Modelling the leadership role of police in controlling COVID-19

Vikram Singh; Shikha Kapoor; Sandeep kumar Gupta; Sandeep Sharma

doi:10.1515/cmb-2024-0010

Article Open Access

Modelling the leadership role of police in controlling COVID-19

Vikram Singh , Shikha Kapoor , Sandeep kumar Gupta and Sandeep Sharma

Published/Copyright: December 6, 2024

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From the journal Computational and Mathematical Biophysics Volume 12 Issue 1

Abstract

During the recent Coronavirus disease (COVID-19) pandemic, different parts of the globe faced indefinite lockdowns. To maintain the lockdown measures, government authorities deployed security forces and police. The primary role of the police is to monitor the movement of the people and advise/guide them to follow the lockdown. In the current work, we propose a nonlinear mathematical model to study the role of police and security personnel in controlling COVID-19. It is observed that the proposed model possesses two equilibria, namely, trivial and non-trivial. We performed the stability analysis of the non-trivial equilibrium point by using the compound matrix technique. Finally, we perform a detailed numerical simulation to investigate the feasibility of the theoretical results. The current study demonstrates that police forces played a significant role in controlling the pandemic.

Keywords: COVID-19; lockdown; police; mathematical model; stability analysis

MSC 2010: 92D30; 34D20

1 Introduction

Outbreaks of different infectious diseases pose a big challenge for the survival of mankind. Despite significant development of medical science, infectious diseases are continuing to occur and damage the social and financial infrastructure of different parts of the globe. The rapidly changing climatic and ecological conditions are providing a conducive environment to the disease-causing agents. The emergence of new infectious diseases makes infectious diseases one of the important research problems of the current time. The same has happened with the recent outbreak of Coronavirus disease (COVID-19). The disease was first reported at Wuhan (China) when 27 cases of COVID-19 were reported on December 12, 2019 [1]. In a short span of time, the disease invaded every part of the globe. The catastrophic impact of the pandemic forced the research community to invest serious efforts to control the exponential spread of COVID-19. During the first year of the pandemic, the entire research community was working on different aspects of the pandemic. On the same line, people working in epidemic modelling came up with a variety of new models to investigate the critical factors involved in the propagation of the disease [2–7,9,13,14, 17,18,25–28,31]. With the help of mathematical models, people investigated the success of different control measures to curtail the pandemic. Some of the studies investigate the applicability of compartmental models (e.g. SIR, SEIR) to COVID-19 pandemic scenario [10,25,31]. On the other hand, some studies proposed new mathematical models by incorporating new compartments to make the model realistic. The success of mass, vaccines in flattening the curve of the epidemic is also studied with the help of mathematical models [26,27]. Further, pandemic scenario varies highly from one continent to another, one country to another due to different geographic, climatic, and health infrastructure conditions. Due to this, a number of models have also been proposed and applied to a particular region and country [8,13,14,28]. Apart from that, models have also been employed to predict the peak of the outbreak. In most cases, the predictions made using an appropriate mathematical model turn out to be helpful to understand the transmission pattern, key parameters, and factors involved in the disease dynamics. Moreover, estimation of the parameters involved in a mathematical model with the help of real data is certainly a boon for the health agency. These estimated parameters are subsequently used to estimate the values of the basic reproduction number ( R 0 ). The value of R 0 is extremely useful to identify the future of the disease. During this pandemic scenario, a number of studies were performed to estimate the value of R 0 for a specific country or for different regions of a particular country [16,23,30]. The estimated values of R 0 also help government and health agencies to identify the region or state of concern. Recently, a new mathematical model has been proposed by incorporating new parameters related to environmental pollution and its possible impact on the spread of COVID-19 [15].

In Table 1, we summarize the region-wise COVID-19 cases and mortalities, and the same is depicted as bar graph in Figures 1 and 2, respectively. The data given in Table 1 clearly underscore the current dismal scenario. The countries of American and European regions are worst affected as they share more than 50% of global cases and mortalities. However, in many countries, the situation is now improving and daily life in most of the countries is getting normal. People are ready to live with new normal and acquiring the changes in their routine life.

