Set-Valued Control Approach Applied to a COVID-19 Model with Screening and Saturated Treatment Function

The purpose of this paper is modelling and controlling the spread of COVID-19 disease in Morocco. A nonlinear mathematical model with two subclasses of infectious individuals is proposed. The population is divided into ﬁve classes, namely, susceptible ( S ), exposed ( E ), undiagnosed infectious ( I nc ), diagnosed patients ( I c ), and removed individuals. To reﬂect the real dynamic of the COVID-19 transmission in Morocco, the real reported data are used for estimating model parameters. Two controls representing screening eﬀort and limited treatment are considered. Based on viability theory and set-valued analysis, a Lyapunov function is constructed such that both exposed and infected populations are decreased to zero asymptotically. The corresponding controls are derived via a continuous selection of adequately designed feedback map. Numerical simulations are presented with three scenarios (cases when each control is used alone and the case when two controls are combined). Our results show that when only one control is to be applied, screening is the most eﬀective in decreasing the number of people in the three infected compartments, whereas combining both controls is found to be highly eﬀective and leads to a signiﬁcant improvement in the epidemiological situation of Morocco. To the best of our knowledge, this work is the ﬁrst one that applies the set-valued approach to a controlled COVID-19 model which agrees with the observed cases in Morocco.


Introduction
Coronavirus disease 2019, known as COVID-19, is a disease caused by a novel betacoronavirus named SARS-CoV-2 that affects the lower respiratory tract [1]. It is a fatal disease for patients who developed various complications including organ failure, septic shock, and pulmonary oedema (see [1,2]). Since its appearance in December 2019 in Wuhan, Hubei Province, China, the virus has extended internationally [3], and the number of reported cases has accelerated around the world. is is why, on 20th January 2020, the World Health Organization (WHO) declared COVID-19 as a public health emergency of international concern [4].
Given the extent of the COVID-19 disease and the urgency to deal with it, mathematical models are a powerful tool for the management and control of this pandemic. Nowadays, the contribution of the epidemiological models in the fight against infectious diseases is indisputable. Indeed, they provide important information on the dynamics of disease transmission, give an overview of the characteristics of an epidemic, and allow predicting its future evolution and evaluating different intervention strategies to find the best control program. For more details on the use of mathematical models in epidemiology, we refer the interested reader to [5][6][7][8][9][10].
Several mathematical modelling studies have been proposed, which aim to understand the dynamics of this disease and provide appropriate responses to the challenges that it presents. For instance, in [11], a mathematical model considering susceptible, exposed, infected, asymptotic, quarantine/isolation, and recovered classes as in the case of COVID-19 disease was developed. Elasticity and sensitivity analyses are performed, and the global stability for the proposed model is studied. e study in [12] proposed a fractional model to describe the dynamics of COVID-19 and parametrized the model using the available infection cases. In [13], the authors proposed a compartmental model taking hospitalized and asymptomatic cases as extra compartments for COVID-19 disease. e local stability of the disease-free equilibrium in terms of the basic reproduction number is investigated, and the sensitivity analysis of the model with respect to the variation of each one of its parameters is provided. Using a generalized SEIR model and based on the public data, Peng et al. [14] estimated the key epidemic parameters and made predictions on the inflexion point and possible ending time for 5 different regions in China. Based on the collected epidemic data in China, the authors proposed in [15] an extension of the classical SEIR model. ey divided the population into 6 different categories and established the time series models based on different mathematical formulas according to the variation law of the original data. Extensions of the classical SIR model by adding time delays were also proposed in [16,17].
