key: cord-0742773-2n3a0ru2
authors: Chaturvedi, Deepa; Chakravarty, U.
title: Predictive analysis of COVID-19 eradication with vaccination in India, Brazil, and U.S.A
date: 2021-03-31
journal: Infect Genet Evol
DOI: 10.1016/j.meegid.2021.104834
sha: 602f13d211e4fda2350cd6dcd23cc25eaefc2b28
doc_id: 742773
cord_uid: 2n3a0ru2

The most important question and concern in these circumstances of COVID-19 epidemic outspread is when will the pandemic end? Vaccination is the only solution to restore life to normalcy in the fastest and safest possible manner. Therefore, we have carried out a predictive analysis for realistic time scale estimates for overcoming the epidemic considering vaccination rate effect on the dynamics of COVID-19 control. In particular we discuss the worst affected large countries like India, Brazil and USA for estimating effect of vaccination rate in expediting the end of the COVID-19 epidemic. We analytically simulated the dynamic evolution of active cases of these countries in the last nine months using the modified SIR model and then included the effect of vaccination to forecast the proliferation dynamics. We hence obtained the transmission parameters, the variation in the reproduction numbers and the impact of the different values of the vaccination shots in the expected curves of active cases in the coming times to predicted the time scales of the end of the epidemic.

The state of emergency and the extent to which the novel corona virus has affected the mankind is unparalleled in human history [1] . Like severe acute respiratory syndrome coronavirus (SARS-CoV) and Middle East respiratory syndrome coronavirus (MERS-CoV), a new noble-coronavirus outbreaks the respiratory disease in the current year 2020 [2, 3] . This new coronavirus is named as "COVID-19" by the World Health Organization (WHO), and the International Committee on Taxonomy of Viruses called it Severe Acute Respiratory Syndrome coronavirus 2 (SARS-CoV-2) [4, 5] . The first patient died with pneumonia caused by the new 2 coronavirus in Hubei province, China, in December 2019 [4, 6] , and it has progressively spread all over the world since then. By 30 th November 2020, 61 million of people have been infected, in which 43.5 million have recovered [7] . Though 1.4 million, 2.3 per cent of total infected people have died from complications arising from Covid-19 related infection [7] . Moreover, this disease has severely affected the world economy and plunged the world into a global recession [8] . The rapid outbreak of COVID-19 in the world happened due to interhuman transmission [9, 10] owing to the large Reproduction factor associated with this epidemic. As a preventive measure various control measures were proposed and implemented like social/physical distancing, frequent hand wash, sanitization, and using mask to inhibit the transmission further [9] . Also, most of the countries has imposed lockdown to impart social distancing to control the transmission, like Italy, China, Spain, India, etc. [11, 12, 13] . Even though lockdown is not the remedial solution of the COVID-19 but still in order to ramp up the health facilities to cope up the upsurge of new infected cases it became an absolute necessity to implement extreme lockdowns despite the heavy loss in economy of the developed as well as the developing countries. There was a prompt response from scientists, doctors, health experts to look for the detection methods, treatment and prevention strategies to combat the disease effectively. Further, COVID-19 has motivated the researchers working in various fields to apply numerical, analytical, statistics analysis to study its transmission rate, reproduction number, and the effect of control measures on the transmission rates. Such studies are of immense help to equip and formulate COVID combat strategies by inhibition and control methods, helps to forecast how much time it would take to eradicate the disease. The transmission of COVID-19 has been studied by SIR (susceptible-infected-recovered) model which was first proposed and investigated by Kermack and McKendrick [14] . Huwen Wang et al. predicted the COVID cases in Wuhan, China by using a modified SEIR model i.e. Susceptible, Exposed, Infectious, and Removed (SEIR) [4] . Qianying Lin et al. studied the outbreak of COVID in China considering individual behavior and governmental actions using SEIR model [6] . Z. Ceylan developed an Auto-Regressive Integrated Moving Average (ARIMA) models to predict the epidemiological trend of COVID-19 prevalence of Italy, Spain, and France [11] . V.Volpert et al. proposed the quarantine model and studied the case of South Korea, China, and Italy [15] . A dynamical analysis of infection spread when vaccination is introduced in the general population is also very important for Planning and Management at the local as well as global level. It is well agreed that herd immunity is not the ultimate solution for seeking respite from this menace, the only solution to restore life back to normalcy in the fastest and safest possible manner is through speedy administration of Vaccine through a massive immunological drive. Various modeling studies have been done to study the role of vaccination on the spread of infectious diseases. In 2011, T.K. Kara and A.Batabyal studied the existence of stability in the disease free and endemic equilibrium, and also discussed the vaccine induced reproduction number to investigate the consequences of providing vaccination [16] . Farrington analyzed the impact of vaccination program on the transmission potential of the infection in large populations [17] . Shulgin In this paper, we have investigated the impact of providing different values of vaccination rate to the progression of active cases of COVID-19. The SIR model is used with two modifications first the transmission rates are modified depending on the response of the people and the governed restrictions; along with this a term for vaccinations also added to the modified SIR model coupled non linear equations. The two forms of transmission parameter [21] have been chosen based on the real time available data for COVID-19 progression from Feb-Nov 2020. In particular the major large population countries like India, Brazil and America that are most severely affected are analyzed and studied in-depth. Also, the variation in the reproduction number in the last 9 months (Feb-Nov 2020) have been studied and discussed and further the effect of vaccination on reproduction number is outlined. In addition to this, the effect of different value of vaccination shots in the progression of active cases of COVID-19 in these countries has been delineated. Furthermore, we have predicted the possible time scales for the end of the epidemic for different values of vaccination rates. A nonlinear dynamics analysis of infection spread and control after the Vaccination drive is administered is of great interest for Planning and Management ofvarious socio economic activities like opening of schools, Malls, Industries, Travelforrestoring life back to normalcy in various countries.

