SEIHR MODEL FOR INDIAN COVID-19: TRUSTWORTHINESS OF THE GOVERNMENT REGULATORY PROCEDURE FOR CORONAVIRUS ASPECTS

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INTRODUCTION
The World Health Organization, or WHO, stated in December 2019 that COVID-19 was a pandemic of worldwide threat after the identification of the disease's fifth human case by China's Wuhan City officials.on June 1, 2022, the virus had caused over 530 million infections and over 6.2 million fatalities globally.The 2nd COVID-19 wave in India has been extremely serious; during the latter week of April 2021, around 30,000 new cases were confirmed daily.According to WHO data, as of December 15, 2022, there were over 6,637,512 confirmed COVID-19 instances and 646,740,524 confirmed cases of the virus.On February 14, 2020, the first COVID-19 case to be reported in Africa took place in Egypt [1].
Since COVID-19 symptoms take at least two to ten days to manifest, it might be challenging to segregate affected people in the early stages of the illness.The main signs and symptoms of COVID-19 are a high fever, respiratory difficulties, and a dry cough.When an infected person coughs or sneezes, respiratory droplets from them may enter the environment and spread the virus.People who have not previously been afflicted by the virus may contract it by breathing in contaminated air or by contacting infected objects.Such human transmission was widespread during the early phases of the COVID-19 outbreak since the general public was ignorant of these SEIHR MODEL FOR INDIAN COVID-19 risk factors and sick people were not quarantined, which allowed them to unintentionally spread the virus to other people [4].
(i) Use soap and water or a hydro alcoholic solution to wash your hands often [5].
(ii) If you cough or sneeze, cover your mouth and nose with the crease of your elbow or a handkerchief; promptly discard the handkerchief and wash your hands [5].
(iii) Steer clear of people who are sick with a fever and cough [5].
(iv) Seek medical attention as soon as you have a fever, cough, or trouble breathing [5].
(v) Avoid unprotected direct contact with live animals and surfaces that have come into contact with animals in markets located in areas where cases of the new coronavirus are currently occurring [5].
(vi) It is best to refrain from consuming raw or undercooked meat or poultry.To prevent cross contamination with raw food, raw meat, milk, or organ meats should be handled carefully in accordance with acceptable food safety practices [5].
Novel virus infection outbreaks in humans are always a cause for concern for the public's health, particularly when little is known about the virus's properties, how it travels among individuals, the severity of the resulting infections, and available treatments.One of the most effective ways to address all of these issues is through mathematical modeling.Effective nondrug treatments, like social exclusion and personal protection, will be essential in managing the outbreak in the absence of COVID-19 vaccines or antivirals [5].(recruitment/birth and death).Pontryagin's maximum concept, which is employed in epidemiological models and outlined in, is used to achieve optimal control.A sensitivity analysis utilizing partial rank correlation coefficients is performed in order to determine which model parameters, when vaccination and treatment are adopted, have a bigger influence on the initial disease transmission R0.The results are graphically displayed.This identification is essential in order to guide policy decisions about which parameters to prioritize for data gathering or to slow the disease's progress [3].The parameter M mathematically stands for the Vaccine administered to the public portion from the category (R(t)), the parameter M stands for the Vaccine administered to the public should depend on time and should provide the best result in formulating an optimal control problem that we must maintain the infected person at a minimum level, regardless of where 0 ≤ M(t) ≤ 1.We have a propensity to draw up the expected COVID-19 scheme in Figure1 followed the above assumptions.The initial states of the system (1) are (0) ≥ 0, (0) ≥ 0, (0) ≥ 0, (0) ≥ 0.

MODEL FORMATIONS
We assume that control parameter N has a fixed value.In this section we mainly focus the study of uniformly boundedness solutions and the basic reproduction number, Stability Criteria for different equilibria and sensitivity analysis etc.

UNIFORMLY BOUNDEDNESS OF INDIAN PANDEMIC COVID-19 EQUATIONS
We verify the boundedness property of the system of nonlinear equations (1).The system of nonlinear equations (1).are uniformly bounded.Based on the assumption that Integrating above inequality by applying Birkhoff and Rota [2] theorem of differential equation we get Hence all the solutions of System of Nonlinear Equations (1) that are commence in { + 5 } are restricted in the region For any  > 0 and for  → ∞

BASIC REPRODUCTION NUMBER OF INDIAN PANDEMIC COVID-19 EQUATIONS
Basic reproduction number (R0) parameter plays an important role within the epidemic model for determinant the character of disease "The number of Secondary individual infected can caused by a single infected individual within the whole time interval" [5,9,22,44] ̇=  − ( +  + )
Currently the Jacobian matrix of  and Ψ at the disease free equilibrium are respectively given by ) The matrix's spectral radius is also known as the basic reproductive number (R0 ) ( −1 ) and is indicated in the current model by

EQUILIBRIA SOLUTION OF INDIAN PANDEMIC COVID-19
Two possible equilibria`s are available for the system of nonlinear equations (1), one is the disease-free equilibrium and the disease disappears in this disease-free equilibrium.And give  0 ( 0 , 0,0,0,  0 ) , Where  0 =