To contain the spread of COVID-19, almost all the countries affected by this pandemic have implemented severe control measures such as complete lockdown, social distancing, isolation, early detection, and the implementation of a robust system to trace contacts. In this context, some authors proposed optimal control frameworks for determining the optimal strategies that can mitigate the negative impact of this disease. e study in [18] considered a mathematical model including a quarantine class and governmental intervention measures such as lockdown, media coverage on social distancing, and improvement of public hygiene. To reduce the infected individuals as well as to minimize the cost of implementing government control measures, an optimal control problem is formulated and solved, where the control considered represents the awareness due to media coverage. e authors in [19] considered two control SEIR-type models describing the spread of COVID-19 in a human population. ey used one control representing the quarantine intensity and tried to find the optimal solution by applying Pontryagin's maximum principle. In [20], a model with five compartments was proposed to highlight the role of infected people without symptoms, infected individuals with mild symptoms, and the complicated cases.
en, an optimal control problem was formulated to lessen the infected individuals by carrying out awareness campaigns, diagnosis, surveillance of airports, and quarantine of infected. In [21], the authors proposed stochastic and deterministic models to investigate the transmission mechanism of 2019-nCoV from 15 January to 5 February 2020 in Hubei province. Stochastic and deterministic analyses were performed, and optimal strategies were provided for the deterministic model. For more examples of works that address the problem of COVID-19 control, we refer to [22][23][24][25][26][27][28].
According to Rismanbaf [29], there is still no vaccine or definitive treatment against the SARS-CoV-2 virus. However, new potential therapies are suggested based on in vitro studies, virtual screenings, and records of their effectiveness on earlier strains of coronavirus, SARS, and MERS. Furthermore, it is well known that lockdown has a colossal economic and social impact; this is why we have to think of alternative solutions. Inspired by the experience of South Korea, which has shown that mass testing is crucial to control COVID-19 disease, a solution to be explored would, therefore, be the extension of screening capacities. In this context, we propose a mathematical model that describes the COVID-19 transmission dynamics, and we investigate the impact of implementing additional treatment protocol and/ or the multiplication of the number of screening tests on the spread of this disease in Morocco. Novelty and the main contributions are as follows.
Firstly, based on the collected epidemic data and analysis of the actual situation in Morocco, we propose an extension of the classical susceptible-exposed-infectious-removed (SEIR) model. In our model, infectious people are divided into two subclasses depending on whether they are diagnosed positive and isolated or they are still undiagnosed. Considering these two categories (diagnosed and undiagnosed patients) will allow us to emphasize the importance of detection effort to fight against the COVID-19 pandemic. Furthermore, by estimating the model parameters using the real data, we obtained a more realistic model that reflects the dynamic of this disease in Morocco. Note that this model could be adapted easily to other countries. en, we consider two controls representing mass screening and limited treatment. For the treatment control, we use a saturated function to take into account the constraints on the health offered in this period of crisis such as insufficient human and material resources and the limitations related to the effectiveness of treatment. To investigate the impact of these intervention strategies on the spread of COVID-19, we formulate a control problem which we solve by a new direct approach. It is important to note that most of the control frameworks indicated above (paragraph 4) used an indirect approach based on the maximum principle of Pontryagin [30], which requires to design a suitable objective functional, prove the existence of an optimal solution, and solve the optimality system consisting of the state and adjoint systems that can be hard to solve. Motivated by this, we propose an alternative approach based on viability theory [31] and set-valued analysis [32]. Our approach enables the user to establish explicit formulas for the feedback controls. More precisely, we construct an appropriate Lyapunov function such that both exposed and infectious individuals decreased to zero asymptotically. e expression of our controls is obtained via a continuous selection of adequately designed feedback map.
Finally, to show the effectiveness of our theoretical results, we provide numerical simulations with different scenarios. To 2 Complexity our knowledge, the present work is the first one that proposes a controlled dynamical model for COVID-19, which is fitted to the existing epidemic data in Morocco, and where the control problem is resolved using a set-valued approach. e remainder of the paper is structured as follows: In Section 2, we introduce the compartmental model of COVID-19. Section 3 will be devoted to treat the control problem of COVID-19 by using a set-valued approach. In Section 4, we present numerical simulations with three scenarios in order to show the efficiency of our control approach. Conclusions are given in Section 5.

