Differential Equations and Applications to COVID-19

Abstract

This paper focuses on the application of the Verhulst logistic equation to model in retrospect the total COVID-19 cases in Senegal during the period from April 2022 to April 2023. Our predictions for April 2023 are compared with the real COVID-19 data for April 2023 to assess the accuracy of the model. The data analysis is conducted using Python programming language, which allows for efficient data processing and prediction generation.

1. Introduction

The COVID-19 outbreak, caused by the SARS-CoV-2 virus, was initially identified in Wuhan, China, in December 2019 [1]. The virus rapidly spread across multiple countries, leading to a significant number of global fatalities. The outbreak was declared a pandemic on 2 March 2020 [2]. As the pandemic is growing, there have been concerns about finding treatments and vaccines, and despite the lockdowns, many researchers develop algorithms to predict new cases. Some researchers use well-known mathematical methods, such as a mathematical logistic growth model [3,4].

Senegal reported its first case of COVID-19 on 1 March 2020, and its COVID-19 data are known from the early stage of the pandemic to nowadays. We are interested in studying the case of Senegal over one year in a retrospective way, from April 2022 to April 2023. The Verhulst logistic function is among the most popular functions to describe a growth phenomenon that is applied to any growth phenomenon. The strength of the model is its capability to estimate the carrying capacity or the maximum level of the growth. We fit the solution of the Verhulst logistic equation to the data from April 2022 to April 2023 using Python 3 version 5.3.0 Since the data are furnished for April 2023, we compare the predictions and the real data.

Our work is organized as follows: Section 2 is about COVID-19. In Section 3, we present the Verhulst logistic equation. Section 4 is about the data analysis. Through this analysis, we seek to uncover the trends and patterns of COVID-19 cases over a specific time frame, specifically from April 2022 to April 2023. The conclusion is given in Section 5.

2. COVID-19

COVID-19, caused by the severe acute respiratory syndrome coronavirus-2 (SARS-CoV-2) [5], emerged unexpectedly in 2019, profoundly impacting human life [6]. The initial outbreak occurred in Wuhan City, China [7], and rapidly spread worldwide, including the United States, Europe, Asia, and other continents. On 30 January 2020, the World Health Organization (WHO) declared it a global public health emergency, and on 2 March 2020, it was declared a pandemic. COVID-19 is the most disruptive infectious disease the world has experienced, ranking second in documented cases after human immunodeficiency virus (HIV). Africa’s response to COVID-19 has exhibited a distinct pattern compared to other regions, with studies indicating a slower rate of exponential growth in the disease burden [8].

As of November 2022, Africa accounted for a relatively small proportion of global COVID-19 cases and fatalities [9]. To mitigate the spread of the virus, African nations have implemented various non-pharmaceutical interventions (NPIs) such as social distancing, mask-wearing, and testing [10]. These measures have played a significant role in reducing the impact of the pandemic, although the emergence of new SARS-CoV-2 variants has presented challenges [11]. Vaccination has become a critical strategy in the battle against the virus and its variants [12].

On 1 March 2020, Senegal confirmed its first case of COVID-19 [13]. In response, the country implemented a range of measures to curb the virus, including travel restrictions, lockdowns, hygiene promotion, and widespread testing. Senegal initiated its vaccination campaign in February 2021, initially relying on the COVAX facility for vaccine supply. The AstraZeneca vaccine was prioritized for healthcare workers, elderly individuals, and those with underlying health conditions. Bilateral agreements and donations also provided additional doses, including the Sinopharm vaccine from China [14]. The vaccination efforts aimed to cover the entire population, particularly focusing on high-risk and vulnerable groups.

Various preventive measures were implemented by governments and stakeholders, including lockdowns, restrictions on gatherings, hygiene promotion, social distancing, and mask mandates in public places. Efforts were made to equip treatment facilities, recruit healthcare professionals, and incentivize frontline workers [15]. Despite these measures, the virus continues to spread, with surges observed in Senegal and other countries worldwide. As of 24 February 2021, Africa had reported 3,872,085 cases and 102,286 deaths [16]. Ongoing statistics indicate a clear upward trend in cases, highlighting the persistent challenges posed by the virus.

