key: cord-0495568-zol1auyu
authors: Sardar, Tridip; Nadim, Sk Shahid; Chattopadhyay, Joydev
title: Assessment of 21 Days Lockdown Effect in Some States and Overall India: A Predictive Mathematical Study on COVID-19 Outbreak
date: 2020-04-07
journal: nan
DOI: nan
sha: 07c7724815245c8855f3c3f4883870ffacb4ead2
doc_id: 495568
cord_uid: zol1auyu

As of April, 6th, 2020, the total number of COVID-19 reported cases and deaths are 4778 and 136. This is an alarming situation as with a huge population within few days India will enter in stage-3 of COVID-19 transmission. In the absence of neither an effective treatment or vaccine and with an incomplete understanding of the epidemiological cycle, predictive mathematical models can help exploring of both COVID-19 transmission and control. In this present study, we consider a new mathematical model on COVID-19 transmission that incorporate lock-down effect and variability in transmission between symptomatic and asymptomatic populations with former being a fast spreader of the disease. Using daily COVID-19 notified cases from three states (Maharashtra, Delhi, and Telangana) and overall India, we assess the effect of current 21 days lock-down in terms of reduction cases and deaths. Lock-down effect is studied with different lock-down success rate. Our result suggest that 21 days lock-down will have no impact in Maharashtra and overall India. Furthermore, the presence of a higher percentage of COVID-19 super-spreaders will further deteriorate the situation in Maharashtra. However, for Tamil Nadu and Delhi there is some ray of hope as our prediction shows that lock-down will reduce a significant number of cases and deaths. in these two locations. Further extension of lock-down may place Delhi and Tamil Nadu in a comfort zone. Comparing estimated parameter samples for the mentioned four locations, we find a correlation between effect of lockdown and percentage of symptomatic infected in a region. Our result suggests that a higher percentage of symptomatic infected in a region leads to a large number of reduction in notified cases and deaths due to different lock-down scenario. Finally, we suggest a policy for the Indian Govt to control COVID-19 outbreak.

As of April 6, 2020, 13,46,004 cases and 74,654 deaths from 2019 novel coronavirus disease , caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), were recorded worldwide [4] . Coronaviruses are enveloped non-segmented positivesense RNA viruses that belongto the Coronaviridae family and the order Nidovirales, and are widely distributed among humans and other mammals [20] . The novel coronavirus, COVID-19 started in mainland China, with a geographical emphasis at Wuhan, the capital city of Hubei province [30] and has widely spread all over the world. Many of the initial cases were usually introduced to the wholesale Huanan seafood market, which also traded live animals. Clinical trials of hospitalized patients found that patients exhibit symptoms consistent with viral pneumonia at the onset of COVID-19, most commonly fever, cough, sore throat and fatigue [3] . Some patients reported changes in their ground-glass lungs; normal or lower than average white lymphocyte blood cell counts and platelet counts; hypoxemia; and deranged liver and kidney function. Most were said to be geographically related to the wholesale market of Huanan seafood [2] . Severe outbreaks occur in USA (3,67,004 cases), Italy (1,32,547 cases), Spain (1,36,675 cases), Germany (103,375 cases), France(98,010), China (81,708 cases) and so many countries and the disease continues to spread globally. This has been declared a pandemic by the World Health Organization. It is the third zoonotic human coronavirus that has arisen in the present century, after the 2002 severe acute respiratory syndrome coronavirus (SARS-CoV), which spread to 37 countries and the 2012 Middle East respiratory syndrome coronavirus (MERS-CoV), which spread to 27 countries.

The 2019 pandemic novel coronavirus was first confirmed in India on 30 January 2020, in the state of Kerala. A total of 4778 confirmed cases, 382 recoveries and 136 deaths in the country have been reported as of 6 April 2020 [5] . The Indian government has introduced social distance as a precaution to avoid the possibility of a large-scale population movement that can accelerate the spread of the disease. India government implemented a 14-hour voluntary public curfew on 22 March 2020. Furthermore, the Prime Minister of India also ordered a nationwide 21-day lockdown at midnight on 24 March to slow the spread of COVID-19, affecting India's entire 1.3 billion population. Despite no vaccine, social distancing has identified as the most commonly used prevention and control strategy [11] . The purpose of these initiatives is the restriction of social interaction in workplaces, schools, and other public spheres, except for essential public services such as fire, police, hospitals. No doubt the spread of this virus outbreak has seriously disrupted the life, economy and health of citizens. This is a great concern for everyone how long this scenario will last and when the disease will be controlled.

