MATHEMATICAL STUDY AIMING AT ADOPTING AN EFFECTIVE STRATEGY TO COEXIST WITH CORONAVIRUS PANDEMIC

In this paper, we propose a discrete mathematical model that describes the evolution of the ”covid-19” virus in a human population and the efforts made to control it. Our objective is to develop a simple, logical and an optimal strategy to reduce the negative impact of this infectious disease on countries. This objective is achieved through maximizing the number of people applying the preventive measures recommended by WHO against the pandemic in order to reduce the infection as much as possible. The tools of optimal control theory were used in this study, in particular Pontryagin’s maximum principle.


INTRODUCTION
According to WHO, Covid-19, which appeared in the fourth quarter of 2019 in China, belongs to the coronavirus family which has a large number of viruses that can cause mild diseases by facial contact.
Most commonly, it causes mild to moderate respiratory illnesses like the common cold with symptoms such as: Á a general feeling of discomfort More seriously it can cause respiratory diseases of the lower respiratory system such as pneumonia or bronchitis, particularly in people with cardiopulmonary disease and those whose immune system is weakened and the elderly.
There is currently no vaccine available to protect against coronavirus. However, scientists at Pasteur Institute are multiplying initiatives to find a vaccine against this virus. They have managed to isolate and culture strains of this new virus and they are working on developing a vaccine within a year and a half, said researchers from the National Reference Center for Respiratory Viruses in a press release dated January 30, 2020. Researchers at the Pasteur Institute now have the virus that causes this infection. This viral isolation opens the way to new diagnostic, therapeutic and prophylactic approaches.
To reduce the risk of coronavirus infection, the WHO recommends the following preventive measures: Á OE washing hands regularly with soap and water for at least 20 seconds (not to mention the handy hydro-alcoholic solutions when you cannot wash your hands) Á avoiding touching the eyes, nose or mouth when hands are not washed Á avoiding contact with sick people Á wearing a mask

BACKGROUND
Since the onset of the coronavirus pandemic, researchers around the world have focused on studying the virus and finding effective strategies to combat it. For example in [5], the authors tried to develop a multiregional model in which they introduced controls representing four different strategies. The first represents the effort of awareness campaigns in region j that aims at introducing health measures to protect individuals from being infected by the virus (covering all cuts, shoes and long sleeve shirts when handling animals, washing hands thoroughly on a regular basis, taking shower and cleaning up work place and home...), the second represents security campaigns and health measures to prevent the movement of individuals from region j to region r , the third represents the effort to encourage the exposed individuals to join quarantine centers in the same region and the fourth is the effort to dispose of the infected animals.
In the article Optimal Control of the COVID-19 Pandemic with Non-pharmaceutical Interventions [1], the authors modeled the evolution of the pandemic by introducing a vaccinated population by ordinary differential equations and they introduced an optimal control aiming to minimize deaths.
In the article A Mathematical Modeling with Optimal Control Strategy of Transmission of COVID-19 Pandemic Virus [7], the authors carried out a continuous model describing the evolution of the disease within the country by introducing three optimal controls, the first of which can be interpreted as the proportion to be subjected to awareness and prevention , the second can be interpreted as quarantine and health surveillance and the third can be interpreted as the proportion to be subjected to diagnosis and monitoring.
In this paper, our work is a little different since we will try to indirectly minimize the number of infected people by preventing their contact with susceptible people and directly the number of people who do not apply preventive measures against the coronavirus. The aim is to find an optimal strategy that will reduce the impact of the coronavirus and at the same time open up all the productive sectors of the country, including tourism.

MODEL FORMULATION
The model that we propose below makes it possible to describe the evolution of the pandemic by distinguishing two populations likely to be infected (S1) and (S2), as well as two hospitalized populations (H1) and (H2).

FIGURE 1. Diagram of the evolution of the Corona pandemic
The different compartments of our model are described below: Compartment S 1 : represents the number of people likely to be infected by the coronavirus and who do not apply preventive measures. It is increased by adding a number Λ 1 each day (the time unit used in this study) and decreases by contact with infected people without symptoms and others with symptoms respectively with reports α 1 S 1 E N and β 1 S 1 I N and also by a natural mortality rate d.
Compartment S 2 : represents the number of people who may be infected by coronavirus and who take preventive measures. It is increased by adding a number Λ 2 each day (the time unit used in this study) and decreases by contact with infected people without symptoms and others with symptoms respectively with reports α 2 S 2 E N and β 2 S 2 I N and also by a natural mortality rate d.
Compartment E: represents the number of infected people not showing symptoms. It increases by the contact with people who apply preventive measures and those who do not apply them respectively by the ratios α 1 S 1 E N and α 2 S 2 E N and also by a contact between these people and infected cases showing symptoms with reports β 1 S 1 I N and β 2 S 2 I N , and decreases by natural mortality d. Logically speaking, these contacts occur before the infected cases are declared "infected" (i.e. before the screening test) and after they will be isolated and hospitalized. Compartment R: represents the number of the recovered people. It increases by the rates θ 1 and θ 2 of people hospitalized and decreases by the rate of natural mortality d.
The previous diagram is mathematically translated by the following nonlinear system: , and it is supposed to be constant.