Model Construction.
e model under consideration in the present paper consists of five coupled nonlinear differential equations which model the dynamic of the COVID-19 disease. We consider that the total population N is divided into five different compartments: (i) Susceptible individuals (S), people who may be infected by the virus (ii) Exposed individuals (E), infected with the virus but without the typical symptoms of infection (iii) Undiagnosed infectious individuals (I nc ), who are infectious but not yet confirmed by a test (iv) Diagnosed infectious individuals (I c ), who are diagnosed as COVID-19-positive patients and are isolated in hospitals (v) Removed individuals (R), people who moved out either by recovery or death We assume that the transmission of COVID-19 occurs following adequate contact between a susceptible and undetected infectious I nc . As the infectious people in I c are isolated, we assume that they have no contact with the susceptible and do not transmit the disease. Due to the nonlinear contact dynamics in the population, we use the incidence function β(I nc /N)S to indicate successful transmission of COVID-19, where β denotes the infection contact rate. We assume that all newly infected individuals enter into the exposed compartment E, where they reside for an average of k − 1 days (k is the rate at which individuals leave the latent class by becoming infectious). A proportion p of the infectious individuals are diagnosed and isolated, and they followed a therapeutic protocol and entered the compartment I c . e remaining infectious patients are considered as free infectious people as they are not yet detected. Among the infectious who are undiagnosed, some of them are diagnosed and isolated at a rate c. We have observed from the clinical data that there was an increasing change within time for the number of individuals who progress from I c to class R; to reflect this, we assume that people in I c enter into the removed class with a time-dependent variable r(t), which is derived from the following logistic function: where t (1/2) represents the time at which r(t) reaches its half value and Δ determines the width of r(t). e parameters r 0 and r f model asymptotic values.
A flowchart of our model is given in Figure 1. e dynamics of the model is governed by the following differential equations: with initial data (2) describes the temporal evolution of human populations, we shall show that the solutions remain nonnegative and bounded. From system (2), we have

Positivity and Boundedness of Solutions. As model
We observe that all the aforementioned rates are nonnegative on bounding planes of the nonnegative cone of R 5 .
us, if we begin in the interior of this cone, we shall always remain in this cone as the direction of the vector field is inward on all the bounding planes. So, all the solutions of system (2) are positive. Furthermore, as we denote the total population by , and by adding equations in (2), it is straightforward to prove that all solutions S, E, I nc , I c , and R are bounded by N(0). Hence, the model is well posed in a biologically feasible domain given by the following positive invariant set:

Parameter Estimation.
Let us now estimate parameters of model (2), as well as the unknown initial conditions E 0 and I nc 0 . Firstly, recall that the progression rate, k, from E to the infectious stage is supposed to be equal to the inverse of the mean incubation period [10]. So, to calculate k, it suffices to know the average incubation period of COVID-19. e study in [33] showed that the incubation period ranges from 2 to 14 days with 95% confidence and has a mean of around 5 days. Another analysis [34] estimated the median incubation period to be 5.1 days (95% confidence interval, 4.5 to 5.8 days). A result from [35] indicated a mean incubation period of 5.2 days (95% CI, 4.1 to 7.0). us, we set the mean incubation period as 5 days and fix k � 1/5 per day. For the remaining parameters and initial conditions E 0 and I nc 0 , they will be estimated by fitting the model with the collected data, communicated by the Ministry of Public Health in Morocco (see Table 1), using the MATLAB routine fminsearch. e resulting estimations are listed in Table 2, and the best model fit to the real data is shown in Figure 2. Note that, for the parameters of the function r(t), the best fit to real data gives r f � 0.2352, r 0 � 0.0096, t (1/2) � 70.3434, and Δ � 10.8262.

Control by a Set-Valued Approach
In this section, we applied a set-valued approach to the control problem of COVID-19. For more details about this approach, we refer the reader to the works in [38,39], where this approach is used to deal with the problem of asymptotic null-controllability. Note that, in [39], the proposed approach is used for epidemic models with only one infectious compartment, while the main added value here is to control the spread of the disease and decrease the number of infected people in several infectious compartments to zero asymptotically.