Table 1

Global distribution of COVID-19 (as on May 25, 2021) [43]

WHO region	Cases	Deaths
Africa	3,446,089	85,964
Americas	6,59,80,739	16,15,127
Eastern Mediterranean	98,63,946	1,97,964
Europe	5,41,10,276	11,34,786
South-East Asia	3,00,88,649	3,72,277
Western Pacific	2,861,544	43,058
Globally	16,63,52,007	34,49,189

Figure 1

Region-wise reported cases. All data have been taken from [43].

Figure 2

Region-wise reported mortalities. All data have been taken from [43].

It is, perhaps, the first instance in the history of medical science that vaccines have been developed and made available in such a short span of time. The vaccination process has already started in many parts of the world. But, the pharma companies find it difficult to meet the demand of the vaccines. Some governments are struggling to provide vaccines to the people of their countries. Certainly, the vaccination process is a long and challenging task in such nations. Thus, in such conditions, the only possible measure which helps in reducing the burden of the ongoing pandemic of COVID-19 is to follow the social distancing. In many countries, governments have implemented nationwide or partial lockdown at different frames of time. In particular, during the year 2020, more than half of the population of the globe was bound to live in a lockdown. During the period of lockdown, many activities came to halt including shutting of almost all businesses, amenities, and places of worship, and people are not allowed to step out from their homes. Only employees and workers involved in essential services, medical staff, media, and police are allowed to function. In the Table 2, we provide a summary of the period of lockdown (implemented during the year 2020) in some of the countries.

Table 2

Summary of COVID-19 lockdown (during 2020) in some major countries [44]

Country	Duration	Type
Brazil	March 17–April 7 (Santa Catarina)	Partial
	March 24–May 10 (S ao Paulo)	Partial
Germany	March 23–May 10	National
India	March 25–May 31	National
Russia	March 30–May 12 (Moscow)	Partial
	March 28–April 30 (Rest of the country)	National
Spain	March 14–May 09	National
Turkey	April 23–April 27	Partial
United Kingdom	March 23–May 13	National
United States	Variable	Partial
Italy	March 08–May 18	National
France	March 16–May 11	National

In many countries, the government deployed police force to ensure the strict implementation of lockdown, while in many countries (e.g India, UK) breaking the lockdown guidelines is declared a criminal offense. In particular, the government of India instructed police officials to take strict legal actions against violators of lockdown. A number of police cases have been registered for violation of lockdown restrictions [36,37]. In the United Kingdom, implementation of lockdown is the key to slow down the spread of COVID-19 in different parts of the country. People are advised to avoid non-essential travel and industrial, academic, and amenities were shut down with an immediate effect. To maintain the lockdown guidelines, police were deployed and empowered to enforce the lockdown. The prime minister of the United Kingdom instructed clearly that people flout the lockdown restrictions will be fined by the police [40,41]. In France, police set up roadblocks across the country to verify exemption declarations of people moving on the road. Up to 7 April, nearly 8 million checks had been made, and half a million fines had been issued for failure to respect rules of confinement [34,35]. As the United States did not implement nationwide lockdown and it has been implemented partially on the local level (i.e. by the governor of a particular state), hence the actions of the police varied significantly. At places where lockdown was voluntarily enforced, police did not arrest offenders. On the other hand, some places are under lockdown enforced by rule and police are free to take strict action against offenders [42]. In Italy, police blocked into and out movement from quarantine centres and did not allow movement in red zones of the country during lockdown [38,39].

Recently, the global research community witnessed a surge in the publications addressing different aspects of COVID-19 (see [8,11,13,19,29,33] and references cited therein). A number of these studies highlighted the role of lockdown in managing the spread of COVID-19. Although, police have played a key role in the success of lockdown in controlling the spread of the disease, yet, none of them investigate the role of police in maintaining the lockdown. In this work, we are using a new non-linear mathematical to study this untouched and interesting issue.

2 Mathematical model

In this section, we propose a new mathematical model to perform our study. In the modelling process, we consider a region with a total population N . Further, we divide the susceptible population into two subclasses: susceptible following lockdown ( S − ) and susceptible not following lockdown ( S + ). The susceptible following lockdown is assumed enough awareness on COVID-19 transmission and social distancing and thus does not exposed to infection. For the sack of simplicity, we call the individuals of S + and S − classes as susceptible and aware individuals, respectively. The infectious individuals are represented by Y , while the population of police is P . The recovered individuals formed a separate class represented by R . The mathematical model consisting of all the aforementioned populations takes the following form:

(1) d S + d t = Λ − λ Y S + − δ S + P − d S + − θ S + + ε S − d S − d t = δ S + P − d S − + θ S + − ε S − d Y d t = λ Y S + − ( γ + α + d ) Y d R d t = γ Y − d R d P d t = Q Y − Q 0 P .