THE OPTIMAL CONTROL PROBLEM
In this paragraph, we aim to introduce two controls that enable us minimizing the number of people who do not apply preventive measures (S1) to protect them from the coronavirus pandemic and decrease the contact of healthy people with infected people. The first control u actually represents an increase in the number of people applying preventive measures against the coronavirus (S2) and a decrease in the number of people who do not apply these preventive measures (S1) thanks to awareness campaigns on media or through direct contact with people on the street. As for the second control v, it represents a rate of increase in the compartments (S1) and (S2) of people likely to be infected and a rate of the decrease in the compartment (E) and (I) of infected persons by applying the measures aiming at reducing social contact. So, the controlled mathematical system is given by the following system of difference equations: (2) The optimal control problem to minimize the objective functional is given by Where the parameters A k > 0 and B k > 0 are selected to weigh, respectively, the relative importance of the cost of awareness campaigns and preventive measures to reduce social contact.
The aim is to find an optimal control, (u * , v * ) such that where U ad is the set of admissible controls defined by The sufficient condition of the existence of an optimal control (u * , v * ) for problem (2) and (3) comes from the following theorem.
Theorem 1. There exists the optimal control (u * , v * )such that subject to the control system (2) with initial conditions. Finally, due to the finite dimensional structure of system (2) and the objective function is an optimal control with corresponding states S * 1 , S * 2 , E * , I * , H * 1 , H * 2 and R * . Therefore, inf
The key idea is introducing the adjoint function to attach the system of difference equations to the objective functional resulting in the formation of a function called the Hamiltonian.
This principle converts the problem of finding the control to optimize the objective functional subject to the state of difference equation with initial condition into finding the control to optimize Hamiltonian pointwise (with respect to the control). Now, we have the Hamiltonian H at time step k, defined by where f i,k+1 is the right side of the system of difference equation (2) of the i th state variable at time step k + 1.
Theorem 2. Given an optimal control (u * , v * ) ∈ U ad and solutions S * 1,k , S * 2,k , E * k , I * k , H * 1,k , H * 2,k and R * k of corresponding state system there exist adjoint functions λ 1 , λ 2 , λ 3 , λ 4 , λ 5 and λ 6 satisfying the following equations: Furthermore, for k = 0, 1, ..., T − 1, we obtain the optimal control (u * , v * ) as Proof. The Hamiltonian H k at time step k is given by So, we have By the bounds in U ad of the control, it is easy to obtain u * k , v * k in the form of (10).

NUMERICAL SIMULATION AND PRESENTATION OF RESULTS
We carried out a numerical simulation to test the reliability of our mathematical model with and without control. The results will be presented in the following.
We start by presenting the results with the different controls by adopting several strategies that will be explained in details : Values  We notice that after having applied the controls, the number of people likely to be infected by the coronavirus has decreased. This outcome was expected due to the implementation of awareness campaigns as well as reducing social contact which make people follow certain preventive measures against the disease and consequently cause a decrease in (S1).

CONCLUSION
The model we formulated to describe the spread of the pandemic (see the system 2) uses logical relationships between the different compartments established from the understanding of spread of the disease by consulting a few WHO reports and checking their reliability using a database containing 500 case records in USA. We conclude by saying that it is possible to achieve the defined objective (which is the effective fight against the pandemic coronavirus) by using two controls. The first represents the awareness campaigns aimed at maximizing the number of people applying the preventive measures against the coronavirus and reduce the number of people who do not apply these measures. The second represents decreasing social contact in public places which aims to reduce the number of infected people due to social contact.
In summary, we say confidently that a global strategy to fight against the pandemic coronavirus must encompasses these two controls, that is to say adopt a strategy based on: Awareness campaigns aimed at getting people to apply the preventive measures against the coronavirus mentioned in the introduction to this paper.
Decreasing in social contact in public places to prevent contamination as much as possible We are therefore making efforts to strengthen the health system, raise awareness and reduce contact in places at risk instead of closing the country. So, in this article, we offer the possibility to see the results of our effective strategy.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.