key: cord-0803264-z06nrzan
authors: Prem Kumar, R.; Basu, Sanjoy; Ghosh, Dipankar; Santra, Prasun Kumar; Mahapatra, G. S.
title: Dynamical analysis of novel COVID‐19 epidemic model with non‐monotonic incidence function
date: 2021-09-02
journal: J Public Aff
DOI: 10.1002/pa.2754
sha: a00c7d45eef027da3c8a48d3a3e34137ad312504
doc_id: 803264
cord_uid: z06nrzan

In this study, we developed and analyzed a mathematical model for explaining the transmission dynamics of COVID‐19 in India. The proposed [Formula: see text] model is a modified version of the existing [Formula: see text] model. Our model divides the infected class [Formula: see text] of [Formula: see text] model into two classes: [Formula: see text] (unknown infected class) and [Formula: see text] (known infected class). In addition, we consider [Formula: see text] a recovered and reserved class, where susceptible people can hide them due to fear of the COVID‐19 infection. Furthermore, a non‐monotonic incidence function is deemed to incorporate the psychological effect of the novel coronavirus diseases on India's community. The epidemiological threshold parameter, namely the basic reproduction number, has been formulated and presented graphically. With this threshold parameter, the local and global stability analysis of the disease‐free equilibrium and the endemic proportion equilibrium based on disease persistence have been analyzed. Lastly, numerical results of long‐run prediction using MATLAB show that the fate of this situation is very harmful if people are not following the guidelines issued by the authority.

model with nonlinear incidence rate and time delay. Yang et al. (2010) formulated a SIR model with vaccination and varying population. Sun and Hsieh (2010) investigated an susceptible exposed infected recovered (SEIR) model with varying population size and vaccination strategy. Zhou and Cui (2011) studied an SEIR epidemic model with a saturated recovery rate. Bai and Zhou (2012) proposed an SEIRS epidemic model with a general periodic vaccination strategy and seasonally varying contact rates. Khan et al. (2015) considered an SEIR model with nonlinear saturated incidence rate and temporary immunity. Elkhaiar and Kaddar (2017) studied the dynamics of an SEIR epidemic model with nonlinear treatment function that takes into account the limited availability of resources in the community. Wang et al. (2018) extended the incidence rate of an SEIR epidemic model with relapse and varying total population size to a general nonlinear form. Tiwari et al. (2017) investigated an SEIRS epidemic model with nonlinear saturated incidence rate. Lahrouz et al. (2012) studied the global dynamics of a SIRS epidemic model for infections with nonpermanent acquired immunity. Tian and Wang (2011) discussed the global stability analysis for several deterministic cholera epidemic models. Samanta (2011) discussed the permanence and extinction of a non-autonomous HIV/AIDS epidemic model with distributed time delay. Cai et al. (2014) investigated an HIV/AIDS treatment model. Gralinski and Menachery (2020) China. Maheshwari et al. (2020) forecasted the epidemic spread of COVID-19 in India using the ARIMA model. Zakharov et al. (2020) predicted the dynamics of the COVID-19 epidemic in real-time using the case-based rate reasoning model. Bonnas and Gianatti (2020) proposed a COVID-19 epidemic model where the population is partitioned into classes corresponding to ages. Roda et al. (2020) demonstrated the reasons for wide variations in numerous model predictions of the COVID-19 epidemic in China. Liu et al. (2020a) developed two differential equations models to account for the latency period of COVID-19 infection. Basnarkov (2021) studied a SEAIR epidemic spreading model of COVID-19. Yang and Wang (2020) proposed a mathematical model for the novel coronavirus epidemic in Wuhan, China. Wang, Lu, et al. (2020) performed the dynamical analysis of a COVID-19 epidemic model. Zlatic et al. (2020) developed a COVID-19 epidemics model spreading on the availability of tests for the disease. Xue et al. (2020) proposed a data-driven network model for the COVID-19 epidemics in Wuhan, Toronto, and Italy. Neves and Guerrero (2020) presented the A-SIR model to predict the evolution of the COVID-19 epidemic. Ndairou et al. (2020) proposed a mathematical model for COVID-19 epidemic with a case study of Wuhan. Jiao and Huang (2020) proposed a SIHR COVID-19 epidemic model with effective control strategies. Zhao and Chen (2020) modeled the epidemic dynamics and control of the COVID-19 outbreak in China. Li et al. (2020) modeled the impact of mass influenza vaccination and public health interventions on COVID-19 epidemics. Pizzuti et al. (2020) investigated the prediction accuracy of the SIR model on networks for Italy. Wang, Zheng, et al. (2020) devised a method to analyze the COVID-19 epidemic. Kantner and Koprucki (2020) computed a strategy for the case that a vaccine is never found and complete containment is impossible. Engbert et al. (2021) presented a Stochastic SEIR epidemic model for regional COVID-19 dynamics by sequential data assimilation.