COVID-19 stands for “Coronavirus Disease 2019”. Let us break down the meaning of each component.

• “CO” denotes “corona”, representing the family of viruses that includes SARS-CoV-2, the virus responsible for COVID-19. Coronaviruses encompass a wide range of viruses, causing various illnesses, from common colds to severe diseases.
• “VI ” stands for “virus”, a tiny infectious agent that can only reproduce within the cells of a living organism. In the case of COVID-19, the virus is known as SARS-CoV-2.
• “D” represents “disease”, referring to the illness caused by the SARS-CoV-2 virus. COVID- 19 specifically affects the respiratory system, leading to a spectrum of symptoms, ranging from mild to severe.
• “19” signifies the year 2019 when the disease was first identified. The initial cases of COVID-19 were reported in Wuhan, Hubei Province, China, in December 2019.

Hence, COVID-19 represents the particular illness triggered by the SARS-CoV-2 virus, originating in 2019 and subsequently disseminating worldwide as a pandemic (Figure 1).

Typical symptoms of COVID-19, which can manifest within 2–14 days after exposure to the SARS-CoV-2 virus, range from mild to severe and commonly include:

• Fever: An elevated body temperature above 38 °C (100.4 °F) is a common symptom of COVID-19.
• Cough: A dry cough is a frequent symptom that may persist and become more severe over time.
• Shortness of breath: Some individuals may experience difficulty breathing or shortness of breath, particularly in severe cases.
• Fatigue: Feeling tired or experiencing extreme exhaustion is a common symptom reported by COVID-19 patients.
• Muscle or body aches: Muscle pain, body aches, or general discomfort can occur with COVID-19.
• Headache: Many individuals with COVID-19 may experience headaches, which can range from mild to severe.
• Sore throat: Soreness or irritation in the throat may be present, often accompanied by a dry cough.
• Loss of taste or smell: An altered sense of taste (dysgeusia) or loss of smell (anosmia) has been reported by some individuals infected with COVID-19.
• Congestion or runny nose: While less common, nasal congestion or a runny nose can occur in some cases.
• Gastrointestinal symptoms: Some individuals may experience symptoms like nausea, vomiting, or diarrhea.

It is important to note that not everyone infected with COVID-19 will experience symptoms, and some individuals may have mild or no symptoms at all. Additionally, symptoms can vary widely between individuals, and some symptoms may overlap with other respiratory illnesses. If you experience any concerning symptoms or have been in close contact with someone who tested positive for COVID-19, it is recommended to consult with a healthcare professional and follow local health guidelines for testing and care.

2.1. How Is COVID-19 Spread?

Transmission of COVID-19 primarily occurs through respiratory droplets that are emitted when infected individuals cough, sneeze, talk, or breathe. The virus, SARS-CoV-2, can be transmitted through the following routes:

• Avoiding direct physical contact, such as touching one another.
• Indirect contact refers to the transmission of the virus through contact with contaminated objects or surfaces. When individuals infected with the virus sneeze, cough, or touch surfaces, they can leave behind infected droplets on these surfaces. The virus can remain viable on these surfaces for varying periods, ranging from several hours to a few days.
• Close contact refers to the situation where individuals are in proximity to an infected person and come into contact with the respiratory droplets emitted from their nose and mouth.
• Transmission through the air is possible in crowded indoor environments with limited ventilation. Small droplets containing the virus can travel greater distances under these conditions. Therefore, meeting people outdoors is considered safer than indoors, even if you maintain a distance of more than 2 m.
• Transmission can also occur through contaminated surfaces. When an individual who is infected with the virus sneezes or coughs, droplets containing the virus can land on nearby surfaces. If you touch these surfaces and then touch your eyes, nose, or mouth, you can potentially become infected as well.
• Inhaling air in close proximity to an infected person who is exhaling droplets containing the virus can also lead to transmission.
• Certain medical procedures can also facilitate the transmission of the virus.
• Aerosol transmission refers to the spread of infected droplets that can remain suspended in the air for extended periods, particularly in indoor environments with limited fresh air circulation.
• Fan and air-conditioner which circulated the infected droplets (especially if they recirculate the air).

