A mathematical model to study the spread of COVID - 19 and its control in India

: In this article, a nonlinear mathematical model is proposed and analyzed to study the spread of coronavirus disease ( COVID - 19 ) and its control. Due to sudden emergence of a peculiar kind of infection, no vaccines were available, and therefore, the nonpharmaceutical interventions such as lockdown, isolation, and hospitalization were imposed to stop spreading of the infectious disease. The proposed model consists of six dependent variables, namely, susceptible population, infective population, isolated susceptible population who are aware of the undesirable consequences of the COVID - 19, quarantined population of known infectives ( symptomatic ) , recovered class, and the coronavirus population. The model exhibits two equilibria namely, the COVID - 19 - free equilibrium and the COVID - 19 - endemic equilibrium. It is observed that if basic reproduction number < R 1 0 , then the COVID - 19 - free equilibrium is locally asymptotically stable. However, the endemic equilibrium is locally as well as nonlinearly asymptotically stable under certain conditions if > R 1 0 . Model analysis shows that if safety measures are adopted by way of isolation of susceptibles and quarantine of infectives, the spread of COVID - 19 disease can be kept under control.

Abstract: In this article, a nonlinear mathematical model is proposed and analyzed to study the spread of coronavirus disease (COVID-19) and its control. Due to sudden emergence of a peculiar kind of infection, no vaccines were available, and therefore, the nonpharmaceutical interventions such as lockdown, isolation, and hospitalization were imposed to stop spreading of the infectious disease. The proposed model consists of six dependent variables, namely, susceptible population, infective population, isolated susceptible population who are aware of the undesirable consequences of the COVID-19, quarantined population of known infectives (symptomatic), recovered class, and the coronavirus population. The model exhibits two equilibria namely, the COVID-19-free equilibrium and the COVID-19-endemic equilibrium. It is observed that if basic reproduction number < R 1 0 , then the COVID-19-free equilibrium is locally asymptotically stable. However, the endemic equilibrium is locally as well as nonlinearly asymptotically stable under certain conditions if > R 1

Introduction
The coronavirus disease (COVID-19) pandemic is of great concern to researchers, governments, and the general public because of its worldwide escalation and high number of deaths associated with it. The COVID-19 disease is an infectious disease caused by coronavirus, which causes illness in animals and humans. On December 31, 2019, a new and unusual case of pneumonia was reported at the World Health Organization (WHO) country office in China. A large number of such cases have taken place in the Chinese city of Wuhan. It was noted that all cases were linked to the Wuhan Seafood Market, which trades in live and fish species, including poultry, bats, marmots, and snakes [21]. It was later discovered that this infection occurred due to a new coronavirus called 2019 novel coronavirus (2019-n COV) [15]. Later, on February 11, 2020, the International committee on viral rehabilitation renamed acute respiratory syndrome coronavirus 2 as SARS-COV-2. Coronavirus SARS-COV-2, which causes COVID-19, is thought to have originated from bats and spread to snakes and pangolins, which is why it has spread to humans, possibly Keeping this aspect in view, the objective of this article is to explore the role of viral density in order to study the spread of COVID-19 pandemic where the viral density is assumed to be proportional to the infected individuals (symptomatic or asymptomatic). A mathematical model is formulated by considering the total population to be variable using simple mass action incidence.

