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1. 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 different 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]. The 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 fight 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 significant successes. The most influential work in the field of mathematics epidemiology was first introduced by Kermack and McKendrick as the SIR model in the year 1927 [2]. Cao et al.[3] discussed a modified model of the SIR (susceptible, infected, recovered) epidemic introduced in order to detect the confirmed 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 time-explicit. We looked at the theory and methods for solving the time-implicit model. Then, 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. The article gives the technique for optimizing the transmission rate sequences. The 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.
1.1. Review Literature
In 2020, COVID-19 is a worldwide emergency. The 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. The 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. The 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. This 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-exposed-infectious-recovered (SEIR) model which has been analyzed extensively by He et al. [11] in the context of the COVID-19 pandemic. The Jacobian method used for the SEIR model yields a biologically reasonable
Table 1
Overview of models studied.
Models | Compartments | Model type | Study area |
SEIR/SLIR | Susceptible (S), exposed/latent (E/L), infectious (I), removed (R) | Deterministic | Europe and North America, New York, Mexico, Zhejiang, Guangdong, Japan, India [31–36] |
SEIQR | Susceptible (S), exposed (E), hospitalized infected (I), quarantine (Q), recovered or removed (R) | Deterministic | India [37] |
SIR-X | Infected (I), susceptible (S), removed (R), quarantined (X) | Deterministic | China [38] |
SIRD | Susceptible (S), infected (I), recovered (R), dead (D) | Deterministic | China, Italy, and France [39] |
SEIHARD | Susceptible (S), exposed (E), symptomatic infectious (I), hospitalized (H), asymptomatic infectious (A), recovered (R), deaths (D) | Deterministic | Washington, New York [40] |
SIRU | Susceptible (S), asymptomatic infectious (I), reported symptomatic infectious (R), unreported symptomatic infectious (U) | Deterministic | China, Hubei, Wuhan [41] |
SEIPAHRF | Susceptible (S), exposed (E), symptomatic (I), super-spreaders class (P), asymptomatic infectious (A), hospitalized (H), recovery (R), fatality (F) | Deterministic | Wuhan [42] |
SEIRU | Susceptible (S), asymptomatic noninfectious (E), asymptomatic infectious (I), reported symptomatic infectious (R), unreported symptomatic infectious (U) | Deterministic | China [43] |
SEIHR | Susceptible (S), exposed (E), symptomatic infectious (I), hospitalized (H), recovered or death (R) | Deterministic | South Korea [44] |
SEIRP | Susceptible (S), exposed (E), infectious (I), removed (R), pathogens (P) | Deterministic | Pakistan [45] |
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. This 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.
The 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. The conclusion of our research work is implemented in the last section 5.
2. Time-Continuous SEIQR Model
The time-continuous SEIQR model is formulated, and its behaviours are described using the Lipchitz condition and Grownwall and Bellman’s inequality in this section.
2.1. 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
(i) Let
Theorem 1 (see [17]).
Let
We give the following: Banach’s fixed point theorem, which will be used to preserve the global uniqueness in time [47, 47].
Theorem 2 (see [48]).
Let
In the following theorem, we present the Grownwall and Bellman inequality, which will be used in the subsequence theorems related to the continuous functions.
Theorem 3 (see [49]).
Let
Theorem 4.
(Method of variation of parameter) For a first-order nonhomogeneous linear differential equation,
2.2. 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
(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
(iii) Each time-varying transmission rate
The 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:
Table 2
Parameters and description.
Parameters | Description |
S(t) | At time t, the number of susceptible people |
E(t) | At time t, the number of exposed people |
I(t) | At time t, the number of infected people |
Q(t) | At time t, the number of quarantined people |
R(t) | At time t, the number of recovered people |
The rate at which susceptible populations migrate to exposed and infected populations | |
The rate at which an exposed population moves to an infected population | |
Transmission rate at which exposed people take outside as isolated | |
Transmission rate at which infected people were added to isolated individual | |
Transmission rate at which isolated persons recovered | |
Natural death rate and disease-related death rate |
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
Applying Theorem 4, and by applying the same procedure to the first-order nonhomogenous linear equation in
We can easily show that
Since
Theorem 5.
For all solution functions of (3), we have
Proof.
The proof follows from
2.4. Global Existence in Time
We arrive at a theorem regarding global existence in the time of (3) based on Theorem 1. For abbreviation, we use the supremum norm
Theorem 6.
The system of nonlinear first-order ODE (3) has at least one solution which exists for all
Proof.
By denoting
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
From the boundedness of our solution functions and the boundedness of our time-varying transmission rates, all requirements of Theorem 1 are fulfilled, and our proof is complete.
2.5. Global Uniqueness in Time
We present the global uniqueness theorem for (3) by utilizing the inductive application of Banach’s fixed point theorem.
Theorem 7.
