Abstract
The COVID-19 (coronavirus disease) pandemic represents a global public health emergency unprecedented in recent history. This explains the growing interest of scientists in this subject. Indeed, the question of the COVID-19 pandemic has led to numerous scientific works, with the aim of estimating the reproduction number, the start date of the epidemic or the cumulative incidence. Their results have contributed to epidemiological surveillance and informed public health policy decisions. In our work, using basic data and statistics from the city of Ouagadougou in Burkina Faso, we first consider a mathematical model of COVID-19 transmission taking into account the possibility of transmission from dead populations to susceptible populations; then, we use another method which is the sentinel’s method of JL Lions to estimate the number of infected populations without any time trying to know the beginning of the epidemic; finally, we highlight the numerical simulation of the considered model solution.
1. Introduction
In practice, this study is taking place within the framework of the fight against COVID-19. The COVID-19 (coronavirus disease 2019) pandemic is a global public health problem, the first cases of which were first reported in Wuhan, China, on 31st December 2019. Our work is important in the strategy fight against COVID-19. COVID-19 is a respiratory infection caused by a Coronovirus called SARS-Cov6-2. A first strategy to control the spread of COVID-19 in the world consisted in confining the population, this made it possible to reduce the number of reproduction. In the theoretical framework, to estimate the number of reproduction, few authors have used mathematical models with spatial diffusion. However, to predict the evolution of the epidemic, researchers have used mathematical models of propagation such as the SIR epidemiological model (see, for example, [1]). In our work, we considered a SCIRD-type model of COVID-19 transmission in which we include mortality terms and spatial diffusion. Indeed, let us assume the most realistic situation of geographical spread of epidemics; moreover, we will consider the case of disturbances and incomplete data. The prediction of unknown data with the confrontation of real parameters is sometimes necessary, and our objective is to show that by using the sentinel method of JL Lions, a mathematical answer can nevertheless be given. More precisely, let be an open and bounded domain of . For the time , set , . We denote by the outer normal on . Then, consider the following problem:
The function represents susceptible persons at risk of contacting COVID-19 at time at the point , the function represents carriers (dead corpse) that transmit the COVID-19 at time at the point , the function describes infective persons capable of transmitting the COVID-19 to persons at risk at time at the point , the function represents recovered persons who have been treated of COVID-19, and the function gives the total number of deaths at time at the point .
It should be noted that this type of mathematical model without spatial diffusion has already been studied by Atangana [2], it uses Italian data to propose a mathematical model of COVID-19 transmission. The COVID-19 problem have been studied by several authors. In [3], Tuite et al. used an age-structured compartmental model to analyze the transmission of COVID-19 in the population of Ontario, Canada. In [4], Zhao et al. modeled the epidemic curve of the 2019-ncov case time series in China while accounting for the impact of variations in disease reporting rate. In [5], Wang et al. estimated the time-varying reproduction numbers in China, using the revolving equation determined by the serial interval (SI) of COVID-19. For the model (1), we have
The parameters of the considered model are presented in Table 1.
In a biological explanation of the model, let be the density of the total population . As there is a term of spatial diffusion , the total population is not constant, and therefore, .
The term represents the disturbance around the infected population whose density is not completely known; is the missing term of the population likely to be infected. We suppose that and also that . It is the real parameters and which are unknown; the other parameters are known. If we add the first five equations of the model (1) we are studying, we get the simplified system: where . Therefore, in the following, system (3) is considered and we assume the following:
(H1):
(H2):
Remark 1. It is assumed that the initial data , , , , and are known function ant belong to . Under the assumptions (H1) and (H2), according to Lions and Magenes [6], system (3) of parabolic type admits a unique solution such that
Now, the problem is as follows: is there a method to get information about the term (so-called pollution term) from the infected population, insensitive to the missing term from the susceptible population (so-called perturbation term)? In other words, we are interested in identifying the unknown parameters in model (3) which has incomplete data; more precisely, we are interested in the identification of pollution in the COVID-19 phenomenon, in the sense that we know little about the initial data as well as certain missing terms. The position (internal) and the nature of pollution (COVID-19) are known, but we do not know its amplitude (number of people infected by COVID-19). Our goal is to find a method to obtain information on these amplitudes.
A partial answer can be obtained using the least squares method which is to take the unknowns, as control variables, and we can not clearly separate and . The sentinel method of Lions [2] provides the right answer to this type of problem. In this paper, we construct sentinels when the supports of the observation function and of the control function are included in two different open subsets of (see Nakoulima [7]).