Mathematical modeling based on system of differential equations may provide a comprehensive mechanism for the dynamics of COVID-19 transmission. Several modeling studies have already been performed for the COVID-19 outbreak [27; 18; 24; 28; 16] . Based on data collected from December, 31 st 2019 till January, 28 th 2020, Wu et al. developed a susceptible exposed infectious recovered model (SEIR) to clarify the transmission dynamics and projected national and global spread of disease [31] . They also calculated around 2.68 is the basic reproductive number for COVID-19. Tang et al proposed a compartmental deterministic model that would combine the clinical development of the disease, the epidemiological status of the patient and the measures for intervention. Researchers found that the amount of control reproduction number may be as high as 6.47, and that methods of intervention including intensive touch tracing followed by quarantine and isolation would effectively minimize COVID cases [28] . For the basic reproductive number, Read et al. reported a value of 3.1 based on the data fitting of an SEIR model, using an assumption of Poisson-distributed daily time increments [19] . A report by Cambridge University has indicated that India's countrywide three-week lockdown would not be adequate to prevent a resurgence of the new coronavirus epidemic that could bounce back in months and cause thousands of infections [25] . They suggested that two or three lockdowns can extend the slowdown longer with five-day breaks in between or a single 49-day lockdown. Data-driven mathematical modeling plays a key role in disease prevention, planning for future outbreaks and determining the effectiveness of control. Several data-driven modeling experiments have been performed in various regions [28; 10] . Currently, there are very limited works that studied the impact of lock-down on COVID-19 transmission dynamics in India.

In the present manuscript, we proposed a new mathematical model for COVID-19 that incorporates the lock-down effect. We also considered variability in transmission between symptomatic and asymptomatic populations with former being a fast spreader of the disease. Using COVID-19 daily notified cases from three highly affected states (Maharashtra, Delhi, and Telangana) and from overall India, we estimated some key parameters of our model. In a large country like India with so much diverse population it is not at all feasible to lock-down (home quarantine) all susceptible population. A certain percentage of the population may be successfully home quarantined during the lock-down period. Thus, we assess the current 21 days (March, 25 th 2020 till April, 14 th 2020) lock-down scenario in India with different lock-down success rates. We studied the effect of different lock-down scenarios in terms of notified cases and deaths reduction for the four locations mentioned above for a certain time period. We estimated the basic reproduction number (R 0 ) for the mentioned four locations. Effective reproduction number is also estimated for different lock-down success rate using our model for these four regions.

Based on the development and epidemiological characteristics of COVID-19, a SEIR type model is more appropriate to study the dynamics of this current pandemic [14; 16; 17; 22] . The model we consider in this paper is an extension of a SEIR model with added asymptomatic A(t) and hospitalized or notified C(t) population (see Fig 1) . As hospitalized or notified persons C(t) are isolated and unable to contact susceptible individuals S(t), we assume that disease transmission only occur in contact with asymptomatic (A(t)) and symptomatic I(t) individuals (see Fig 1) . We also assume variability in disease transmission in asymptomatic and symptomatic population with later being a fast spreader of infection with variability factor ρ. Recovery rate for asymptomatic, symptomatic and hospitalized or notified populations are assumed to be different with rates γ 1 , γ 2 , and γ 3 , respectively. We considered disease related deaths (at a rate δ) only for hospitalized or notified population. Based on these assumptions, we develop the following compartmental model for COVID-19 outbreak:

where, N (t) is the total human population at time t. A model (2.1) flow diagram is provided in Fig 1. Biological interpretations of model (2.1) parameters are provided in Table 1 . Lock-down population are fraction of the susceptible individuals that are home quarantined. Particularly, when the level of infection is high, people will be encouraged to take the required steps to avoid contact with the infected individuals in order to protect themselves and their families, leading to a reduction in transmission levels. These rates of transmission also reflect a strong steps to control disease. We assume that a portion of the susceptible population become isolated due to lock-down at a success rate l. Since population which are isolated due to lock-down do not contact the infection therefore average effective contact rate of a single symptomatic and asymptomatic indi-

and

, respectively. We also assume that population in the lock-down compartment L(t) again become susceptible after the lock-down period ( 1 ω ) is over. Based on these assumptions model (2.1) with lock-down become:

Here, l and 1 ω are lock-down success rate and lock-down period respectively. Remaining parameters of the model (2.2) are given in Table 1 .