Mathematical model
To formulate a pandemic model, let ( ) N t be the total human population in a region under consideration affected by coronavirus disease (COVID-19) at any time t. The population is divided into five subpopulations, namely, susceptible population ( ) X t , infective population ( ) Y t , isolated susceptibles population ( ) X t i who are aware of undesirable consequences of the COVID-19, population of quarantined infectives (symptomatic) ( ) Y t i , and recovered class with density ( ) R t . Let ( ) V t be the cumulative density of COVID-19 virus present in the environment, emitted from infectives at any time t.
The following assumptions are made to model the dynamics of the transmission of the coronavirus disease (COVID- 19), (i) The recruitment rate of susceptibles A is constant.
(ii) Susceptibles become infected directly by coming in contact with infectives with a contact rate β (i.e., βXY ). (iii) Susceptibles become infected indirectly by coming in contact with contaminated surfaces having coronaviruses with a contact rate λ (i.e., λXV ). (iv) Isolated aware susceptibles obtain infected by a rate upon contact with infectives (i.e., γεX Y i , ≤ ≤ ε 0 1). (v) Isolated aware susceptibles also obtain infected by a rate ( upon contact with coronavirus present on the contaminated surfaces (i.e., ( In view of the aforementioned assumptions an considerations, the transmission dynamics of the spread of COVID-19 is expressed in the following system of nonlinear differential equations: As mentioned earlier, susceptibles get infected by rates βXY and λVX, which increase the growth rate of infective individuals. The constants β and λ are the transmission rates of disease directly by infectives Y and indirectly by coming in contact with contaminated surfaces having viral density V , respectively. It is assumed that knowing the undesirable consequences of the pandemic, aware susceptibles keep themselves in isolation. The constant λ 1 is the rate of transfer of susceptible people to isolated aware susceptible class. It is reasonable to assume that some of the isolated susceptible people may become susceptible again joining susceptible population class. Thus, λ 11 is the rate at which people from the isolated aware susceptible class move to susceptible class. The constant d is the natural death rate coefficient of human population. Further, the isolated susceptible people may also get infected upon casual contact with infectives and contaminated surfaces filled with coronaviral density, thus increasing the growth rate of infectives. The constant γ, ( ) < γ β λ min , , is the rate of transfer of isolated susceptibles to infected class (i.e., γX Y i ). A fraction of this transmission (i.e., γεX Y i , ≤ ≤ ε 0 1) is due to interaction of susceptibles with infectives and the other (i.e., ( ) − γ ε XV 1 i ) is due to coming in contact with virus density present on the contaminated surfaces. It is observed that if proper safety precautions (such as use of face cover/mask, comply with social distancing, not touching the surfaces, and use of soap and/or sanitizer for hand wash) are taken by both susceptibles and infectives, then the possibility of the susceptibles to get infected will reduce and some of the infectives may get well and join recovered class. Further, if infected individuals are found COVID-19 positive, they are quarantined in hospitals for proper treatment for a prescribed time period. Thus, infected individuals are hospitalized by a rate β 1 and are recovered by a rate δ 1 , thus joining quarantined and recovered class, respectively. After undergoing proper treatments during hospitalization and getting well, these people join recovered class by a rate δ 2 . The constant α is the disease-induced death rate. It is assumed that when an infected person (known or unknown COVID-19-positive individuals) touches a surface, the surface also gets infected with COVID-19 virus. Therefore, it is reasonable to assume that the growth rate of COVID-19 viral density is directly proportional to the infected human population. The constant θ represents the growth rate of COVID-19 virus on the surfaces, θ 0 being its natural depletion rate.
Using the assumption = + + + + N X Y X Y R i i , Models (1)-(6) reduces to the following system: Now, we will proceed with reduced Models (7)-(12).
Remark 1. From the model system, it is noted here that > β γ and > λ γ.  3 Equilibrium analysis and basic reproduction number Models (7)-(12) have the following two equilibria: implies that in the absence of infective individuals and COVID-19 virus density deposited on surfaces, areas, etc., the disease will not persist.
To determine basic reproduction number R 0 [7], it is sufficient to take equations (8), (10), and (12) satisfying necessary constraints: From the aforementioned system, the infectious matrix F, corresponding to new infections in the population at disease-free equilibrium, is The nonsingular matrix V , denoting the transfer terms at disease-free equilibrium, is Now, basic reproduction number R 0 is the spectral radius of the next-generation matrix − FV 1 and is given as 1 .
implies that in the presence of infective individuals and surfaces contaminated with COVID-19 virus, disease will always persist. Some infective individuals, which are kept under proper treatment in the form of isolation and hospitalization, may get recovered.
The solution of endemic equilibrium ( is given by the following system of simultaneous equations: Using equations (13) and (15)- (18) in equation (14), we obtain It is noted from the model system (7)- (12), corresponding to the endemic equilibrium * E , that ( ) ( , which implies that as the recovery rate of the infectives due to various processes involved in the treatment of infectives (such as efforts made by the government) increases, then the equilibrium level of infectives in the society decreases. In view of this result, it can easily be shown that , which implies that as the recovery rate of the infectives due to various processes involved in the treatment of infectives (such as hospitalization) increases, then the equilibrium level of quarantined infectives decreases.  2 and the remaining two are given by the following equation:

Nonlinear stability analysis
In this section, the stability [22] behavior of the endemic equilibrium * E is studied for whole region of attraction Ω near the equilibrium * E .
Theorem 3. The endemic equilibrium * E is nonlinearly asymptotically stable provided the following conditions hold inside the region of attraction Ω:  (See Appendix C for proof).

