OPTIMAL NIPAH VIRUS (NIV) SPREAD CONTROL DYNAMICS

. In this work, we look at the vaccine and condom Nipah virus model using an optimum control analysis. We implemented measures to limit infection dissemination and control. We examine four distinct controls: personal protection, rapid testing, burying infected pigs, and therapy control. We construct an ideal control model and demonstrate the mathematical results using the suggested controls. The results of optimum control suggest that measures can be useful in decreasing infected individuals and increasing the health of society


INTRODUCTION
The Nipah Virus (NiV) is a transmissible from animal virus which may trigger deadly encephalitis and serious lung conditions in humans.It is a contagious virus that primarily affects Pteropodidae fruit bats, but it can also impact swine and humans.During an epidemic among Malaysian swine producers in 1999, the Nipah virus was detected.There have been no new cases recorded in Malaysia since 1999.It was discovered in Bangladesh in 2001, and outbreaks have occurred practically every year thereafter.East India has also been ravaged by the illness on a regular basis.The virus has been discovered in the recognized native reservoir (Pteropus bat species) as well as a variety of additional bat species in Cambodia, Ghana, Indonesia, Madagascar, the Philippines, and Thailand [14].
Humans are affected with the virus through interaction with sick animals especially pigs or their bodily secretions, or through ingestion of contaminated food items.Close contact with an infected person may result in from person to person.spread, particularly in hospital settings.
According to Tan et al [15] ,patients hospitalized to the University Hospital in Kuala Lumpur and their family members served as the study's subjects.110 people from 14 houses were analyzed.Asymptomatic Nipah infection was seen in 30 out of 110 (27%) of the household members.
Although no drugs or vaccines are presently available for treatment, a phase 1 clinical research of a Nipah virus vaccine candidate (HeV-sG-V) began in February 2020 [13].The vaccine will be administered to normal individuals aged 18 to 49 to test its effectiveness and safety in triggering an immune response [13].
During NiV epidemics in South Asian nations, interactions between animals, people, and the environment were very important [2].Numerous variables influence human-animal interactions, including prolonged droughts, reduced animal habitats as a result of deforestation, manmade fires that destroy forests in Indonesia, and pig rearing mixed with farmland.[1] .
For the study of the dynamics of disease spread, researchers have developed mathematical models like the one in [7], [8], [6], which has changed our knowledge of these contagious illnesses and aided in the development of sufficient and effective countermeasures.Biswas investigated disease mechanisms by employing the SIR fundamental mathematical model.He investigated this model further and investigated potential control and avoidance techniques with an optimum solution [10], [9] and [3].Sultana [4] used optimum control theory to create a quantitative study of Nipah viral infections.Control tactics in their work included raising consciousness and providing treatment.They proved the presence of optimum controls, and the ideal controls are defined using Pontryagin's utmost concept.According to the numerical modeling, the optimum control method is far more effective at reducing infectious people and the associated costs of both measures.Furthermore, Mondal et al [16] propose and evaluate a mathematical model for Nipah viral management with vital mechanics.They [11], [16] include the quarantine of contagious people based on the existence of isolation facilities alongside monitoring coverage.
According to the numerical modeling, better sanitation and confinement of infectious people are sufficient to contain Nipah viral spread.As Nipah virus control plans,Omede et al [12] weighed avoidance measures to keep people in the vulnerable area from coming into contact with the virus, Nipah virus treatment for people in the area that is infectious compartment, and Nipah virus treatment for people in the curative area.
The fundamental purpose of this research is to investigate the four control techniques; personal prevention(pp), rapid testing, burying infected piglets, and therapy control in the vaccines and condoms Nipah virus model.This paper is structured as follows: Section 2, Methodology of NiV model.In Section 3, Existence of the optimal control of NiV.Section 4, optimal control numerical simulation.Section 5, discussion and conclusion.