Mathematical properties of the model (2.1) We studied the positivity and boundedness of solution of the model (2.1) (See Appendix A). The system (2.1) demonstrates two equilibria, that is, the disease-free equlibrium and an unique endemic equilibrium (See Appendix A). The disease-free state is locally asymptotically stable whenever the corresponding basic reproduction number (R 0 ) is less than unity (See Appendix A). By using a nonlinear Lyapunov function, it is also seen that the disease-free equilibrium is globally asymptotically stable whenever R 0 < 1 (See Appendix A). In addition, the model (2.1) has an unique endemic equilibrium if R 0 exceeds unity. Furthermore, using the central manifold theory, the local stability of the endemic equilibrium is established whenever R 0 > 1 (See Appendix A).

Daily COVID-19 reported cases from Maharashtra, Delhi, Tamil Nadu and whole India for the time period March, 10 th 2020 till March, 29 th 2020 (for Maharashtra), March, 17 th 2020 till April, 2 nd 2020 (for Delhi), March, 18 th 2020 till April, 1 st 2020 (for Tamil Nadu), and March, 2 nd 2020 till March, 31 st 2020 (for India) are considered for our study. These three states are the most affected regions in India due to current COVID-19 pandemic [5]. Daily COVID-19 notified cases were collected from [5].

In Table 1 , we listed the key parameters of the model (2.1) that are estimated from the data. We also estimated some unknown initial conditions of the model (2.1) from the data. MATLAB based nonlinear least square solver f mincon is used to fit simulated and observed daily COVID-19 notified cases for these three states and the whole country during the mentioned time period. Delayed Rejection Adaptive Metropolis Hastings [12] algorithm is used to sample the 95% confidence region. An elaboration of this model fitting technique is provided in [23] .

In a large country like India with so much diverse population it is not at all feasible to lock-down (home quarantine) all susceptible population. A certain percentage of the population may be successfully home quarantined during the lock-down period. Thus, we assess the current 21 days (March, 25 th till April, 14 th ) lock-down scenario in India with different success rate namely, 20%, 40%, 60%, and 80%, respectively. For example, a 20% lock-down success in Maharashtra means that 20% of the susceptible population in this state is successfully home quarantined during the lock-down time period.

To predict COVID-19 notified cases and as well deaths (daily and total) with different lock-down success rate, we undertake the following procedure.

• We run model (2.1) (without lock-down) using sample values of the estimated parameters and initial conditions (See Appendix B) till March, 25 th 2020. Using our simulation result, we predict the notified cases and deaths during this time duration.

• Using values of the state variables onset of the lock-down starting day on March, 25 th 2020 as initial condition, we simulate the model (2.2) (with lock-down) up to April, 14 th , 2020. Proceeding as earlier, we predict the notified cases and deaths during this time span.

• Again using values of the state variables onset of the lock-down ending day on April, 14 th , 2020 as initial condition, we run the model (2.1) (without lock-down) up to desired prediction period. Using result of this model run, we predict the notified cases and deaths during this time span.

• Finally, we ensemble all predicted cases and deaths during the whole time period.

2.1.5. Estimation of R 0 , and R t The Basic reproduction number (R 0 ) [29] for the model (2.1) is given as follows (see Appendix A):

The effective reproductive number (R t ) is defined as the expected number of secondary infection per infectious in a population made up of both susceptible and non-susceptible hosts [21] . If R t > 1, the number of new cases will increase, for R t = 1, the disease become endemic, and when R t < 1 there will be a decline in the number of new cases.

Following [21] , the expression of R t is given as follows :

where,ŝ(t) is the fraction of the host population that is susceptible at time t. R 0 can easily be estimated by plugin the sample values of the unknown parameters in the expressions (2.3).

Following procedure is adapted to estimate R t for different lock-down scenario:

• We first estimate R 0 using the estimated parameter values (see Table B5 in Appendix B) for the model (2.1).

• Proceeding as earlier, we run the models (2.1) and (2.2) at different time clusters (with and without lock-down). Using our simulation results, we estimateŝ(t) and R t .

Model (2.1) fitting to daily COVID-19 notified cases for Maharashtra, Delhi, Tamil Nadu and India is Depicted in Fig 2. Estimated values of the model (2.1) parameters and unknown initial conditions are provided in Appendix B, Table B1 and Table B2 , respectively. In Maharashtra, estimate of symptomatic influx fraction (κ) suggesting low number (about 3%) of symptomatic infected in the population (See Table B1 in Appendix B). However, higher value of the transmission rate (β 1 ), and low value of the transmission variability factor (ρ) indicates that existence of potential super-spreaders among symptomatic infected (see R 0 in equation (2.3) and Table B1 in Appendix B). Low value of κ (see Table B1 in Appendix B) indicates there may be a large number of undetected COVID-19 cases in the population of Maharashtra. However, low value of ρ indicates contribution of these asymptomatic infected towards new infection is very low (see expression of R 0 in equation (2.3) and Table B1 in Appendix B). To further investigate on super-spreaders, we varies κ (from 0.1 to 0.5) in the expression of R 0 . We found that around 60% -92% increase in the current value of R 0 (see Table B5 ) if 10% to 50% new infection become symptomatic. This is an alarming situation as in coming days there may be a possibility of higher percentage of symptomatic infection and which leads to more and more super-spreaders in the population of Maharashtra.

We predicted notified cases as well deaths in Maharashtra for different lock-down situation (see Methods section) using our models with and without lock-down (see equations (2.1) and (2.2)) starting from March, 9 th 2020 till May, 7 th 2020 (60 days).

Predicted total notified cases and deaths under different (no lock-down, 20%, 40%, 60% and 80%) lock-down scenario in Maharashtra during the mentioned period is provided in Table B3 and Table B4 , respectively. Furthermore, predicted daily notified cases and deaths under different (20%, 40%, 60% and 80%) lock-down scenario in Maharashtra during the mentioned period is depicted in Fig 3 and Fig 4, respectively. Our predictive study suggest that only 0.1% notified case reduction and (0.3% − 0.5%) death reduction are possible under different lock-down circumstances in Maharashtra (see Table B3 and Table B4 ). Thus, 21 days lock-down will not improve current COVID-19 situation in Maharashtra.

For further investigate the lock-down effect in Maharashtra, we estimate the effective reproduction number (R t ) (see Methods section) for the period March, 9 th 2020 till May, 7 th 2020 under different lock-down scenario (no lock-down, 20%, 40%, 60% and 80%). Dynamics of R t under different lock-down situation is depicted in Fig 5. Our result on R t suggest that there may be a decrease in new cases during the lock-down period in Maharashtra (see Fig 5) . However, after the lock-down is over on April, 15 th 2020 a sharp increase in R t indicates that influx of new notified cases will rise again (see Fig 5) . This result further support our claim that lock-down in Maharashtra will not help its current COVID-19 situation. Lock-down scenarios are implemented with success rate 20%, 40%, 60% and 80% respectively. A lock-down success 20% in Maharashtra means that current implemented lock-down will successfully home-quarantine 20% of the total susceptible population. Figure 5 : Effective reproduction number for the period 09/03/2020 till 07/05/2020 in Maharashtra with different lock-down scenario. Yellow interval is current 21 days lock-down period (25/03/2020 till 14/04/2020) in Maharashtra. Lock-down scenarios are implemented with success rate 0%, 20%, 40%, 60% and 80% respectively. Here, 0% lock-down success means no lock-down, and 20% represent current implemented lock-down will successfully home-quarantine 20% of the susceptible population in Maharashtra and similarly for other scenarios.

In Delhi, estimate of ρ (see Table B1 in Appendix B) indicates as like Maharashtra here also contribution of asymptomatic infected in COVID-19 new infection is low. Higher value of κ (see Table B1 in Appendix B) indicate that a higher percentage of symptomatic infected in Delhi. However, as estimate of disease transmission rate (β 1 ) is moderate (see Table B1 in Appendix B) therefore there may be lower percentage of super-spreaders in Delhi. Estimate of R 0 in Delhi is provided in Table B5 . Higher value 4.62 (3.70 -5.40) of R 0 (see Table B5 ) in Delhi may be due to presence of high percentage symptomatic infected in the population (see Table B1 in Appendix B).

Notified cases as well as deaths in Delhi are predicted for different lock-down situations mentioned earlier (see Method section) staring from March, 17 th 2020 till May, 15 th 2020 (60 days). Predicted total notified cases and deaths under different (no lock-down, 20%, 40%, 60% and 80%) lock-down scenario in Delhi for the mentioned period is provided in Table B3 and Table B4 , respectively. Daily predicted cases and deaths during the mentioned period in Delhi under different lock-down scenario is depicted in Fig 6 and  Fig 7, respectively. Prediction of cases and deaths in Delhi (see Table B3 and Table B4) suggest that (30% -33%) case reduction and (39% -52%) death reduction are possible under different lock-down scenario. Thus, 21 days lock-down in Delhi will be effective and should be extended for few weeks to further case and death reduction.

