key: cord-1002536-q6rcxn7m
authors: Basu, Sanjoy; Kumar, R. Prem; Santra, P. K.; Mahapatra, G. S.; Elsadany, A. A.
title: Preventive control strategy on second wave of Covid-19 pandemic model incorporating lock-down effect
date: 2022-01-11
journal: Alexandria Engineering Journal
DOI: 10.1016/j.aej.2021.12.066
sha: 874503ffe84b381b0bb0942739aa3ee468ab6446
doc_id: 1002536
cord_uid: q6rcxn7m

This study presents an optimal control strategy through a mathematical model of the Covid-19 outbreak without lock-down. The pandemic model analyses the lock-down effect without control strategy based on the current scenario of second wave data to control the rapid spread of the virus. The pandemic model has been discussed with respect to the basic reproduction number and stability analysis of disease-free and endemic equilibrium. A new optimal control problem with treatment is framed to minimize the vulnerable situation of the second wave. This system is applied to study the effects of vaccines and treatment controls. Numerical solutions and the graphical presentation of the results predict the fate of India’s second wave situation on account of the control strategy. Lastly, a comparative study with control and without control has been analysed for the exposed phase, infective phase, and recovery phase to understand the effectiveness of the controls. This model is used to estimate the total number of infected and active cases, deaths, and recoveries in order to control the disease using this system and studying the effects of vaccines and treatment controls.

The novel Covid-19 causes from the strain of a large family of viruses, i.e., coronavirus develops illnesses ranging from the common cold to acute respiratory syndromes. Common symptoms of the SARS CoV-2 strain include respiratory symptoms such as fever, dry cough, tiredness, congestion, sore throat, vomiting or diarrhoea and shortness of breath according to the WHO [1] . Some infected people are also not getting any smell. Since its outbreak, most of the countries across the world are perceiving a surge in their daily Covid-19 disease tally. Several lockdowns, announcing several guidelines, aggressive testing, and timely provision of medicines are some of the ways in which governments are trying to avoid the continuous spread of the disease and to break the chain. Some organizations, either independently or in collaboration with government authorities, have come up with the announcement of vaccines, which are undergoing several stages of trials and approval process early in the year 2021. Besides that, the modern biological mathematical model and control system are essential to understand the mechanism of viral disease transmission and to control the spread of the virus. The worldwide different mathematical models of this epidemic that take into account the different situations in different countries help the scientists of medical science, microbiologists, and virologists to fight against Covid-19.

The second wave of Covid-19 has been overwhelming in India as more than thirty thousand new cases were confirmed daily over the last week of April 2021. The country faces extreme shortages of beds in the hospital, oxygen supplies, medicines and life support equipment. But they arranged all the facilities very fastly so that it has been under control after the peak situation in May and June in the same year. At that time Government has given the topmost priority to vaccinate the people in massive quantities in a war situation, whereas the density of the population in India is the main challenge. People have already been learned to adapt and overcome the economic challenges linked to lockdowns, thereby limiting the financial impact.

The Indian economy showed its ability to bounce back quickly after the first wave of Covid-19 and seems to pull this off once more.

The rapid spread of epidemics along the interactions was analysed using the Susceptible-Infected-Recovered (SIR) model [2] , or SEIR model [3] [4] [5] . Global analysis for the general epidemiological model with vaccination control strategy has been discussed by Yang et al. [6] and Sun et al. [7] . Tian and Wang [8] investigated the global stability of the cholera epidemic model, and the HIV/AIDS epidemic model has been described by Samanta [9] and Cai et al. [10] . The transmissibility of a novel coronavirus mathematical model has been prescribed by Cheng et al. [11] . Another model of airborne viral diseases with risk analysis and features of Covid-19 has been discussed by Adekola et al. [12] , and Anirudh [13] explained the dynamics transmission in envisaging the Covid-19 and the future of this pandemic. The case study of the transmission impact of coronavirus on preventive measures in Tanzania has been discussed by Mumbu et al. [14] .