Control Problem Statement.
In Morocco, as in several countries around the world, government authorities have opted for severe confinement as a strategic measure to limit the spread of COVID-19 in the country. Of course, this strategy is effective to avoid large losses in human lives, but it can lead to serious economic and social consequences, especially if the duration of the confinement is long. Other measures such as mass screening and treatment can also be implemented to limit the damage and allow a safe return of economic activity. To study the impact of mass screening and treatment on the dynamics of COVID-19 in Morocco, we introduce two controls u 1 and u 2 into model (2), where (i) u 1 represents all the efforts which allow the multiplication of the number of screening tests such as the widening of diagnostic sites in professional environments in commercial and industrial units, the detection of contact subjects in family clusters, and the increase of the number of laboratories authorized to carry out COVID-19 tests. (ii) u 2 represents the treatment of people confirmed to be COVID-19 positive and who are placed in isolation in hospitals. It is true that there is not yet a specific drug or vaccine for COVID-19, but some studies have shown that protocols combining several drugs have proven their effectiveness (a review on potential treatments for COVID-19 can be found in [29]). It is often assumed that when the resources for treatment are sufficient, the term of treatment is a linear function to infected individuals. However, in the case of COVID-19, as there are constraints and limitations in terms of medical resources, drugs, hospital beds, adequate isolation spaces, and the effectiveness of treatment, it would be more appropriate to consider a saturated treatment function where the control u 2 is the treatment rate and d ≥ 0 is the saturation factor that measures the effect of the infected being delayed for treatment. For examples on the use of the saturated treatment function in epidemiological modelling, we refer to [40][41][42].

Remark 1
(i) It is easy to see that the solution of system (9) remains nonnegative and bounded for all t ≥ 0; therefore, we do not need to impose (in the context of our approach) this constraint on the state of the system. (ii) It is also noteworthy to mention that the notion of the invariant set coincides with the viability one when there exists a unique solution of system (9). We refer the interested readers to [31].

Set-Valued
Approach. Now, we return to problems (11a) and (11b), which we will treat in the context of viability theory and set-valued analysis stated in Appendix. erefore, in order to establish our control method, first we need to extend the dynamics of COVID-19 model (9) with the control dynamics governed by the following differential equation: where v is the new control that has values in the closed subset with α 1 and α 2 being positive numbers. If the initial state of dynamical system (12) belongs to U, i.e., u 0 � u(0) ∈ U and v ∈ V, then the solution of differential equation (12) also belongs to U, i.e., u(t) ∈ U for all t ≥ 0, since the solution of the differential equation satisfies the following inequality: for i � 1, 2. According to Definition A.2, a convenient R 3 + -Lyapunov function related to problems (11a) and (11b) can be given by φ(y, z) � 〈v, y〉 +〈δ, z〉, for all (y, z) ∈ R 3 × R 3 , (15) where ] and δ denote vectors of R 3 + whose both coordinates are positive. Indeed, if a C 1 function w: R + ⟶ R 3 + satisfies φ(w(s), _ w(s)) ≤ 0, for all s, then the real nonnegative function q � δ 1 w 1 + δ 2 w 2 + δ 3 w 3 is such that As a result, q(s) ⟶ 0 at infinity. ereby, both functions w i do so for all i � 1, 2, 3. en, condition (11b) can be written as viability of the subset under the augmented control system Ultimately, consider the feedback map G φ , for each (x, y, u) ∈ D φ : Let where π 1 denotes the mapping (x, y, u) ⟶ (x, y). en, we are ready to show the following result.

Theorem 1. Any continuous selection of the map G φ provides a solution of problems (11a) and (11b) for each
Proof. For such a selection ζ, it follows that Since D φ is closed, f, h, and ζ are continuous; then, by using Lemma A.1, we deduce that subset D φ is locally viable under system (18). Since both f and h have linear growth and ζ is bounded, subset D φ is viable under system (18). Now, let (x 0 , y 0 ) belong to subset Ω φ and u 0 be such that (x 0 , y 0 , u 0 ) ∈ D φ ; then, system (18) anks to equation (17), this solution satisfies φ(y(s), _ y)(s) ≤ 0, for all s ≥ 0.
By taking (x 0 , y 0 ) ∈ R 6 + and using Gronwall inequality, we get y(s) ⟶ 0 as s ⟶ ∞. Furthermore, since ζ(x, y, u) ∈ V, thanks to inequality (14), u(s) ∈ U for all s ≥ 0. Ending the proof, problems (11a) and (11b) have the solution from Ω φ ∩ R 6 + . In what follows, we will provide the explicit expression of the feedback map G φ (·) of equation (19) in order to get its continuous selection as required by eorem 1. Now, let us set ϕ(x, y, u) ≐ φ(y, h(x, y, u)). (23) en, according to equation (15), we get and let

