins and the option ‘density=true '. As a result, we were able to calculate the probability density (dashed-blue curve). The dashed-green is the curve of power law $y=\alpha x^k$ where $\alpha =1,k=-1.23$.

In parallel, the global situation in Chad is illustrated in Figure 2; as in the Senegalese case, part (H) represents the instantaneous evolution of the number of active, infected, and infection-recovery rate. The x-axis represents time in days where the initial time $t_0=0\left(2020/03/19\right)$ corresponds to start of pandemic (14 days after the beginning of pandemic in Senegal) and the final time $t_{\mathrm{\infty }}$. The population of Chad is estimated at 15.95 million people, the rest of the details are similar to those in Senegal's case except that data about severe, contact, untraceable, imported, and vaccinated cases are missing. We can see the totals and their proportions in part (K), the cumulative density in part (I), and the infected histogram in part (J), where the dashed-green is the power law curve $y=\alpha x^k$ where $\alpha =1$ and k = −1.86. Since the Chadian authorities stopped reporting daily data in mid-October 2021, we decided to present the data on a monthly basis. Thus, Figure 3 is the monthly equivalent of Figure 2.

Figure 2. Chadian global situation. Description as shown in Figure 1.

Figure 3. Tchad-Covid-19 monthly outbreak. Description as shown in Figure 1.

Although reading the data in Figures 1–3 shows us a global view of the pandemic, the curves representing them are not smooth. Therefore, the result of interpolation via the fast Fourier transform is shown in part (L) of Figure 4, where solid curves represent filtered densities and dotted curves are the original data; the values ${\delta }_A,{\delta }_S,{\delta }_I,{\delta }_R$ are the amplitude thresholds for active, susceptible, infected, and removed. Part (M) demonstrated the PSD of susceptible and infected individuals as a function of frequency; this is the foundation for calculating threshold amplitudes. Thus, we filtered the data to easily observe, for each outbreak, the evolution of removed versus infected, which shows the cyclical nature of the pandemic (part (N)). While the Fourier transformation can be used to smooth curves and refine analyses, it requires regular sampling, as is the case in Senegal, unlike Chad. This is why we could not do it for the latter.

Figure 4. Fourier analysis of the Senegalese case.

Figure 5. Chad-Senegal monthly mean case.

Nevertheless, by noting the natural regularity of active curves in the two countries, we found some characteristics of outbreaks: duration, size, peaks, and minima (Table 4). Thus, Senegal has four waves that last an average of six months, whereas Chad has three waves that last an average of five months. The area of the active curve is the size of a wave, and its percentage is calculated from the overall area. Finally, we grouped by month the averages of active, infected, and removed cases. This allowed us to observe that in Senegal, July and December are the months when the epidemic is growing, whereas April and October are relaxation months. However, due to a lack of data from the Chadian side, we are unable to reach such a judgment.

The above-mentioned results show that the more data there are, the more precise the observation of the characteristics of the pandemic will be.

From this perspective, the contrast between the Chadian and the Senegalese cases lies in the sampling rate. To date, 7% of the Senegalese population has been tested, compared to less than 1% in Chad. Despite this, there is an apparent similarity between the cumulative data following a logistic distribution and the probability of daily infections following a power law. We also note the regression or even extinction of the pandemic since the beginning of 2022.This might be the outcome of barrier measures, a vaccination program, collective natural immunity, or other factors (Figures 1 and 2).

4. Conclusion

In essence, we present data on the COVID-19 pandemic in Senegal and Chad from its beginning to the present day (about 2 years). The major indication is the number of active persons (patients at time t); the evolution curve of this indicator demonstrates that there are two nearly equal-sized waves (outbreaks) every year. For Senegal, the average of peaks and minima is 7514 and 43, respectively, while in Chad, we have 457 and 19. In both countries, disease characteristics are similar; for example, infection and recovery rates are nearly constant, and the distribution of infected people is identical (approximating the power law). When we compare the amount of persons examined (1% in Chad against 7% in Senegal), we see a paucity of data on the Chadian side, which explains the difference in the regularity of the curves, the best of which are on the Senegalese side.

Because we only covered the issue through data collecting and graphical analysis, extra work may be done by modelling these processes. Existing models can be validated as well.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Vaccination started at the beginning of March 2021; it's one year after the first case.