To further investigate the lock-down effect in Delhi, we estimate R t (see Method section) for the period March, 17 th 2020 till May, 15 th 2020 under different lock-down scenario. Time-series of R t for Delhi under different lock-down scenario is depicted in Fig 8. Estimate of R t shows that it fall rapidly during the lock-down period (25/03/2020 till 14/04/2020) and shows some increment after lock-down is over (see Fig 8) . However, R t become saturated and remain below unity after the lock-down period is over (see Fig 8) . This indicates that the number of new notified cases will decrease after the lockdown period (25/03/2020 till 14/04/2020) is over in Delhi. This result further support our claim that 21 days lock-down will be effective in Delhi and should be extended further for more cases and deaths reduction. Lock-down scenarios are implemented with success rate 20%, 40%, 60% and 80% respectively. A lock-down success 20% in Delhi means that current implemented lock-down will successfully home-quarantine 20% of the total susceptible population. Lock-down scenarios are implemented with success rate 0%, 20%, 40%, 60% and 80% respectively. Here, 0% lock-down success means no lock-down, and 20% represent current implemented lock-down will successfully home-quarantine 20% of the susceptible population in Delhi and similarly for other scenarios.

In Tamil Nadu, estimate of ρ (see Table B1 in Appendix B) indicate that here asymptomatic infected population contribute higher in producing new COVID-19 cases in compare to Maharashtra and Delhi (see expression of R 0 in equation (2.3) and Table B1 in Appendix B). As percentage of symptomatic infected (see Table B1 in Appendix B) in the population is also very high in Tamil Nadu, therefore, the estimate of R 0 , 8.44 (5.20 -13.23), for Tamil Nadu found out to be much higher than Maharashtra, Delhi and overall India (see Table B5 ).

Notified cases and deaths in Tamil Nadu are predicted for different lock-down scenario as mentioned earlier (see Method section) starting from March, 18 th 2020 till May, 16 th 2020 (60 days). Predicted total notified cases and deaths under different (no lock-down, 20%, 40%, 60% and 80%) lock-down situation in Tamil Nadu for the mentioned time duration are provided in Table B3 and Table B4 , respectively. Predicted daily notified cases and deaths for the mentioned time interval under different lock-down circumstances are depicted in Fig 9 and Fig 10, respectively. Predicted cases and deaths in Tamil Nadu (see Table B3 and Table B4 ) suggest that (21% -29%) notified case reduction and (32% -48%) notified death reduction are possible under different lock-down scenario. Therefore, our prediction result suggest that 21 days lock-down in Tamil-Nadu will be effective if it is extended further for few more weeks.

For further investigation in lock-down situation in Tamil Nadu, we estimate R t (see Method section) for the time period March, 18 th 2020 till May, 16 th 2020 under different lock-down scenario. Dynamics of R t for Tamil Nadu under different lock-down scenario for the mentioned time duration is depicted in Fig 11. R t falls rapidly during the lockdown period (25/03/2020 till 14/04/2020) and after that it become almost stagnant (below unity). This shows that number of new cases in Tamil Nadu will fall after lockdown period is over. Therefore it is high time to implement few weeks lock-down again in Tamil Nadu after the current lock-down period is over. Figure 9 : Model prediction of Daily COVID-19 cases for the period 18/03/2020 till 16/05/2020 in Tamil Nadu with different lock-down scenario. Yellow interval is the current 21 days lock-down period (25/03/2020 till 14/04/2020) in Tamil Nadu. Lock-down scenarios are implemented with success rate 20%, 40%, 60% and 80% respectively. A lock-down success 20% in Tamil Nadu means that current implemented lock-down will successfully home-quarantine 20% of the total susceptible population. Figure 10 : Model prediction of Daily COVID-19 deaths for the period 18/03/2020 till 16/05/2020 in Tamil Nadu with different lock-down scenario. Yellow interval is the current 21 days lock-down period (25/03/2020 till 14/04/2020) in Tamil Nadu. Lock-down scenarios are implemented with success rate 20%, 40%, 60% and 80% respectively. A lock-down success 20% in Tamil Nadu means that current implemented lock-down will successfully home-quarantine 20% of the total susceptible population. Figure 11 : Effective reproduction number for the period 18/03/2020 till 16/05/2020 in Tamil Nadu with different lock-down scenarios. Yellow interval is the current 21 days lock-down period (25/03/2020 till 14/04/2020) in Tamil Nadu. Lock-down scenarios are implemented with success rate 0%, 20%, 40%, 60% and 80% respectively. Here, 0% lock-down success means no lock-down, and 20% represent current implemented lock-down will successfully home-quarantine 20% of the susceptible population in Tamil Nadu and similarly for other scenarios.