In the aforementioned model, Λ is the constant recruitment rate and λ is the disease transmission rate. The security forces converted susceptible individuals into aware individuals through counselling or by using the force administered by law at a rate δ . θ is the rate by which the susceptible individuals join the aware class by self. d is the natural death rate. It is observed that aware individuals knowingly or unknowingly become susceptible as they go out for some of their important work or to perform their routine work. To incorporate this shift, we include ε as the rate at which aware individuals leave the aware class and become susceptible. γ is the recovery rate for infected individuals and α is the disease-induced mortality rate. The security forces deployed at a constant rate of Q which is assumed proportional to the infected population. It is also assumed that security force diminishes due to transfer, illness, or mortality with rate Q 0 .

Since the variable R does not appear in any other equation of the model system 1, therefore analysis of the following reduced system will serve the purpose

(2) d S + d t = Λ − λ Y S + − δ S + P − d S + − θ S + + ε S − d S − d t = δ S + P − d S − + θ S + − ε S − d Y d t = λ Y S + − ( γ + α + d ) Y d P d t = Q Y − Q 0 P .

The positivity of all the parameters involved in the model is also assumed, and the model is investigated under the following initial conditions:

(3) S + ( 0 ) > 0 , S − ( 0 ) ≥ 0 , Y ( 0 ) ≥ 0 , P ( 0 ) ≥ 0 .

3 Basic properties

In this section, we will obtain the results on the positivity and boundedness of the solution of the proposed model system (2). First, we will discuss the positivity followed by the boundedness of the solution.

3.1 Positivity of solutions

The following theorem ensures the positivity of solutions of model system (2).

Theorem 1

Solutions of the model system (2) with positive initial condition (3) are positive for all t > 0 .

Proof

Let t = Sup { t > 0 ∣ S + > 0 , S − > 0 , Y > 0 , P > 0 } . Now, we obtain the following from the first equation of the model system (2)

d S + d t ≥ Λ − ( λ Y + δ P + d + θ ) S + .

Next, we obtain

d d t S + ( t ) exp ∫ 0 t ( λ Y + δ P ) d ϕ + ( d + θ ) t ≥ Λ exp ∫ 0 t ( λ Y + δ P ) d ϕ + ( d + θ ) t .

Further,

S + ( t ) exp ∫ 0 t 1 ( λ Y + δ P ) d ϕ + ( d + θ ) t 1 − S + ( 0 ) ≥ ∫ 0 t 1 Λ exp ∫ 0 Φ ( λ Y + δ P ) d ϑ + ( d + θ ) ϑ d ϑ .

Therefore,

S + ( t ) ≥ S + ( 0 ) exp − ∫ 0 t 1 ( λ Y + δ P ) d ϕ + ( d + θ ) t 1 + exp − ∫ 0 t 1 ( λ Y + δ P ) d ϕ + ( d + θ ) t 1 × ∫ 0 t 1 Λ exp ∫ 0 Φ ( λ Y + δ P ) d ϑ + ( d + θ ) ϑ d ϑ .

The bounds for the other components of the solution can be obtained in a similar manner.□

3.2 Boundedness of solutions

The following theorem ensures the boundedness of the solutions of model system (2).

Theorem 2

All solutions of model system (2) are bounded.

Proof

From the first three equations of the model system (2), we obtain

( S + + S − + Y ) ′ = Λ − d ( S + + S − + Y ) − ( α + γ ) Y ≤ Λ − d ( X 1 + X 2 + Y ) .

Hence, we obtain

lim t → ∞ Sup ( X 1 + X 2 + Y ) ≤ Λ d .

Now, from the last equation of the model system 2, we deduce that

d P d t ≤ Q Λ d − Q 0 P .

Next, we can obtain

P ≤ Q Λ Q 0 d .

From the aforementioned discussion, it is evident that all solutions are bounded.

Therefore, the region of attraction for model system (2) is given by

□ Π = ( S + , S − , Y , P ) ∈ R + 4 ∣ 0 ≤ S + , S − , Y ≤ Λ d , 0 ≤ P ≤ Q Λ Q 0 d .