2.2. Treatment

At present, there is no approved treatment specifically for COVID-19. The management of the disease depends on its severity, ranging from supportive care for mild cases to more intensive interventions for severe cases. The primary focus is on providing symptomatic and supportive care, including monitoring vital signs, ensuring adequate oxygen levels, maintaining blood pressure, and addressing complications like secondary infections or organ failure. Given the potential fatality of COVID-19, numerous investigational treatments are currently being studied:

• Symptom management: Many COVID-19 cases involve mild symptoms that can be managed at home.
• Hospital care: For severe cases, hospitalization may be necessary. This can involve providing supplemental oxygen therapy to maintain adequate oxygen levels, administering intravenous fluids, and monitoring vital signs closely.
• Corticosteroids: In severe cases, corticosteroids like dexamethasone are used to reduce inflammation and suppress an overactive immune response. They have been shown to improve outcomes and reduce mortality rates in hospitalized patients requiring oxygen therapy or mechanical ventilation.
• Immune-based therapies: monoclonal antibodies, convalescent plasma, and other immune- based therapies are used in specific cases to help boost the immune response and neutralize the virus. These treatments are typically reserved for individuals at high risk of severe disease or those with moderate symptoms.
• They have been shown to improve outcomes and reduce mortality rates in hospitalized patients requiring oxygen therapy or mechanical ventilation.
• In severe cases of COVID-19 with significantly low oxygen levels, respiratory support such as mechanical ventilation or extracorporeal membrane oxygenation (ECMO) may be required to provide adequate oxygen therapy.

It is crucial to acknowledge that the treatment approach for COVID-19 is continuously evolving with ongoing research and clinical trials. Healthcare professionals should be consulted to determine the most suitable treatment options based on factors such as patient characteristics, disease severity, and the latest insights.

2.3. Prevention of Rapid Spread and Disease Progression of COVID-19

In order to prevent infection and promote effective immune responses to the virus, it is crucial to ensure adequate indoor air circulation, practice proper cough etiquette, and maintain good hand hygiene. Additionally, implementing social and physical distancing measures among individuals is essential to prevent overwhelming the healthcare system. It may be necessary to temporarily accommodate an increased number of patients, as long as it remains within the capacity of the current healthcare infrastructure. These strategies align with appropriate healthcare policies, such as the pursuit of herd immunity as observed in Sweden. By implementing these measures, there is a reasonable expectation that we can successfully navigate the projected second wave anticipated in the northern hemisphere during the late fall and winter seasons. Here are some key strategies:

