March 2020, is still spreading around the world up to date 26 September 2020. The number of people infected is beyond 32 million on 26 September 2020 with 989,380 deaths [24] . In Senegal, the number of cumulative cases is currently 14839 with 2624 individuals undergoing treatment on 25 September 2020 [13] . The first cases, from Wuhan, were notified to WHO on 31 December 2019 [25, 26] while, Senegal notified its first case on 02 March 2020 [13] . Because of its limited resources, as in many sub-Saharan African countries, it is therefore valuable to understand the growth and the timing in responding to the logistic needs of their health system. We note that several developed countries that nevertheless have high-capacity health structures have been overwhelmed and this considerably impacted negatively in the combat against the pandemic. In [14] it is discovered that the growing number of COVID-19 cases in the United States could gravely challenge the critical care capacity, thereby exacerbating case fatality rates. As pointed out in [23] , under-resourced health systems may pose a threat to patient care as well as the safety and well-being of health care workers. From the appearance of the first cases of COVID-19, the Senegalese authorities decided to This work was completed with the support of the NLAGA project. arXiv:2011.06278v1 [q-bio.PE] 12 Nov 2020 anticipate by increasing gradually the capacity of its hospitals to take care of patients suffering from COVID-19. One of the aims of the present paper is to study different scenarios of involution of the number of virus patients by varying the growth rate and the number of, say, beds or rooms. As an approach, we use a mathematical model that integrates the different stages of individuals (Susceptible, Exposed, Asymptomatic, Symptomatic, Removed) with the parameters depending on the health resources availability. To this cope, we consider a modified SEWIR model [16] in which we include a time-dependent carrying capacity K(t) representing the evolution of the Senegalese hospitals' capacity in response to the growing number of cases in Senegal. Our analysis highlights the importance of anticipation and timing to avoid overwhelming in the health system. More precisely, our analysis shows, in particular, that, while the epidemic growths exponentially fast, by respecting a certain logistic growth (at which the number of beds is being added) in the health system, the epidemic might remain under control. As a consequence, a non-negligible number of expenses can be saved. Finally, a machine learning approach to project forward the cumulative number of virus infected cases is provided. Let us mention that several studies have been conducted in different contexts to assess the impact of hospital saturation in COVID-19. We refer for examples to [5, 7, 8, 14, 21] and the references therein. Our work complements several studies that have been carried out on the spread of Sars-CoV-2 in Senegal [2, 3, 4, 6, 15, 16, 17, 18] . It should be noted that the aforementioned studies did not address the impact of saturation on patients care in Senegal. However, some studies on the prediction of the number of new cases for Senegal have already been carried out using machine learning [2, 3, 15, 16] and on the impact of contaminated objects using compartmental models [6] . The paper is organized as follows. In section 2, we describe in detail our model as well as the parameters involved. It is followed by Section 3 concerning the estimation of the unknown parameters. In section 4 we derive the time-dependent effective reproduction number which is a key function that informs about the impact of the various decisions taken by the authorities during the epidemic. In Section 5, we first present the calibration of the model to the cumulative number of reported cases. We then present numerical simulations on the impact of the dynamics of health capacity on the patients care. Finally, in Section 6, we use machine learning to project forward the number of cumulative cases until 1st December 2020.
