Dynamics of COVID-19 Using SEIQR Epidemic Model

e major goal of this study is to create an optimal technique for managing COVID-19 spread by transforming the SEIQR model into a dynamic (multistage) programming problem with continuous and discrete time-varying transmission rates as optimizing variables. We have developed an optimal control problem for a discrete-time, deterministic susceptible class (S), exposed class (E), infected class (I), quarantined class (Q), and recovered class (R) epidemic with a nite time horizon.e problem involves nding the minimum objective function of a controlled process subject to the constraints of limited resources. For ourmodel, we present a new technique based on dynamic programming problem solutions that can be used to minimize infection rate and maximize recovery rate. We developed suitable conditions for obtaining monotonic solutions and proposed a dynamic programming model to obtain optimal transmission rate sequences. We explored the positivity and unique solvability nature of these implicit and explicit time-discrete models. According to our ndings, isolating the aected humans can limit the danger of COVID-19 spreading in the future.


Introduction
Mathematical models are useful for determining how an infection behaves when it enters a population and determining whether it will be eradicated or continue under di erent settings. COVID-19 is currently causing tremendous concern among researchers, governments, and the general public due to its rapid spread and a high number of deaths [1]. e transmission of this disease is caused by the tiny particles or droplets called aerosols that carry the virus into the atmosphere caused by a contaminated person while sneezing, coughing, or exhaling. Many researchers and scientists are continuously working to reduce the transmission of this vicious disease throughout the world. Infectious diseases are the disciplines that focus on the study of the dynamics of infectious diseases as well as the relationship between these diseases and the various factors involved in their appearance and evolution, in order to implement a ght against this spread. Despite its youth, mathematical modeling is a valuable tool for understanding disease transmission mechanisms, is playing an increasingly important role in epidemiology, and has already contributed to signi cant successes. e most in uential work in the eld of mathematics epidemiology was rst introduced by Kermack and McKendrick as the SIR model in the year 1927 [2]. Cao et al. [3] discussed a modi ed model of the SIR (susceptible, infected, recovered) epidemic introduced in order to detect the con rmed number of infected cases and consecutive burdens on isolation wards and ICUs. Also, Nesteruk [4] developed the variables used in the proposed model by introducing a SIR epidemic model and explaining how to dominate the spread of the disease. To restore the pandemic with the involvement of social distancing and lockdown, Gerberry and Milner presented a data-driven susceptible, exposed, infected, quarantined, and recovered (SEIQR) model in [5]. From the publication of Zeb et al. [6], epidemiological's SEIQR model with isolation class in 2020 and their mathematical epidemiology has expanded in numerous directions, involving biology and computer science by Zima et al. [7], Zhou et al. [8], and Kermack and McKendrick [9,10]. Some recent studies have focused on this area of research by He et al. [11], Rahimi et al. [12], Hussain et al. [13], Youssef et al. [14], Prabakaran et al. [15], and Youssef et al. [16]. In this current paper, we implement the discrete type of SEIQR model and discuss the solvability of both continuous and discrete type SEIQR model. We examined the behaviour of the time-continuous model. We have developed two time-discrete models: time-implicit and timeexplicit. We looked at the theory and methods for solving the time-implicit model. en, to control and anticipate the dynamics of COVID dissemination, we establish appropriate transmission rate limits. To do this, we devised a dynamic programming problem to optimize transmission rate sequences under arbitrary beginning conditions. We propose safety guidelines and essential precautionary measures based on the optimized rate sequences to control COVID spread. e article gives the technique for optimizing the transmission rate sequences. e epidemic models and their time-discrete variations have been studied by Allen [17] and Ghosh et al. [18]. Several approaches towards fractional-order mathematical models of COVID-19 were studied by the authors Alqhtani et al. [19], Valliammal and Ravichandran [20], Nisar et al. [21], Vijayakumar et al. [22], and Alderremy et al. [23]. However, the aforementioned studies and references mostly contain explicit approaches with respect to time-discrete epidemic models.