4 Equilibrium analysis

In this section, we will determine the possible equilibrium points of the model system (2). Through the equilibrium analysis, we have obtained the following equilibrium points:

The trivial equilibrium point E 0 = ( Λ d , 0 , 0 , 0 ) .
The non-trivial equilibrium point E * = ( S + * , S − * , Y * , P * ) .

The existence of E 0 is trivial. Here, we discuss the existence of E * in detail. The values of S + * , S − * , Y * , and P * can be obtained by solving the following system of equations:

(4) Λ − λ Y * S + * − δ S + * P * − d S + * − θ S + * + ε S − * = 0 δ S + * P * − d S − * + θ S + * − ε S − * = 0 λ Y * S + * − ( γ + α + d ) Y * = 0 Q Y * − Q 0 P * = 0 .

Now, from the last equation of system (4), we obtain P * = Q Y * Q 0 and the third equation gives S + * = ( γ + α + d ) λ .

By using the values of S + * and P * in the second equation, we obtain

S − * = ( γ + α + d ) ( δ Q Y * + Q 0 θ ) Q 0 λ ( d + ε ) .

Finally, using the values of S + * , S − * , and P * in the first equation of the system yields

Y * = Λ λ Q 0 ( d + ε ) + ε Q 0 θ ( γ + α + d ) − Q 0 ( d + θ ) ( d + ε ) ( γ + α + d ) λ Q 0 ( d + ε ) d δ Q

Therefore, the non-trivial equilibrium point E * exists under the following condition:

(5) Λ λ ( d + ε ) + ε θ ( γ + α + d ) ( d + θ ) ( d + ε ) ( γ + α + d ) > 1 .

We can define Λ λ ( d + ε ) + ε θ ( γ + α + d ) ( d + θ ) ( d + ε ) ( γ + α + d ) as the basic reproduction number.

5 Stability analysis

In this section, we derive the conditions for stability of the equilibrium points obtained in the section (4). First, we discuss the stability of the trivial equilibrium point followed by the stability analysis of the non-trivial equilibrium points.

Consider the Jacobian of the model system (2) about the trivial equilibrium point:

M 0 = − ( d + θ ) ε − λ Λ d − δ Λ d θ − ( ε + d ) 0 δ Λ d 0 0 λ Λ d − ( γ + α + d ) 0 0 0 Q − Q 0 .

It is trivial to check that all the eigenvalues of the matrix M 0 are negative if λ Λ d < ( γ + α + d ) .

Next, we study the stability of the non-trivial equilibrium point E * . To obtain the condition for local stability of E * , consider the Jacobean matrix about E * :

M * = M 11 M 12 M 13 M 14 M 21 M 22 0 M 24 M 31 0 M 33 0 0 0 M 43 M 44 ,

where

M 11 = − ( λ Y + δ P + θ + d ) , M 12 = ε , M 13 = − λ S + , M 14 = − δ S + M 21 = δ P + θ , M 22 = − ( ε + d ) , M 24 = δ S + M 31 = λ Y , M 33 = λ S + − ( γ + α + d ) , M 34 = 0 M 43 = Q , M 44 = − Q 0 .

Now, the characteristic equation of the matrix M * is

Θ 4 + P 1 Θ 3 + P 2 Θ 2 + P 3 Θ + P 4 = 0 ,

where

P 1 = M 22 − M 11 − M 33 + M 44 P 2 = M 11 M 33 − M 12 M 21 − M 11 M 22 + M 13 M 31 − M 11 M 44 − M 22 M 33 + M 22 M 44 − M 33 M 44 P 3 = M 11 M 22 M 33 + M 12 M 21 M 33 + M 13 M 22 M 31 − M 11 M 22 M 44 − M 12 M 21 M 44 + M 11 M 33 M 44 + M 13 M 31 M 44 + M 14 M 31 M 43 − M 22 M 33 M 44 P 4 = M 11 M 22 M 33 M 44 + M 12 M 21 M 33 M 44 − M 12 M 24 M 31 M 43 + M 13 M 22 M 31 M 44 + M 14 M 22 M 31 M 43 .

Now, by using the classical Routh-Hurwitz criterion, we state the following result on the local stability of the non-trivial equilibrium E * .