COVID-19 INDIAN PANDEMIC FOR LOCALLY ASYMPTOTICALLY STABLE
We planned to examine the local asymptotic stability of both infected Free and Endemic Equilibria, If R0 < 1 then the disease free symmetry E0 is locally asymptotically stable if R0 > 1 then the disease free equilibrium E0 is locally asymptotically unstable.[12,21,19,44] The disease free equilibrium of the system of Nonlinear Equation ( 1) is given by the following Jacobian Matrix The characteristic equation of the system Nonlinear equations (1) at its disease free equilibrium is given by ( + ) 2 ( + ( + ))( + ( +  + ))( + ( +  +  + )) = 0 From the above Jacobian matrix if all the eigen value are non-positive only when R0 < 1.Then the system is locally asymptotically stable if R0 < 1 and it is unstable if R0 > 1.
Note 1: If R0 increases to its value greater than 1 then the disease free equilibrium E0 losses its stability.
Note 2: When R0 = 1 the system of nonlinear equations (1) permits through a Transcritical bifurcation around its disease free equilibrium.When R0 < 1 the disease free equilibrium happens and locally asymptotically stable if R0 > 1 is the beginning criteria for both Present and asymptotic stability for endemic equilibrium point, At the beginning R0 > 1 then the disease free equilibrium reduces to unstable in nature.Then there is a change of feasibility besides stability occurs at R0 = 1.[12,21,19,44].The characteristic equation of the system (1) around its endemic equilibrium E2 is

If
Equation (5) shows that the first pair of roots are positive real and rest of the roots are Quadratic polynomial and all the other parametric values are positive.We accomplish that equation ( 1) is locally asymptotically steady everywhere its endemic equilibrium E1 in the view of Routh-Hurwitz criterion.

SENSITIVITY ANALYSIS
In From the above system of equations   is positive, while   ,   ,   ,   are all negative, t is all negative.This clearly shows that increasing the value of  will increase the value of R0 while increasing the value of , , ,  will decrease the value of R0, A highly sensitive parameter should be carefully analysed since tiny changes in the system might result in big numerical changes R0.

OPTIMAL CONTROL FOR COVID-19 INDIAN PANDEMIC
Here M(t) stands for the Vaccine administered to the public it is one of the time varying control strategy implemented by Indian government, which gave the awareness of Vaccine administered to the public.Recent days it is very important strategy implemented by Indian Government which minimizes the virus spread between the peoples.In order to determine such a strategy, we can use the optimal control theory to minimise the spread of the virus between people, the main objective of the optimal control problem.We may construct the goal in the following way: Subject to the model proposed (1).Where c1 is defined for the infected population and c2 is restricted to the control.The above functional aim is linear and regulated in bounded states.
We can also use standard results to ensure optimum control and appropriate optimum states [8].
The aim is to determine the optimal control value N(t) in such a way such that By Using Pontryagin`s Maximum Principal [8,28,37,44] to develop the required conditions for our optimal control and corresponding states.Then the Lagrangian is given by The Hamiltonian is defined as follows (, , A vice parameter of the Optimal Control M(t) exists according to the states S, E, I, H and R: The transversally conditions were satisfying by the adjoin variables We try to minimize the Hamiltonian using the control variable M(t), More over the Hamiltonian is linear in the control parameter, if we consider the optimal control is singular.Then the switching function as If switching function vanishes on non-trivial interval of time then the singular control occurs, and then the optimal control is to take its upper bound or its lower bound according as Using the system of equations ( 2) and ( 5) we obtain

𝐻𝑒𝑟𝑒 𝐾 = 𝐴 − 𝛽𝑆𝐸 − 𝛾𝑆 + 𝜖𝐸
The control parameter M will not exist for the fore said equations, hence calculation of second order derivative with respect to time is essential.
The above equation can be written in the form  numerically.When R0 <1 and the solutions of model ( 1) converge to the DFE, as shown in Fig. 2, the numerical result is verified.
India, Maharashtra, Kerala, Karnataka, Andhra Pradesh and Tamil Nadu are the top five hotspot of COVID-19 virus spreads, In Maharashtra as on 09 th December 2023 totally 81,71,942 COVID-19 cases are conformed in that 8023379 are recovered and 148563 are Dead due to COVID-19, In Kerala as on 09 th December 2023 totally 69,07,976 COVID-19 cases are conformed in that 6835931 are recovered and 72045 are Dead due to COVID-19, In Karnataka as on 09 th December 2023 totally 40,88,956 COVID-19 cases are conformed in that 4048597 are recovered and 40359 are Dead due to COVID-19, In Andhra Pradesh as on 09 th December 2023 totally 23,40,677 COVID-19 cases are conformed in that 2325944 are recovered and 14733 are Dead due to COVID-19, In Tamil Nadu as on 09 th December 2023 totally 36,10,774 COVID-19 cases are conformed in that 35,72,693 are recovered and 38,081 are Dead due to COVID-19 these data are published in MoHFW [23].Indian Government imposed various disease control strategy such as Lockdown, Social Distancing, Washing Hands Frequently Wearing Mask, Sanitize hands with alcohol base sanitizer and vaccine administer etc., In these Lock down helps us to make self-isolation by staying home and it helps us to recover from mild infection of COVID-19, While maintaining six feet Social Distancing helps to avoid the disease spread through the tiny droplets in Public places meanwhile wearing mask will prevent the spread of tiny droplets through air.In light of the global rollout of COVID-19 vaccinations, we develop and examine a COVID-19 model that accounts for treatment options, immunization of susceptible persons, and hospitalized/infected patient care.Several important biological and epidemiological aspects of COVID-19 are included in our suggested model, such as demographic characteristics

Fig 5 :Fig 6 :
Fig 5: Government control strategy of Vaccine admistred to the Public at 0%

Fig 7 (
Fig 7 (a): Plot for Change in Susceptible Rate at disease free equilibrium point

Fig 8 (
Fig 8 (a) Plot for Change in Susceptible Rate at endemic equilibrium point

Table II :
Parameter meanings for model