Review Literature.
In 2020, COVID-19 is a worldwide emergency. e first cases occurred in December 2019, and as of 6 : 34 pm CEST, 28 July 2022, there have been 571,198,904 confirmed cases of COVID-19, including 6,387,863 deaths, reported to WHO. As of 25 July 2022, a total of 12,248,795,623 vaccine doses have been administered. e rapid spread of COVID-19 has already caused great public attention and many heated discussions, and the Chinese mass media have been reporting relevant information about the virus and the outbreak.
Ming et al. [24] show that effective public health measures are required to be implemented in time to avoid the breakdown of the health system, and the media can certainly play a crucial role in conveying updated policies and regulations from authorities to the citizens. e finding that SARS-2-S exploits ACE2 for entry, which was also reported by Kermack and McKendrick [9] while the present manuscript was in revision, suggests that the virus might target a similar spectrum of cells as SARS-CoV. However, upon its outbreak, various research, including but not limited to Okhuese [25], began to predict the scale that the virus would hit the world; the ratio of the death to recovery rate has seemingly been a positive proportion. Allen [17] studied about time-discrete SI, SIR, and SIS epidemic models, and its properties. Kermack and McKendrick [10] analyzed an outbreak such as the one in Hubei is captured by SIR dynamics where the population is divided into three compartments that differentiate the state of individuals with respect to the contagion process: infected (I), susceptible (S) to infection, and removed (R) (i.e., not taking part in the transmission process). Mathematical modeling has been influential in providing a deeper understanding on the transmission mechanisms and burden of the ongoing COVID-19 pandemic, contributing to the development of public health policy and understanding. Most mathematical models of the COVID-19 pandemic can broadly be divided into either population-based, SIR (Kermack-McKendrick)type models, driven by (potentially stochastic) differential equations proposed by Nesteruk [4] in which individuals typically interact on a network structure and exchange infection stochastically. is point emerges also clearly from a number of recent model-based contributions that have extended the basic SIR model to account for key insights from economic theory, namely by allowing for peoples' (rational) adjustment of work, consumption, and leisure activities in the face of infection risk. More generally, the idea is to model explicitly the exposure to the virus (of those people who are susceptible), as in the susceptible-exposedinfectious-recovered (SEIR) model which has been analyzed extensively by He et al. [11] in the context of the COVID-19 pandemic. e Jacobian method used for the SEIR model yields a biologically reasonable R 0 , but for more complex compartmental models, especially those with more infected compartments, the method is hard to apply as it relies on the algebraic Routh Hurwitz conditions for stability of the Jacobian matrix. An alternative method proposed by the authors Van den Driessche and Watmough [26] gives a way of determining R 0 for an ODE compartmental model by using the next generation matrix. Batista [27] applies logistic growth regression model to predict the final size of the Covid-19 epidemic. Basically, NSFD is an iterative method in which we get closer to solution through iteration was given by Mickens [28]. e authors Vijayakumar et al. [22] discussed about approximate controllability results for fractional Sobolev type Volterra-Fredholm integro-differential systems of order 1 < r < 2. Finding the variants that predict severe disease, we developed a collaboration of four international computational centers (Iran, Italy, Malaysia, and Greece). In [29], the authors Bairagi et al. have introduced a mathematical model for controlling the outbreak of COVID-19 by augmenting isolation and social distancing features of individuals and also solved the utility maximization problem by using a nonco-operative game. In 2021, the multidisciplinary approach was necessary to address the multidimensional aspects of COVID-19 infection by established collaborations discussed by the authors Rahimi et al. [12]. e study by the authors Prabakaran et al. [15] looked into the evolving geographic diversity of the SARS-CoV-2. We then consider how positive factors like social distancing measures and detrimental factors such as delays in testing onset affect optimal testing strategies and outbreak controllability. roughout, Youssef et al. [14] focus their analyses on empirically supported parameter values including realistic testing rates. While many existing COVID-19 SIR-like compartmental models explore the effects of testing with forms of isolation like quarantine or hospitalization, the majority of these studies assume simple linear equations for the rates at which tests are administered and individuals are isolated. e authors Hussain et al. [13] discussed about the complex systems and network science approaches, along with technological advances and data availability, are becoming instrumental for the design of effective containment strategies. In a nonsense region, Hilfer's neutral fractional derivative provided controllability results using Monch's method, Banach's contraction principle, fractional calculus, and semigroup property was studied by the authors Nisar et al. [21]. Some recent updates regarding the modeling of the coronavirus, the authors Alderremy et al. [23] constructed a mathematical model based on the fuzzy fractional derivative and obtained the results. e authors Valliammal and Ravichandran [20] are discussed in detail the fractional integro-differential equation with different conditions in various spaces. In [30], the authors Awal et al. proposed a framework that uses Bayesian optimization to optimize the hyperparameters of the classifier and adaptive synthetic (ADASYN) algorithm to balance the COVID and non-COVID classes of the dataset. In 2022, the authors Youssef et al. considered a modified model to analyze the disease dynamics of the coronavirus infection by taking the real cases from Saudi Arabia [16]. Alqhtani et al. [19] analyzed about spatiotemporal dynamical patterns arising from subdiffusion reaction-diffusion systems of predator-prey interaction are modeled in the sense of the Caputo fractional operator. Ghosh et al. [18] have studied about discrete-time epidemic model for the analysis of transmission of COVID-19 based upon data of epidemiological parameters.In this article, we have considered the epidemic model published in [6,30]. en, we have extended the idea of the article [1] to the considered model. As a result, we recap and extend certain conclusions on the features of the time-continuous classical SEIQR model, and we suggest an implicit time-discrete version of this classical SEIQR model, proving that it retains many of the qualities of the time-continuous version. As a result, the goal of this research is to propose a nonautonomous SEIQR model, investigate the properties of its time-continuous formulation, and design an implicit numerical solution approach that preserves the time-continuous variant's primary properties. e goal of this article is to propose, analyze, and optimize COVID-19 using the SEIQR epidemic model. According to our investigations, COVID-19 outbreaks might be caused by human-to-human interaction. As a result, isolation of the infected humans can reduce the COVID-19 spread in the future. Literature review and comparison of various of these models are presented in Table 1.
More precisely, our main contributions can be summarized as follows: (i) First, we suggest a time-continuous SEIQR model modification with time-varying transmission and recovery rates. (ii) Second, we draw the conclusion that the formulation of our time-continuous problem is well-posed. is comprises continuous reliance on initial conditions and time-varying rates, global existence in time, and global uniqueness in time, all of which are based on an inductive application of Banach's fixed point theorem.
(iii) In the case of the time-discrete implicit model, we provide unique solvability, monotonicity properties, and an upper error bound between the solution of the implicit time-discrete problem formulation and the solution of the time-continuous problem formulation. (iv) In order to maximize transmission rate sequences under arbitrary beginning conditions, we have developed a dynamic programming problem. Based on the optimal rate sequences, we suggest safety guidelines and important safety precautions to control COVID spread. e paper is arranged as follows: Section 1 is dedicated to the introduction. In section 2, we present the time-continuous and time-discrete SEIQR model. In section 3, we give the monotonicity properties and long-time behaviour. An error analysis is given in section 4. e conclusion of our research work is implemented in the last section 5.