Several authors studied the sentinel problem. We refer to [7–9] and the references therein. In [10], Ainseba et al. used the method of sentinels to identify parameters of pollution in a river. Bodart and Demeestere applied it in [11] to identify an unknown boundary. In [12], Soma and Sawadogo studied the boundary sentinels with given sensitivity in population dynamics problem. In [9], Mophou and Nakoulima studied the problem of sentinels with given sensitivity. Sawadogo introduced in [13] the distributed sentinels into the equation of the dynamics of populations to study a population subject to a migratory phenomenon. Recently in [14], Hamadi et al. have studied the existence of construction of a sentinel for the epidemiological model SIR with spatial diffusion. In this paper, we apply the sentinel method to identify parameter in a mathematical model of COVID-19 transmission with spatial diffusion and incomplete data which takes into account the possibility of transmission from dead populations to susceptible populations. The problem is as follows: given , find a control w in such that if is solution of (3) and is defined by (6); then (7) and (8) hold.
The remainder of this paper is as follows: in Section 2, we briefly explain the concept of the sentinel In Section 3, we establish the equivalence between sentinel problem and null controllability problem. Sections 4 and 5 are devoted, respectively, to preliminary results and proof of main result. In Section 6, we formulate the information given by sentinel. In Section 7, we highlight the numerical simulation of the considered model solution.
2. Sentinel Method
The sentinel concept relies on the following three objects: some state equation (for instance), some observation function, and some control function to be determined.
A state equation represented here by (3) and we denote by depends on two parameters and the unique solution of (3).
An observation on nonempty open subset is called the observatory set. The observation is in , for the time . We denote by this observation
A function called “sentinel”. Let . Let on the other hand be some open and nonempty subset of such that .
For a control function , we define the functional
We say that defines a sentinel for the problem (1) if there exists such that is insensitive (at first order) with respect the to missing terms , which means where here corresponds to and is of minimal norm in . That is,
3. Null Controllability Problem
We show in this section that the existence of the sentinel comes to null controllability property. We begin by transforming the insensibility condition (7).
Set
Then, the function is solution of
Problem (10) is linear and has a unique solution . The insensibility condition (7) holds if and only if
We can transform (11) by introducing the classical adjoint state. More precisely, we define the function as the solution of the backward problem:
Since and , the adjoint problem admits a unique solution given by . The function depends on the control that we shall determine:
Indeed, if we multiply the first equation in (12) by and we integrate by parts over , we obtain
So, condition (7) (or (11)) is equivalent to
Thus, sentinel problems (6), (7), and (8) are equivalent to the following null controllability problem: given , find a control in such that if is the solution of (12), then (8) and (14) hold.
In the following, we set
4. Existence of a Sentinel
For the study of the existence of a sentinel, we use a method developed in [15].
We begin with some observability inequality. Using (15), we have the following:
Proposition 2. Let be ; then, there exists a positive constant such that where positive with bounded
Proof. See Fursikov and Imanualov [16].
According to the RHS of (16), we consider the space endowed with the bilinear form defined by
According to Proposition 2, this symmetric bilinear form is a scalar product on .
Let be the completion of with respect to the norm
Then, is a Hilbert space for the scalar product and the associated norm.
Remark 3. We can precise the structure of the elements of . Indeed, let be the weigthed Hilbert space defined by endowed with the natural norm This shows that is imbedded continuously in : .
Now if and , then from (16) and the Cauchy-Schwartz inequality, we deduce that the linear form defined on by is continuous. Therefore, from the Lax-Milgram theorem, there exists a unique solution of the variational equation:
Proposition 4. Assume that , and let be the unique solution of (21). We set and
Then, the pair is such that (12)–(14) hold (i.e., there is some insensitive sentinel defined by (6) and (7)).
Moreover, we have where is a constante and .
5. Construction of the Sentinel
Having shown the existence of a sentinel, our goal now is to build this sentinel for the (1). Note that the existence of this sentinel justifies the existence of the optimal control.
5.1. Existence of the Optimal Control
For the following, we will consider the following optimization problem:
Theorem 5. There is a unique couple solution of the problem .
Proof. By Proposition 4, the domain is nonempty. On the other hand, the map is continuous, coercive, and strictly convex; then, we deduce that there is a unique solution for the problem (P)
5.2. Penalization Method
In this subsection, we are concerned with the optimality system for . Since a classical way to derive this optimality system is the method of penalization due to Lions [8], here, we use this method. For this, we introduce the penalized cost function: where .
We consider the following problem : with
The following proposition gives the existence of the solution for .
Proposition 6. It is assumed that the assumptions of the previous section are satisfied. Then, there is a couple a as unique solution of the problem .
Proof. Since and , consequently, is nonempty and is closed. Moreover, is continuous, coercive, and strictly convex. Then, the problem has a unique solution denoted by .
Now, we study the convergence of the couple when . For this, we have the following result:
Proposition 7. Let the unique solution of the problem . Then,
Proof. As is the solution of , then, especially for . Inequality (30) is written which implies Thus, From (33), we conclude that is bounded in . So On the other hand, because we have and according to the inequality (34), we have With Given solution of (36), then . Moreover, we have Since , then, where is a constant. Therefore, there is a subsequence denoted by , such that in . As the injection of into is compact and by passing to the limit, we finally find On the other hand, is convex and continuous; then, Using in (41), the estimate We get Finally, as is the unique solution of , then, . On the other hand is solution of (40) and by uniqueness of the solution for the heat equation, we deduce that .