METHODOLOGY OF NIV MODEL
From the humans and pigs people that are capable of transmitting the virus to others, C(t): they are people who have been infected with the virus but do not develop any symptoms of the disease, and still carry and transmit the virus to others, I i (t): they are isolated individuals undergoing treatment who are capable of transmitting the virus to others, I t (t): they are not isolated but undergoing treatment as individuals that are capable of transmitting the virus to others, R(t): they are individuals who have recovered from the Nipah virus and are capable of contacting the virus again, D(t): the bodies of those who died due to the virus.
We assume the following: Natural sickness recovery can take place because of powerful antibodies [12].Casual touching of the dead bodies will expose the individuals to the virus [17].There is an interaction between the farmer and the infectious pigs [5].Since they are continuously watched, medical personnel safeguard themselves against the virus, and infection can happen in a therapy class, isolated individuals do not aid in the spread of NiV [12].The general public has easy access to and can afford condoms, an infected isolation facility, and vaccinations [5].After some time, people who have recovered become susceptible to infection once more.We assume the following: Natural sickness recovery can take place because of powerful antibodies [12].Casual touching of the dead bodies will expose the individuals to the virus [17].There is an interaction between the farmer and the infectious pigs [5].Since they are continuously watched, medical personnel safeguard themselves against the virus, and infection can happen in a therapy class, isolated individuals do not aid in the spread of NiV [12].The general public has easy access to and can afford condoms, an infected isolation facility, and vaccinations [5].After some time, people who have recovered become susceptible to infection once more.
The above formulations and assumptions result in the deterministic system of nonlinear ordinary differential equations that characterize the dynamics of Nipah virus infection spread in two groups.
2.1.Optimal Control Procedure.The Nipah viral model is subjected to optimum management theory given in (2.1).We examine four distinct controls to reduce Nipah viral infection and further spread in the population.The following definition applies to these controls: The safety control u 1 calls for reducing interaction with contagious persons, including dead victims of the Nipah virus, cleaning hands frequently, using hand sanitizer, and wearing masks.Further, Avoid meeting and limit their movement in areas with a high number of cases.Rapid testing of people in the vulnerable stage to detect Nipah virus carriers (C) and symptomatic individuals constitutes the second control u 2 .To further reduce the infection, the people should be isolated or confined to their house after recognition through testing.The control u 3 is burying infected piglet.The fourth control u 4 is the therapy control.The use of Ribavirin and any other substances that can boast immune system especial different species of vegetables.Also bitter kola could also be very relevant.Following is a system of optimal control issue that may result from the talk above: Our goals are to decrease the overall number of contagious people in the community and the costs associated with protective gear, vaccine production and dissemination, quick testing, and treatment at predetermined intervals.We reduce the target function provided by, (2.3)The weights k 1 , k 2 , k 3 , k 4 which are constant parameters for u 1 , u 2 , u 3 , u 4 will standardized using the optimal control condition.
The variables defined above have bounds on Lebesgue integrable functions.We strive for optimum settings u * i for i=1,...4, such that J(u * i ) = min(J(u i , Ω 1 )) where u i ∈ U the control set defined as We explain the terms in the integrand in (2.3) above as follows: The term indicates the expense involved with tracking infectious people.The term reflects the expense of all kinds of protection against infectious people and pigs.The term k 2 u 2 2 reflects the expense of all types of rapid testing of exposed people.The term k 4 u 2 4 indicates the total expense of all kinds of therapy for all infected people in the infectious groups.

EXISTENCE OF THE OPTIMAL CONTROL FOR NIPAH VIRUS MODEL
3.1.Existence of the State.From the equation (2.2), we have Solving (3.1), we obtain where the N(0) = N 0 is the initial value of the total populations.As t −→ ∞ in (3.3) Therefore, Finally the system (2.2) is bounded and the solutions exist.

3.2.
Existence of the Objective Functional.Pontryagin's optimal concept reduces the state system (2.2) to an issue of reducing the Langrangian L and Hamiltonian H with regard to (u 1 , u 2 , u 3 , u 4 ) at each position.The integrand of the goal functional makes up the Langragian of the management problem, which is the Hamitonian enhanced with penalty terms for control restrictions and is given by We look for the lagrangian's minimal number.This can be done by specifying the Hamiltonian H for the control problem, comprised of the integrand of the function with the goal and the result of the inner product of the right hand sides of the state equations and the co-state variables or adjoint variables λ i , i = 1, ..17 with respect to state variable as The system's status and control variables (2.2) have positive values.Convex and closed characterize the control set U. The presence of optimal control is demonstrated by Corollary 4.1 in [18] because the integrand of the objective cost function U given by (2.2) is a convex function of (u 1 , u 2 , u 3 , u 4 ) on the control set U. As a result, there are positive integers ξ 1 , ξ 2 and varsigma such that where ξ 1 , ξ 2 > 0, and ς > 1 Since the answers are bounded, the state system's Lipschitz condition is satisfied.Considering that the state variables are constrained, this proves the presence of an optimum control.