Mathematical models and projections in relation to climate and geographical change, environment, humanitarian disasters, and global health play a critical role in producing evidence in response to every viral outbreak such as Ebola, MERS, SARS, and currently SARS CoV-2, and in eliminating chronic infections such as viral Hepatitis, HIV, and Tuberculosis. Many researchers have recently investigated the current scenario of SARS CoV-2 in greater detail using the effects of fractional order models, reaction diffusion systems, and complex networks , effect of lock-down [45, 46] and predicted different control strategies [47] [48] [49] [50] [51] [52] [53] [54] . Impact of isolation disobedience and movement restrictions on CoV-19 pandemic has been studied by Stipic et al. [55] and mathematical model of CoV-19 disease on the basis of latency and age structure has been considered by Blyuss and Kyrychko [56] . Kalman filter in the epidemiological model with a robust approach to predict CoV-19 outbreak in Bangladesh has been discussed by Islam et al. [57] . James et al. [58] and Glass [59] studied the second wave mortality of CoV-19 in Europe and US. The negative geographic correlation of estimated incidence between the first two waves of COVID-19 in Italy has been analysed by Carletti and Pancrazi [60] . Ershkov and Rachinskaya [61] studied the new approximation of mean-time trends for the CoV-19 second wave in the key six countries. An SEIR model of SARS CoV-2 second wave in France and Italy has been explained by Faranda and Alberti [62] and Ghanbari [63] forecasted the second wave of covid-19 in Iran. Lastly, the second wave of SARS CoV-2 due to interactions between social processes and dynamics disease has been discussed by Pedro et al. [64] . Another method for studying epidemic models is to use a fractional dynamical system. Recently, The mathematical model and dynamics of a novel coronavirus with ABC and CF fractional derivatives are investigated in [65] . In [66] , the fractional model and numerical algorithms for predicting COVID-19

with isolation and quarantine strategies are explored. In the absence of a vaccine, the need for proper quarantine without lockdown for 2019-nCoV is investigated in [67] . A nonsingular fractional-order model of the dynamics of the novel coronavirus is established in [68] .

The goal of this paper is to develop some mathematical pandemic models based on the current situation for the second wave of the Covid-19 outbreak with a lock-down and control strategy without lock-down in the Indian population. For the estimated hypothetical parameter values that demonstrate the virus's effect, the graphical presentation of the basic reproduction number has been described. The effect of the Covid-19 mathematical model on reproduction number and stability analysis has been discussed. Lastly, a comparative study with and without control was conducted in the Indian population for the exposed phase (E), infective phase (I), and recovery phase (R) to determine the efficacy of the control strategy.

For the proposed Covid-19 pandemic with lock-down effects mathematical model, the following are our assumptions:

i. S(t) denotes the susceptible population consists of people who have not yet been infected with the COVID-19 virus.

ii. L(t) denotes lock-down population is a fraction of the susceptible population and those are home quarantined.

iii. E(t) denotes the exposed population who are infected but who have not been detected by testing.

iv. T (t) denotes the infected population that has been identified and is being treated through testing.

v. I(t) denotes the population that is asymptomatically infected.

vi. R(t) denotes the recovered population, which is thought to be immune to re-infection.

Also, the biological meanings of the parameters are described below as follows:

Λ : the recruitment rate at which new individuals (including immigrants and newborns) enter the susceptible compartment in the Indian population, α 1 : the rate at which susceptible compartment moves to lock-down compartment, α 2 : the rate at which lock-down compartment moves to a susceptible compartment, β : effective contact rate of infective individuals, d : the natural death rate of each compartment, γ 1 : the rate at which the exposed compartment moves to an infected compartment under treatment, γ 2 : the rate at which the exposed compartment gets infected but asymptomatic, δ 1 : the rate at which the infected compartment gets treatment and does not spread the disease, δ 2 : the rate at which the infected compartment gets treatment and moves to the recovered compartment, σ 1 : the rate at which asymptomatically infected population does not spread disease, σ 2 : the rate at which asymptomatically infected population gets recovered without treatment, σ 3 : the rate at which asymptomatically infected population gets treatment and does not spread disease. the transmission process With these assumptions, the mathematical model of the second wave of Covid-19 outbreak with lock-down is formulated by the following system of differential

where S(t), L(t), E(t), T(t), I(t) and R(t) are the densities of susceptible population, lockdown population, exposed population, infected population under treatment, asymptomatically infected population and recovered population respectively at the time t. In this model, the lockdown compartment is a fraction of the susceptible population and those are home quarantined.