SIR (susceptible-infected-recovered) epidemic model is a very popular and effective model to understand the dynamics of the spread of any epidemic disease [22] . The SIR model can be easily modified to include the vaccination term, if s V is the vaccination shots per day then the coupled set of equations of SIR model can be modified [22] as follows- 

and it can be written as

di t i t s t i t dt

The transmission parameter,  is a dynamic parameter and dependent over the infected and recovered fraction of people at any time t . Thus, we have taken two different forms of  as discussed in paper [21] , i.e. From Eq. (7), we obtain that the infection increases exponentially at the early epidemic growth phase if we assume that the susceptible population is equal to the total population of any city, Thus, from the above expression, we found that the number of secondary cases generated per primary case depends on the ratio of 0  and 0  , which is called a reproduction number 0 R , and is given by [22] 

Thus, the secondary cases increase if 0 R is greater than 1 and it decays when it is less than 1. Hence the 0 R is the special parameter to study the transmission of infection in any area. But the number of susceptible people decreases with time due to the increase in infection, and the effective reproduction number over time , t R is given by the product of 0 R and the proportion of susceptible individuals in the population [22] 

The above expression shows that the effective reproduction number [15] . Also, the disease will not spread if 0 R is less than 1 [23] . (13)). t R reaches first to 0.2 for the higher inhibition strength parameter. It means the epidemic eradicates faster for higher value of inhibition strength parameter. Fig. 1(b) has been plotted to show the variation in t R with days for different sets of response term, c and m. Initially, the steepness of the curve decreases with m (promptness of response parameter) and then increases as m reaches J o u r n a l P r e -p r o o f Journal Pre-proof 6 near to 0 value. Thus, the promptness of response decides the steepness or rate of decrease of t R curve.

We also plotted the variation in t R for different values of c when m is 1, and for different sets of response term in Fig. 2(a) and Fig. 2(b) , when the transmission parameter is

We see that the lower part of the curve shifts upward for nonzero values of c, when m is 1 (see Fig. 2(a) ), and as m decreases, the lower part of the curve shifts downward (see Fig. 2(b) ). It means that when infection inhibition strength parameter is zero, then t R reduces to 0.2, whereas it lies near to 1, when infection inhibition strength parameter is non zero. As the promptness of response decreases, the t R tends to shift downward. The epidemic eradicates faster for the case either when c is 0 or when m or c tend towards 0.

In current scenario, the active cases of COVID-19 are highest in the countries like USA, India and Brazil. So, we have fitted and analytically simulated the active cases of these countries, and found the transmission parameters, and hence studied the variation in the reproduction numbers. This analysis is extremely crucial for understanding, government and societal response for disease control. The vaccination applied to a country will definitely gradually make a progressive control over the epidemic. However, the timescales to bring the situation under control or eradication of the epidemic completely will rely on the vaccination rate superimposed over the response parameters, b, c and m. Therefore, a quantitative estimate for a realistic determination of disease eradication requires estimation of the current values of b, c and m parameters from the available data of COVID-19 progression and then superimposing vaccination rates on the coupled nonlinear differential equations mentioned above.