Time-Continuous SEIQR Model
e time-continuous SEIQR model is formulated, and its behaviours are described using the Lipchitz condition and Grownwall and Bellman's inequality in this section.

Mathematical Background Material.
Here, we revisit the Lipschitz continuity of a function on Euclidean spaces, the local Lipchitz condition, Banach's fixed point theorem, and the method of variation of the parameter, which will be used in the subsequent sections.
Definition 1 (see [46]). Let q 1 and q 2 be two positive integers and D ⊂ R q 1 (i) Let U ⊂ R q 1 be an open set and H: U ⟶ R q 2 . en, H is called as locally Lipchitz continuous if for every element y 0 ∈ U there exists a neighborhood V of y 0 such that the restrictions of H to V are Lipchitz continuous on V.In a more general framework, we consider a nonlinear initial T is vectorial function with initial point z 0 ∈ R n . e following theorem, which is a direct consequence of Gronwall's lemma, can be used to prove global existence in time.
holds for all x, y ∈ X. en, the mapping T has a unique fixed point.
In the following theorem, we present the Grownwall and Bellman inequality, which will be used in the subsequence theorems related to the continuous functions.
holds for all t ∈ I, we have

Theorem 4. (Method of variation of parameter) For a firstorder nonhomogeneous linear differential equation, y
and A is an arbitrary constant.