Complexity
According to Lemma A.2, we can easily get the expression of the contingent cone T D φ (·) as follows.
Hence, we are ready to determine a useful expression of the feedback map G φ (·) given by (19). To that end, we need to define the following functions and map: for all (x, y, u) ∈ R 8 . It is noteworthy here that the control affine structure of the system involves that function ℓ φ is affine in control u, and function m φ depends only from state (x, y). Consequently, we are ready to state the following result.

Lemma 2. We have, for each
Proof. We can easily get expression (30) by using Lemma 1 and considering equations (19) and (29).
Recall that, in eorem 1, we have proved the case where problems (11a) and (11b) have a solution from Ω φ ∩ R 6 + . In the next theorem, we will treat these problems from R 6 + . ereafter, we need to introduce the following map, for each σ > 0: for each (x, y, u) ∈ R 8 . □ Theorem 2. Assume that, for some σ > 0, the map C σ φ given by (31)
Consequently, the control given by u on [0, s 1 ] and w on (s 1 , ∞) leads to the solution of problems (11a) and (11b) from (x 0 , y 0 ) .

Numerical Simulation
In this section, we provide some numerical simulations using MATLAB to validate the theoretical results obtained in the previous section. To investigate the impact of controls u 1 and u 2 on the spread of COVID-19 in Morocco, we will consider three different control scenarios. e first one is when the control u 1 (mass screening) is used alone. e second scenario is when only treatment (control u 2 ) is applied. e last one is when controls u 1 and u 2 are coupled. For our numerical simulations, we consider parameter's values and initial conditions given in Table 2. e control period is from March 20, 2020, to May 12, 2020 (t f � 53 days). It should be noted that in order to give the continuous selection of feedback map C σ φ (·), for each scenario, as expressed in equation (31), we first derive expressions of the functions ℓ φ and m φ given, respectively, by equations (27) and (28). In what follows, r ′ (t) represents the time derivative of r(t), and its expression is given by Scenario 1. Control with mass screening.
In this case, we assume that only mass screening, u 1 , is implemented, whereas the control u 2 is set to zero. e functions ℓ φ and m φ are given by where m φ,1 � (δ 3 − δ 2 )cI nc . en, a continuous selection that provides a solution of problems (11a) and (11b) is given by In Figures 3(a)-3(c), we depict the evolution of the exposed individuals (E) and infectious individuals in compartments (I nc ) and (I c ) over time in both cases with and without control. We observe in these figures that mass screening significantly reduces the number of the infected people. At the end of the 53-day period, the number of the infected individuals in E, I nc , and I c is reduced by 82.7%, Complexity 9 84.3%, and 63.4%, respectively. Figure 3(d) shows the control u 1 as a time-dependent function. We can see, in this figure, that the control u 1 should be implemented with full effort over 40 days.

Scenario 2. Control with treatment.
With this scenario, only treatment is employed as a control strategy, i.e., u 1 � 0. In this case, expressions of the functions ℓ φ and m φ are given by

10
Complexity where m φ,2 � −δ 3 (I c /(1 + dI c )). en, a continuous selection that provides a solution of problems (11a) and (11b) is given by Figures 4(a)-4(c) display the dynamics of the state variables E, I nc , and I c with and without control u 2 . It is clear from these figures that the control u 2 effectively reduces the number of people who are diagnosed and isolated, while the exposed and the undiagnosed individuals are unaffected by this control. e result in Figure 4(c) indicates that the number of people detected who are isolated decreases with treatment by 19.5% at the end of the control period. us, recovery of infective via treatment can reduce the burden on the health system. In addition, from Figure 4(d), one can note that, in order to reduce the number of the isolated individuals, the control u 2 should be intensively used during almost the whole control period.  12 Complexity en, a continuous selection that provides a solution of problems (11a) and (11b) is given by Figures 5(a)-5(c) show a comparison among the number of the infected individuals in the compartments E, I nc , and I c in the uncontrolled case and when the two controls are combined. We observe, in Figures 5(a) and 5(b), that the curves are almost similar to the case when only the control u 1 is applied (Scenario 1). However, we can note from Figure 5(c) that the result obtained for the confirmed and isolated cases (I c ) is more interesting than that in Figures 3(c) and 4(c). us, at the end of the 53 days, the number of people in the compartment I c , when the controls u 1 and u 2 are used together, is reduced by 81.6% instead of 63.5% in the first scenario and 19.5% in the second scenario. Furthermore, it is important to note that when two controls are activated, the control u 1 should be used with a full effort, while the control u 2 has to be applied with its maximum level during 36 days with a gradual reduction at the end of the control period (see Figure 5(d)).
Overall, the numerical results under various interventions show that if we have to choose only one control, the screening control is the most useful because it allows a significant decrease in all the infected compartments. In addition, combining the two controls contributes positively to the improvement of the final results, particularly by reducing the number of people in the I c compartment.