An Optimal Control's Characteristics. The essential criteria for Pontryagin's Optimal
Concept serves as the foundation for the important standards for optimal management [19].The systems (2.2) and (2.1) are reduced by this idea.to the problem of reducing a Hamiltonian H pointwise with respect to the constraints u i , : i = 1, 2, 3, 4. The set of adjoint equations can be derived using the correct partial derivatives of H with regard to the state variables.
Theorem 3.1.The given optimal controls u * i and solutions S(t) * , S v (t) * , S u (t) * , S vc (t) * , S vn (t) * , S uc (t) * , S un (t) * , E(t) * ,C(t) * , I(t) * , I it (t) * , I t (t) * , R(t) * , D(t) * , S p (t) * , E p (t) * , I * p of the control system (4.409)-(4.425)that minimizes J(u * i ) over U. Then there exists adjoint variables λ i satisfying with the transversality conditions, λ i (T ) = 0 (3.9) i = S(t) , S v (t), S u (t), S vc (t), S vn (t), S uc (t), S un (t), E(t),C(t), I(t), I it (t), I t (t), R(t), D(t), S p (t), E p (t), I p The optimality condition is given by Furthermore, we have the controls u * i , Proof.The findings from the [18] cases provide proof that an optimal control issue exists.Differentiating H with regard to each state variable results in the system ruling the adjoint variables.
Consequently, the adjoint system is expressed as, For t ∈ [0, T ], the optimal controls u 1 , u 2 , u 3 , and u 4 can be solved from the optimality condition where and solving for the optimal control (u * 1 , u * 2 , u * 3 , u * 4 ) we obtain Hence, the required optimal control condition is obtained as

OPTIMAL CONTROL MODEL NUMERICAL SIMULATIONS
The Python programming language is used in this part to implement numerical answers to the optimum system.We looked at four factors that rely on the neighbors of the state variables:

Conclusion and Recommendation. Public health policymakers, especially those in
Cambodia, Ghana, Indonesia, Madagascar, the Philippines, and Thailand who are at risk for infection due to confirmation that the virus has been discovered in known natural reservoirs (Pteropus bat species) and many other bat species, will benefit from the availability of methods that can be used to determine the best way to stop the spread of Nipah Virus [14].
In this study, we investigated the optimal combination of personal protection, quick testing, burying sick pigs, and therapy control methods to restrict infection spread in the vaccine and condom Nipah virus model.The goal function of the optimal control issue was then defined.
We create a vaccine and condom Nipah virus model and display the pictorial results.The four control strategy have been examined in the form of personal protection control(prevention) which involves minimizing the contact among the infectious people including death body of

4 ]
dt where i = 1, 2, 3, 4, and T is final time, Ω 1 stands for the collection of all contagious compartments, A i , i = 1, .., 6 represents the sections' non-negative weight values E,C, I, I it , I t , D respectively and k 1 , k 2 , k 3 , k 4 are the weight constant for the control variable u 1 , u 2 , u 3 , u 4 respectively.

(A) Protection control u 1 FIGURE 4 . 2 .FIGURE 4 . 3 .FIGURE 4 . 4 .Figure 4 . 1
FIGURE 4.2.Simulations Showing The Effect of Controls u 1 , u 2 , u 3 , u 4 on Infectious Human Population I Figures 4.2, 4.3 and 4.4 depict the impact of various control measures, including therapy controls, burying sick piglets, fast testing, and protection, on the infectious population I and NiV carriers (asymptomatic) population C, respectively.Each measure helps to contain the spread of NiV by lowering the population's proportion of NiV carriers (asymptomatic) and infectious individuals and dead bodies population.However, as shown in figure 4.2(E), 4.3 (e) and 4.4 (e), there is a substantial effect when combining the four controls at once.Such results are also experienced when apply to the exposed E as it drastically reduces the exposed individuals.
population, we have S p (t): the number of pigs who are not yet infected with the NiV at time t, but may get it if they come into contact with other infected pigs or eat contaminated fruits, E p (t): the number of pigs that have come into contact with the infectious agent or pathogen that causes the Nipah virus, I p (t): the number of pigs that

TABLE 2 .
Γ 4 Infection force on S un Γ 1 Infection force on S vc Γ 2 Infection force on S vn κ The rate at which an exposed population becomes infected θ The proportion of the exposed population who becomes a NiV carrier ε Rate of susceptibility among recovered persons δ 1 Illness-related death rate in NiV-Carriers δ 2 Illness-related death rate in infectious population δ 3 Illness-related death rate in infectious isolated people undergoing treatment δ 4 Illness-related death rate in infectious people undergoing treatment δ d Illness-related death rate in infectious pigs µ d The rate at which deceased bodies are disposed of (burial/cremation) µ p Pig mortality rate µ Natural death rate min 1, max 0, Iλ 10 + I it λ 11 + I t λ 12 − λ 13 (I + I t + I it ) k 4

TABLE 4 .
[9] ,and u 4 .We run the model both with and without supervision, and then we contrast the outcomes.Because the limits are not completely effective, we took into account their number values between zero(0) and one(1): u 1 , u 2 , u 3 , and u 4 .The parameter values used in the models are displayed in Tables 4.1 and 4.2.The visually optimized system's numerical answer, obtained using Python software, is displayed below.Unlike state variables, which have starting conditions, adjoint variables have final conditions [[5],[9]].
FIGURE 4.1.The solution of the model 2. Description of the parameter