Theorem 3

The endemic equilibrium point E * is locally asymptotically stable provided P i > 0 , where i = 1 , 2 , 3 , 4 , and P 1 P 2 P 3 > P 3 2 + P 1 2 P 4 , P 1 P 2 > P 3 .

To derive the conditions for global stability of the non-trivial equilibrium point, we use the compound matrix method described in [20–22,32]. The method is widely used in the study of the three-dimensional system but limited application to the four-dimensional system. The present study involves the four-dimensional model system, and, thus, application of the technique is a bit tedious.

Consider, the Jacobian matrix about an arbitrary point:

J = − λ Y − δ P − d − θ ε − λ S + − δ S + δ P + θ − ( ε + d ) 0 δ S + λ Y 0 λ S + − ( γ + α + d ) 0 0 0 Q − Q 0 .

The second compound matrix can be obtained as follows:

J [ 2 ] = j 11 0 δ S + λ S + δ S + 0 0 j 22 0 ε 0 δ S + 0 Q j 33 0 ε − λ S + − λ Y ( δ P + θ ) 0 j 44 0 − δ S + 0 0 ( δ P + θ ) Q j 55 0 0 0 λ Y 0 0 j 66 ,

where

j 11 = − λ Y − δ P − ( ε + θ + 2 d ) j 22 = − λ Y − λ S + − δ P − ( γ + α + θ + 2 d ) j 33 = − λ Y − δ P − ( s + θ + Q 0 ) j 44 = λ S + − ( ε + γ + α + 2 d ) j 55 = − ( ε + d + Q 0 ) j 66 = λ S + − ( γ + α + d + Q 0 ) .

Consider

D = 1 Y 0 0 0 0 0 0 1 Y 0 0 0 0 0 0 0 1 Y 0 0 0 0 1 P 0 0 0 0 0 0 0 1 P 0 0 0 0 0 0 1 P .

Now, by using J [ 2 ] and B , we obtain the following matrix:

A = D f D − 1 + D J [ 2 ] D − 1 ,

which takes the form

A = j 11 − Y ′ Y 0 λ S + δ S + P Y δ S + P Y 0 0 j 22 − Y ′ Y ε 0 0 δ S + P Y − λ Y ( δ P + θ ) j 44 − Y ′ Y 0 0 − δ S + P Y 0 Q Y P 0 j 33 − P ′ P ε − λ S + 0 0 Q Y P ( δ P + θ ) j 55 − P ′ P 0 0 0 0 λ Y 0 j 66 − P ′ P .

The block representation of the matrix A is

B = B 11 B 12 B 13 B 14 B 21 B 22 B 23 B 24 B 31 B 32 B 33 B 34 B 41 B 42 B 43 B 44 ,

where

B 11 = j 11 − Y ′ Y B 12 = ( 0 , λ S + ) B 13 = δ S + P Y , δ S + P Y B 14 = 0 B 21 = ( 0 , − λ Y ) T B 22 = j 22 − Y ′ Y ε ( δ P + θ ) j 44 − Y ′ Y B 23 = 0 0 0 0 B 24 = δ S + P Y , − δ S + P Y B 31 = ( 0 , 0 ) T B 32 = Q Y P 0 0 Q Y P B 33 = j 33 − P ′ P ε ( δ P + θ ) j 55 − P ′ P B 34 = ( − λ S + , 0 ) T B 41 = 0 , B 42 = ( 0 , 0 ) B 43 = ( λ Y , 0 ) B 44 = j 66 − P ′ P .

Next, we define the following norm on R 6 [12]

∣ ( z 1 , z 2 , z 3 , z 4 , z 5 , z 6 ) ∣ = max { ∣ z 1 ∣ , ∣ z 2 ∣ + ∣ z 3 ∣ , ∣ z 4 ∣ + ∣ z 5 ∣ , ∣ z 6 ∣ } .

Now, the Lozinskiǐ measure σ of the matrix A is defined as follows:

σ ( A ) = lim h → 0 + ∣ I + h A ∣ − 1 h .

