Mathematical Model of the Spread of the Coronavirus Disease 2019 (COVID-19) in Burkina Faso

In this paper, we develop a mathematical model of the COVID-19 pandemic in Burkina Faso. We use real data from Burkina Faso National Health Commission against COVID-19 to predict the dynamic of the disease and also the cumulative number of reported cases. We use public policies in model in or-der to reduce the contact rate, this allows to show how the reduction of the daily report of infectious cases goes, so we would like to draw the attention of decision makers for a rapid treatment of reported cases.

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Currently, many mathematical models of the COVID-19 have been developed, mainly for the Coronavirus epidemic in China (see [1] [3]- [8]). We have developed a model describing the Coronavirus epidemic in Burkina Faso, focussing on the effects of public policies imposed by the government to contain this epidemic and the number of reported occurred cases.

Mathematical Model
Considering the known characteristics of the Coronavirus disease 2019 (COVID-19) pandemic, we assume that each person is in one of the following compartments:  S (Susceptible) means the number of persons who are not infected by the disease pathogen at time t, so who are susceptible to infection.  E (Exposed) represents the number of persons who are in the incubation period after being infected by the disease pathogen. These persons have no visible clinical sign of the disease. They can infect other people but with lower probability than people in the infectious comportments.
 I (Infectious) means the number of persons who start developing clinical signs, these persons are asymptomatic infectious cases.
 I r (Reported symptomatic infectious cases) represents the number of persons who are reported and isolated at time t.  I u (Unreported symptomatic infectious cases) stands for the number of unreported symptomatic persons at time t.  R (Recovered) represents a person who has survived the disease, is no longer infectious and has developed a natural immunity to the disease pathogen. This leads to the following transfer diagram (Figure 1). The mathematical model consists of the following system of ordinary differential equations:  η is the transition rate of a person in state r I to the state R (day −1 ).  θ is the transition rate of a person in state u I to the state R (day −1 ).  N is the number of people in the territory before the start of the pandemic.
The initial data of the system is supplemented by The time t is in day, the asymptomatic infectious individuals ( ) I t are infectious for an average time period of 1 α days. We also suppose that the population is constant i.e.
Proposition 2.1. The basic reproduction number for the model system (1) is defined by Proof. We use the method of next generation matrix in [9] to compute the reproduction number 0 R .
We get We have   E  I  I  I   E  I  I  I   E  I  I  I   E  I  I A. Guiro et al. and Thus, we obtain The basic reproduction number is defined as the dominant eigeinvalue of the Therefore, The basic reproduction number 0 R is defined as the number of cases that one infected person generates on average during his infectious period, in an un- Since

Data
In this part, we use real data generated by the Burkina Faso National Health With these data, we can see the daily dispersion of the infected case ( Figure   2), and the cumulative reported infected cases (Figure 3).

Model Parameters for COVID-19 in Burkina Faso
Some of the parameters used in the simulations presented in Section 5 are from the literature. However, due to specificity of our real data, we fit some other parameters mainly to adjust the cumulative curve , .

bt C t a c t t = − ∀ ≥
We use the cumulative number of reported symptomatic infectious cases from

Simulation and Comments
The data we use here in Table 3 Table 3.
For the parameters used in this model, we present some important parameters and threshold values related to the Coronavirus Epidemic in Burkina Faso. In particular, we observe that the basic reproduction number 0 4.9 R = is bigger than other reproduction number values reported in the literature [3] [10] [12]. This could be explained by the fact that we have taken into account unreported infected persons.
We assume that the exponential increase phase of the epidemic is intrinsic to the population of each region. Also, the Susceptible population ( ) S t is not significantly reduced over the time. We suppose that the entire population of Burkina Faso at the date 0 t are susceptible so, ( ) 0 20000000 S t = , the exposed population at the same date

Without Any Public Policies until March 26, 2020
We use the cumulative curve to fit the parameter γ . From the beginning of the epidemic until March 9, 2020, we assume that there was no public policy so we fit 0.7 γ = . This could be corroborated by the fact that the cumulative infected data curve fits well with the component of the reported infectious. Figure 4 shows the evolution of the cumulative infected reported cases and the forecast A. Guiro et al.

With Pubic Policies Started on March 27, 2020
From this date, we decrease the rate of contact ( ) Depending on the public measures taken, µ increases, so the contact rate

Conclusion
In this paper, we have developed a mathematical model of COVID-19 for Burkina Faso, inspired by models in [3] and [10]. We have been able to estimate some parameters which have made it possible to fit the model to real data from the start of the Epidemic up to March 27, 2020 (when public policies were introduced). It emerges from this model that the most important parameter here is the contact rate which is a time dependent function (with respect to the public policies taken). A drastic reduction of the contact rate can lead to a considerable reduction in the number of infectious and of the duration of the epidemic.