Remark 5.
The aforementioned theorems imply that if β and λ are very small, then the possibility of the satisfying local and nonlinear stability conditions is more plausible. Thus, it can be speculated that the transmission rate of disease directly from the infectives to the susceptibles and the rate of contact of susceptibles with COVID-19 virus density present on the contaminated surfaces have a destabilizing effect on the model system. For small values of β and λ, the equilibrium values of * E obtain stabilized.

Numerical simulation
The numerical simulation of the model system (7)- (12) is given here to show the existence of equilibrium and the feasibility of stability conditions. We integrate the model system by the fourth-order Runge-Kutta method using MATLAB with the following set of parameter values given in Table 1 The graphical representation of the results of numerical simulation is displayed in Figures 1-4.
In Figure 1(a), the comparison between the confirmed reported values of infected people in India by WHO [6,23] and that predicted by the mathematical model using the aforementioned set of parameter values is shown. It is seen from the figure that the model system fits well with the real data. For simulation, the fraction of the population initially is assumed as follows: Figure 1(b), the variation of confirmed infective population density is displayed with time t for distinct values of λ, the transmission coefficient of disease to susceptibles due to virus density present in the environment. It is observed from the figure that as the value of λ increases, the infective population increases, which indicates the faster spread of the disease with an increase in the indirect contact of susceptibles with contaminated surfaces.      Figure 2 represents the variation of infective population and virus density with time for different values of λ 1 , the rate of transfer of susceptible population to isolated aware susceptible class. It is seen from Figure 2(a) that the infective population decreases with an increase in the value of λ 1 . This is due to the fact that as more people isolate themselves, the spread of disease decreases, which decreases the number of infective population. This decline in the infective population ultimately decreases the density of virus in the environment ( Figure  2(b)). Thus, if more susceptible people isolate themselves from direct interaction with infectives or from indirect interaction with contaminated surfaces, the spread of disease decreases, which in turn decreases the density of virus present in the environment (Figure 2(b)).
The variation of quarantined infectives and virus density, respectively, with time is displayed in Figure 3 for distinct values of β 1 , the rate at which infected individuals are quarantined. It is seen that with an increase in the value of β 1 , the quarantined infective population increases (Figure 3(a)), and this increment in quarantined infective population, not spreading the diseases anymore, in turn decreases the density of the virus (Figure 3(b)). Thus, if the infective population is quarantined, then the density of virus decreases. Figure 4 displays the variation of quarantined infectives and recovered population, respectively, with time for different values of δ 2 , the recovery rate of quarantined infectives. It is noted from Figure 4(a) that the quarantined infectives decreases with an increase in the value of δ 2 , as more people obtain recovered during quarantine period. Thus, the spread of disease decreases, which leads to a decline in the infective population and, as such, the recovered population increases (Figure 4(b)).

Conclusion
In the article, a nonlinear mathematical model has been proposed and analyzed to study the spread of coronavirus disease COVID-19 in the human population with constant immigration of susceptibles. In the modeling process, the total human population is divided into five subclasses, namely, susceptibles, infectives, isolated aware susceptibles, quarantined symptomatic infectives, and the recovered population. The effect of some critical parameters on the spread of the disease is studied. The analysis of the model has been performed using the stability theory and numerical simulation, and some inferences have been drawn by establishing the local and nonlinear stability results. The model results indicate similar pattern as seen with the reported data. The model analysis shows that if more people are exposed to virus deposited on contaminated surfaces or areas, the infective population increases. However, if the rate of transfer of susceptible population to isolated aware susceptible class increases, the infective population declines, leading to a decreased virus density in the environment. This also highlights the importance of social distancing, use of face cover/mask, avoidance of social gatherings, etc., to make effective isolation so that disease spread is decreased. The increased density of virus present in the environment, however, leads to faster spread of the deadly coronavirus disease COVID-19 if proper safety measures are not adopted.
Acknowledgement: The authors are thankful to the reviewers and the editor for their useful comments that helped us in finalizing the manuscript.
Funding information: This research received no specific grant from any funding agency, commercial, or nonprofit sectors.
From equation (7), we note that From equation (9), we note that 1 11 . Similarly, it can be proved that ( ) , which implies that Similarly, it can be proved that ( ) Hence, the lemma.

B Proof of Theorem 1
To determine the local stability behavior of * E , the linearized system of the model (7) is expressed as follows: Now, consider the following positive definite function: where ( ) = … k i 1 6 i are the positive constants to be chosen appropriately. Differentiating U with respect to "t" along the solutions of linearized system of (7)-(12), we obtain  where ( ) = … m i 1 6 i are the positive constants to be chosen appropriately. Differentiating W with respect to "t" along the solutions of system (7)