A SCENARIO TO FIGHT MONKEYPOX USING A MATHEMATICAL MODEL

. In this paper, we consider a scenario to attack the infectious disease monkeypox taking advantage of the experience we have with the Corona epidemic to reduce its negative effects on humanity and the world economies. This by proposing three control strategies. Pontryagin’s maximum principle is applied in order to characterize the optimal controls, and the optimality system is resolved using an iterative approach. At last, numerical simulations are executed to verify the theoretical analysis using MATLAB.


INTRODUCTION
The recent outbreak of monkey pox virus has now occurred in various countries and almost on continents [1]; humanity froze in anticipation of the next lock-down; the (MPXV) infectious disease can be transmitted by airborne droplets, began to spread around the world, and considered as a rare viral zoonotic disease that is transmitted to humans from animals [2]. The monkey pox virus is a relative of natural black smallpox, belongs to the same genus orthopoxvirus, transmitted by airborne droplets [12]. Most cases are related to sexual contact and contact with body fluids [13]. Unlike coronavirus, monkey does not form aerosols (small droplets that create an air suspension and can travel long distances). In addition, monkeypox is not a disease that is transmitted during the asymptomatic phase, and even if it is mutated, there is already a proven vaccine that can be modified if necessary. And new variants of the monkeypox virus may not appear as quickly as in COVID-19, due to the biology of the virus itself. Thus, it should be borne in mind that the virus may at some point mutate in such a way that it begins to be effectively transmitted from person to person. This probability is what causes the greatest concern [14,2].
According to the WHO, there are two main branches of strains: the West African which is considered as milder, with a mortality rate of 3.6% of cases, and the Central African (Congolese), considered more severe, with a fatality rate of 10.6%. The pathogenesis of monkeypox is slightly understood. It is known that after entering the human body, the virus stays for a long time in the regional lymph nodes, where it multiplies, and then spreads hematogenically and lymphogenically through the organs. The virus has tropicity to the epithelium, increases the production of pro-inflammatory interleukin-10; Pathohistological changes are characterized by epidermal necrosis with progressive epithelial hyperplasia, expansion of the boundries of necrotic changes; by the time the sebaceous glands and hair follicles are destroyed [15,16].
The incubation period is 7-21 days, usually about two weeks [17]. The disease begins acutely with headache, weakness, chills, fever up to 39.5-40°C. The lymph nodes located near the site of the pathogen's penetration become inflamed, increase in size, and become painful when touched. Due to severe intoxication, there may be a decrease in appetite, nausea, vomiting. On day 3-4, the fever subsides to 38.5°C or less, a rash appears on the face, feet, palms, and then the trunk. Monkey pox rashes go through several stages sequentially [18]. First, a spot with a diameter of up to 1 cm is formed, which turns into a bump. The next stage is the appearance of a bubble with transparent, and then cloudy contents, then a crust is formed with the outcome on the scar. During the last three stages, the temperature rises again to 39°C, chills are observed, and the state of health worsens. Due to the presence of rashes on the pharyngeal mucosa, there is a cough, a feeling of rawness, and dryness in the throat [19,20]. Up to 70% of patients have a rash on the oral mucosa, which is manifested by pronounced discomfort when chewing food, increased salivation. In 30% of cases, the genitals are affected, in 20% the eyelids, which is accompanied by sharp pain in the genitals and eyes, respectively. With the addition of pyoenic flora, an increase in fever, a decrease in blood pressure, an increased heart rate, and impaired consciousness are detected-harbingers of infectious and toxic shock [20,21].
The main complications of monkeypox are purulent-necrotic lesions. These include bronchopneumonia, which is characterized by the formation of necrotic foci, respiratory distress syndrome and high mortality. Rare complications include damage to the eyes and gastrointestinal tract, sepsis. Post-infection corneal scarring can lead to blindness. In the second week of the disease, profuse diarrhea, vomiting and progressive dehydration may occur. Septic conditions were recorded with a large number of rash elements (over 4500) [22,2].
Most monkeypox infections recover with no medical treatment, especially gastrointestinal symptoms (e.g., diarrhea, vomiting) will require oral/intravenous rehydration to reduce gastrointestinal fluid losses [23]. Etiotropic antiviral drugs may be effective in treating monkeypox patients, yet these drugs were approved for the management of smallpox implicated on animal models. Studies concerning these drugs have been conducted in humans, however the efficiency of these agents has not been thoroughly developed [22]. The main measures in the treatment of monkeypox are infusion and oral detoxification, anesthesia and other means of symptomatic therapy. The use of antibiotics is indicated in the presence of purulent complications. It is recommended to irrigate the mucous membranes after each meal and toilet visit with antiseptic properties. Wet bandages should be used to prevent scarring, especially on the face. However, the use of cidofovir, originally licensed for the treatment of cytomegalovirus retinitis in HIV infection, is currently in clinical trials . The development of oral modifications worked by inhibiting the viral DNA polymerase for the treatment of this infection is underway in New York and North Carolina. These drugs have completed the second phase of human clinical trials, proving their safety [24,25].
Multiple studies believed that those who were vaccinated against smallpox in childhood can be protected from monkey pox, since the vaccine gave a sterile immunity, that is, the vaccinated person ceased to be a source of new viruses and vaccination gave lifelong immunity. According to WHO, vaccination against smallpox is 85% effective against smallpox infection in monkeys.
The two viruses are similar, so the immune response formed as a result of using the smallpox vaccine is effective against monkeypox [26]. Nevertheless, it is impossible to talk about significant collective immunity, since the vast majority of people under the age of 40 are not vaccinated with the smallpox vaccine. However, there are two smallpox vaccine available in the US: ACAM2000 and Jynneos, the first one was approved by the US Food and Drug Administration (FDA) in 2007; this vaccine is based on a live smallpox virus that can be transmitted to people who are in close contact with the vaccinated person, and the common side effects include lymphadenitis, malaise, fever, myalgia, and headache. These effects are less common in re-vaccinated individuals than in those who received the vaccine for the first time. The Jynneos vaccine was approved by the FDA in 2019, and made on the basis of a weakened virus. It is intended for the prevention of smallpox and monkeypox in adults aged 18years and older. The latter is suitable for those who cannot be vaccinated with ACAM2000 due to contraindications, for example, atopic dermatitis, weak immunity, as well as during breastfeeding or pregnancy. This is the only FDA-approved monkey pox vaccine that is intended not only for military personnel, but also for civilians [27,28,29]. The manuscript is arranged as follows. In Section 2, we briefly describe our proposed model. In section 3, we provide some basic properties of the model. In section 4 we introduce the problem with control, and we give some results on the existence and characterization of the optimal control using Pontryagin's maximum principle.
The numerical simulation on MATLAB confirms the theoretical results in section 5. Finally we conclude our work in section 6.