Several researcher investigated the dynamics of COVID-19 using fractional order models (Askar et al., 2021; Awais et al., 2020; Rezapour et al., 2020) . Rihan et al. (2020) analyzed a stochastic SIRC epidemic model with time-delay for COVID-19. Bambusi and Ponno (2020) explained the linear behavior in COVID-19 epidemic as an effect of lockdown. Alberti and Faranda (2020) presented statistical predictions of COVID-19 infections by fitting asymptotic distributions to actual data. Abbasi et al. (2020) discussed the Optimal control for Impulsive SQEIAR Epidemic model on COVID-19 epidemic. Lobato et al. (2020) identified an epidemiological model to simulate the COVID-19 epidemic. Khan and Atangana (2020) modeled the dynamics of novel coronavirus with fractional derivative. Alshammari and Khan (2021) analyzed the dynamics of modified SIR model with nonlinear incidence and recovery rates. Pal et al. (2021) presented a COVID-19 model with optimal treatment of infected individuals and the cost of necessary treatment. Khan et al. (2021) focused on the novel coronal virus model to understand its dynamics and possible control. Khajanchi and Sarkar (2020) This paper determines the fate of coronavirus infective individuals introduced into the population in India. The dynamics of the nonlinear system have been considered in the study with reinfection turned off. The basic reproduction number (BRN) ℜ 0 is estimated and analyzed as a threshold parameter for the stability analysis of the disease-free equilibrium (DFE) and endemic equilibrium. The uniform persistence of the disease near the threshold parameter is also determined.

The proposed COVID-19 model involves a specific postulate considered for developing mathematical modeling in the Indian perspective.

Hypothetically, we imagine unknown infected peoples are spreading the diseases. Known infected peoples are isolated, so they are not able to spread the diseases. In the model, susceptible individuals enter into the unknown infected population by adequate personal contact with the unknown infected individuals given by non-monotonic incidence function αSIu The known infected individuals entered the recovered class after recovered from the COVID-19. Here in this model, we consider the recovered class and the reserved class as the same and denote the density at time t by R t ð Þ. The model of the study has been taken in the following form

The

where S t ð Þ, I u t ð Þ, I k t ð Þ and R t ð Þ are the densities at the time t of susceptible population, unknown infected population (incubate the illness but do not have any symptoms and not identified), known infected population (in the isolated ward), and recovered or reserved population, respectively, and the parameters Λ, α, β, δ, r 1 , r 2 , and and those individuals who move to the reserved or safer areas from the susceptible compartment (S).

3.1 | Non-negativity of solutions Theorem 1. Every solution of the system (1) with initial conditions (2) are non-negative for every t ≥ 0.

Proof. The right hand side of the dynamical system (1) is completely continuous and locally Lipschitzian on C 1 and hence the solution S t ð Þ, I u t ð Þ, I k t ð Þ, R t ð Þ ð Þ of the system (1) with the initial conditions (2) exists and is unique on the interval 0, l ½ Þ with 0 < l ≤ ∞. From the first equation of the system (1) with initial condition S 0 ð Þ> 0, we get

Integrating the second equation of the system (1) with initial condition I u 0 ð Þ≥ 0, the solution can be written in the form as

From the third and fourth equations of the system (1) with initial conditions I k 0 ð Þ≥ 0 and R 0 ð Þ≥ 0, we have

This completes the proof of the theorem.■

Theorem 2. All solutions of system (1) which lies in R 4 þ are uniformly bounded and are confined to the invariant region

The time derivative of Equation (3) is

Applying the theory of differential inequality (Birkhoff & Rota, 1989) , we find that

Thus, all solutions of the COVID-19 system (1) which initiate in R 4 þ are uniformly bounded and confined to the region Δ,

Hence the feasible region Δ with initial conditions (2) is positively invariant region under the flow induced by the system (1) in R 4 þ .■

Remark. All solutions of system (1) have non-negative components, given non-negative initial values in Δ and stay in Δ for t ≥ 0 and globally attracting in R 4 þ with respect to the system (1). Therefore, we restrict our attention to the dynamics of the system (1) in Δ. Thus the system (1) with initial conditions (2) defined on Δ ¼

is well-posed mathematically and epidemiologically and it is sufficient to study the dynamics of the dynamical system (1) with initial conditions (2) defined on Δ.