Continuous Problem Formulation.
At first, let us assume the following assumptions [50,51] for the upcoming calculations.
(i) Let the population size varies over time be N is varying over time (i.e., population size � μN(t) for all t ∈ [0, ∞)).
(ii) We divide the population into five homogeneous subgroups, namely susceptible people (S), exposed (E), infectious (I), quarantined (Q), and recovered (R). We can clearly assign every individual to exactly one subgroup. Hence, we obtain S, E, is Lipchitz continuous and continuously differentiable, and there exist constants x min and x max such that 0 < x min ≤ x(t) ≤ x max for all t ≥ 0 and x ∈ π, β, c, σ, θ, μ . e choice of time-dependent transmission rates is possible because the countermeasures such as lockdowns, social distancing, or other political actions like curfews and different medical treatments reduce possible contact between susceptible and infectious people.Our equations of the time-continuous SEIQR model read as follows: SEIRP Susceptible (S), exposed (E), infectious (I), removed (R), pathogens (P) Deterministic Pakistan [45] with initial conditions e detailed parameters and description are given in Table 2 2.3. Nonnegativity and Boundedness of Solutions. Now, we prove the boundedness of the solution to (3). For this purpose, we modify ideas given in [51,52] deriving the following lemmas, so consider the bounded, time-varying transmission rates given above.

Lemma 1. Each solution of system (3) is bounded below by zero.
Proof. Consider, the first relation of (3), By taking (dS(t)/dt) � S ′ (t), equation (4) can be expressed as a first-order nonhomogeneous linear differential equation in S(t) as Applying eorem 4, and by applying the same procedure to the first-order nonhomogenous linear equation in E(t), We can easily show that E(t) ≥ 0 for all t ∈ [0, ∞). Proceeding like this, we can show that I(t), Q(t), and Since , and total population is finite, S(t), E(t), I(t), Q(t), and R(t) are bounded above, and hence, the proof is complete.

Theorem 5. For all solution functions of (3), we have
Proof. e proof follows from

Theorem 6. e system of nonlinear first-order ODE (3) has at least one solution which exists for all
Clearly, G is Lipchitz continuous, due to the continuity of each components.
Assuming the supremum norm on our Euclidean space, and with the help of triangle inequality, we arrive Table 2: Parameters and description.

Parameters Description S(t)
At time t, the number of susceptible people At time t, the number of exposed people At time t, the number of infected people At time t, the number of quarantined people At time t, the number of recovered people β e rate at which susceptible populations migrate to exposed and infected populations π e rate at which an exposed population moves to an infected population c Transmission rate at which exposed people take outside as isolated where k � μ max + β max + π max + c max + σ max + θ max . From the boundedness of our solution functions and the boundedness of our time-varying transmission rates, all requirements of eorem 1 are fulfilled, and our proof is complete. (1) Consider the time interval [0, τ] is applicable to Banach's fixed point theorem (2) For x 1 , x 2 , y 1 , y 2 ∈ R, by triangle inequality, we have en, the second equation in (3) becomes Since it is a first-order nonhomogenous linear equation in Similarly, we can easily show the following inequalities: Summing implies  (3) to the fully explicit discrete scheme (14) as given as follows: and a fully implicit scheme (15) as where Since the fully explicit scheme (14) simply reduces to a linear system, our main interest is in a fully implicit discrete scheme because it preserves the nonlinear structure of the continuous problem.

Journal of Mathematics 7
In a similar way, we can easily find the other parameters as where (16) uniquely solvable for all j ∈ 1, 2, . . . , M − 1 { }, and we have where A, B, and C are given in upcoming equations in (27)- (29).