MATHEMATICAL MODEL
We consider a mathematical model SEIQHI c R that describe the dyanamic of transmission of Monkeypox virus in a given population. We divide the population into seven compartments.
Compartment E is representing the number of individuals infected by the virus but not yet symptomatic (according [11] to the disease necessarily passes through this phase). The compartment E is increasing by β 1 SI N and β 2 SE N . This Compartment decreasing by the amount µ (natural mortality) and the the amount σ E (the rate of normally infected people with symptoms) Compartment I is representing the number of individuals infected with symptoms, increasing by the amount σ E and decreasing by the amount µ , δ 1 ( the induced death rate of Monkeypox), α 1 (The rate of infected people is usually with symptoms and they need quarantined at home), α 2 ( The rate of people who developed the virus rapidly ) and α 3 (The rate of people suffering serious complications after the virus has destroyed a large part of their lungs) Compartment Q the number of people who have been quarantined in their homes, increasing by α 1 I and decreasing by the amount µ, γ 1 and θ 1 .
Compartment H the number of people who have been quarantined in the hospital. This compartement increase by α 2 I(t) and θ 1 Q and decrease by the amount µ, δ 2 (The rate of people dying under quarantine in hospitals), γ 2 (The rate of people with a quarantine in hospitals recovered from the virus ) and θ 2 (The rate of hospitalized people suffering serious complications after the virus has destroyed a large part of their lungs) Compartment I c : the number of patients with severe complications after the virus destroyed a large part of their lung, decreasing by the amount µ, δ 3 and γ 3 . In addition, this compartment is increasing by α 3 I and θ 2 H Compartment R the number of individuals who have recovered, decreasing by the amount µ and increasing by γ 1 Q, γ 2 H and γ 3 I c we obtain the following model The rate of people who were infected by contact with exposed The rate of people who were infected by contact with infected. σ The rate of normally infected people with symptoms.
The rate of infected people is usually with symptoms and they need quarantined at home .
The rate of people who developed the virus rapidly.