To evaluate the equilibrium points of the system (1), we have to study the zero growth isoclines and the point of interaction. The possible steady-state boundary equilibrium point is

Since the last equation of system (1) does not depend on other equations, we simply study the reduced

where g I u ð Þ¼ αIu 1þδI 2 u is increasing when I u is small and decreasing when

The endemic equilibrium point of the system (4) is

To eradicate the disease from a varying size population, the more stringent way requires that the total number of the virus-infected population I u t ð ÞþI k t ð Þ ð Þ ! 0, while a weaker requirement is that proportion sum of the same tends to zero (Busenberg et al., 1991) . Thus we need to find the conditions for the existence and stability of the DFE P 0 Λ r1þd1 ,0,0 and the endemic equilibrium P * S * , I * u , I * k À Á . Therefore, P 0 Λ r1þd1 ,0,0 is the DFE of (4), which exists for all positive parameters.

The BRN is the average number of secondary infections generated by a single infection and is one of the most vital threshold quantities which mathematically represent the spreading of the virus infection.

The Jacobian matrix of system (4) at an arbitrary point P S, I u ,

The stability of P 0 is equivalent to all the eigenvalues of the characteristic equation of J P ð Þ at P ¼ P 0 being with negative real parts, which can be assured by the BRN (ℜ 0 ) obtained by the next-

where ℜ 0 is the epidemiological threshold parameter.

Then the system (1) can be written as

The Jacobian matrices of F x ð Þ and V x ð Þ at the DFE P 0 are given by

from the above two matrices we get the matrices F and V as below

The spectral radius of the matrix FV À1 is ρ FV À1 À Á which is the

Þβþd1 ð Þ > 0. From this observation, it is obvious that if the transmission coefficient α from the susceptible population (S) to unknown infected population (I u ) decreases, then the BRN ℜ 0 also decreases and therefore reduces the burden on the infection. Otherwise, if α increase, then ℜ 0 would also increase, and thus, the transmission of virus infection will also rise; therefore, the scenario will be very harmful to society. Now, the BRN ℜ 0 has been presented graphically in Figure 2 with respect to related estimated or hypothetical parameter values given in Table 1 . From Figure 2 , it is observed that as the value of α increases, ℜ 0 also increases simultaneously and become greater than unity after a certain value of α. Therefore, it is said that up to a certain value of α, the DFE point is stable (Theorem 3) and beyond that value of α, the endemic equilibrium point is stable (Theorem 6).

This section will discuss the parameter restrictions of the local stability of DFE.

Theorem 3. The disease free equilibrium P 0 Λ r1þd1 ,0,0 of the system (4) is locally asymptotically stable if ℜ 0 < 1.

Whereas, it is unstable if ℜ 0 > 1.

Proof. The variational matrix of the system (4) at P 0 Λ r1þd1 ,0,0 is given by

The eigen values of the characteristic equation of

For stability, all eigen values must be negative. So,

Hence, the DFE is locally asymptotically stable if ℜ 0 < 1.■

In this section, we will discuss the parameter restrictions of the global stability of DFE.

Theorem 4. When ℜ 0 < 1, the disease free equilibrium n o is the singleton set P 0 , when ℜ 0 < 1. By using Lasalle's invariance principle (LaSalle, 1976) , the DFE P 0 is globally asymptotically stable in Δ 1 , when ℜ 0 < 1.■

In this subsection, an effort is made to understand the uniform persistence of the dynamical system (4) for the threshold parameter by applying the acyclicity theorem (Sun & Hsieh, 2010) .

Definition 1. The system (4) is said to be uniformly persistent (Butler et al., 1986) if there exists a constant σ > 0 such that all solutions S t ð Þ,I u t ð Þ, I k t ð Þ ð Þ with positive initial S 0 ð Þ, I u 0 ð Þ, I k 0 ð Þ satisfy the following inequality min lim

Let X be a locally compact metric space with metric d, and let Γ is a closed non-empty subset of X with the boundary ∂Γ and interior Γ 0 .

Clearly, ∂Γ is a closed subset of Γ and let Φ t be a dynamical system on Γ.

Then set Y in X is said to be invariant if Φ Y, t ð Þ¼Y.

