Mathematical analysis of epidemic model to assess the impact of lockdown on COVID-19

: COVID-19 and its variants, have been a worst pandemic, the entire world has witnessed. Tens of millions of cases have been recorded in over 210 countries and territories as part of the ongoing global pandemic that is still going on today. In this paper, we propose a SEI mathematical model to investigate the impact of lockdown to the controlling and spreading of infectious disease COVID-19. The epidemic model incorporates constant recruitment, experiencing infectious force in the latent period and the infected period. The equilibrium states are computed. Under some conditions, results for local asymptotic stability and global stability of disease-free and endemic equilibrium are established by using the stability theory of ordinary differential equations. It is seen that when the basic reproduction number 𝑅𝑅 0 < 1 , the dynamical system is stable and diseases die out from the system and when 𝑅𝑅 0 > 1 , the disease persists in the dynamical system. When 𝑅𝑅 0 = 1 , trans critical bifurcation is appeared. The numerical simulations are carried out to validate the analytical results.


Introduction
Coronavirus disease 2019 (COVID- 19) is an infectious disease caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2).The disease was first identified in the city of Wuhan, Hubei Province, China at the end of 2019 and spreaded globally, resulting in the ongoing 2019-2020 coronavirus pandemic.As per WHO situation reports on 4th of October 2020, the COVID-19 has claimed 1,030,738 lives, along with 34,804,348 confirmed cases worldwide [1] .On 11th of March 2020, Novel COVID-19 outbreak had been declared as a pandemic by WHO and made the call for countries for taking quick actions and scaling up the response to treat, detect and reduce dynamical transmission to save people's lives was reiterated.India stood at an important turning point in its challenge in opposition to COVID-19.To protect the country and to control the spread of COVID-19 outbreaks in India, bold and decisive steps were taken on the 24th of March 2020, by announcing a 21-day nationwide lockdown.Despite control policies such as to find, isolate, test, treat and trace the infected individuals and apart from a complete ban on all ages of people from stepping out of their homes, closure of commercial and private establishments, suspension of all research institutions, training, educational, closure of all worship places, suspension of non-essential private and public transport, the prohibition of Since, interventions such as the role of the media, advertisements, lockdown, etc. have a massive impact on the course of infectious diseases and hence have been discussed via mathematical models for various diseases, for example, a mathematical epidemiological model of COVID-19 cases in Italy [17] , modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy [18] , the effect of therapy and awareness campaigns on a SIR model [19] , a mathematical model via non-linear systems describe the spread of the COVID-19 virus [20] , fundamental theory of infectious disease transmission using straightforward compartmental models based on ordinary differential equations, such as the straightforward Kermack-McKendrick epidemic model [21] and a fresh COVID-19 epidemic model with media coverage [22] , a nonlinear SEIRS type epidemic model with media impact for transmission dynamics of infectious diseases [23] , reproduction numbers of infectious disease models [24] , and reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission [25] .In this paper, we demonstrate theoretically how the preventive measure of lockdown impacts the dynamical transmission of coronavirus disease by proposing the SEI mathematical epidemic model, in which illness is infectious not only in the infected period but in the latent period too.Our model is like that proposed by Liu and Cui [26] , but there are fundamental differences between the two models.The model proposed by Liu and Cui is the classical SIR model where the disease is infectious in the infected period only.
The paper is organized as follows: In section 2 an SEI model incorporating the impact of lockdown on the spreading of COVID-19 is proposed with the consideration that the disease is infectious in the latent period too.In section 3, the threshold parameter R0 is obtained, and based on so obtained number, the feasibility of equilibria has been discussed.Section 4 deals with discussion and analysis of local and global stability of disease-free equilibrium (DFE) along with local stability of endemic equilibrium (EE) by Routh-Hurwitz criterion and constructing a suitable Lyapunov function.To illustrate the viability of theoretical analysis, the numerical simulations of the proposed model are carried out by using MATLAB in section 5.The last section 7 is dedicated to the discussion and conclusion of the paper.