The following proposition gives the optimality system for the pair .
Proposition 8. The problem admits an optimal solution if and only if, there exists a unique function such that is the solution of the following optimality system: with and with
Proof. Since is the unique solution of then, by applying the Euler-Lagrange, optimality conditions, we find
or
From the definition of the functional and the linearity of the operator , is written:
So
There remains
Passing to the limit, we get
So
A calculation analogous to that above in equation (48) gives
We introduce the following adjoin state defined by
So (53) and (54) become, respectively,
and
We deduce
From where
Thus, for in , we deduce that
On the other hand, with ; then by application of the Lions-Magenes theorem, the trace function exists.
We multiply (60) by , and we integrate by part over ; we get
In particular for and the fact that we have (54), then (61) becomes
Finally,
5.3. Optimality System (OS)
In this subsection, we will state an important result which gives the optimality system for the problem , characterizing the optimal control, which also implies the construction of the sentinel.
Theorem 9. The pair is the unique solution of , if and only if there exists a function such that the triplet is the solution of the following optimality system:
Proof. From the following equality, and from the fact satisfies the inequality (33), it comes and since we deduce that According to (66) and (68), we conclude that is bounded in . Therefore, there are a subsequence and a function in such that Thus, On the other hand, from the previous proposition, we have By uniqueness of the limit, we deduce that
6. Information Given by the Sentinel
We assume that the population density is observed on , so
Because of (7), we can write
In (6), is observed and using (3)
So (7) becomes where .
From (6), we have where here and denote the characteristic functions of and , respectively.
The derivative only depends on and other known data. Consequently, the estimate (76) contains the information on (see for details Remark 10 below).
Remark 10. The knowledge of the optimal control provides informations about the pollution term , and let be the solution of Multiplying (78) by and integrating by parts over , we get So from (16) and (77), we deduce
7. Numerical Simulation of the Studies Model Solution
In this part, the idea is to highlight the numerical simulation of the solution of model (1), for . The purpose is to present the method of lines (MOL) solution of the COVID-19 modeling equations, a system of five partial differential equations (PDEs), describing the interaction resulting between susceptible persons at risk of contacting COVID-19, carriers (dead corpse), infective persons, recovered persons, and number of deaths. The method of lines is a semidiscrete method which involves reducing an initial boundary value problem to a system of ordinary differential equations (ODEs) in time through the use of a discretization in space. The resulting ODE system is solved by applying solver ode15s of MATLAB. This method provides very accurate numerical solution for linear or nonlinear PDE’s in comparison with other existing methods. For initial conditions, let us look at the case of Ouagadougou city, Burkina Faso, where population is estimated to 3 million. Notice that Burkina Faso population is around 20 million. Towards the beginning of March 2020, the number of people infected was around 3. In order to make our model fit that data, the initial conditions (day zero) are , , ; ; . One can note that parameter values are not for Burkina Faso excepted rate of natural death parameter. One must note that for all the following figures that the legend is data 1 for , data2 for , data 3 for , and data 4 for Using the values of the parameters in Table 1, we obtain the following graphs.
Figure 1 shows that susceptible persons’ number decreases progressively with a peak at [day].
Figure 2 shows that the peak of the pandemic is reached on the tenth day, before beginning to decrease.
According to the numerical simulation of Figure 3, the number of infected persons evolves in the same manner as those of susceptible persons and carriers.
Figure 4 shows that number of people cured increases.
Figure 5 show that The number of deaths linked to COVID6-19 is decreasing.
The results are due to the fact that one does not have accurate data on COVID-19 death rate.
Remark 11. According to observations made after the numerical simulation of the COVID-19 model, it is very important to have real data that will allow us to do a good simulation with a good interpretation of the results in order to make them useful for decision makers.
8. Conclusion
In this paper, using a mathematical model of COVID-19 transmission, we have determined the number of people infected with COVID-19 by the sentinel method. Indeed, we first studied the existence and the construction of a sentinel for the system (3). We have shown that the search for a sentinel is indeed equivalent to the study of controllability. The null controllability is ensured by an observability inequality obtain through Carleman’s estimates. Then, we used the constructed sentinel to obtain information on the number of people infected with COVID-19. Finally, we made a numerical simulation of the studied model solution. In perspective, in the concrete applications of singular nonlinear problems in epidemiology, the problem of controllability takes on its full meaning for epidemiological models. For example, consider the following system: as well as the functional and we will then seek to determine the control in such a way that it brings the number of infected to zero after a time T.
Data Availability
No underlying data was collected or produced in this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.