Theorem 5. Suppose the conditions H1 and H2 holds true for the dynamical system Φ t H1: The system Φ t has a global attractor. 

which is possible,as ℜ 0 < 1 Then the dynamical system Φ t is uniformly persistent with respect to Γ 0 .

Proof. For the modified COVID-19 system (4), we assume that

So it is concluded that M ¼ P 0 f g and ω x ð Þ ¼ P 0 f g for all x ∈ M ∂ which proves (i) and (ii) of H2. From Theorem 3 the DFE P 0 is unstable When ℜ 0 > 1 and also W s M ð Þ¼ ∂Γ which indicates that (iii) and (iv) of H2 are satisfied.

Since all system (4) solutions are uniformly bounded, a global attractor exists and hence H1 holds true for the F I G U R E 3 Time series plot of S (a), I u (b), I k (c) when ℜ 0 < 1 for different initial conditions using parameter values from Table 1 F I G U R E 4 Time series plot of S (a), I u (b), I k (c) when ℜ 0 > 1 for different initial conditions with parameter values from Table 1 with α ¼ 2 Â 10 À9 system (4). Hence the system (4) is uniformly persistent with respect to Γ 0 , when ℜ 0 > 1.■

From Theorem 4, it is already observed that DFE is globally asymptotically stable when ℜ 0 < 1 which implies that there is no endemic equilibrium when ℜ 0 < 1. To analyze the existence of nontrivial interior equilibrium of system (4), it should satisfy the following conditions:

with S,I u , I k > 0 and the above Equations (9) leads to the following solu-

Hence the nontrivial interior equilibrium P * of system (4) exists if ℜ 0 > 1.

Theorem 6. The endemic equilibrium P * of the system (4) is locally asymptotically stable in Δ 1 if ℜ 0 > 1.

F I G U R E 5 Time history of the unknown infected population (I u ) and known infected population (I k ) for α ¼ 15 Â 10 À11 ; α ¼ 25 Â 10 À11 , and α ¼ 35 Â 10 À11 F I G U R E 6 Time history of the unknown infected population (I u ) and known infected population (I k ) for α ¼ 13 Â 10 À11 ; α ¼ 23 Â 10 À11 , and α ¼ 33 Â 10 À11

Proof. The variational matrix of the system (4) at

The characteristic equation of J P * À Á is

where B 1 ¼ À a 11 þ a 22 ð Þand B 2 ¼ a 11 a 22 À a 21 a 12 . One of the eigenvalues of J P * À Á from the Equation (10) is a 33 which is negative. The

Routh-Hurwitz conditions state that the quadratic equation 

Obviously B i > 0 for i ¼ 1,2. Hence, the positive equilibrium point P * S * , I * u , I * k À Á is locally asymptotically stable if ℜ 0 > 1.■ F I G U R E 7 Time history of the unknown infected population (I u ) and known infected population (I k ) for α ¼ 23 Â 10 À11 and α ¼ 25 Â 10 À11 F I G U R E 8 Long run prediction of the unknown infected population (I u ) and known infected population (I k ) for α ¼ 23 Â 10 À11 and α ¼ 25 Â 10 À11

To investigate the globally stability of the endemic equilibrium of system (4) when ℜ 0 > 1, we apply here a geometric approach (Li & Muldowney, 1996) . We begin the preliminary discussion on the geometric approach by formulating the local version of the C 1 closing lemma of Pugh (Hirsch, 1991) . Consider the differential

simply connected

and f ∈ C 1 G ! R n ð Þ .

Let A y ð Þ be an n 2 Â n 2 matrix valued function which is C 1 in The following quantity q 2 ¼ lim

well defined. If there exists a compact absorbing set κ & G and the system (11) has a unique equilibriumŷ in G, then the unique equilibriumŷ of (11) is globally asymptotically stable in G if q 2 < 0.

Theorem 7. Assume that ℜ 0 > 1. Then there exist β > 0 such that the unique endemic equilibrium P * is globally asymptotically stable in the interior of Δ 1 when β ≥ β.