An SEI model with lockdown impact
Coronavirus disease continues to race around the world at a worrisome and fierce pace.Governments, organizations, and people are yelling to find out the safeguards to fight back effectively.To get these answers, accurate and comprehensive models are needed that are good enough to depict as many as possible aspects of the disease.Every mathematical epidemic model emphasizes certain components of actual real phenomena and ignores others due to limitations of relevant mathematical theories and complexities involved therein.Likewise, it is almost impossible to reflect all aspects of genuineness for the disease COVID-19 via a single mathematical model [27] .In this section, we propose a mathematical model for infectious coronavirus disease in which an infected person does not become infective for some time.Such a person who is infected but not infective is called exposed.The total population N is partitioned into three compartments: S(t) the class of susceptible, E(t) the class of exposed, I(t) the class of infectious.In most cases, a host must go through a latent stage after the original infection before becoming contagious.To the best of our knowledge, in the SEI models related to COVID-19 proposed till now, the disease has not been considered infectious in the latent period.But, keeping in view the transmitting feature of COVID-19, here we consider the following SEI model with the feature that the disease is infectious in a latent period as well, and our model is depicting the effect of lockdown on the dynamical transmission of the disease.
The Schematic flow chart diagram of the system (1) is given in Figure 1.The model is grounded on mass action incidence and positive parameters with below epidemiological interpretations: •  is the constant rate at which the susceptible are recruited in the population.•  is the average natural death rate across all groups.•  is the rate of efficient contact in the latent period before lockdown.

•
is the effective contact rate prior to lockdown during the infected time.

•
is the rate of transmission between infected and exposed people.
is the rate of disease caused deaths from an exposed population.

•
is the rate of disease caused deaths from an infectious population.

•
is the rate of segregating after disease is constant.
The control measures implemented by public health officials during a disease outbreak, such as lockdowns, restaurant closures, school closures, isolating infected people, postponing conferences, etc., can have an impact on the homogeneous incidence rate.The contact per unit of time t is typically decreased by these required actions.This causes a high number of infected people but a smaller chance of infection per contact, which could cause non-linearity in dynamic transmission rates.In the proposed model (1), () =  + and ℎ() =  + are the rate of contacts reduced as the impact of lockdown in the infected and latent period respectively, where  can be thought of as the rate of implementation of lockdown.Here  and n reflect the reactive velocity of lockdown and people in infected and latent periods respectively.Thus  − () and  − ℎ() are the frequency of effective contact during the latent and infected time during lockdown.Further, it is assumed that  <  .The functions () and ℎ() satisfy the following: Since lockdown cannot prevent the disease completely, we take ,  > .Note that when the reported exposed and infectious number arrive at  and  respectively, then in each case, corresponding reduced value of transmission equal to  2 . Here +  +  = .Therefore, we have . From biological point of views, we discuss the system (1) in the following feasible region  = �(, , ) ∈  3 :  =  +  +  ≤   �.Thus, the total population remains bounded for all future  ≥ 0.Here the domain  is non-negative invariant as no solution paths leave through any boundary.The right-hand side (RHS) of each of the equations in the system (1) is smooth and continuously differentiable so that the initial value problems have singe solutions that exists on maximal intervals.Since paths can't leave , solutions exist ∀  > 0.
Thus, the model is epidemiologically and mathematically well posed.