The evolution in active cases of COVID-19 in Brazil since Feb 15, 2020, is shown in Fig.  3 (a) when only 0 cases were reported [7] . It shows that the reported data (solid black curve) till Nov 17, 2020, is well simulated by the numerical model (dashed red and blue curves). Here, the form of infection transmission parameter which matched well with the reported data is parameter is taken for a period of 9 months for the curve, 0  = 0.5. This infers that the government had taken strict actions to control the initial increase in cases and to restrain the transmission. In figure, we see that the fitting of active case curve is done with the two sets of J o u r n a l P r e -p r o o f 7 transmission parameter. The second blue curve ((c, m) is (0.461, 1/25)) has lowered the maxima in comparison to the red curve ((c, m) is (0.471, 1/20)). It infers that both the government and people are able to control the growth of COVID-19.

To understand the transmission of COVID-19, it is necessary to study the dynamics of reproduction number with days, so we plotted t R in Fig 3(b) . Initially, t R is maximum and it is equal to 5 due to 0 R 00 ()  at 0 th day. t R starts decreasing due to decrease in the number of susceptible populations as shown in Eq. (13) . We see that the second set of transmission parameters (blue dashed curve of Fig. 3(a) ) has shifted the t R curve down due to change in the strength of infection inhibiting parameter. The slope of the t R curve is reduced due to decrease in the promptness of response parameter from 1/20 to 1/25. It means that the people are actively responding to control the disease. Figure shows that the t R is reduced to 1 at nearly 176 th day (9 th August 2020) when the active case is maximum and thereafter it continues to reduce slowly. The t R is 0.8699 at 275 th day of study. So, it is inferred that the epidemic in Brazil is under control as t R is less than 1.

In USA, 12 cases of COVID-19 were reported on Feb 15 2020. The Fig. 4(a) shows the progression in Active cases of COVID-19 in USA since Feb 15 2020 [7] . The active case first increases and get steady for small period after 95 th day, i.e. from 20 th May to 14 th June, and thereafter again it started increasing at 120 th day due to relaxation in restrictions imposed (stay at home) in different states. The second rise in COVID-19 also get steady after 190 th day to 240 th day and again rising again thereafter. The figure shows that the reported data is fitted by using 5 simulated curves with the form of the infection transmission parameter, 0 ( ) ( ) m t ci t   implying the socio/economic activities are still allowed to function amidst strict restrictions and therefore a flattening of active cases is expected. This can be predicted that the USA government is attentive and has taken continuous steps in last 9 months to control the pandemic while easing out economic activities vigilantly. So, we took an average value of the effective infection transmission (EIT) parameter of the curve 0  as 0.4, with the change in the response parameter value (c, m) from (1.225, 1/4) to (0.4, 1/15), (0.325, 1/32), (0.288, 1/64), and finally to (0.276, 1/64) .   Fig 4(b) shows the variation in the effective reproduction number with days. t R is equal to 0 R and it is maximum at t=0 day. t R decreases at the faster rate as the value of the infection inhibition strength parameter is 1.225 and reduced to 1.001 at 125 th days due to good governance by the American government. We observe a sharp hike in t R at 126 th day due to a large drop in the infection inhibition strength parameter from 1.225 to 0.4, and also the slope is reduced as the promptness response parameter reduces from ¼ to 1/15. This change in t R is mainly due to the impact of releasing the restriction by the government. The figure shows that the t R is reduced to 1 at nearly 224 th day (26 th September 2020), thereafter it continues to reduce to 0.9878 till 240 th day. Again t R becomes greater than 1 after 241 st day. t R is 1.22 on 275 th day of study. It infers J o u r n a l P r e -p r o o f Journal Pre-proof 8 that the epidemic in USA is rising. So, people have to become more attentive towards the COVID-19, so that they control the rise in active cases.

The Fig. 5(a) shows the progression in Active cases of COVID -19 in India since Feb 15 2020, when 0 active cases were reported [7] . It shows that the active cases increase slowly initially in India in comparison to the other two countries, Brazil and America. This is mainly due to the strict central lockdown imposed by the government, in which all services, educational institutes, transportation were closed. But we observe that the growth in the active cases does not follow any symmetric trend due to different phases of unlocking by the Indian government. So, the reported data is simulated with nine sets of transmission parameter taking an average value of effective infection transmission parameter as 0  = 0.33. The analytical form of the infection

The active case is maximum at the 215 th day (i.e. 17 th September 2020).