Proof. To prove this result, we find the second additive compound matrix J 2 ½ from the Jacobian matrix J P ð Þ of (5) for the reduced system (4) at P * in the following form:

F I G U R E 9 Unknown infected population (I u ) with time for corresponding the value of β ¼ 0:25 and β ¼ 0:30 F I G U R E 1 0 Known infected population (I k ) with time with the recovery rate r 2 ¼ 3 Â 10 À2 and r 2 ¼ 5 Â 10 À2

where

Therefore, the matrix Q can be written in the following block form as

Suppose μ be the Lozinskii measure with the above defined norm, so as described in (Martin, 1974) and it follows the condition as μ Q ð Þ≤ sup f 1 ,f 2 ð Þ, where

and Q 21 j j are the matrix norm with respect to the L 1 vector norm. If μ 1 denotes the Lozinskii measure with respect to L 1 norm. Using the second equation of system (4), we have

F I G U R E 1 1 Day wise infected, recovered, and deceased population from 1st to 20th March 2020 in India

Using the third equation of system (4), we have

Since system (4) is uniformly persistent when ℜ 0 > 1, there exist σ > 0 and T 1 > 0 such that t > T 1 implies I u t ð Þ ≥ σ, I k t ð Þ ≥ σ, and 1 t ln

which implies that q 2 < 0. Thus, the endemic equilibrium P * of the reduced system (4) is globally asymptotically stable, when ℜ 0 > 1.■

In this section, firstly we consider the case when the BRN ℜ 0 ¼ 0:4805 < 1 by utilizing the parameter values as in Table 1 . For different initial conditions, the dynamics of the system (4) is represented in Figure 3 . These figures illustrate that the susceptible population (S) persists and tends to S 0 ¼ 7:6231 Â 10 7 as t ! ∞ and the unknown infected population (I u ) and the known infected population (I k ) tends to zero as t ! ∞, that is, the system (4) approaches the DFE P 0 7:6231 Â 10 7 ,0,0 . This numerical simulation supports the result stated in Theorem 4. Next, we consider the case when ℜ 0 ¼ 3:8442 > 1 by utilizing the values of parameter from Table 1 with α ¼ 2 Â 10 À9 . For various initial conditions, the dynamics of the system (4) is represented in Figure 4 . These figures illustrate that the susceptible population (S), the unknown infected population (I u ) and the known infected population (I k )

all persist, that is, the system (4) tends to endemic equilibrium P * 7:27579 Â 10 7 ,5:4120 Â 10 4 ,8:35830 Â 10 5 .

To determine the outbreak of infected individuals of COVID-19 disease in the Indian population, we present the numerical simulation of the proposed dynamical system (1), using MATLAB for simulation experiments, based on the SARS-CoV-2 virus-infected cases in the time frame in India. Here, we employ the nonlinear least-squares curve fitting method with the help of "fminsearch" function from the MATLAB Optimization Toolbox to obtain the best-fit parameters for INDIA. The procedure looks for the set of initial guesses and preestimated parameters for the model whose solutions best fit or pass through all the data points by reducing the sum of the square difference between the observed data and the model solution, that is, If a theoretical model t↦Γ t,q 1 , q 2 , …, q n ð Þ is attained and depend on a few unknown parameters q 1 ,q 2 , …, q n and a sequence of actual data points t 0 , y 0 ð Þ, …, t j , y j À Á is also at hand then the aim is to obtain values of the parameters so that the error E calculated can attain a minimum,

where E≔ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P j i¼0 Γ t, q 1 , q 2 , …, q n ð Þ À y i ð Þ 2 q .

For simulation, we assume the initial values are S 0 ð Þ¼8 Â 10 8 , Table 1 ). The values of the parameters are fixed based on the following real-time data of India, as shown in Table 2 , the spread of the COVID-19 disease during the lockdown period is recorded as follows (MoHFW, 2021) Figure 5 has been drawn for the unknown infected population (I u ) and known infected population (I k ) for α ¼ 15 Â 10 À11 ;

α ¼ 25 Â 10 À11 , and α ¼ 35 Â 10 À11 for the specified period from March 25 to April 14, 2020 during the first lockdown. This figure is fascinating because it is observed that for α ¼ 25 Â 10 À11 , the known infected population curve of our proposed system fits to the curve of the real confirmed infected individuals in India during the above said period. Figure 6 shows the time history of the unknown infected population (I u ) and known infected population (I k ) for α ¼ 13 Â 10 À11 ;

α ¼ 23 Â 10 À11 , and α ¼ 33 Â 10 À11 for the period from April 15 to April 20, 2020 during the second lockdown situation. Figure 10 is very interesting because it is observed that for α ¼ 23 Â 10 À11 , the known infected population curve of our proposed COVID-19 system fits to the curve of the real confirmed infected individuals in India during the above said period. and known infected (I k ) populations of the proposed system corresponding to α ¼ 23 Â 10 À11 and α ¼ 25 Â 10 À11 fits to the curve of the real confirmed infected individuals in India during the lockdown period from March 25 to April 20, 2020.