We consider compartmental epidemic model that incorporates a time-dependent carrying capacity K(t) of health structures for the treatment of COVID-19 patients at the hospitals. The model reads as follow
The state variable S(t) is the number of susceptible individuals at time t, E(t) the number of exposed individuals at time t and W (t) the number of waiting cases for confirmation (with infectiousness) at time t. The number of confirmed and monitored cases at time t is denoted by I(t). The variable K(t) describes the effective carrying capacity of the health structure, which is the maximum number of I(t) individuals that can be correctly monitored. Finally R(t) represents the removed individuals at time t due to death or recovery. The figure below represents the diagram flux of the model. The term εβ 1 (t)I(t)S(t) represents the transmission of the disease within the health structures where ε > 0 can be interpreted as the probability that a susceptible individual is working in the health system. More precisely we set ε = N H /S 0 with N H the number of health care workers and S 0 the initial number of susceptible individuals. The quantity of 1/β 2 is the average incubation period. We assume that the average waiting time for confirmation is 1/β 3 days. After the waiting time, a fraction 1 − α 1 is monitored while the remaining fraction α 1 is removed due to death or recovered. The removal rate of the confirmed cases is described by t → γ (1 − I(t)/K(t)) + + η in order to take into account the saturation effect of the public health structures. The term γ (1 − I(t)/K(t)) + + η which is a decreasing function of I(t) describes the efficiency of the health structures. The maximal efficiency is γ + η while the minimal efficiency η is reached when the number of confirmed cases becomes greater than the carrying capacity K(t). In our modeling procedure, we have introduced a time-dependent carrying capacity which is motivated by the fact that in Senegal, the carrying capacity of the health structures was gradually increased in response to the increase in the number of infected individuals. The dynamics of the increase in the number of beds remains unknown because only announcements have been made. We chose a logistic equation to mimic the growth of the public health capacity to its saturation. Indeed, from the onset of the epidemic, the authorities announced a carrying capacity of 30 which gradually increased according to the number of cases. The maximum capacity of the structures remains unknown but in view of the announcements made, we assume that it is approximately 3000. The growth rate r = 0.1 of t → K(t) is chosen such that it increases from 30 on 02 March to approximately 3000 on 10 June and that the number of hospitalized individuals is below the effective carrying capacity.
There are other possibilities of choosing the growth rate r but the value r = 0.1 appears to give the best fit to the data. The figure below describes the evolution of K(t) with respect to time and the evolution of the number of hospitalized individuals. The evolution of the public health capacity and number of monitored are illustrated in Figure 2 . More precisely the dynamic of the carrying capacity is given by:
with K 0 = 30, r = 0.1 and K max = 3000. The parameter K sa will serve to explore different scenarios in which the efforts of increasing health capacities are aborted. In model (2.1)-(2.4), we use a time-dependent transmission rate to take into account the various events that have impacted the spread of disease. Indeed several government actions such as partial lockdown, social distancing, interurban traffic ban, border closure have impacted the progression of the disease. The time-dependent transmission rate has the following form
where the parameters are estimated by using the data on reported cases. More precisely we have β 11 = max β 10 e −θ1(t2−t1) , θ 2 β 10 and β 12 = min(β 11 e θ1(t3−t2) , θ 3 β 10 )
with t 0 = 02 March the date at which the first case was reported [13] , t 1 = 24 March, t 2 = 12 April and t 3 = 20 April. We distinguish three phases. Phase 1 takes place between the dates t 0 = 02 March, the beginning of the epidemic, until the first government actions at date t 1 = 24 March. Phase 2 which begins at t 1 , the date on which decisions such as public closing, interurban traffic ban, border closure were taken, is ended at date t 2 = 12 April. The date t 2 corresponds to a sudden increase in the number of cases with a decline on approximately date t 3 = 20 April where Phase 3 began and cover the period of 20 April to 28 June the date at which we stop our study. The cause of the sudden increase in 12 April remains unknown to us at this time. More accurate data would be needed in order to understand the advent of this phenomenon. However, we can follow the evolution of the dynamics of the epidemic using the time-dependent transmission rate. The parameter θ 1 describes the intensity of the government actions or sudden events that impacted the transmission rate. We refer to [10] where such interpretation has been used to describes the government actions. The parameters θ 2 , θ 3 and θ 4 allow us to measure the impact of each phase on the transmission rate β 10 . The values of the parameters θ i , i = 1, 2, 3, 4 estimated in Section 3 are
and the estimated value of β 10 in Section 3 is
The figure below describes the transmission rate t → β 1 (t). Next we summarize the Table 1 are given in the next 
In this section, will use the approach developed in [9, 10, 11, 12] in order to estimate the parameters from the early phase of the epidemic. Let C(t) be the cumulative number of reported cases at time t be defined by
with t 0 = 02 Mars 2020 the date of the first reported cases in Senegal so that C(t 0 ) = 1. The early phase of the epidemic corresponds to the period from the announcement of the first case on 02 Mars 2020 to the first government actions at t 1 = 24 March 2020.