Monotonicity Properties and Long-Time Behaviour
In this section, we develop a suitable atmosphere in which our implicit scheme obeys the monotonic properties as in the continuous case. For this, we give the following lemmas and finally provide a nonlinear programming problem to optimize the transmission sequences.
n�1 becomes a decreasing sequence.
Proof. Taking y � μ j+1 NΔt j and (16). We arrive the relation From (30), we obtain (1 + x)S j+1 � y + S j and which and the proof is complete.
is a decreasing sequence.
becomes an increasing sequence.

Remark 1.
Since R j is monotonic and bounded above by total population, then it will converge and lim j⟶∞ R j � R * exists, and (E j ) and (S j ) are decreasing sequence, and we easily observe that lim j⟶∞ I j � 0.

Formulation and Discussion.
Due to the ongoing nature of the COVID-19 pandemic, it was impossible to fully comprehend the short-or long-term implications of this global disruption. In our study, when there is no quarantine, a single infected individual can spread the infection to about two other people; however, when quarantine is imposed, there is a chance of preventing further transmission of infection. However, some of the exposed individuals may avoid quarantine due to fear of stigma and death. In other words, this does not achieve zero infection in the population, implying that additional interventions are required to eradicate the virus. If there are no adequate interventions in place, the virus will remain in the population for a long time, but it will eventually drop over time. But, still, there will be a small number of sick people who have the ability to start another outbreak even after measures like quarantine and public health education/awareness raise the number of exposed and infected people dramatically but not to zero. According to this, COVID-19 will not be completely eradicated even with prompt development of measures.In order to study the effects of isolation, quarantine, and the percentage of exposed people who will be quarantined, we did numerical simulations. Many authors have developed numerous mathematical models to limit the spread of viruses. Here, we have developed the optimization technique to control the spread of the virus. is approach enables us to control a viruses future spread and predict how it will spread in the future.After summarizing all of the prior principles, the problem is transformed into a dynamic programming problem model with constraints imposed by the previous lemmas by keeping S k , E k , I k , Q k , and R k are constants and β k , π k , μ k , c k , σ k , and θ k are variables at each level of the optimization. Note that the following dynamic programming problem preserves the monotonic properties. e dynamic programming problem is given by Min(β k + π k + μ k ) − (c k + σ k + θ k ) and subject to the constraints: Since S 0 , E 0 , I 0 , Q 0 , and R 0 are known initial conditions, we get a dynamic programming problem for level-0 time if we keep k � 0 in the above model. We shall obtain an optimal (feasible) solution (β * 0 , π * 0 , μ * 0 , c * 0 , σ * 0 , θ * 0 ) by employing the optimization technique in operation research.
en, these values are assigned as We will receive the values for finding S 1 , E 1 , I 1 , Q 1 , and R 1 . We will receive a dynamic programming problem for level-1 time if we keep k � 1 in this model. We will achieve the ideal solution as (β * 1 , π * 1 , μ * 1 , c * 1 , σ * 1 , θ * 1 ) using the optimization technique. ese values are assigned as We will receive the values for S 2 , E 2 , I 2 , Q 2 , and R 2 . If we keep going in this direction, we will end up with transmission rate sequences , and (θ k ) ∞ k�1 which provide a sufficiently viable stable solution for the situation, correspond to S 0 , E 0 , I 0 , Q 0 , and R 0 . After a certain stage, each transmission rate becomes constant. We must follow the appropriate standard operating procedures and safety precautions in order to acquire these sequences in practice. e isolation class, it appears, plays a significant role in achieving this possible solution.

Error Analysis
Now, we will set an upper limit for error propagation. We need to construct certain assumptions for our convergence analysis before proving the required statements. e following is a list of them: We get the following theorem under these conditions, in which we adopt notions from the error analysis of an explicit-implicit solution algorithm.

Theorem 9.
e difference between the solutions of the timecontinuous system formulation (3) and the time-discrete system (16) fulfills if the aforementioned assumptions are met, then Proof. Since this is technical proof, we will start with a brief description of our technique. e first step is the estimation of local errors between time-continuous and time-discrete solutions. After that, we look at error propagation over time.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.