Concluding Remarks and Discussion
In this work, we were interested in modelling and controlling the COVID-19 pandemic in Morocco. We have proposed an extension of the SEIR model that is adapted to the epidemiological data in Morocco. us, in addition to the susceptible (S), exposed (E), and removed (R) compartments, we have divided infectious patients into two classes. e first class (I c ) involves infectious people who are confirmed by a test and placed in isolation in hospitals. e second class (I nc ) contains infectious individuals who are not yet tested. is distinction between detected and undetected people seems more realistic than considering a single compartment that includes all the infectious people. It highlights the role of screening in the dynamics of COVID-19 transmission. Also, it helps to differentiate the isolated infectious who no longer contribute to the transmission of the disease from those who are still free and participate positively to the spread of COVID-19. Furthermore, to have a model that reflects as much as possible the epidemiological situation of Morocco, we have estimated the parameters of our model from official data published by the Ministry of Public Health in Morocco from March 20, 2020, to May 12, 2020.
To reduce the number of infected people in the compartments E, I nc , and I c , we formulate a control problem based on the proposed model to which we added two controls representing mass screening and limited treatment.
is control problem is solved by a new approach based on results from the viability theory which presents the advantage of directly finding the expression of the considered controls. More precisely, we have proposed a set-valued approach based on viability theory and set-valued analysis that allows deriving controls via continuous selections of the designed feedback map.
Numerical simulations are carried out for three different scenarios, namely, the application of mass screening only, the adoption of a control strategy based on treatment only, and finally, the combination of these two intervention measures (screening and treatment). e results obtained have shown that when treatment is applied alone, it reduces only the number of people hospitalized, whereas mass screening has a significant effect on the three infected compartments E, I nc , and I c . Our results show that when screening is implemented alone, the number of E + I nc + I c at the final time decreased by 71%, while there is only a reduction by 19.5% when treatment is used alone. Hence, we conclude that between applying screening only and treatment only, screening is the most effective control intervention. Furthermore, combining screening and treatment is found to be better since the reduction of the total number of the infected at the final time reaches 83%. g(z) ∈ T D (z), for each z ∈ D.

(A.3)
In addition, if g has a linear growth, then D is viable under system (A.1).
Recall that function g mapping a subset D of R n into R n is said to be of linear growth if there exists a constant k > 0 such that |g(z)| ≤ k(1 +|z|), for all z ∈ D. (A.4) Remark A.1. Nevertheless, such a condition is not required when function g is bounded on subset D or, in particular, when the latter subset is bounded. Next, we present a result which was established by Aubin and Frankowska [32]. It will be fundamental for building our control laws. Let ϕ: R n ⟶ R be a differentiable function, and consider the subset D≐ z ∈ R n |ϕ(z) ≤ 0 .
(A.5) en, we have the following result.
A selection of the set-valued map Q is a mapping ζ such that ζ(z) ∈ Q(z) for all z. Ultimately, the Lyapunov function considered in this paper is introduced as follows [38].
Definition A.2. Let Y stand for a nonempty subset of R N and φ: Y × R N ⟶ R N , a C 1 real-valued function. We call φ an Y-Lyapunov function if the following holds:

Data Availability
e data used to support the findings of this study are included within the article.