The initial conditions for the compartments of the system (1) are as follows:

For biological purposes, the initial conditions exist and are unique in the interval [0,∞) per unit of time.

In this section, some properties of the epidemic model (1) and the invariant region where all solutions of the system (1) exist have been described. The basic reproduction number for the equilibrium point of the system has also been analysed here. The following two theorems are used to demonstrate the Positivity and Boundedness of the model's solution based on [69] .

Theorem 1. All solutions of the system (1) with initial condition (2) are non-negative and unique for all t≥0.

Proof. From the system of equations (1), it is observed that

As all compartment functions of system (1) are completely continuous and the solutions of (1)

Therefore, all the solutions of the system (1) are non-negative and unique ∀ t > 0.

All solutions of the system (1) which lies in R 6 + are uniformly bounded and confined to the region

Proof. Let us define N = S + L + E + T + I + R as the total number of the high-risk human population at time t. From the system (1), it is observed thaṫ

We find by using the differential inequality [69] as

. Therefore, all the solutions of (1) are uniformly bounded and confined to the invariant region

For the equilibrium points of the system (1), we have to find the point of interaction at the zero growth isoclines. Here, we get a unique boundary equilibrium point which is given by

To eradicate the virus disease from the population of varying sizes, we have to search the stringent way to make the virus-infected population tend to zero [70] . Therefore, ε 0 is the disease-free equilibrium (DFE) point of the system (1) for all positive parameters.

To evaluate the basic reproduction number (BRN), we arrange the equations of the reduced system of (1) in the following manner

where as the last compartment, R(t) does not depend on other equations of system (1). To establish the stability of ε 0 on the system (1) by using next generation operator method [71] , we assume y = (E, I, T, S, L) on the system (3) such that dy dt = f − v, where f is a transmission part which expresses the production of new infections and v is an evolution part. Both f and v are given by

In this section, the local and global stability analysis [69, 73] of DFE of the system (1) has been studied.

Theorem 3. The system (1) is locally asymptomatically stable at the DFE point ε 0 if 0 < 1,

Proof. The Jacobian matrix of the system (1) at DFE point ε 0 is given by

The characteristic equation of (4) is

The eigen values of (5) is given by λ = −d, − (δ 1 + δ 2 + d) and

where

all eigen values are negative if 0 < 1. Therefore, the system (1) is locally asymptomatically stable at DFE point ε 0 if 0 < 1, and unstable if 0 > 1.

In this subsection, we discuss the global stability of the DFE of the system (1) by using the technique of Castillo-Chavez et al. [72] , where as we already know that DFE point ε 0 is locally asymptomatically stable if 0 < 1, and unstable if 0 > 1. The system (1) can be written in the following form:

where χ = (S, L, R) ∈ R 3 represents the number of individuals which are not infected at present, and W = (E, T, I) ∈ R 3 denotes the number of infected individuals including latent and asymptomatic cases. In the following technique, the global asymptomatically stability of the disease-free equilibrium is guaranteed by the following two conditions as follows: Proof. In this case,

for (χ, W ) ∈ Ω 1 . Clearly, A is an M-matrix and χ 0 = (S 0 , L 0 , 0) is a GAS equilibrium of the system dχ dt = F (χ, 0). Therefore, the conditions (A1) and (A2) are satisfied. Hence, the following theorem is proved.

In this part, we will discuss the existence and stability behaviour of the endemic equilibrium (EE) for the system (1) at the equilibrium point ε * .

To investigate the existence of EE, the solution ε * (S * , L * , E * , T * , I * , R * ) of the equilibrium equations at steady state for the system (1) can be determined in terms of S * . Let m 1 =

Let us define λ * = βγ 2 (E * + T * + I * ).