• Vaccination: Getting vaccinated against COVID-19 is highly effective in preventing severe illness, hospitalization, and mortality. Vaccines have been developed and approved for emergency use in many countries. Following the recommended vaccination schedule and receiving booster shots, if necessary, can provide significant protection against the virus.
• Practice proper hygiene by consistently washing your hands with soap and water for at least 20 s, particularly after being in public spaces or touching surfaces. If soap and water are not accessible, utilize hand sanitizer that contains at least 60% alcohol. Avoid touching your face, especially your eyes, nose, and mouth, as this can lower the chances of viral transmission.
• Use masks: When in indoor public spaces or crowded outdoor settings, wear a mask that covers your nose and mouth. Masks play a crucial role in preventing the spread of respiratory droplets that may contain the virus. Adhere to local guidelines and regulations regarding mask usage.
• Maintain physical distance: Practice social distancing by keeping a distance of at least 1 m (3 feet) from others, especially in situations where physical distancing may be challenging, such as crowded places or close-contact settings.
• Ventilation and air circulation: Ensure proper ventilation in indoor spaces by opening windows or using mechanical ventilation systems. Good airflow can help dilute and remove viral particles from the air, reducing the risk of transmission.
• Avoid large gatherings: Limit or avoid gatherings, especially in enclosed spaces, where the risk of transmission is higher. Instead, opt for virtual meetings or outdoor activities that allow for better physical distancing.
• Stay home if unwell: If you have symptoms related to COVID-19, such as fever, cough, or breathing difficulties, it is important to stay at home and consult with a healthcare professional. Follow local guidelines on testing, self-isolation, and quarantine protocols.
• Follow public health guidelines: Stay informed about the latest guidance and recommendations provided by local health authorities, including travel advisories, testing protocols, and quarantine requirements. Adhere to these guidelines to protect yourself and others.
• Enhanced cleaning and disinfection: Regularly clean and disinfect frequently touched surfaces, particularly in areas with high foot traffic. Use approved disinfectants that are effective against COVID-19.
• Support contact tracing efforts: Cooperate with contact tracers if you test positive for COVID-19 or have been in close contact with someone who has. Providing accurate information can help identify potential transmission chains and prevent further spread.

These preventive measures work in conjunction with one another to reduce the spread of the virus. It is important to adapt and follow the specific guidelines and recommendations provided by health authorities in your region, as they may vary depending on the local epidemiological situation and vaccination rates.

3. Verhulst Logistic Equation

The Verhulst differential equation, known as the Verhulst logistic equation, is a first-order differential equation due to Pierre François Verlhust and of the form

$${\displaystyle \frac{dP}{dt}}=aP\left(1-{\displaystyle \frac{P}{E}}\right),$$

(1)

where $P\left(t\right)$ represents the population size, a is the population rate, which is the Malthusian parameter, and E is the carrying capacity of the environment [17]. The story of the Equation (1) is rooted in the work of Adolphe Quetelet in 1835 on “Man and the Development of his faculties”, where he introduced the concept of obstacles limiting long-term geometric population growth. Queterlet proposed that these obstacles create resistance proportional to the square of the population growth rate, which inspires Verlhust to propose a differential equation, (1), to model population growth as an alternative approach. When the population $P\left(t\right)$ in the Equation (1) is significantly smaller than E, an approximate equation

$${\displaystyle \frac{dP}{dt}},$$

(2)

can be used resulting in exponential growth

$$P\left(t\right)=P\left(0\right)e^{at}.$$

(3)

When $P\left(t\right)$ is negative, the growth rate would become negative. In 1847, Verlhust introduced a modified differential equation

$${\displaystyle \frac{dP}{dt}}=a(P-b)\left(1-{\displaystyle \frac{P-b}{E}}\right).$$

(4)

Despite Verlhust’s initial indecision regarding model equations, the logistic equation reemerged independently several decades later from various sources. The term “carrying capacity” represents the maximum population, denoted as E, in the logistic equation. The Verlhust Equation (1) can be solved in a few steps

• First step: dividing the Equation (1) by $p^2$, one obtains

  $${\displaystyle \frac{d}{dt}}\left[{\displaystyle \frac{1}{p}}\right]=a\left({\displaystyle \frac{1}{E}}-{\displaystyle \frac{1}{P}}\right).$$

  (5)
• Defining $x\left(t\right)={\displaystyle \frac{1}{P\left(t\right)}}$, we have

  $${\displaystyle \frac{d}{dt}}x\left(t\right)=-a\left(x\left(t\right)-{\displaystyle \frac{1}{E}}\right).$$

  (6)

The solution to the Equation (6) is given by

$$x\left(t\right)=\left({\displaystyle \frac{1}{p\left(0\right)}}-{\displaystyle \frac{1}{E}}\right)e^{-at}+{\displaystyle \frac{1}{E}}.$$