The nonlinear ODE system (3) has a unique solution that exists for all
Proof.
Consider the system of equations given in (3):
(1) Consider the time interval
(2) For
(3) We assume that
Beginning the proof by letting
Then, 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
By choosing
2.6. Time-Discrete Implicit SEIQR Model
Assume that our time interval
2.7. Discussion and Formulations
Here, we are transforming the continuous system (3) to the fully explicit discrete scheme (14) as given as follows:
Observe that
2.8. Implicit Time-Discrete Problem Formulation
In this section and subsequent sections, we derive the recurrence type of solutions for the implicit scheme (15). For that, we assume that
2.9. Unique Solvability
In this section, we provide the method for finding the solution of (16). Letting
when
when
Now, substituting
In a similar way, we can easily find the other parameters as
Theorem 8.
Assume
Proof.
Substituting
Again, substituting the
We get the quadratic equation in the form when we solve equation (24)
The proof is completed by taking the roots of equation (26).
Similarly, by substituting
Since
3. 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.
Lemma 2.
If
Proof.
Taking
From (30), we obtain
Clearly,
Lemma 3.
If
Proof.
Taking
Here,
Lemma 4.
If
Proof.
Taking
Since
Lemma 5.
If
Proof.
Taking
Lemma 6.
If
Remark 1.
Since
3.1. 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. This 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
The dynamic programming problem is given by
Since
4. 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. The following is a list of them:
(i) Let
(ii) Allow the time-continuous and time-discrete models beginning circumstances to coincide
(iii) Let
(iv) Allow the time-varying transmission rates
(v) Allow the time-varying transmission and recovery rates to be bounded by 0 and 1
(vi) Choose
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.
The difference between the solutions of the time-continuous 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. The first step is the estimation of local errors between time-continuous and time-discrete solutions. After that, we look at error propagation over time. Finally, we look into the accumulation of these errors. At the same time, time-discrete solutions are expressed as
1) For the purpose of examining local errors, we assume that
a) For the time-discrete solution
Now, add and subtract
By applying the mean value theorem, there exists
This yields
Substituting equation (42) in the above and then solving the above relation, we get
Now, applying the triangle inequality in
By the mean value theorem, there exists
Hence,
Using the triangle inequality again in
From equation (52), we can easily find that
Now, substituting
Now,
Similarly, for
Proceeding through the similar steps, we can find
Using equations (52) and (61),
Now, the term
From (65) and (66), we get
Similarly, from (62) and (66),
Now, applying the procedure similar to the case
Similarly, substituting the
We can also easily find a
Finally,
We can also easily arrive
Using the same procedure given above, we get
Substituting equations (83)–(85) in
Substituting equation (86) in (73), we obtain the case.
c) For the time-discrete solution
Using the triangle inequality and mean value theorem, we can easily find
Applying
Solving the above equations, we get
The case (c) is verified by substituting equation (93) in (89).
d) Let the time-discrete solution for
Simplifying the above equation, we arrive
Substituting equation (98) in (96), we get the quarantine case.
e) Let the time-discrete solution for
which yields recovered instances by substituting equation (103) into (101).
(69) can be rewritten in the following form:
Definition 2.
Define
2) In reality,
3) The upper bound between the time-discrete solution and the time-continuous solution is verified below.
For
For
Now, applying the geometric series in (119), we obtain
Assuming
Hence, we get (39).
5. Conclusion
The present paper is devoted to the analysis and optimization of the SEIQR epidemic model containing an isolation class. We derived both continuous and discrete schemes of the SEIQR model as well as the global existence of solutions and nonnegativity bounded properties for both schemes. Along with this, we have illustrated the solving technique for the discrete scheme and proposed a new optimization technique with the help of a dynamic programming problem. In addition, we have analyzed the error between continuous and discrete schemes in the last section. Finally, we conclude that the isolation class plays a vital role in controlling the COVID-19 pandemic situation. More interestingly, the results also reveal that COVID-19 can exhibit oscillatory behaviour in the future. On the other hand, social distancing methods, quarantine efficiency, and isolation can be used to keep it under control. Future research could look into the effects of current coronavirus mutations like Delta and Omicron on the COVID-19 pandemic’s dynamics. We also suggested an alternative dynamic model of the SEIQR class, which can theoretically be generalized to generate continuous and discrete-time models such as SEIR, SEIRS, SIRS, SEI, SEIS, SI, SIS, SEIQRS, SIDARTHE, and others in future work.
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Abstract
The 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 finite time horizon. The problem involves finding the minimum objective function of a controlled process subject to the constraints of limited resources. For our model, 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 findings, isolating the affected humans can limit the danger of COVID-19 spreading in the future.
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1 Department of Mathematics, Sacred Heart College, Tirupattur 635601, Tamilnadu, India
2 Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysore 570005, Karnataka, India
3 Department of Mathematics, Ibb University, Ibb, Yemen