All expression of endemic equilibrium ε * (S * , L * , E * , T * , I * , R * ) shows that the system (1) satisfy the following linear equation in terms of λ * from (8):

A 0 > 0 as m 3 > 0 and m 5 > 0, it is observed that the system (1) has a unique EE point ε * whenever 0 > 1 and there is no positive EE point whenever 0 < 1. we assume that β is the bifurcation parameter and get the critical value of β = β * at 0 = 1,

The Jacobian J(ε 0 ) (4) of the system (1) at β = β * , has a right eigenvector corresponding to the zero eigenvalue given by u = (u 0 , u 1 , u 2 , u 3 , u 4 , u 5 ) T , where

Similarly, the Jacobian J(ε 0 ) (4) of the system (1) at β = β * , has a left eigenvector v =

Let us assume that S = x 0 , L = x 1 , E = x 2 , T = x 3 , I = x 4 , R = x 5 and dx k dt = f k (k = 0, 1, 2, 3, 4, 5). By Castillo-Chavez and Song [74] , we have

As a < 0 and b < 0 at β = β * , the unique EE ε * is locally asymptomatically stable for 0 > 1 [74, 75] and a transcritical bifurcation happens at 0 = 1.

In this context of mathematical modelling, it is framed a new optimal control problem so that this vulnerable situation of the second wave of Covid pandemic can be minimized. Our Covid-19 epidemic model with treatment control becomes:

with initial conditions S(0) > 0, E(0) ≥ 0, I(0) ≥ 0, R(0) ≥ 0, wherew(t) = w 1 (t)+w 2 (t)+w 3 (t).

The aim of this pandemic model with a control strategy is to minimize the susceptible, exposed and asymptomatically infected population, and to maximize the number of recovered population by using the feasible minimal control variables w 1 (t), w 2 (t) and w 3 (t). The susceptible compartment induces an optimal vaccine control w 1 (t) before the spread of infection. The optimal treatment controls w 2 (t) and w 3 (t) should be provided to exposed and asymptomatically infected populations, respectively, whereas the total population must be constant.

The effects of infection on susceptible, exposed, and asymptomatically infected compartments are opposite to the recovered compartments around them, thus it is necessary to minimize them.

According to these optimal control strategies, the objective functional [76] [77] [78] is considered as

Here, 1 , 2 and 3 are positive weight factors that represent a patient's level of acceptance of the vaccination and treatment on the exposed and asymptomatically infected population, respectively. The square of the control variables reflects the severity of the side effects of the treatment. Now, the goal of the problem (11) is to minimize the objective functional (12) so that all types of infected individuals due to coronavirus as well as the cost of treatment can be minimized.

It is assumed that S(t), E(t) and I(t) be state variables with control variable setw(t) =

1, 2, 3} be the admissible control set such that 0 ≤w(t) ≤ N . The system (11) can be written in the following form:

where

Equation (13) is a nonlinear system with a bounded coefficient.

Let us assume,

and F (Z) of (14) satisfies

where the constants σ 4 > 0 and σ 5 > 0 are independent of the variables S(t) and I(t) which are less than N respectively.

Thus, the function G(Z) is uniformly lipschitz continuous.

Therefore, a solution of the system (13) exists [69] from the existence of W and the restriction on S(t), E(t), I(t) and R(t) ≥ 0. To describe the necessary conditions for the optimal control, the Hamiltonian (H) is defined for (11) to (12) as follows:

where λ i for i = 1, 2, 3, 4 be the adjoint functions of t to be determined later.

There exists an optimal control solution set w 2 , w 3 ) of the control system (11) .

Proof. Here, the state and control variables are always positive. By using the results in [78] , the convexity of the objective functional in w 1 , w 2 and w 3 is satisfied in this minimizing problem (11) . By definition, the set of all control variables w = (w 1 (t), w 2 (t), w 3 (t)) ∈ W is closed and convex. The optimal control system is bounded, which indicates the compactness of w.

Again, it has been observed that the integrand of (12) i.e.,

is convex on the control set W . For some positive numbers l 1 and l 2 , there exist a constant n > 1 such that To find the solution and necessary conditions for the optimal control, the Pontryagin's Maximum Principle [79] has been used in the next theorem.

Let us assume that S * (t), E * (t), I * (t) and R * (t) be optimal solutions for the optimal control problem (11)- (12) associated with the optimal control variables w * 1 (t), w * 2 (t) and w * 3 (t). Then there exist four adjoint variables λ 1 , λ 2 , λ 3 and λ 4 which satisfy the following results

where the boundary conditions are given as

Hence, the optimal control triples are

variables is obtained as

where the boundary conditions are given by λ i (t end ) = 0 for i = 1, 2, 3, 4. Using the optimality condition of the Pontryagin's Maximum Principle [79] , it is obtained that

By using the bounds for w, the optimal control variables w * 1 (t), w * 2 (t) and w * 3 (t) can be obtained as

Furthermore, the control system (11) is converted to the following system:

with the following Hamiltonian:

To determine the effect of the optimal control and state variables, this is required to solve equations (19) and (20) .