Following [12,24], we define the following estimate

σ ( A ) ≤ sup { g 1 , g 2 , g 3 , g 4 } ,

where

g 1 = σ 1 ( B 11 ) + ∣ B 12 ∣ + ∣ B 13 ∣ + ∣ B 14 ∣ , g 2 = σ 1 ( B 22 ) + ∣ B 21 ∣ + ∣ B 23 ∣ + ∣ B 24 ∣ , g 3 = σ 1 ( B 33 ) + ∣ B 31 ∣ + ∣ B 32 ∣ + ∣ B 34 ∣ , g 4 = σ 1 ( B 44 ) + ∣ B 41 ∣ + ∣ B 42 ∣ + ∣ B 43 ∣

in the expression of g i s , ∣ B i j ∣ ( i ≠ j and i , j = 1 , 2 , 3 , 4 ) represent matrix norms with respect to l 1 vector norm and σ 1 is the Lozinskiǐ measure with respect to l 1 norm [24].

Further, from the third and fourth equations of the model system (2), we obtain

(6) Y ′ Y = λ S + − ( γ + α + d ) P ′ P = Q Y P − Q 0 .

Now, by using equation (6), we calculate σ 1 ( B i i ) and ∣ B i , j s ∣ as follows:

σ 1 ( B 11 ) = − λ Y − δ P − λ S + − ( ε + θ + d − γ − α ) σ 1 ( B 22 ) = − λ Y − λ S + − d σ 1 ( B 33 ) = − λ Y − Q Y P − d σ 1 ( B 44 ) = λ S + − Q Y P − ( γ + α + d ) ∣ B 12 ∣ = λ S + ∣ B 13 ∣ = δ S + P Y ∣ B 14 ∣ = 0 ∣ B 21 ∣ = λ Y ∣ B 23 ∣ = 0 ∣ B 24 ∣ = δ S + P Y ∣ B 31 ∣ = 0 ∣ B 32 ∣ = Q Y P ∣ B 34 ∣ = λ S + ∣ B 41 ∣ = 0 ∣ B 42 ∣ = 0 ∣ B 31 ∣ = λ Y .

Finally, we obtain the expressions for g i s , i = 1 , 2 , 3 , 4 :

(7) g 1 = − λ Y − δ P + δ S + P Y − ( ε + θ + d − γ − α ) g 2 = λ Y + δ S + P Y − d g 3 = λ S + − d g 4 = λ S + + λ Y − ( γ + α + d ) .

Next, we define

Ω 1 = inf t ∈ ( 0 , ∞ ) − λ Y − δ P + δ S + P Y − ( ε + θ + d − γ − α ) Ω 2 = Sup t ∈ ( 0 , ∞ ) λ Y + δ S + P Y − d Ω 3 = { Sup t ∈ ( 0 , ∞ ) ( λ S + ) − d } Ω 4 = { Sup t ∈ ( 0 , ∞ ) ( λ S + + λ Y ) − ( γ + α + d ) } .

Consider a positive constant, Ω , such that

(8) max { Ω 1 , Ω 2 , Ω 3 , Ω 4 } < − Ω .

Further, we have

q ¯ = lim t → ∞ Sup Sup x 0 ∈ Σ 1 t ∫ 0 t σ ( A ) d s ≤ − Ω < 0 .

Finally, we state the following theorem on global stability of the non-trivial equilibrium point E * .

Theorem 4

The non-trivial equilibrium point E * is globally asymptotically stable provided there exists a constant Ω satisfying equation (8).

6 Numerical simulation

In this section, we perform numerical experiments to validate the theoretical results. Further, we will also investigate the impact of variation of certain parameters on the system dynamics. To perform our numerical study, we will consider the following set of parameter values

A = 200 , λ = 0.00001 , d = 0.006 , δ = 0.02 , γ = 0.02 , α = 0.002 , θ = 0.04 , ε = 0.04 , Q = 0.02 , Q 0 = 0.8 ;

For the aforementioned set of parameter values, the components of the non-trivial equilibrium points are obtained as follows:

S + * = 2,800 , S − * = 26797.6878 , Y * = 800.4954 , P * = 20.0123 .

It can also be checked that for the aforementioned set of parameter values, the condition for existence of non-trivial equilibrium point (5) is satisfied.

First, we study the non-linear stability of the proposed model system 1 in the S + − S − plane (Figure 3). We plotted trajectories started from different positions, and from Figure 3, it is easy to note that all the trajectories converged to the same point.

$Figure 3 Global stability of the non-trivial equilibrium point E * {E}^{* } in the S + − S ‒ {S}_{+}-{S}_{‒} plane.$

Figure 3

Global stability of the non-trivial equilibrium point E * in the S + − S ‒ plane.