In this phase we assume that the number of infected individuals growth exponentially. Hence with such assumption, the cumulative number of cases has the following form
The Figure 4 below shows the comparison between the t → C(t) and the data from 02 March until 24 March 2020 that allows us to estimate the parameters χ 1 and χ 2 to χ 1 = 0.1612 and χ 2 = 1.979. and β 1 (t) = β 10 , it follows that we have the following approximation for a short time period
(3.4) Furthermore differentiating (3.1) and (3.2) with respect to t we obtain
Solving the second and the third equation of (3.4) we get:
Plugging (3.5) and (3.6) into the first equation of (3.4), it follows that
Therefore using (3.3) together with (3.5)-(3.7) and the values of the parameters of Table  1 we obtain E 0 = 1.18, W 0 = 4.55, I 0 = 1.39 (3.8) and β 10 = 3.8593 × 10 −8 .
(3.9) In order to obtain the values of the parameters θ i , i = 1, 2, 3, 4 that appear in the transmission rate t → β 1 (t) we perform a curve fitting by using the data from t 1 = 23 March 2020 until 28 June 2020. The estimated values are listed below 
The effective reproduction number, R e , is the expected number of secondary cases produced by one typical infection joining a healthy population during its infectious period while the time-dependent effective reproduction number R e (t) is the instantaneous transmissibility of the disease at time t. The time-dependent effective reproduction number allows to follow the evolution of the epidemic as the time evolves in particular how R e (t) is far above or far below 1. In order to obtain R e (t) we note that β 1 (t)S(t) is the number of new infections per unit of time generated by one infectious individual. Thus assuming β 1 (t)S(t) approximately constant during 1/β 3 , it follows that the term
is the average number of new infections generated by one infectious individual (W) during its period of infectiousness 1/β 3 . Moreover one infectious individual (W) generates on average
infectious individual(s) (I). Those 1−α 1 infectious (I) has a maximum mean infectiousness period 1/(γ + η) so that they generate
We finally define the time-dependent effective reproduction number as
The figure below gives the evolution of the effective reproduction number. We observe that the effective reproduction number is larger than 1 (approximately 3) during the early phase of the epidemic that is from 02 March to 24 March and is smaller than 1 during the second Phase. We can therefore observe that the first actions of the government had a positive effect on the spread of the epidemic because it tended to disappear if the effective measures would have been kept in the same trend. However, the sudden increase in cases on April 12th impacted the effective reproduction number to bring it to around 5. It goes down until it is close to but greater than 1 from April 20th. The cause of this increase is unknown to us but a better knowledge of the data may allow us to find an explanation.
In this section we perform numerical simulations of the dynamics of the spread of the COVID-19 epidemic based on our proposed model (2.1)-(2.4). We consider the regions of Dakar, Thies and Diourbel which concentrate around 95% of the reported cases. To do so we assume that the initial number of susceptibles is the population of Dakar, Thies and Diourbel, that is to say S 0 = 7857353 [1] . The values of the parameters of the model are listed in Table 2 . The time-dependent transmission rate is given in (2.5)-(2.7) while the estimated initial conditions in Section 3 are E 0 = 1.18, W 0 = 4.55, I 0 = 1.39. In Figure 6 below we compare the cumulative number of cases t → C(t) defined in (3.1) with 95% of the cumulative number of reported cases showing that our model agrees with the data. In order to illustrate the impact of the saturation of health structures on the number of monitored individuals, we consider several scenarios. Indeed, we consider the same evolution of the reception capacity with interruptions in growth at predefined values. More precisely we consider the following dynamic