In this part, we may be able to apply our model to smaller populations and make region-specific predictions with the availability of reliable data [80] [81] [82] . The numerical investigation has been explained based on the control strategy and without control strategy for the second wave of the Covid-19 pandemic system. A comparison is also made between the present model and India's current situation within a period. The characteristics of the parameters β and γ 2 based on 0 is presented through figures 1 and 2. 

The BRN graph has been drawn in figure-1 based on the real field value of parameters from table-1. From figure-1, it is cleared that if β and γ 2 increase, the basic reproduction number ( 0 ) increases. Therefore, our basic aim is to control the contact rate of infected individuals (β) and the rate at which the exposed population gets infected but asymptomatic (γ 2 ); otherwise, the pandemic system is unstable, and hence, this situation will become very harmful to our civilization. Figure-2 show the combined effect of the sensitive parameters β and γ 2 on the basic reproduction number ( 0 ). From figures 2(A) and 2(B), it is observed that as β which is the contact rate of infected individuals, increases and there is a high chance of population gets infected in contact with asymptomatic infected individuals, and as a result, the 0 value approaches towards unity and above. Hence, the spread of the disease would be increased. We have not used any mathematical method for parameter estimation; we use the trial and error initial conditions, respectively, as given in Table-1 is the susceptible population. In this part, we predicted that the disease is not spread all over the country due to lockdown. The optimal control is applied to three different strategies for the system (11), namely, (i) induce an optimal control vaccine w 1 on the susceptible compartment before the spread of infection, (ii) the first treatment control w 2 on exposed populations, and (iii) second treatment control w 3 on asymptomatically infected populations. Figure-7 represents the optimal control using the values of the different parameters given in Table-2 . Therefore, this graph shows that 1st control is essential at the initial stage of the outbreak than when it prevails. with all controls and without control is shown in Figure 8 . It has also been presented the same kind of comparative time-series diagrams incorporating the effect of controls for exposed phase (E), infective phase (I), and recovery phase (R) in Figures 8. All graphs with controls show that it is a very useful model to control this pandemic. These simulations help to understand that using controls effectively reduces the spread of the disease, and the three controls together yield the best result to control the outbreak.

The proposed epidemic models have been studied on the outbreak of SARS CoV-2 disease in the Indian population. Firstly, in the lock-down model, it has been described that the BRN increases if the contact rate of infective individuals (β) and the rate at which the exposed population gets infected but asymptomatic (γ 2 ) increases. Therefore, it is very harmful to our society if these two parameters β and γ 2 increase. This system has a unique disease-free equilibrium, which is globally stable when 0 < 1. The unique endemic equilibrium is locally asymptomatically stable for 0 > 1, and a transcritical bifurcation occurred at 0 = 1. According to the sensitivity analysis of the basic reproduction number 0 , identifying the rate of infection from susceptible zone to infected zone may reduce the death rate and the number of infected people. Lastly, a control strategy with respect to vaccination and treatment has been applied to the same system except the lock-down compartment to minimize the susceptible, exposed and asymptomatically infected populations. This shows a path to reduce the spread of this virus. A comparative study has been analysed on the models with control and without control, respectively, for exposed phase (E), infective phase (I), and recovery phase (R) to understand the effectiveness of using controls. The study concluded that staying at home as much as possible and keeping infected people in an isolated area would help to slow the spread of COVID-19. In addition, we must provide appropriate treatment for those infected with SARS-CoV-2, as well as vitamins, tonics, and supplements to protect those who are not infected. Advice has been provided in detail to assist the Indian population in slowing the spread of COVID-19. Finally, the proposed model can provide useful information for understanding their dynamics, which is critical for predicting the transmission and widespread application of various epidemics around the world. This model is useful for predicting the total number of infected, active cases, and deaths, which provides a more accurate representation of the infection rate and may be useful in the future for COVID-19

prevention and control. This helps us make future decisions so that we can control or limit the spread of the epidemic.

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