(7)

Since $x\left(t\right)={\displaystyle \frac{1}{p\left(t\right)}}$, the solution $P\left(t\right)$ to the Equation (1) is given by $P\left(t\right)={\left[x\left(t\right)\right]}^{-1}$, which is explicitly

$$P\left(t\right)={\displaystyle \frac{E{P}_0}{e^{-at}[E-P_o]+P_o}}.$$

(8)

4. Data Analysis

This study focuses on Senegal, a country located in West Africa, which coordinates approximatively between 12° N and 17° N latitude 17° W and 16° W longitude. Senegal has land area of around 196,712 square kilometers and an estimated population of about 18,015,360 people as of 22 May 2023. Senegal borders with Mauritania, Mali, Guinea, Guinea-Bissau, and the Gambia, with the Atlantic Ocean forming its western border. It has a tropical climate with distinct wet and dry seasons. The average daily temperature ranges from 25 °C to 35 °C, and the relative humidity varies between 50 percent and 90 percent. The rainy season occurs from June to October, while the dry season spans from November to May. The Verhulst logistic model was employed to fit the time series of total COVID-19 cases.

The calculations were performed using Python. In our data analysis, we used a variety of Python libraries to ensure comprehensive and robust data processing and prediction generation. Below are the specific libraries and tools employed:

• Urllib: Utilized urlopen to download datasets from online sources, specifically from the Our World in Data GitHub repository.
• Pandas: Used for data manipulation and analysis, including loading data, cleaning, trans- forming, and merging datasets.
• Seaborn: Used for statistical data visualization, simplifying the creation of informative and attractive visualizations.
• NumPy: Essential for data manipulation and numerical computations, facilitating operations on large datasets, including matrix operations and array manipulations.
• Matplotlib: Utilized for basic data visualization, providing control over plot aesthetics to create detailed and customizable graphs.
• SciPy: Employed for advanced mathematical functions and optimization, such as curve fit for fitting curves to data points.
• Plotly: Used for interactive and offline data visualization, enabling dynamic and interactive plotting through plotly.express, plotly.graph objs, and plotly.subplots. Facilitated offline plot creation with plotly.offline and iplot.
• Warnings: Utilized to suppress unnecessary warnings for cleaner output.

For the data sources and collection, we sourced our data from the COVID-19 dataset available at the Our World in Data GitHub repository. This dataset includes comprehensive information on COVID-19 cases, deaths, testing, and vaccination data across various countries. There are 395 observations for each variable in the daily COVID-19 data, which were taken from (https://github.com/owid/covid-19-data/tree/master/public/data (accessed on 25 March 2023)) (Figure 2).

Data from COVID-19 are available. Location, date, total cases, new cases, total deaths, and new deaths are the six variables that make up the COVID-19 data that was used for the study. The information covered the period from 1 April 2022, through 30 April 2023. There are 395 observations for each variable in the daily COVID-19 data, which were taken from (https://github.com/owid/covid-19-data/tree/master/public/data (accessed on 25 March 2023)).

The Figure 3 displays the trend of the number of new cases and fatalities each day, as well as the number of cases and deaths overall, for COVID-19 in Senegal through time. It offers information on the rate of infection and aids in comprehending the peaks and valleys in the quantity of new cases. The combination of these four subplots offers a complete picture of the COVID-19 situation in Senegal (Figure 4, Figure 5 and Figure 6).

4.1. Data Analysis over the Period April 2022 to April 2023

4.1.1. Descriptive Statistics and Distribution of the Considered Data Set

The

Table 1. Descriptive statistics of COVID-19 cases.

Variables	Mean	Sd	Median	Min	Max	Se
Total cases	87,923.12	1192.00	88,601.00	85,895.0	88,997.00	59.90
New cases	7.87	12.48	3.00	0.00	83.00	0.62
Total deaths	1968.46	1.86	1968.00	1964.00	1971.00	0.09
New deaths	0.01	0.18	0.00	0.00	3.00	0.01

below provides an overview of the descriptive statistics for the COVID-19 cases used in the study.