Steady states and reproduction number (𝑹𝑹 𝟎𝟎 )
The system (1) has a DFE  0 = �   , 0, 0�.Since the local stability of the DFE of compartmental models is governed by  0 of the system, referring to Rafiq et al. [7] and Liu and Cui [26] , we compute the  0 by the next-generation matrix method [28,29] .We have the following two vectors ℱ and  to represent the new infection terms and remaining transfer terms respectively.
Proof.For EE, set the RHS of system (1) equal to zero.Note that the EE  1 (, , ) satisfies  > 0,  ≥ 0,  > 0, we have Adding Equations ( 2), ( 3) and ( 4 Since  = 0 corresponds to disease free equilibrium, therefore cancelling  throughout, and using the above equation takes the form Case (i) When  0 > 1 and  < , we have Therefore, the product of roots of Equation ( 8) i.e., −  ′  ′ > 0 therfore there are two possibilities: either one root is positive or all the three roots are positive.But the sum of roots i.e., −  ′  ′ < 0, therefore, we cannot have all roots positive.Henece there is only one positive root i.e., unique EE of the system (1)   which is given by Therefore, Equation (8) becomes ( ′  2 +  ′  +  ′ ) = 0. Since  ′ = 0 corresponds to disease free equilibrium, therefore for endemic equilibrium, we consider  ′  2 +  ′  +  ′ = 0 which is a quadratic equation with sum of roots −  ′  ′ < 0 and the product of roots  ′  ′ > 0. Therefore, no positive root of this equation exists and hence no EE exists for  0 = 1.□

Stability analysis
In this Section we will evaluate the local stability of the model (1).
Proof.The Jacobian matrix  = [  ]of the linearization of system (1) at point arbitrary point (, , ) is ( 2 + ), which are real and negative.So,  0 is marginally locally stable.More precisely, as  0 increases through 1, there is an exchange of stability between DFE and the EE (which is biologically meaningless if  0 < 1 ).Hence the equilibrium infective and infectious population size depend continuously on  0 and there is a forward, or transcritical bifurcation in equilibrium behavior, at  0 = 1.
Proof.Since,  = +   − , its derivative along the solutions of the system (1) is Clearly, as   ≤ 1 , for  0 ≤ 1,  ′ ≤ 0 .Also  ′ = 0 only if  =  = 0 .Therefore, the maximum invariant set in  is the singleton set {E 0 } and hence the global stability of E 0 when  0 ≤ 1 follows from LaSalle's invariance principle.□ In the next theorem, we will prove the asymptotic stability of the EE  1 ( * ,  * ,  * ) and for that, the following notations have been used.1) is LAS if the following conditions hold true: , the rate constant of segregating after disease and hence  is chosen satisfying: Let us choose  and  31 such that where, Proof.Let  =  −  * ,  =  −  * ,  =  −  * be small perturbations about the  1 .Using these new variables, we linearize the model ( Note that Equations (i) and (ii) follows by the fact that  1 ,  2 ,  3 < 0 and hypothesis (12).Also, if we select it can be seen easily that (iii) and (v) are valid.

Numerical simulations and discussions
In this Section, we present computer simulation results for model Equation (1) by using MATLAB 16.0.
For the same values of parameter , i.c.'s and  * as used for S.N.1 of Table 2, we see that the endemic equilibrium point for set of values of S. N. 2 of Table 2 and Figure 3 exists as  1 ( * ,  * ,  * ) =  1 (36630.24,82.08925, 1287.675).Using the parameters in Table 1, S. N. 2 along with the i.c.'s stated above in this subsection, variational curves of ,  and  with  in the Figure 5 validate the results of theorem 4.2 when  0 = 1.

Discussion and conclusion
In this manuscript, we have formulated an SEI mathematical epidemic model combined with the impact of lockdown for COVID-19, taking into consideration the fact that the disease is infectious in a latent period too.For this model, we have found the  0 and shown that if this parameter is less than 1, then DFE is GAS.Under certain restrictions, the existence of an EE for  0 > 1, followed by conditions ensuring local asymptotic stability of EE is presented.By the means of the results of the work and the numerical simulation done in section 5, it has been verified that the lockdown significantly decreases the transmission rate of the disease.Since  0 is independent of ,  and , it can be said that the lockdown does not change the reproduction number.

Figure 3 .
Figure 3. Plots of Infectious, Exposed and Susceptible population v/s time for S. N. 2 of Table2.

Table 1 .
Parameters for the simulation.
Figure 2. Plots of Infectious, Exposed and Susceptible population v/s time for S. N.1 of Table 1.5.2.Impact of Implementation of Lockdown in case of EE i.e.,   >

Table 2 .
Parameters for the simulation.