RESULTS
In this section we will give some preliminary results the solutions S(t), E(t), I(t), Q(t), H(t), I C (t) and R(t) of the proposed system (8) are positive for all t ≥ 0.
Proof. We have from the first equation of the system that We multiply this inequation by exp t 0 β 1 By integrating this inequality from 0 to t we obtain Then S(t) ≥ 0 for all t ≥ 0. Similary, we can also prove that E(t), I(t), Q(t), H(t), I C (t) and R(t) are positive for all t ≥ 0. Proof. Let N = S + E + I + Q + Q + H + I C + R, then Hence This proves that the set Ω is positively invariat for the proposed system (8). Proof On the other hand, let Y =

MODEL WITH CONTROL
Considering the waste left by the Corona epidemic and its impact on the world's economies and on the health structures of various countries, and to avoid similar results of monkeypox we propose four control strategies: The first control strategy ( control u 2 ) can be interpreted as the proportion using smallpox vaccine because the monkeypox virus is closely related to the virus that causes smallpox, so the smallpox vaccine can protect people from smallpox [30]. The second control strategy (control u 1 ) can be interpreted as the proportion to be submitted to the awareness and prevention of individuals. the third control strategy (control u 3 ) can be interpreted as hospitalization and medical management of patients to spare them from the health problems of the disease. the fourth control strategy (control u 4 ) can be interpreted as hospitalization, intensive care and recovery room for those infected with serious health problems If we take S = S N , E = E N , I = I N , Q = Q N , H = H N , I C = I C N and R = R N then we obtain the normalized system (10) this system could be rewritten as follows

Description of Optimal Control.
In this section, we minimize the total number of exposed and infected humans to Monkeypoxin the population using the control variables u 1 (t), u 2 (t) and u 3 (t) in the model. We define the objective function as subject to model (19), where a 1 , a 2 and a 3 are positive weight constants of exposed, infected humans and infected with complication respectively. b 1 , b 2 , b 3 and b 4 are positive weight constants of control variables u 1 (t), u 2 (t), u 3 (t) and u 4 (t) respectively.
The purpose is to find an optimal control of u * 1 , u * 2 , u * 3 and u * 4 such that is the control set subject to the model (19) with initial conditions.

Existence of the optimal control.
In this section we present the theorem that proves the existence of an optimal control u * = (u * 1 , u * 2 , u * 3 , u * 4 ) that minimizes the cost function J .
Theorem 4.1. There exists an optimal control (u * Proof. The existence of optimal control may be obtained by utilizing a result of Fleming and Rishel [31,32].
The existence of optimal control of the model (19) would be considered by applying the theorem.
Therefore, we can now apply the necessary conditions to the Hamiltonian, H, in eq.(15).
with the transversality conditions Proof. We have according to the theorem of Pontryagin [33,34] +λ 7 (t)u 2 (t) In other words we have this completes the demonstration.

NUMERICAL SOLUTION OF THE OPTIMAL CONTROL
We present numerical simulations for the aforementioned optimization problem in this section. We make the program in MATLAB (5.1), and we simulate our work with diverse data.
The optimization systems are solved using a discretized iterative process that converges after a proper test analogous to the FBSM. First, the state system is solved with the initial hypothesis forward in time, and then the adjoint system is solved backward in time due to the transversality conditions. Then, we update our optimal control values with the state and co-state resources derived in the above steps. Finally, we run the preceding steps until the standard tolerance is reached.

Discussion.
Due to the impact that the Corona epidemic has had on global health systems [35] and economies [36], any major spread of monkeypox will have catastrophic consequences.
Therefore, we have developed a vision for confronting this disease and avoiding its impact on humanity and economies through the four control strategies above ( section (4)).
These strategies proved to be effective in reducing the number of infected people and individuals who have serious health problems (See fig. (2) and (3)), which will reduce the strain on health systems and avoid the scenario of the Corona epidemic.
Our control plans have also proven to increase the number of people recovering (See fig. (5)), thereby reducing any potential impact of this disease on individuals and the economy.

CONCLUSION
In this study, we propose a scenario for combating the infectious disease monkeypox, drawing on lessons learned from the Corona outbreak to minimize the disease's serious effects on humanity and global economies. This is accomplished through the use of three different control strategies. The optimal controls are characterized using Pontryagin's maximum principle, and the optimality system is resolved using an iterative approach. Finally, using MATLAB, numerical simulations are run to verify the theoretical analysis.

DATA AVAILABILITY
The disciplinary data used to support the findings of this study have been deposited in the Network Repository (http://www.networkrepository.com).