It is also clear that α as a representative of lack of following the good practices such as proper hand wash, sanitizing the places, nasal, and oral covering with a mask, social distancing has an effect in the proposed coronavirus model and an increase in α means many individuals are not following the good practices as said above and as a result, many individuals gets infected and move to the unknown infected population (I u ). Long run prediction of the time history of the known infected population (I k ) for α ¼ 23 Â 10 À11 and α ¼ 25 Â 10 À11 is drawn

in Figure 8 , which shows that the disease gets diminished may be due to the availability of a Vaccine.

From Figure 8 , it is observed that the active cases will decrease and the COVID-19 disease will persist in the society for a long period.

Furthermore, it is noticed that the COVID-19 disease diminishes after a long period if people do not strictly follow the government guidelines and vaccination is not found at the earliest as we have considered that geographical and climatic factors do not have any impact on this virus infection.

From Figure 9 , it is noticed that if β increases, which is a representative of the identification process of the infected individuals, then the unknown infected population reduces and hence we can identify the individuals for quarantine. Thus, the lockdown period can be reduced and people can come back to a normal situation. Figure 10 shows that if the recovery rate r 2 increases, which representing the quality of treatment and cooperation of the patient to the treatment, then the known infected population (I k ) tends to zero in the long run. Hence, the rapid spread of COVID-19 disease is reduced drastically. It is also observed that the day-wise recovered cases are also increasing and day-wise deceased cases are very low. From the above discussions, it can be said that SARS-COV-2 is threatening to the society, but not deadly dangerous until now in Indian perspective.

The proposed model has determined the outbreak of COVID-19 disease in the Indian population. The BRN ℜ 0 is the threshold limit that determines the dynamical proliferation. The reproduction number ℜ 0 decreases if the transmission coefficient α decreases. If α increases, then the transmission of COVID-19 disease increases and is very harmful to society. The system has a unique DFE P 0 , which is globally stable if ℜ 0 < 1 which symbolizes that the disease diminishes eventually. When ℜ 0 > 1, the system is uniformly persistent under some conditions, a unique endemic equilibrium is globally stable. The study

shows that the infected population who incubate the illness of our proposed system fits well to the real confirmed infected individuals in India during the lockdown period.

By observing the time graph of the known infected class (I k ) for various values of α, we conclude that if people do not strictly follow the guidelines and lockdown measures imposed by the Government of India to prevent the rapid spread of the infection, the situation will be out of control.

We are grateful to the editor and anonymous referees for their careful reading, valuable comments, and helpful suggestions which have helped us to improve the presentation of this work significantly.

The authors declare no conflicts of interest.

Analysis and draft the paper: R. Prem Kumar and Sanjoy Basu. Collected the data, conceived and designed the analysis, perform the analysis tool for the paper: D. Ghosh and P. K. Santra. Supervised, designed the analysis and wrote the paper: G. S. Mahapatra.

Available at https://www.who.int, www.mohfw.gov.in, www.

covid19india.org

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We, the undersigned, give our consent for the publication of identifiable details, which can include figures and details within the text ("Material") to be published in the above Journal and Article. Therefore, anyone can read material published in the Journal. 

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Prem Kumar is an Assistant Professor of Mathematics at Avvaiyar Government College for Women, Karaikal-609602 in the Union Territory of Puducherry. He is pursuing his Ph.D at National Institute of Technology Puducherry

Basu has been involved in teaching and research for more than twelve years and has published more than seven research articles in various International, National journals and proceedings to his credit

Ghosh has been involved in research for more than four and half years and has published more than five research articles in various international journals and proceedings to his credit

Prasun Kumar Santra holds the degrees of M.Sc. in Applied Mathematics from

Santra has been involved in research for more than five years and has published more than twelve research articles in various international journals and proceedings to his credit. His research interest field is Mathematical Biology

Shibpur, presently IIEST Shibpur, India. Dr. Mahapatra has PREM KUMAR ET AL. been involved in teaching and research for more than seventeen years and has published more than ninety research articles in various International, National journals and proceedings to his credit

Dynamical analysis of novel COVID-19 epidemic model with non-monotonic incidence function

introduction of the uniform persistence theorem . This quantity is defined as per the Geometric approach method to investigate the global stability of the endemic equilibrium