where K sa , the saturation level, is respectively 3000, 2500, 2000, 1500 and 1000 and is illustrated by the following figure. In Figure 8 we plot the corresponding number of monitored individuals that is t → I(t) when the dynamics of the carrying capacity is given by (5.1) with K sa = 3000, 2500, 2000, 1500 and 1000 while the growth rate is fixed at r = 0.1. We see that the epidemic is under control, until 28 June only when K sa = 3000 and K sa = 2500. This shows that the maximum value of the carrying capacity, K sa , may impact significantly the efficiency of public health structures by slowing down the rate at which individuals are removed from the hospital. In Figure 9 we look at the influence of the growth rate r on the number of monitored individuals. We see that increasing r from 0.1 to 0.5, does not impact significantly the number of monitored individuals and the dates at which overwhelming occurs. This is the case for all values of K sa . For r = 0.05, Figure 10 , we see that the number of monitored individuals becomes larger than the maximum number of available resources at the early stage of the epidemic, around 20 April. We also observe that I(t) remains unchanged for different values of K sa . Thus our model captures and validates the obvious fact that the faster you increase the number of beds, the better you can handle the hospitalized people. Figure 10 emphasizes more the sensibility of the growth rate r which can also be seen/interpreted as the timing in anticipating overwhelming. Indeed we observe that for the maximal value of resources K sa = K max = 3000, K(t) stay larger than I(t) until 28 June for r ≥ 0.06 while for r ≤ 0.05 we see that the number monitored becomes larger than available resources from mid-April to the end of June. In view of our discussion, it would be important to have a condition that guarantees the unsaturation of health structures. It is contained in the following Lemma 5.1. Assume that K 0 > I 0 . Assume in addition that t → K(t) is increasing and differentiable with respect to t. If
In this section, we present a machine learning approach for the forecasting of the cumulative number of confirmed cases C(t). First, we collect the pandemic data from [13] , from March 02, 2020, to October 22, 2020. Then, we perform a forecast with Prophet to predict the final size of coronavirus epidemy The numerical tests are performed by using Python with the Panda library [20] , and were executed on a computer with the following characteristics: intel(R) Core-i7 CPU 2.60GHz, 24.0Gb of RAM, under the UNIX system.
Prophet [19, 22] , is a procedure for forecasting time series data based on an additive model where non-linear trends are fit with yearly, weekly, and daily seasonality, plus holiday effects. It works best with time series that have strong seasonal effects and several seasons of historical data. Prophet is robust in dealing with missing data and shifts in the trend and typically handles well outliers. For the averaging method, the forecasts of all future values are equal to the average (or "mean") of the historical data. If we let the historical data be denoted by y 1 , ..., y T , then we can write the forecasts aŝ y T +h|T =ȳ = (y 1 + y 2 + ... + y T )/T
The notationŷ T +h|T is a short-hand for the estimate of y T +h based on the data y 1 , ..., y T . A forecasting interval gives an interval within which we expect y t to lie on, with a specified probability. For example, assuming that the forecast errors are normally distributed, a 95% forecasting interval for the h-step forecast iŝ
where σ h is an estimate of the standard deviation of the h-step forecast distribution.
6.1.1. Diagnostics. Here, we make some diagnostics by using the cross validation (see Table 3 ) and the performance metrics (see Table 4 ) using mse, rmse, mae and mape. The Figure 15 illustrate these cross validation metrics, making 10 forecasts with cutoffs between 2020-10-08, 00:00:00 and 2020-10-17, 00:00:00 (initial='220 days', period='1 days', horizon = '5 days'). Table 3 , by comparing the values obtained on October 13, 2020 (column y=15307) with the predicted one (columnŷ=15666.025418), we see that the error is 2.29%. The predicted value is always within the confidence interval. So, Prophet seems to give us good value. The Prophet forecasting of confirmed cases, with trend changepoints, is given by Figure  13 , and the trends and weekly increase are given by Figure 14 .
With Prophet, at ∼ November 12, 2020 we may obtain > 16330 confirmed cases and > 16950 confirmed cases at ∼ December 01, 2020 (see Tables 5 and 6 , respectively). The forecastes of confirmed cases are illustrated in Figures 15a and 15b . Table 5 . Prophet: predicted cumulative confirmed cases ∼November 12, 2020. Table 6 . Prophet: predicted cumulative confirmed cases ∼December 01, 2020. 