Overall in India, estimate of κ (see Table B1 in Appendix B) suggest that there is large number of undetected cases in the population. Furthermore, the estimate of ρ (see R 0 in equation (2.3) and Table B1 in Appendix B) suggest that these large undetected cases contribute a notable amount in producing new COVID-19 cases. Estimate of β 1 (see Table B1 in Appendix B) in overall India found out in resemblance with its values for Delhi and Tamil Nadu. Estimate of the basic reproduction number (R 0 ) for overall India is provided in Table B5 .

Notified cases as well as deaths in India are predicted for different lock down situation (see Method section) mentioned earlier starting from March, 2 nd 2020 till May, 7 th 2020 (67 days). Predicted total notified cases and deaths under different lock-down scenario (no-lock-down, 20%, 40%, 60%, and 80%) in India for the mentioned time span are provided in Table B3 and TableB4, respectively. Predicted daily notified cases and deaths in India for the mentioned time duration under different lock-down scenario are depicted in Fig 12 and Fig 13, respectively. Predicted cases and deaths in India (see Table B3 and TableB4) suggest that (0.2% -0.6%) notified case reduction and (0.2% -0.5%) notified death reduction are possible under different lock-down circumstances. Thus, 21 days lock-down will not improve current COVID-19 situation in overall India.

To further investigate in this direction, we estimate R t (see Method section) for the time period March, 2 nd 2020 till May, 7 th 2020 under different lock-down scenario. Dynamics of R t under different lock-down scenario for the mentioned time duration is depicted in Fig 14. Dynamics of R t over the mentioned time duration suggest that there may be decrease in new cases during the lock-down period in India. However, sharp increase in R t after the lock-down period indicates that daily notified cases will rise again (see Fig 14) . This result further support our claim that 21 days lock-down will not improve current COVID-19 situation in overall India. Our claim in resemblance with a recent COVID-19 study on overall India [25] . Figure 12 : Model prediction of Daily COVID-19 cases for the period 02/03/2020 till 07/05/2020 in India with different lock-down scenario. Yellow interval is the current 21 days lock-down period (25/03/2020 till 14/04/2020) in India. Lock-down scenarios are implemented with success rate 20%, 40%, 60% and 80% respectively. A lock-down success 20% in India means that current implemented lock-down will successfully home-quarantine 20% of the total susceptible population. Figure 13 : Model prediction of Daily COVID-19 deaths for the period 02/03/2020 till 07/05/2020 in India with different lock-down scenario. Yellow interval is the current 21 days lock-down period (25/03/2020 till 14/04/2020) in India. Lock-down scenarios are implemented with success rate 20%, 40%, 60% and 80% respectively. A lock-down success 20% in India means that current implemented lock-down will successfully home-quarantine 20% of the total susceptible population. Figure 14 : Effective reproduction number for the period 02/03/2020 till 07/05/2020 in India with different lock-down scenarios. Yellow interval is the current 21 days lock-down period (25/03/2020 till 14/04/2020) in India. Lock-down scenarios are implemented with success rate 0%, 20%, 40%, 60% and 80% respectively. Here, 0% lock-down success means no lock-down, and 20% represent current implemented lock-down will successfully home-quarantine 20% of the susceptible population in India and similarly for other scenarios.

Up to April, 6 th , 2020, total number of reported COVID-19 cases and deaths in India are 4778 and 136, respectively [5]. This tally rises with few hundred new notified cases every day reported from different locations in India [5]. This is an alarming situation as with a huge population within few days India will enter in stage-3 of COVID-19 transmission. In the absence of neither a effective treatment or vaccine and with an incomplete understanding of epidemiological cycle, predictive mathematical models can help strengthen our understanding of both COVID-19 transmission and control [22] .

In this present study, we consider a new mathematical model on COVID-19 transmission that incorporates the lock-down effect. In our model, we also considered transmission variability between symptomatic and asymptomatic population with former being a fast spreader of the disease. Using daily notified cases from three states (Maharashtra, Delhi, and Tamil Nadu) and form whole India, we studied the effect of 21 days lock-down (25/03/2020 till 14/04/2020) on notified cases and deaths reduction in those regions. Our result suggest that Lock-down will have no effect in Maharashtra and overall India. Furthermore, presence of higher percentage of COVID-19 super-spreaders will further deteriorate the situation in Maharashtra. However, for Delhi and Tamil Nadu there is some ray of hope as our prediction shows that lock-down will reduce significant number of notified cases and deaths in these two locations. Further extension of lock-down may place Delhi and Tamil Nadu in a comfort zone. To find the answer of the question why lock-down have some effect in Delhi and Tamil Nadu but no effect on overall India and Maharashtra? We closely look at the parameters sample table for these four locations (see Table B1 ) and found that in Tamil Nadu and Delhi there is a large percentage of symptomatic infected exists in the population. Whereas, in Maharashtra and overall India percentage of symptomatic infected population is very low (see Table B1 ). Thus, there may be a possibility that larger proportion of symptomatic infected in a population can help epidemic curve to reach its peak quickly. To further investigate our claim, we simulate new notified cases for these four locations with mentioned prediction period without any lock-down (see Fig 15) . It can be easily seen from Fig 15, that in Delhi and Tamil-Nadu notified new cases reaches the epidemic peak and whereas for Maharashtra and overall India notified cases slowly increases and still increasing during the mentioned time duration. As new notified cases in Delhi and Tamil Nadu grow faster therefore, 21 days lock-down has some significant impact in reducing cases and deaths in these two locations. Thus, 21 days lock-down may be effective in reducing significant amount of cases and deaths in those location in India where percentage of symptomatic infected population is higher. Finally, we provide a suggestion for the Indian Govt. and Policy makers to do the following steps:

1. An extensive survey to find the percentage of symptomatic infected in different states and regions.

Focus implementing extensive lock-down in those locations only where the percentage of symptomatic infected is very high.

3. Provide relaxation in lock-down in other locations for some time. This will increase the percentage of symptomatic infection.

Repeat step-2, when a region has a sufficient percentage of the symptomatic infected. 

The DFE of the model (2.1) is given by

The local stability of ε 0 can be established on the system (2.1) by using the next generation operator method. Using the notation in [29] , the matrices F for the new infection and V for the transition terms are given, respectively, by

It follows that the basic reproduction number [13] , denoted by R 0 = ρ(F V −1 ), where ρ is the spectral radius, is given by

Using Theorem 2 in [29] , the following result is established. The threshold quantity, R 0 is the basic reproduction number of the disease [13; 8; 7] . This represent the average number of secondary cases generated by a infected person in a fully susceptible population. The epidemiological significance of 4.3 is that when R 0 is less than unity, a low influx of infected individuals into the population will not cause major outbreaks, and the disease would die out in time. Proof. Consider the following Lyapunov function

We take the Lyapunov derivative with respect to t,

Since all the variables and parameters of the model (2.1) are non-negative, it follows thaṫ L ≤ 0 for R 0 ≤ 1 withL = 0 in diseases free equilibrium. Hence, L is a Lyapunov function on Ω. Therefore, followed by LaSalles Invariance Principle [15] , that Hence, it follows from the fifth equation of the model (2.1) that, for t > max{B 1 , B 2 },

Therefore using comparison theorem [26] 

Similarly (by using lim t→∞ inf A(t) = 0 and lim t→∞ inf I(t) = 0, it can be shown that

Thus, it follows from above two relations Therefore by combining all above equations, it follows that each solution of the model equations (2.1), with initial conditions ∈ Ω , approaches ε 0 as t → ∞ for R 0 ≤ 1.

