key: cord-0785216-u6t0mxd7 authors: Chowdhury, Sourav; Roychowdhury, Suparna; Chaudhuri, Indranath title: Universality and herd immunity threshold : Revisiting the SIR model for COVID-19 date: 2021-06-09 journal: International Journal of Modern Physics. C, Physics and Computers DOI: 10.1142/s012918312150128x sha: 81eaf08a3a922c02b40fd9e4cf10a59b4bbe5d0d doc_id: 785216 cord_uid: u6t0mxd7 COVID-19 pandemic has been raging all around the world for almost a year now, as of November 1, 2020. In this paper, we try to analyze the variation of the COVID-19 pandemic in different countries in the light of some modifications to the susceptible-infected-recovered (SIR) model. The SIR model was modified by taking time-dependent rate parameters. From this modified SIR model, the basic reproduction number, effective reproduction number, herd immunity, and herd immunity threshold are redefined. The re-outbreak of the COVID-19 is a real threat to various countries. We have used the above-mentioned quantities to find the reasons behind the re-outbreak of this disease. Also, the effectiveness of herd immunity to prevent an epidemic has been analyzed with respect to this model. We have also tried to show that there are certain universal aspects in the spread and containment of the disease in various countries for a short period of time. Finally, we have also analyzed the current pandemic situation in India and have attempted to discuss the possibilities in order to predict its future behaviour using our model. Nowadays, the COVID-19 pandemic is spreading throughout various countries. Due to this pandemic, a total of 213 countries have been affected, 47 million people are already infected and 1.2 million people have died worldwide, as of November 1, 2020. 1 Also in India, a total of 8.2 million people have been infected and 0.12 million people died, during this period. 1 The worrying thing is that these numbers are still increasing on a daily basis. Also, some countries like Italy, Russia, Germany have faced a re-outbreak of this disease. COVID-19 pandemic is not only affecting public health but also affecting social health, world economy, and countries' economy. Therefore, it is very important to understand the behavior of this pandemic to prevent such widespread damage. In this article, we mainly take the basic susceptible-infected-recovered (SIR) model. 2 SIR model is a well-known and widely used model. The SIR model and its successors like SEIR, SIRS, SEIRD, SEIAR have been used before in various epidemics like influenza, dengue, Ebola. [3] [4] [5] [6] [7] [8] [9] These models are also being used to predict and understand the behavior of the COVID-19 pandemic. [10] [11] [12] [13] [14] [15] [16] [17] [18] There are already various papers that are present on this topic, [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] still we have tried to understand this pandemic from a different point of view. Here, we modify the basic SIR model by taking the time-dependent rate parameters, (i) Infection rate (β), (ii) Removed rate (γ). From this modified SIR model we re-define the four important quantities, which are, (i) Basic reproduction number (R 0 ), (ii) Effective reproduction number (R i ), (iii) Herd immunity of the population (HI i ), (iv) Herd immunity threshold of the population (HIT ). The herd immunity of a population (HI i ) is the fraction of immune people, against a disease, present in there. Herd immunity of a population is a key to decrease an epidemic. Herd immunity can increase in two different ways, (i) naturally, by increasing the number of infected people, and (ii) artificially, by vaccines. Nowadays when an epidemic starts in a country, the administration of that country usually intervenes and imposes various restrictions in the form of partial or full lockdowns as in the COVID-19 case. As a result of this, the herd immunity of the population is forced to remain low. Despite the low herd immunity, it is also seen that, as in the case of COVID-19, the number of infected cases reached some peak value and then started to fall off in many countries. This requires a different explanation from the ones that have been previously proposed. We have tried to explain this aspect with the modified SIR model and have used the data of COVID-19 from various countries to verify them. The data used for this analysis is taken from "The Humanitarian Data Exchange". 23 However, such low herd immunity of a population can bring a resurgence to the epidemic. This phenomenon is often termed in literature as reinfection or re-outbreak. Various authors have addressed this issue earlier. 24, 25 Here, we have tried to discuss this fact with the help of Effective reproduction number (R i ), Herd immunity of the population (HI i ), and Herd immunity threshold of the population (HIT ). As we have mentioned earlier, governments resort to several methods to mitigate a pandemiclike situation, as seen from the present COVID-19 situation. Thus the behavior and the dynamics of the COVID-19 pandemic could have common features in different countries. We have tried to address this aspect by searching for universal features in the parameters of our model. The paper is organized as follows: Section 2 deals with a discussion of the basic and modified SIR models along with the reproduction numbers, herd immunity, and herd immunity threshold. Section 3 consists of various simulations of the modified SIR model by considering imaginary parameter values and conditions. In section 4 we try to find the time variation of the model parameters in light of the COVID-19 data for several countries. In section 5 we present a brief study of the COVID-19 behavior in India and the last section, we present our concluding remarks. Here we discuss a simple epidemic model which is known as SIR model. 2, 26 If we assume that the population is homogeneously mixed then the model equations can be written as, S(t) = Number of susceptible people at time t. I(t) = Number of active cases or the number of infectious people at time t. R(t) = Number of removed people at time t. Removed persons are those who were previously infected but now recovered or dead. N = Total population in a community or a region or a country. β = Infection rate. γ = Removed rate of the infected people. Hence, S(t) + I(t) + R(t) = N = Total number of people in a population. I tot (t) denotes the total number or the cumulative number of infected cases at time t. So, I tot (t) = I(t) + R(t). Hence the rate of increase of I tot (t) can be written as, Also, we know that This equation looks like the logistic equation of population growth. However, the main difference is that the increment rate of the total infected cases depends on the number of active cases rather than on itself. An epidemic will start to spread if the number of infected people in a population starts to increase with time, which means, Hence, from equation 2 we can write, Similarly, an epidemic will start to decrease when, Also, at the peak of an epidemic, The reproduction number tells us about the average number of susceptible people who can be infected by an infectious individual. Basic reproduction number R 0 is a very important quantity that tells us whether an epidemic will start or not. R 0 can be defined as the number of secondary infections due to a single primary infected individual at the initial time of an epidemic. If at any time t, S(t) and I(t) are the numbers of susceptible and infected people respectively then the number of people who will be infected newly is β S N I per unit time. Now initially the number of infected people in a population is very less, so we can assume S(0) ≈ N . Hence initially an infected people can infect β number of susceptible people per unit time. The removed rate is γ, hence on average, an infected person will remain infected for 1 γ unit of time. So, in this period, an infected person can spread the disease to β γ number of people. Hence this is the number of secondary infections due to a single infected individual. Thus R 0 = β γ . So, from equation 7 we can say that initially, an epidemic will start to grow if, or, So, if R 0 > 1 then a disease will start to spread, and if R 0 < 1then the spread of the disease will die down. R 0 describes only the initial behavior but not the late-time behavior of an epidemic. So, we introduce another important quantity which is known as the instantaneous reproduction number or effective reproduction number and denoted as, R i (t). At any time t, the number of susceptible people can be infected by a single infected individual is β S N per unit time. Hence on average, a single infected individual can infect a total of β γ S N number of people until that individual is removed. Thus, Now if we assume that the infection rate β and recovery rate γ are constants then, Like the previous way, we can find that an epidemic will increase if R i (t) > 1 and will decrease if R i (t) < 1. At the peak of an epidemic, the effective reproduction number, R i (t) = 1 (from equation 9). Immune people in a population are those who are currently infected (I) and who were infected, which means the removed people (R). So, at any time t total number of immune people in a population is I(t) + R(t) = I tot (t). Herd immunity in a population is defined as the fraction of the population which is immune to a disease. 27, 28 So, to stop the spread of the disease there have to be a minimum fraction of immune people in a population and which is called the "Herd immunity threshold (HIT )". From equation 8, we can say that an epidemic will start to decrease when, . Now fraction of immune people at time t is HI i (t) is the instantaneous herd immunity of a population at any time t. So from equation 14, we can write, Hence an epidemic will stop if, Hence herd immunity threshold defined as, If β and γ are constant of time then we can write, So, if HI i (t) < HIT then the epidemic will increase and will decrease if, HI i (t) > HIT . So, at the peak of an epidemic, HI i (t) = HIT (from equation 9). In this section, we present certain modifications to the SIR model. The spread of an epidemic depends on the infectivity of a disease which is an universal property of the virus. However, it also depends on the social behavior and social interaction of a region, which can vary with time. Also, when an epidemic starts, governments in those regions try to contain the spread of the disease by locking that region down or part of it, making rules to prevent infections (like maintaining social distancing, wearing masks, and spreading awareness among the people). Hence the rate of infection β will not be constant always. Here we assume that the rate of infection β is a time-dependent quantity. Also removed rate γ can be time-dependent. It is mentioned earlier that increment of the removed rate depends not only on the recovery of infected people, also on the death rate or fatality rate of the disease. So, in general, the removed rate can be changed if the recovered rate or death rate is changed. Initially, knowledge among the doctors and scientists is limited for a new disease. But as time goes by the knowledge about that disease will increase and this may help to improve the recovered rate. On the other hand, there are factors such as medical infrastructure, infectivity of the disease etc. which directly affect the recovered rate and the death rate. Thus it is difficult to disentangle these two effects in the current purview of our model and it would be only possible to predict the total increment of the removed rate as a function of both. We would be interested to extend this model and study these two effects separately in future to better predict the behaviors of the deceased rate and the recovered rate. Due to these modifications, the relation of basic reproduction number becomes R 0 = β(0) γ(0) because the basic reproduction number is a quantity that is measured initially. Also, the instantaneous reproduction number or effective reproduction number is given by, R i (t) = β(t) γ(t) S(t) N . As we discussed in subsection 2.1, an epidemic will increase if R i (t) > 1 and will decrease if R i (t) < 1. At the peak of the epidemic R i (t) = 1. The instantaneous herd immunity of a population, HI i (t) is given by, HI i (t) = 1 − S(t) N . Now, for the basic SIR model, herd immunity threshold (HIT ) is a constant in time since β and γ are constant quantities. But here herd immunity threshold is a time-dependent quantity and is given by, HIT (t) = 1 − γ(t) β(t) . Hence the maximum possible value of herd immunity threshold is As we discussed in subsection 2.2, an epidemic will increase if HI i (t) < HIT (t) and will decrease if HI i (t) > HIT (t). At the peak of the epidemic HI i (t) = HIT (t). From the basic SIR model, we can describe only one way to prevent an epidemic in terms of herd immunity and herd immunity threshold. This is, • An epidemic can be prevented by increasing herd immunity (HI i (t)) of the population than the herd immunity threshold value (HIT ). However, in the modified SIR model, the herd immunity threshold is a time-dependent quantity. So, we can describe two different ways to prevent an epidemic. The first way is same as the previous one. The second way is, • An epidemic can be prevented by decreasing the herd immunity threshold (HIT (t)) to the current herd immunity of the population (HI i (t)). Herd immunity threshold (HIT ) can be reduced by decreasing the infection rate (β(t)) or increasing the removed rate (γ(t)). From equation 2 and equation 3 we can write, where, I = Number of infected or active cases at time t. R = Number of removed cases at time t. From equation 3 we can write, Hence, from equations 20 and 21, we can estimate β(t) and γ(t). From these time dependent rate parameters, we can easily estimate the variation of HIT (t) and R i (t). Let a region initially contain a total of N = 10000 people where a disease starts to spread with infection rate β = 0.7 day −1 . Also, let the removed rate for this disease be γ = 0.15 day −1 . Hence, the basic reproduction number R 0 has a value, R 0 ≈ 4.6667 > 1, implying an epidemic will start to grow in this region. Let there initially be only one infected person in the region, which means I(0) = 1. Thus, the initial number of susceptible people is S(0) = N − I(0) = 9999 and removed people is R(0) = 0. If we simulate the SIR system with the above-mentioned values of the parameters and the initial conditions, we get the results which are shown in Figure 1 . In this system, the herd immunity threshold has value HIT = 0.7857. Thus, when 78.57% of people of this population are immune from the disease, the epidemic will start to decrease. From figure 1 , we further see that the effective reproduction number R i (t) is reduced from R 0 to unity, and then it is reduced further. Similarly, we can see that herd immunity HI i (t) is increased from zero to HIT , and then it is increased further and has a value greater than HIT . Recalling that an epidemic will start to decrease after I(t) attains the peak (as previously discussed). Let, at t = t p , I(t) attains its peak. So, from figure 1 we can see that at t p , R i (t p ) = 1.0 and HI i (t p ) = HIT = 0.7857. Here, t p = 19.47 days. Hence, approximately 20 days are needed to start to decrease the spread of the disease. At the end of the epidemic, which means, at t → ∞, the number of susceptible persons left in the population is S(∞) ≈ 97. This is very small compared to the whole population. Also at, t → ∞, R i (∞) = 0.0457 and HI i (∞) = 0.9902. Hence, at the end of the epidemic, the herd immunity of the population is increased than the HIT which has the value 0.7857 and the effective reproduction number reduced from one. Thus, there is no chance of a re-outbreak in this population. The picture of row two shows us the variation of herd immunity HI i (t) with time. So, we can see that almost all of the people are infected once during this epidemic. If the fatality rate of this disease is very low then technically fewer people will die. But the problem is that the large number of infected cases will create pressure on the medical facilities, which can make the situation worse. Thus many people can die because of not getting proper treatment. But if the fatality rate is large then the situation will be very grave. In this situation, the government and local authorities will intervene and will put some restrictions and safety rules to prevent the epidemic. Due to these restrictions and rules, the infection rate will decrease and thus fewer people will be infected during this epidemic. Let, at time t = t I = 10 government starts to intervene. Here we consider two cases of the government interventions, (1) Intervention-1 : At, t I = 10, β decreases from 0.7 day −1 to 0.25 day −1 . Means, β(t) = 0.7 day −1 when t < t I and β(t) = 0.25 day −1 when t ≥ t I . (2) Intervention-2 : At, t I = 10, β decreases from 0.7 day −1 to 0.35 day −1 . Means, β(t) = 0.7 day −1 when t < t I and β(t) = 0.35 day −1 when t ≥ t I . From the I(t) vs. t plot of figure 2 we can see that due to the government's intervention, the peaks are flattened than the no intervention case. Also, it is seen in figure 2 that due to government's intervention at t = t I = 10, the curve of R i (t) decreases fast as compared to no intervention. We found that at the end of the epidemic, the number of susceptible people left in the population is approximately 3111 and 1290 for intervention-1 and intervention-2 respectively. So, interventions help to decrease the total number of infections. Also from figure 2, we can see that if the intervention is more strict then the epidemic will take more time to reach the peak. intervention-2 (right). Figure 3 shows the variation of herd immunity HI i (t) for the two interventions. But we can see that there are some basic differences between these herd immunity plots. For both cases, the maximum value of herd immunity threshold is HIT max = 0.7857. Due to the intervention, the herd immunity threshold value is reduced from the maximum value to 0.4 and 0.5714 for intervention-1 and intervention-2 respectively. Here, the reduced value of the herd immunity threshold due to the intervention is represented by HIT in . As, the difference between the HIT max and HIT in2 for intervention-2 is small, thus the herd immunity at the end of the epidemic (HI i (∞)) not just goes beyond the HIT in2 also it crosses HIT max . But for intervention-1, the herd immunity of the population crosses HIT in1 but could not able to cross the HIT max . For intervention-1 For intervention-1, the herd immunity of the population HI i (∞) does not cross the HIT max , so there can be a re-outbreak of the epidemic because of the increment of β. β can be increased due to the negligence of the safety rules by the population or maybe easing the restrictions by the government. We assume that negligence starts at t N . So, there can be two possibilities : (a) β increases to such an amount that HIT (t N ) > HI i (∞), (b) β increases to such an amount that HIT (t N ) < HI i (∞). Figure 4 shows the two possible scenarios of the first case. As, in this case HIT (t N ) > HI i (∞), thus there will always be a chance of re-outbreak of the epidemic. Here we consider two examples and it has been assumed that the negligence starts at t N = 55 and t N = 90 respectively. Also, let β increased from 0.25 to 0.6 in both examples. From figure 4 we can see that the second peak occurs at t p2 = 64.12 and t p2 = 123 for the two consecutive examples. After the second outbreak, we can see that the herd immunity of the population goes beyond the HIT max for both examples. The right picture of row one shows how effective reproduction number R i (t) vary with time. The picture of row two shows us variation of herd immunity HI i (t) and herd immunity threshold HIT (t). Here, we have to remember that the accurate value of HIT max of an epidemic is impossible to find. Because HIT max not only depends on the virus characteristics, it also depends on the social behavior of a region. Also, HIT max can change from country to country and between various mutation states of a virus. Hence, after these outbreaks, there will be no chance of re-outbreak again. For this example, herd immunity reaches HIT max after the second outbreak. But there can be a system where herd immunity reaches HIT max after several outbreaks. Also, it is hard to say when a re-outbreak will start because re-outbreak starts whenever the population will start to neglect the rules or the government will lift the restrictions. Here, a re-outbreak can also occur when the epidemic is almost at the end (I(t) ≈ 0). Of course, this re-outbreak cannot occur due to the internal infected people. However, if somehow a small number of susceptible people become infected (maybe for migration, from the tourists, or by some external carrier) then a re-outbreak can again start. Figure 5 shows an example of this kind. Here we assume that the negligence starts at t N = 150 and the β value increased to 0.65. But at the time t N , I(t N ) ≈ 0 thus no outbreak can possible. Let, at time t f = 175, ten susceptible people are somehow infected. Hence I(t f ) = 10. Because of this small disturbance, we can see that a second outbreak occurs and the peak of it is at t p2 = 248.2. So, we can say, until a population crosses the HIT max there will always be a chance of re-outbreak. The right picture of row one shows how effective reproduction number R i (t) varies with time. The picture of row two shows us variation of herd immunity HI i (t) and herd immunity threshold HIT (t). Figure 6 shows an example of the second case. Now, for this second case we assume that t N is a little bit far from the peak time. At this time herd immunity of the population HI i (t N ) ≈ HI i (∞). As, for the second case, HI i (∞) > HIT (t N ) hence, HI i (t N ) > HIT (t N ). So, there is no possibility of re-outbreak. But if t N is close to t p then there can be a re-outbreak if HI i (t N ) < HIT (t N ) < HI i (∞). Here we assume that t N = 75 and from this time t N , β increased from 0.25 to 0.4. For intervention-2 the herd immunity of the population very quickly crosses HIT max after crossing HIT in . So, there is no chance of re-outbreak. The right picture of row one shows how effective reproduction number R i (t) vary with time. The picture of row two shows us variation of herd immunity HI i (t) and herd immunity threshold HIT (t). Here we have assumed that the negligence time is t N = 55 and due to this negligence β increases to 0.6. Figure 7 shows that there is no increment of active cases, which means no re-outbreak. So, the differences between the two interventions are (1) As intervention-1 is stronger than the intervention-2, thus fewer people are infected for intervention-1 than the intervention-2 during this epidemic. (2) For intervention-2, herd immunity of the population crosses the maximum herd immunity threshold value but for intervention-1, herd immunity of the population does not cross the maximum herd immunity threshold value. So, we can see that there can be various types of interventions, different from strictness. Let, reduced value of the infection rate is represented by β in . If β in is close to the β max then we will say that intervention is a loose one. However, if β in lies a little bit far from β max , then we will say the intervention is a strict one. Now for both no intervention case and loose intervention case, we saw that the herd immunity of the population crosses the maximum herd immunity threshold. But fewer people are infected in the loose intervention case than the no intervention case. So, we can say that if the government and local authorities decide to fight an epidemic by increasing the herd immunity of the population, then loose intervention will be a good option than the no intervention one. Now for the strict intervention case, we saw that number of infections is small compared to the others. But problem is that there will always chances of re-outbreak in the population when strict interventions are lifted or loosen (As it is impossible to maintain strict interventions for a very long time). So, from our model, we can say that, if the goal is to decrease the number of infections then strict intervention will be a good option. Also, if the goal is to increase the herd immunity of the population beyond the maximum herd immunity threshold value then loose intervention will be a good option. However, we have to remember that in real life situation the herd immunity and maximum herd immunity depends on many factors. So, these interventions can have different impacts on the population and might not work well. In this section, we take COVID-19 pandemic data 23 of various countries and fit that data by a piece-wise continuous function. From these fitted functions, we calculate the infection rate (β(t)), removed rate (γ(t)), and herd immunity of the population (HI i (t)). Also, we calculate the herd immunity threshold (HIT (t)) and effective reproduction number (R i (t)) from fitted active cases (I), removed cases (R), and estimated infection rate (β), removed rate (γ). Here data are taken from that date when the recovered cases and deaths both start to increase steadily with time (except for Brazil). There are large discontinuities in Brazil's data at the initial time and around the end (01/11/2020) of the data. Thus we have rejected the initial part of the data. We have retained the end part of the I and R data. However, the end part of the active cases data has not been used for the analysis. Table 1 shows such range of dates for various countries. For a country, the first date of the range is considered as the initial time t=0. 9 show how the active cases and removed cases of various countries are varying with time. In figures 8 and 9, the black dashed lines indicates the 95% prediction interval of the fitted data. We don't show this interval in some plots or some portions of the plot because the intervals are very close to the best fit line. Figure 8 shows that the active case curves have different behaviors for different countries. Active cases curves are increasing for the countries like USA and are decreasing in countries like India, Bangladesh, Colombia. Also, we find that a re-outbreak of the disease has started in Italy, Russia, Brazil and has ended in Peru. Figures 10 and 11 show how the infection rate (β) and removed rate (γ) vary with time in different countries. In these figures, the gray dashed lines represents the discontinuities that come because of the piece-wise continuous fitting of I and R data and not from the model. So, the discontinuities do not have any physical significance. The variation of removed rate is different for different countries. But removed rates are varying in the order of ∼ 10 −2 . Besides, we can roughly say that the removed rates are linearly varying with time. Countries like Russia, Bangladesh, Italy, and Brazil have removed rate which is approximately constants with time. Also for India and Colombia, we can approximate that the removed rate is increasing linearly with time. In the USA removed rate is approximately decreasing linearly with time. In the case of Peru, we will also get a linear region in a particular period of time. The variation of β is also different for different countries. However, the initial variation of β for different countries are approximately the same. We can see that initially, all the βs have an approximately exponential decrease. Here we take an exponentially decreasing function f (t) = ae −b(t−t0) + c. We fit this function to that portion of the β in which it is approximately decreasing exponentially. Table 2 shows the parameter values of the fitted function in different countries. We can see that initially, the fitted function has values of order ∼ 10 −1 for all of the countries. Then the values of this function quickly decreased to an order of ∼ 10 −2 (except Italy). So, we can see that there lies a universal feature among the infection rates (β) for a limited period of time. Figures 12 and 13 show how herd immunity (HI i (t)), herd immunity threshold (HIT (t)), and effective reproduction number (R i (t)) vary with time in different countries. In figures 12 and 13, the gray dashed lines again represent the discontinuities that appear because of the same reason as discussed earlier. We can see that herd immunity of the population is very close to zero and this is true for all of the countries. This happens because the total number of infected cases in a country is very small compared to the country's total population. Now, there are many reasons behind the small number of infected people compared to the country's population. First of all, as we mentioned earlier, countries take various actions and restrictions to reduce the spread of the disease. Besides that, we only work with the reported cases. However, there can be many unreported cases due to the lack of testing or awareness, which are not taken care of in our model. [29] [30] [31] [32] Under-repoting and asymptomatic cases can affect our work and results. This is one of the major drawbacks of our model. The variation of the herd immunity threshold is different between the countries. This happens because of the different variations of both β and γ between the countries. Table 3 shows the maximum values of herd immunity threshold (HIT max ) in different countries. We can see that the values of HIT max lie between 0.9 to 1.0, except for Peru, Colombia and Brazil. Figure 13 shows that, initial value of R i (t) varies so much. R i (t) is not directly calculated from the data. It is estimated from the model. Now, in our model, initial value of R i (t) is approximately related to the HIT max . Also, R i (t) is sensitively depends on the HIT max . So, when HIT max varies from ∼ 0.7 to 0.97 then R i (t) varies from ∼ 3.3 to 33.3. From figures 12 and 13 we can infer: • In USA, HI i (t) < HIT (t) and R i (t) > 1. So, infection is still increasing for these countries. • Whereas in countries like Colombia, Bangladesh, and India HI i (t) > HIT (t) and R i (t) < 1. So, the infection is in a downward trend. • In Italy, Russia, and Brazil, HIT (t) and R i (t) goes below HI i (t) and 1.0 respectively. So, the infection has started to decrease. But after some time HIT (t) and R i (t) goes above HI i (t) and 1.0. Hence the infection starts to rise again and the re-outbreak occurs. A similar thing happens for Peru. In some plots of herd immunity threshold, HIT (t) goes into the negative region. This happens because at that time removed rate (γ) becomes larger than the infection rate (β). We can also say that a negative herd immunity threshold of a population means there is no need for any immune people to prevent the disease. The disease will automatically decrease with time. So from these discussions, we can see that the herd immunity threshold (HIT (t)) is a major player than the herd immunity HI i (t) to decrease the COVID-19 infections. As, governments of the countries, intervene in the situation and imposed restrictions on the population, so, infection rates are reduced. Thus, herd immunity threshold (HIT (t)) is changed and becomes the main quantity which describes the decrease of infections. Later, if this restrictions are eased or lifted then a re-outbreak of the disease can happen (like Italy, Russia, and Brazil). In this section, we will try to analyze the current COVID-19 pandemic situation in India. Also we will try to find the possibility to predict the future behavior of the COVID-19 pandemic by effective reproduction number (R i (t)) and herd immunity threshold (HIT (t)), as has been studied by several authors. [33] [34] [35] Figure 13 shows that, in India, the effective reproduction number initially decreases quickly from a large value. But then it starts to decrease very slowly and becomes less than unity. Also, figure 12 shows that, in India, the herd immunity (HI i (t)) of the population is very low. But herd immunity threshold (HIT (t)) goes below the herd immunity (HI i (t)) of the population and then into the negative region. Thus the infection is decreasing. If, in future the herd immunity threshold (HIT (t)) and the effective reproduction number (R i (t)) remains below than the herd immunity (HI i (t)) and 1.0 respectively, then the infection will decrease continuously. However, the herd immunity of the population is very low thus a reoutbreak can occur at any time in countries like Russia, Brazil, and Italy. We can't speculate the time of the re-outbreak because we don't know how the infection rate (β) and removed rate (γ) will vary with time. So, only in future we will able to get the actual variation of the COVID-19 pandemic in India and the rest of the world. In this paper, we have modified the basic SIR model by considering the infection rate (β) and the removed rate (γ) both as time-dependent parameters. We have redefined the four important variables, basic reproduction number (R 0 ), effective reproduction number (R i (t)), herd immunity (HI i (t)) of the population, and herd immunity threshold (HIT ), according to the modifications that are suggested in the SIR model. Since infection rate and removed rate are both time-dependent parameters, the herd immunity threshold (HIT ) is also time-dependent. The salient points of this study are presented below : • The herd immunity threshold (HIT ) is related to β and γ as, HIT (t) = 1 − γ(t) β(t) . Hence, HIT (t) can be varied with the variation of β and γ. For example, herd immunity threshold (HIT ) can be reduced by decreasing infection rate, β(t) or by increasing removed rate, γ(t). • An epidemic will start decreasing when HI i (t) > HIT (t), which means herd immunity has greater value than the herd immunity threshold. Thus we propose two different ways of reducing the spread of a disease : (i) by increasing herd immunity of the population over the herd immunity threshold, and (ii) by decreasing the herd immunity threshold below the herd immunity of the population. • We fit the COVID-19 data of various countries with piece-wise continuous functions. From these fitted functions, we have estimated the time variation of the infection rate (β) and the removed rate (γ) of the SIR model. Using these rate parameters ( β and γ), we have estimated the variations of the effective reproduction number(R i (t)), the herd immunity of the population (HI i (t)), and the herd immunity threshold (HIT (t)). • For this COVID -19 pandemic, we see that the herd immunity of the various countries', is close to zero. Also, we have found that the initial value of the herd immunity threshold is ∼0.9. However there are many countries, where COVID -19 pandemic is decreasing. This means, the herd immunity threshold went below the herd immunity at a certain time. • We observe that the infection is currently decreasing in the countries like Colombia, Bangladesh, India and had decreased sufficiently like Italy, Russia. and Brazil. In these countries, the herd immunity threshold went below the herd immunity of the population at a certain time. We have also found that the removed rate γ(t) does not change much in these countries. However, infection rate β(t) is greatly decreased from the initial value and thus herd immunity threshold (HIT (t)) is also reduced. We also observe that the infection rate β(t) can be decreased by imposing restrictions, like lockdown, or increasing awareness about the disease. • Hence, the countries are trying to prevent COVID -19 pandemic by the second method that is taking measures so that HIT (t) goes below HI i (t). However, for the same reason the herd immunity of the population also remains low. Thus, a country might face a re-outbreak of the disease very easily. • In countries like Italy, Russia and Brazil, we see that there a re-outbreak of the COVID-19 disease has occurred now. Here, the herd immunity threshold has increased again due to the increment of the infection rate (β) or decrement of the removed rate (γ). We understand that infection rate (β) will increase if people start to neglect restrictions or if the government decides to lift lockdown. Thus imposing restrictions and lockdown is a good way to prevent the spread of the disease. However, there always remains a chance of re-outbreak of the disease and this is the major drawback of this method. • In COVID-19 pandemic we see that the epidemic is prevented mainly by the second method, where herd immunity threshold (HIT ) goes below the herd immunity (HI i (t)) of the population. This makes herd immunity of the population (HI i (t)) insignificant. In addition to these, we have also tried to find some universal features between the variations of these components in different countries. • We have found that initially there is an exponential drop in the infection rates of various countries. Infection rates decrease from a value of the order of ∼ 10 −1 to a value of order ∼ 10 −2 exponentially in a very short period of time. • Also, we have found that the maximum value of herd immunity threshold in different countries has a value between 0.9 to 1.0. • In the end, we have tried to discuss the upcoming COVID-19 behavior in India from the point of view of effective reproduction number and herd immunity threshold. We see that the COVID-19 pandemic is currently decreasing in India. This is because the herd immunity threshold (HIT ) has gone below the herd immunity (HI i (t)) of the population. Since herd immunity (HI i (t)) of the population is close to zero, we predict that with the smallest of fluctuations, there is a possibility of re-outbreak in India. We would like to conclude that COVID-19 pandemic might be a curse on humanity, but we can defeat it together. We will hope that our future will be COVID-19 free very soon. The fitted functions of the active cases and the removed cases for the various countries are given below. USA For USA the fit function for active cases is, The The value of the parameters are, For India the fit function for active cases is, The The value of the parameters are, a11 = 50, b11 = 13, c11 = −2.287, d11 = 0.2378, f 11 = 1.789 × 10 −5 . a21 = 4.724 × 10 5 , b21 = 2.139 × 10 4 , c21 = 262.5, d21 = 5.476, e21 = −0.04073. For Peru the fit function for active cases is, The value of the parameters are, For Russia the fit function for active cases is, The fit function for removed cases is, The value of the parameters are, a11 = 72, b11 = 11.13, d11 = 0.7236, e11 = −0.0298, f 11 = 0.0005763. a21 = 3.494 × 10 4 , b21 = 7121, c21 = 20.55, e21 = −0.003328, f 21 = 1.51 × 10 −5 . For Colombia the fit function for active cases is, For Bangladesh the fit function for active cases is, a + b × t + d × t 3 + e × t 4 0 ≤ t ≤ 94 a1 + b1 × (t − 94) + c1 × (t − 94) 2 + d1 × (t − 94) 3 94 < t ≤ 131 +e1 × (t − 94) 4 The value of the parameters are, a11 = 66, b11 = 31.08, f 11 = 4.687 × 10 −5 , g11 = −3.551 × 10 −7 . a21 = 1.238 × 10 5 , b21 = 2559, c21 = −121, d21 = 4.979, e21 = −0.05306. a21 = 2.376 × 10 5 , b21 = 2819, c21 = −48.14, d21 = 0.8254, e21 = −0.005229. For Italy the fit function for active cases is, The value of the parameters are, a11 = 62, b11 = 5, d11 = 1.051, f 11 = −0.0003997, g11 = 3.796 × 10 −6 . The value of the parameters are, a11 = 1.4 × 10 4 , b11 = 2155, c11 = −54.36, d11 = 2.628. a21 = 1.785 × 10 5 , b21 = 6869, c21 = 452.7, d21 = −1.939. Live update of COVID-19 situation in different countries-Worldometers The Humanitarian Data Exchange, COVID-19 database Modeling Infectious Diseases in Humans and Animals Proceedings of the National Academy of Sciences We would like to thank Dr. Indrani Bose, Dr. Tapati Dutta, and Dr. Sujata Tarafdar for their useful comments and suggestions. We also like to thank the Department of Physics, St. Xavier's College, Kolkata for providing support during this work. Last but not the least, the authors would like to acknowledge the anonymous referee for his/her valuable comments and suggestions.