The average number of new COVID-19 cases per day was 7.87, with a standard deviation of 12.48, while the average daily death rate was 0, with a standard deviation of 0.18. The synopsis also revealed that the distribution of COVID-19 deaths and newly observed daily cases is positively biased. In Senegal, the number of total cases is 88,997 (Figure 7).

4.1.2. Results and Discussion

Mathematical models of various complexities are employed to understand and predict the spread of epidemics. The Verhulst logistic equation, a non-linear first-order ordinary differential equation (ODE), is historically the first model used for this purpose [18]. It is commonly utilized to describe population growth and advertising performance. In the context of COVID-19, this model has been applied in previous studies [19]. The analysis of COVID-19 data suggests different growth patterns in different regions, with exponential growth observed in some countries and power law growth in others [20]. To capture these dynamics, the generalized logistic equation is utilized [21,22]. Despite its mathematical simplicity, the logistic equation provides valuable insights. Various statistical tests were employed to examine the COVID-19 responses in Senegal.

A p-value is a measure used in statistical hypothesis testing to help determine the strength of the evidence against the null hypothesis, which generally represents a default assumption or a statement of no effect or no relationship and depends on the test one is performing. To evaluate the randomness of COVID-19 responses in Senegal, the Ljung-Box test was conducted using Python software. The test results indicated that the p-values (0.06864417 for COVID-19 responses and 0.7643388 for deaths) were both greater than the significance level of 5 percent. Therefore, we cannot reject the null hypothesis that the responses and deaths are random, suggesting no autocorrelation in Senegal. Keenan’s one-degree test for non-linearity was employed to analyze the autoregressive nature of the daily confirmed cases and daily deaths time series in Senegal. Python statistical software was used for this purpose. The test yielded a p-value of 0.00 for the daily confirmed cases, which is below the significance level of 0.05. Hence, we reject the null hypothesis and conclude that the daily confirmed cases in Senegal do not exhibit an autoregressive process. Similar conclusions were drawn for the number of death cases, as the p-value $\left(0.00\right)$ was also below the significance level (

Table 2. Ljung-Box test on COVID-19: Total Cases in Senegal.

Total Cases of COVID-19
Statistics	17.2678
p-value	0.0686441

,

Table 3. Ljung-Box test on COVID-19: Total Deaths in Senegal.

Total Deaths of COVID-19
Statistics	6.580794
p-value	0.7643388

,

Table 4. Keenan’s one-degree test for non-linearity on COVID-19: New Cases in Senegal.

New Cases of COVID-19
Statistics	−15.66869
p-value	0.00

and

Table 5. Keenan’s one-degree test for non-linearity on COVID-19: New Deaths in Senegal.

New Deaths of COVID-19
Statistics	−30.73934
p-value	0.00

).

4.2. Trends over the Period April 2022–April 2023

In summary, the logistic model effectively captures the progression of COVID-19 cases over time, despite its simplicity. The logistic function defined as “logistic(t, a, K)” solves the standard logistic differential equation given by the Equation (1). This equation is commonly used to model population growth or the spread of a phenomenon over time. Figure 8 shows the plots of the real COVID-19 data and the logistic curve.

From Figure 8, we observed that the logistic function does not fit the data well. This is likely because COVID-19 cases do not start from zero; there is an existing baseline level. Looking for the best fit of the data, we add a constant term “b” to the logistic function denoted as “logistic offset(t, a, K, b, N0)”. It solves a modified version of the logistic Equation (4)

$${\displaystyle \frac{dP}{dt}}=a(P-b)(1-{\displaystyle \frac{P-b}{E}}),$$

(9)

where the term “b” introduces the additional constant and allows for an offset or baseline value in the logistic function. This modified equation incorporates an offset or baseline value, represented by the constant “b”. It is particularly useful when the phenomenon being modeled has a baseline or an initial value that is different from zero. Figure 8 shows the plots of the real COVID-19 data and the logistic curve with the additional constant term.

From the Figure 9, we observed that the logistic function with the additional constant term, which incorporates the offset or baseline value, provides a good fit to the COVID-19 data. This modification allows for a more accurate representation of the initial conditions and external influences affecting the COVID-19 spread in Senegal. Based on the Figure 8 and Figure 9, it can be inferred that incorporating an additional constant term in the logistic function provides a better fit for modeling COVID-19 data in Senegal compared to the logistic function without the constant term. The inclusion of the constant ‘b’ enhances the representation of initial conditions and baseline, resulting in improved alignment with the observed data.

4.3. Comparison/Discussion with Predicted Results from April 2022 to 2023

To assess the accuracy and reliability of the logistic model, we compared the predicted COVID-19 cases with the actual data. Performance metrics like mean squared error (MSE) and coefficient of determination (R-squared) were used in this study to evaluate the model’s goodness-of-fit and predictive ability.

• The average of the squared differences between predicted and actual values is calculated using the mean squared error (MSE). It indicates the overall difference between the model’s predictions and reality. The squared discrepancies between each predicted value and its matching actual value are averaged to produce the MSE.

  $$\mathrm{MSE}={\displaystyle \frac{1}{n}}\sum {\left(y_{\mathrm{true}}-y_{\mathrm{pred}}\right)}^2$$

  (10)
  A smaller MSE value indicates superior model performance, indicating minimal discrepancies between predicted and actual values. An MSE of zero represents a perfect fit of the model to the data. It is important to consider that outliers can have a significant impact on the overall score of MSE, as their squared differences exert a greater influence.
• R-squared, or the coefficient of determination, indicates the percentage of variance in the dependent variable that can be accounted for by the independent variable(s) in the model. It evaluates the degree to which the model fits the data. R-squared is calculated as:

  $$R-\mathrm{squared}=1-{\displaystyle \frac{S{S}_{\mathrm{residual}}}{S{S}_{\mathrm{total}}}}$$

  (11)

  where: SS residual is the sum of squared residuals (differences between the actual and predicted values), and SS total is the total sum of squares, which measures the total variance of the dependent variable. The evaluation of the logistic model’s performance was conducted on the COVID-19 data for Senegal from April 2022 to April 2023. The

  Table 6. Evaluation Metrics for Logistic Model Predictions.

  Evaluation Metric	Value
  Mean Squared Error (MSE)	152,054.68
  Coefficient of Determination (R-squared)	0.89

  displays the result of the different performance metric values on the cases of COVID-19 in Senegal. From the Table 6, it can be observed that the mean squared error (MSE) is equal to 152,054.68 and the coefficient of determination (R-squared) is 0.89. As the R-squared value is closer to 1, it indicates a stronger relationship and better predictive performance of the model.

Finally, we calculated the mean absolute percentage error (MAPE) to assess the model’s accuracy. The average percentage difference between the expected and actual values is quantified. The following equation can be used to determine MAPE.

$$\mathrm{MAPE}={\displaystyle \frac{1}{n}}\times \sum |{\displaystyle \frac{\mathrm{Actual}-\mathrm{Predicted}}{\mathrm{Actual}}}\times 100,$$

(12)

where n is the number of data points, Actual represents the actual values, and Predicted represents the predicted values. MAPE is expressed as a percentage, indicating the average percentage error in the predictions. It provides a relative measure of the accuracy, allowing comparison across different datasets and models. In this thesis, the mean absolute percentage error (MAPE) is calculated using Python to evaluate the accuracy of the predictions. The MAPE value obtained is $0.0019$, which is close to $0\%$. This result indicates a very accurate prediction, meaning that the predicted values are very close to the actual values (the reality of April 2023).

5. Conclusions

The pandemic has significantly impacted the global population, leading to widespread health, social, and economic consequences. Indeed, the social and economic consequences of COVID-19 have been widespread and significant, affecting individuals, communities, and economies around the world. The pandemic has caused significant disruptions to daily life, including school closures, job losses, and reduced social interactions. The impact has been particularly pronounced for vulnerable populations, including those with pre-existing health conditions, low-income households, and marginalized communities.

In terms of the economy, COVID-19 has had a significant impact on businesses, with many small businesses closing permanently due to decreased revenue and increased costs associated with adapting to the pandemic. The pandemic has also led to disruptions in global supply chains, reducing the availability of goods and increasing prices for certain products. The COVID-19 pandemic has highlighted and worsened pre-existing social and economic disparities. Vulnerable communities have been disproportionately impacted, experiencing higher rates of illness, death, job losses, reduced access to support systems, and decreased educational opportunities. Additional key areas affected by the pandemic include:

Mental Health Impact: The COVID-19 pandemic has had a profound effect on mental well-being. Social isolation, fear, anxiety, and grief associated with the virus have contributed to a rise in mental health disorders. The strain on healthcare systems has also limited access to mental health services, exacerbating the mental health crisis. The COVID-19 pandemic has underscored the significance of social cohesion and community resilience. Many communities have come together to support each other through mutual aid networks, volunteering, and community initiatives. This crisis has emphasized the significance of collective action.

Unemployment and Income Loss: The economic impact of COVID-19 has resulted in a significant rise in unemployment rates. Many people have lost their jobs or experienced reduced working hours, leading to income loss and financial instability. This has disproportionately affected vulnerable populations, including low-income workers, informal sector workers, and those in precarious employment.

Poverty and Inequality: The pandemic has exacerbated existing social inequalities and pushed many people into poverty. The loss of livelihoods and reduced access to basic services have left vulnerable populations struggling to meet their basic needs. Women, children, the elderly, and marginalized communities have been particularly affected by the socio-economic consequences of the pandemic.

In this work, we explored the application of differential equations, specifically the Verhulst logistic equation, to the study of COVID-19. By utilizing mathematical modeling and differential equations, we aimed to gain insights into the dynamics and trends of COVID-19 cases in Senegal over a certain time period (April 2022 to April 2023). First, we provided an overview of COVID-19, discussing its origins, transmission, and impact on public health. We highlighted the importance of mathematical modeling in understanding the spread and control of infectious diseases, particularly in the case of COVID-19.

Next, we introduced the Verhulst logistic equation, a widely used mathematical model for population growth. We adapted this model to analyze the spread of COVID-19, considering factors such as population size, total cases, total deaths, new cases, and new deaths. By fitting the Verhulst logistic equation to the COVID-19 data in Senegal, we were able to study the trend and dynamics of the disease over time. Through our analysis, we observed the trend of COVID-19 cases in Senegal from April 2022 to April 2023. We examined the growth rate, peak periods, and fluctuations in case numbers, providing valuable insights into the progression of the disease. We also compared two variations of the logistic function: one without an additional constant term and the other with an additional constant term. We demonstrated the relevance and effectiveness of applying differential equations, specifically the Verhulst logistic equation, to the study of COVID-19. By analyzing the COVID-19 data in Senegal, we gained valuable insights into the trends, dynamics, and impact of the disease. Our analysis revealed that a shift of b in the parameter P in the logistic function provides a better fit for the COVID-19 data in Senegal. This research contributes to the broader field of epidemiology and provides valuable information for public health decision-making and future pandemic preparedness. As further research and data become available, it is crucial to continue refining mathematical models, incorporating new variables, and considering the evolving nature of the pandemic. By combining rigorous mathematical modeling with empirical data, we can continue to enhance our understanding of COVID-19 and contribute to effective strategies for disease control and prevention. Since machine learning is an innovative approach that has extensive applications in prediction, an extension of this work is to use the Verhulst equation to predict and apply machine learning techniques to identify patients at high risk.