In this section, the existence of the endemic equilibrium of the model (2.1) is established. Let us denote

Let ε * = (S * , E * , A * , I * , C * , R * ) represents any arbitrary endemic equilibrium point (EEP) of the model (2.1). Further, define

It follows, by solving the equations in (2.1) at steady-state, that

Substituting the expression in (A-4) into (A-3) shows that the non-zero equilibrium of the model (2.1) satisfy the following linear equation, in terms of λ * :

Since a 0 > 0, µ > 0, k 1 > 0, k 2 > 0, k 3 > 0 and k 4 > 0, it is clear that the model (2.1) has a unique endemic equilibrium point (EEP) whenever R 0 > 1 and no positive endemic equilibrium point whenever R 0 < 1. This rules out the possibility of the existence of equilibrium other than DFE whenever R 0 < 1. Furthermore, it can be shown that, the DFE ε 0 of the model (2.1) is globally asymptotically stable (GAS) whenever R 0 < 1. From the above discussion we have concluded that Theorem 4.2. The model (2.1) has a unique endemic (positive) equilibrium, given by ε * , whenever R 0 > 1 and has no endemic equilibrium for R 0 ≤ 1. Now we will prove the local stability of endemic equilibrium.

Theorem 4.3. The endemic equilibrium ε * is locally asymptotically stable if R 0 > 1.

Proof. The Jacobian matrix of the system (2.1) J ε 0 at DFE is given by

Here, by taking β 1 as a bifurcation parameter, we use the central manifold theory method to determine the local stability of the endemic equilibrium [9] . Taking β 1 as the bifurcation parameter and gives critical value of β 1 at R 0 = 1 is given as β * 1 = (µ + σ)(γ 1 + τ 1 + µ)(γ 2 + τ 2 + µ) [κσ(γ 1 + τ 1 + µ) + (1 − κ)ρσ(γ 2 + τ 2 + µ)] The Jacobian of (2.1) at β 1 = β * 1 , denoted by J ε 0 | β 1 =β * 1 has a right eigenvector (corresponding to the zero eigenvalue) given by w = (w 1 , w 2 , w 3 , w 4 , w 5 , w 6 ) T , where w 1 = − µ + σ µ w 2 , w 2 = w 2 > 0, w 3 = (1 − κ)σ γ 1 + τ 1 + µ w 2 , w 4 = κσ γ 2 + τ 2 + µ w 2 , w 5 = τ 1 (1 − κ)σ(γ 2 + τ 2 + µ) + τ 2 κσ(γ 1 + τ 1 + µ) (γ 1 + τ 1 + µ)(γ 2 + τ 2 + µ)(δ + γ 3 + µ) w 2 , w 6 = γ 1 (1 − κ)σ µ(γ 1 + τ 1 + µ) + γ 2 κσ µ(γ 2 + τ 2 + µ) + γ 3 τ 1 (1 − κ)σ µ(γ 1 + τ 1 + µ)(δ + σ 3 + µ) + γ 3 τ 2 κσ µ(γ 2 + τ 2 + µ)(δ + γ 3 + µ)

Similarly, from J ε 0 | β 1 =β * 1 , we obtain a left eigenvector v = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) (corresponding to the zero eigenvalue), where v 1 = 0, v 2 = v 2 > 0, v 3 = ρβ * 1 γ 1 + τ 1 + µ v 2 , v 4 = β * 1 γ 2 + τ 2 + µ v 2 , v 5 = 0, v 6 = 0.

Selecting the notations S = x 1 , E = x 2 , A = x 3 , I = x 4 , C = x 5 , R = x 6 and dx i dt = f i . Now we calculate the following second-order partial derivatives of f i at the disease-free equilibrium ε 0 and obtain

Now we calculate the coefficients a and b defined in Theorem 4.1 [9] of Castillo-Chavez and Song as follow

Replacing the values of all the second-order derivatives measured at DFE and β 1 = β * 1 , we get a = − β * 1 v 2 µ Π H (ρw 3 + w 4 )(w 2 + w 3 + w 4 + w 5 + w 6 ) < 0 and b = v 2 (ρw 3 + w 4 ) > 0

Since a < 0 and b > 0 at β 1 = β * 1 , therefore using the Remark 1 of the Theorem 4.1 stated in [9] , a transcritical bifurcation occurs at R 0 = 1 and the unique endemic equilibrium is locally asymptotically stable for R 0 > 1. 

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Location S(0)

The Funder had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

The authors declare that they have no conflicts of interest. This subsection is provided to prove the positivity and boundedness of solutions of the system (2.1) with initial conditions (S(0), E(0), A(0), I(0), C(0), R(0)) T ∈ R 6 + . We first state the following lemma...., n} [32] .Proof. By re-writing the system (2.1) we haveB(X(t)) = (B 1 (X), B 1 (X), ..., B 6 (X)) T We note thatThen it follows from the Lemma 4.1 that R 6 + is an invariant set. We observed from the system thatHence the system (2.1) is bounded.