The prime objective of the current work is to study the role of police in controlling the COVID-19 pandemic during the lockdown. To demonstrate the same, we perform a number of numerical experiments. We plotted different graphs depicting variation of infected population with respect to different parameters. The initial condition used for this purpose is (2,800, 26,797, 700, 20).

In Figure 4, we study the variation in the infected population with the change in Q . From the figure, it is easy to observe that the size of the infected population decreases significantly as we increase the value of Q . This clearly demonstrates that the disease can be controlled by deputing more police to maintain the lockdown. In Figure 5, we study the impact of the variation of δ on the infected population. It is clear from the figure that the infected population decreases if police control the lockdown violators and counsel them to follow lockdown. Next, we study the variation in infected population when θ varies (Figure 6). From the figure, one can easily observe that the infected population decreases as θ increases, which clearly reflects that if people follow the lockdown guidelines then disease can be controlled. Now, it is interesting to study the fluctuation in the infected population, if people break the government guidelines of lockdown. To investigate this, we vary the parameter ε and plotted the corresponding change in the infected population (Figure 7). The figure clearly depicts the disastrous output if people do not follow the lockdown guidelines. Further, we also calibrate the outcome of a reduction in police deputed to maintain the lockdown. To achieve this goal, we vary the parameter Q 0 and plotted the corresponding graphs for the infected population in Figure 8. From the figure, one can easily observe that the infected population rises significantly if the sufficient police force is not deputed on a particular location. Finally, in Figure 9, we plotted a surface plot by varying δ and Q simultaneously. From the figure, it is evident that if the rate of deputing police at a particular site and the rate of maintaining lockdown by the police increases, then the spread of infection can be control.

Figure 4

Variation in infected population for different values of Q .

$Figure 5 Variation in infected population for different values of δ \delta .$

Figure 5

Variation in infected population for different values of δ .

$Figure 6 Variation in infected population for different values of θ \theta .$

Figure 6

Variation in infected population for different values of θ .

$Figure 7 Variation in infected population for different values of ε \varepsilon .$

Figure 7

Variation in infected population for different values of ε .

$Figure 8 Variation in infected population for different values of Q 0 {Q}_{0} .$

Figure 8

Variation in infected population for different values of Q 0 .

$Figure 9 Surface plot showing simultaneous effect of δ \delta and Q Q on infected population.$

Figure 9

Surface plot showing simultaneous effect of δ and Q on infected population.

7 Conclusion

The COVID-19 pandemic was one of the major threats of recent times to the survival of mankind. During the initial phase of the pandemic, health agencies did not have a clear understanding of the transmission mechanism of COVID-19. Due to this, following the measures of social distancing and maintaining sanitization was the primary requirement to reduce the burden of COVID-19. To ensure the follow-up of social distancing, the government and local administration took help from the police. Further, breaking lockdown guidelines was declared a crime in many countries and, thus, the role of police comes into the picture. Police had been empowered with special powers by law. To investigate the role of police, we formulated a new non-linear mathematical model. The proposed model possesses two equilibrium points, namely, trivial E 0 and non-trivial E * . The local stability conditions for both equilibria have been obtained using the popular Routh-Hurwitz criterion. To establish the global stability of the non-trivial equilibrium point, we use the geometric approach. The numerical experiments have also been carried out to validate the qualitative results as well as to study the role of various parameters on the long-term behaviour of the system. The results obtained through numerical simulation demonstrate that police played an important role in reducing the stress of COVID-19. In particular, the success of the lockdown in controlling the spread of the disease largely depend upon the efforts of the police.

Acknowledgements

We gratefully acknowledge suggestions made by learned anonymous reviewers whose comments help us improve the quality of the current work.

Funding information: The authors state no funding involved.
Author contributions: Vikram Singh and Shikha Kapoor involved in writing draft and editing. Sandeep kumar Gupta reviewed and edited the manuscript. Sandeep Sharma discussed the idea, supervised, and reviewed the draft.
Conflict of interest: The authors state that they have no competing interests to declare.
Ethical approval: This research did not require ethical approval.
Data availability statement: This article does not involve data sharing, as no datasets were generated or analyzed during the current study.

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Received: 2023-09-04

Revised: 2024-08-10

Accepted: 2024-08-13

Published Online: 2024-12-06

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/cmb-2024-0010

Keywords for this article

COVID-19; lockdown; police; mathematical model; stability analysis

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