In Fig 5(b) , we study the dynamic of reproduction number with days. It shows that t R is maximum at day 0 according to Eq. (13). t R reduces to 1.272 from 2.872 (day 1) in 111 days due to the promptness strength parameter as 1/8. It shows that the first phase of lockdown is very effective to control the transmission. But in the case of second transmission parameter set, the promptness strength parameter is reduced to 1/128, and thus the slope of the t R curve is reduced. Similarly, the slope of the next part is low due 1/64 value of promptness strength parameter. So, we see ups and downs or rise and fall due to different values of promptness strength parameter. It infers that t R does not follow the same trend of reduction due to different phases of lockdown. Also, the rise and fall in t R of India shows the sporadic, inconsistent and unpredictable responses. t R is reduced nearly to 1 at approx. 218 th day (20 th September 2020) and again it becomes greater than 1 at 221 th day and again reduces back below to 1 near 223 th day. t R is 0.828 at 275 th day of study. It shows that the epidemic is under control in India.

In the above section, we analytically simulated the reported data of COVID-19 of countries, Brazil, USA and India. We have seen that the effective reproduction number for Brazil and India is reduced to near to 1. But in the case of USA, the effective reproduction curve is increasing due to increase in the active cases. So, to have a better look over the transmission of COVID-19, we study the expected curves of active case and the effective reproduction number till the end of the next year 2021 in Fig. 6 . We found that the expected rate of decrease in the active case of India (Fig 6(c) ) is highest than that of USA (Fig 6(b) ) and Brazil (Fig 6(a) ) (denoted by solid black dashed curve). When we study the expected dynamic of reproduction number (blue solid curve), we found that after 275 days, 9 after being controlled. Thus, vaccination is the only solution to completely eradicate the disease from the world.

So, we study the impact of different values of vaccination shots in the expected curve of COVID-19. For this, we have used the Eqs. (6) to (8) , and the vaccination shots ( s V ) given per day are 0, 0.5 lakh, 1 lakh, and 1.5 lakh and it is shown in Fig. 7 . When the s V is zero, the respective curve (red solid curve) just lies over the expected curve. The figure shows that as we increase the value of s V , the rate of decreasing of active cases becomes faster in all the three cases. Also, we plotted the variation in the respective dynamics of reproduction number for different values of vaccination shots in Fig. 8 . We observe that the t R is lowered at faster rate for higher values of vaccination shots. Thus, the epidemic would be under control or eradicate faster if we increase the values of t R . In Fig. 9 , we study the approximated days (T end ) for different values of vaccination shots when the active case is reduced to 1% of the maximum active cases of the respective countries. T end decreases with the increase in the vaccination shots. So, T end is inversely proportional to the value of the vaccination shots given per day. When we compare, the Figs 9(a) and 9(c) of Brazil and India respectively, Brazil needs higher value of s V than India. When we see the curve of USA, we found that USA needs even higher value of s V to get control over COVID-19 due to its highest number of active cases and an approach of dynamic disease control with relaxation rather than inhibitive disease eradication approach.

In conclusion, we have predicted the possible time scales for the end of the epidemic for different values of vaccination rates. We have done the predictive analysis of vaccination for COVID-19 in India, Brazil and the USA. This analysis of infection spread and control after the Vaccination drive is administered is of great interest for Planning and Management of various socio-economic activities like opening of schools, Malls, Industries, Travel for restoring life back to normalcy in various countries.

1. It is found that we require more than one sets of transmission parameter to fit the active cases of Brazil due to change in the response of the people while lockdown and on releasing the restrictions. While the active case of India is fitted by using nine sets of response term which shows the different phases of lockdown and unlock by the Indian government, and the active case of USA is fitted by using five sets of response term. We have taken an average of the effective infection transmission parameter for the last nine months. 2. We have discussed the variation in the effective reproduction term ( t R ) with respect to the response term of the transmission parameter. We found that t R reduces at the faster rate for higher values of effective infection transmission parameter (c) whereas the rate decreases as the promptness response parameter decreases. 3. It is found that COVID-19 is under the control in all the three countries as the expected effective reproduction term ( t R ) is less than 1.

J o u r n a l P r e -p r o o f 4. We found that the USA require very large number of vaccination shots to eradicate COVID-19 than India and Brazil due to its highest number of active cases as well as disease control regime with dynamic and lenient response. 5. Even with an active vaccination administration it may take a year or so for overcoming COVID epidemic thus social distancing, hygiene practices and restrictions on educational institutions and travel may have to be continued in the year 2021 as well. Highlights of the paper  Active cases are fitted with more than one sets of transmission parameter.  Effective reproduction number ( tR ) reduces at the faster rate for higher values of effective  infection transmission parameter (c).  The slope of effective reproduction number depends on the promptness response parameter  (m). 

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Please check the following as appropriate:All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version. This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue.The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript