key: cord-0954043-1f2luunz authors: Pal, D.; Ghosh, D.; Santra, P. K.; Mahapatra, G. S. title: Mathematical Analysis of a COVID-19 Epidemic Model by using Data Driven Epidemiological Parameters of Diseases Spread in India date: 2020-04-29 journal: nan DOI: 10.1101/2020.04.25.20079111 sha: 0f41b7435ff7b4331ca397eb452a2e505cac0873 doc_id: 954043 cord_uid: 1f2luunz This paper attempts to describe the outbreak of Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) or named as novel coronavirus (COVID-19) via an epidemic model. This dangerous virus has dissimilar effects in different countries in the world. It is observable that the number of new active coronavirus cases is increasing day by day across the globe. India is now in the second stage of COVID-19 spreading, and as a densely populated country, it will be an epidemic very quickly if proper protection / strategies are not under-taken based on the database of the transmission of the disease. This paper is using the current data of COVID-19 for the mathematical modeling and its dynamical analysis. As an alternative of the standard SEIR model, we bring in a new representation to appraise and manage the outbreak of infectious disease COVID-19 through SEQIR pandemic model, which is based on the supposition that the infected but undetected by testing individuals are send to quarantine during the incubation period. During the incubation period if any individual be infected by COVID-19, then that confirmed infected individuals are isolated and the necessary treatments are arranged in proper way so that they cannot taint the other residents in the community. Dynamics of the SEQIR model is presented by basic reproduction number R and the comprehensive stability analysis. Numerical results are depicted through apt graphical appearances using the data of five states and India. India. In Maharashtra (Con…rmed cases-335 with deaths-16), Kerala (Con…rmed cases-286 with deaths-02), Tamil Nadu (Con…rmed cases-309 with deaths-01), Delhi (Con…rmed cases-219 with deaths-04), Telangana(Con…rmed cases-158 with deaths-07), Uttar Pradesh (Con…rmed cases-172 with deaths-02), Rajasthan (Con…rmed cases-167 with deaths-0) it grows very rapidly (as of 04-04-2020) [16] . It grows moderately in the states like Assam (Con…rmed cases-16 with deaths-0) , Bihar (Con…rmed cases-29 with deaths-01),Chandigarh (Con…rmed cases-18 with deaths-0), Ladakh (Con…rmed cases-14 with deaths-0) , West Bengal (Con…rmed cases-53 with deaths-03) (as of 03-04-2020) [16] . Where as in the states like Odisha (Con…rmed cases-05 with deaths-0), Puducherry (Con…rmed cases-05 with deaths-0), Mizoram (Con…rmed cases-01 with deaths-0) , Himachal Pradesh (Con…rmed cases-06 with deaths-0 1), Goa (Con…rmed cases-06 with deaths-0) it grows very slowly ) (as of 03-04-2020) [16] . The central government of India and all state governments have taken proper precautionary actions to restrain the spreading of COVID-19 in India from 21 March 2020. A 14-hour intended public curfew was performed in India on 22 March 2020 [17] . Also, the prime minister of India has declared complete lockdown for twenty one days starting from 24 March, 2020 [17] . The Government has take numerous procedures, like maintaining a certain social distance, encouraging social consensus on self-protection such as wearing face mask in public area, quarantining infected individuals, etc. Moreover, till the con…rmed COVID-19 cases in India are growing day by day until the concluding of the current manuscript. Now, not only medical and biological research but also mathematical modeling approach also cooperates and plays a crucial role to stop COVID-19 outburst. By mathematical modeling we may forecast the point of infection and …nishing time of the disease. It also helps us to make proper decision about the necessary steps to restrain the spreading the diseases. In 2019, COVID-19 virus spreads rapidly worldwide. Therefore, it is an alarming situation for becoming a global pandemic [18] due to the severity of this virus. Therefore, real world epidemiological data is necessary for increasing situational consciousness as well as notifying involvements [19] . So far, when a pandemic such as SARS, the 2009 in ‡uenza pandemic or Ebola ( [20] [21] [22] [23] ) outbursted, during the …rst few weeks of that outburst real situation analysis paid attention on the severity, transmissibility, and natural history of an budding pathogen. Again mathematical modeling supported by the dynamical equations ( [24] , [26] ) can a¤ord detailed characteristics of the epidemic dynamics same as statistical methods ( [27] - [28] ). At the early stage of COVID-19 pandemic, many researches have been done in statistical approach as well as in mathematical modeling to guess the main endemic parameters such as reproduction number, serial interval, and doubling time ( [29]- [30] ). Leung et al. [31] calculated number of con…rmed cases transferred from Wuhan to other major cities in China. Wu et al. [32] proposed a SEIR model structure to predict disease spread world wise based on data traced from 31 December 2019 to 28 January 2020. Read et al. [25] studied a SEIR model based on 3 COVID-19. Imai et al. [33] presented a person to person transmission COVID-19 disease and predicted the dimension of the outburst of COVID-19 in Wuhan city, China [33] . Volpert et. al. [34] studied a SIR mathematical model to restrict the increase of coronavirus via initiating …rm quarantine procedures. In this current paper, we develop a COVID-19 epidemic disease model …tted in Indian situation. In this model system we divide Indian population into …ve subpopulations such as susceptible population, Infected but not detected by testing population, quarantined population, Con…rmed infected population who are in under treatment in isolation word, and the population who are lived in secured zone not a¤ected by COVID-19 virus. Then by using dynamical modeling analysis, our aim is to forecast the con…rmed Indian COVID-19 cases in future speci…cally in di¤erent States in India. We also analyze the proposed disease model mathematically to understand transmission dynamics of the COVID-19 virus amongst humans. Several researchers have previously devised mathematical model for spread of infectious diseases ([35]- [38] ). India is also a¤ected by this imported disease and the number of active COVID-19 patients increases day by day right now. Depending on the recent situation, India Government has taken some strategies to stop spreading COVID-19 virus. This section presents a SEQIR model of COVID-19 based on the current situation of the disease in Indian environment. We espouse an alternate that reproduces several key epidemiological properties of COVID-19 virus. The present model structure of COVID-19 describes the dynamics of …ve sub-populations of Indians such as susceptible (S(t)), Infected but not detected by testing population (E(t)), quarantined (Q(t)), Con…rmed infected population who are in under treatment in isolation word (I(t)) as well as population who are lived in secured zone not a¤ected by COVID-19 virus (R(t)). We assume total population size of India is N (t) and N (t) = S(t) + E(t) + Q(t) + I(t) + R(t). In this model, quarantine refers to the separation of infected individuals from the common Indian population when the populace is infected but not infectious. By Infected Indian population, we guess that the Indian individual who have con…rmed infected by the COVID-19 virus. Again by population in secured zone, we presume those Indian individuals who have not a¤ected by corona virus disease. To make our proposed SEQIR model more realistic, we include several demographic e¤ects by supposing a proportional natural death rate d 1 > 0 in each of the …ve Indian sub-populations. Furthermore, we incorporate a net in ‡ow of susceptible Indian individuals into the county (India) at a rate (> 0) per unit time. comprises of new birth of Indian child, immigration and emigration from and in India. The ‡ow diagram of the COVID-19 infection model in present situation of India is depicted through Figure 1 4 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted April 29, 2020 Modeling of susceptible population ( S(t)): By recruiting individuals into the region (India) at a rate , the susceptible population is augmented and condensed by natural death d 1 . Also the susceptible population decreases through interaction between a susceptible individual and infected but not detected by testing individual. This population is also decreased by constant rates 1 and 1 respectively to be converted into quarantined individual, as well as recovered individual. It is a real fact in India especially in the districts (Purulia, Murshidabad, Birbhum (West Bengal), Gaya, Bhagalpur (Bihar), etc.) that the susceptible population is directly sent to secured zone population due to fear e¤ect among the inhabitants. This situation arises due to lack of proper or adequate treatment or testing facility for the large number of population in India. Therefore, the rate of change of susceptible population are governed by the following di¤erential equation: Modeling of Infected but not detected by testing population ( E(t)): The infected but not detected by testing population indicates those individuals who are infected but their infection is not detected due to inadequate testing facility. This population increases at a rate by the interaction between a susceptible individual and infected but not detected by testing individual. This population decreases due to quarantine at a rate 2 and due to natural death rate d 1 . Due to very high population density, it is very di¢ cult for the Indian Government to isolate some infected but not detected by testing individual, and send them 5 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted April 29, 2020. . https://doi.org/10.1101/2020.04. 25.20079111 doi: medRxiv preprint for quarantine period. Keeping this fact in mind, let this population also directly decrease by infected population at a rate r 1 . Therefore, the rate of change of infected but not detected by testing population is governed by the following di¤erential equation: Modeling of quarantine population ( Q(t)): Incubation period for COVID-19 is 2 days to 14 days. This period is very crucial for disease transformation from one individual to another individual. Therefore, we have to isolate those individual from susceptible and infected but not detected by testing individual for 14 days to control spread of COVID-19 in India. This mentioned population is known as quarantined population. Quarantined population is increased at a rate 1 and 2 from susceptible as well as infected but not detected by testing population respectively. This population is decreased at a rate r 2 and 2 due to infected population and population in secured zone correspondingly. Let the natural death rate be d 1 of this population, hence the rate of change of quarantine population is as follows: Modeling of con…rmed infected population ( I(t)): The infected population who have con…rmed positive report by the COVID-19 test, is increased by infected but not detected by testing at a rate r 1 (infected but not detected by testing, such all population are not possible for quarantined due to lack of space or other reasons. Infected but not detected by testing individual may become illness for COVID-19, and subsequently their test report becomes positive. So they enter directly to the infected population, this particular case is very harmful to protect spreading COVID-19 in India), and also increased at rate r 2 from quarantined population as usual. Infected population is decreased at rate 3 ; and d 1 due to recovered population respectively. Let the natural death rate be d 1 and to make it more realistic d 2 is the rate of death for infection, and hence the rate of change of infected population is governed by the following di¤erential equation: Modeling of secured zone population not a¤ ected by COVID-19 ( R(t)): We assume that susceptible, quarantine as well as infected individuals recover from the disease at rates 1 ; 2 and 3 respectively and enter in secured zone population. This population is reduced by a natural death rate d 1 . Thus, rate of change of secured zone population is not a¤ected by COVID-19 virus is governed by the following di¤erential equation: Combining equations ((1)-(5)) our wished-for model structure takes the following form: 6 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted April 29, 2020. . https://doi.org/10.1101/2020.04. 25.20079111 doi: medRxiv preprint with initial densities: All the parameters and corresponding biological meaning are presented in Table 1 given below. Table 1 : Explanation of parameters exploited in our proposed SEQIR model structure The recruitment rate at which new individuals enter in the Indian population The transmission rate from susceptible population to infected but not detected by testing population 1 The transmission coe¢ cient from susceptible population to quarantine population 2 The transmission coe¢ cient from infected but not detected by testing population to quarantine population 1 The transmission rate from susceptible population to secured zone population 2 The transmission coe¢ cient from infected but not detected by testing population to secured zone population 3 The transmission rate from quarantine population to secured zone population The transmission rate from infected but not detected by testing population to infected population for treatment r 2 The transmission rate from quarantine population to infected population for treatment d 2 Death rate of Infected population due to Covid-19 infection Natural death rate of all …ve sub-populations The above SEQIR model formulation (6) can be rewritten as where 7 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 29, 2020. Proof. As the right hand side of SEQIR model structure (7) is completely continuous and locally Lipschitzian on C, the solution (S(t); E(t); I(t); R(t)) of (7) with initial conditions exists, and is unique on From system (7) with initial condition, we have The second equation of system (7), implies dE dt BE(t) Using initial condition we obtain Again from third and fourth equations of the system (7) with the help of initial condition provide Furthermore, last equation of (7) provides dR Therefore, we can see that S(t) > 0; E(t) 0; I(t) 0; R(t) > 0; 8t 0: This concludes the proof of the theorem. Theorem 2 All solutions of the SEQIR model structure (7) that initiate in R 5 + are bounded and enter into a region de…ned by = n (S; E; Q; I; R) 2 R 5 where (S(t); E(t); Q (t) ; I(t); R(t)) SEQIR model structure (7) . Di¤erentiating both sides with respect to t, we have After substitute the values of dS(t) dt , dE(t) dt , dQ(t) dt , dI(t) dt and dR(t) dt from equation (7), we obtain CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 29, 2020. where = min fd 1 ; d 1 + d 2 g. Now by using comparison theorem, we obtain . Therefore, all solution of the model structure (7) enter in the region . Hence, we have completed the required proof. The diseases-free equilibrium (DFE) of the proposed SEQIR model (7) To …nd the BRN of our proposed SEQIR model structure (7), we take the assistance of next-generation matrix method [41] formulation. Assume y = (E; Q; I; R; S) T , then the system (7) can be rewritten as where z(y) = z is known as transmission part which, articulates the production of new infection and is known as transition part, which explain the alter in state. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 29, 2020. . https://doi.org/10.1101/2020.04. 25.20079111 doi: medRxiv preprint The Jacobian matrices of z(y) and (y) at the DFE E 0 is given by Therefore, F V 1 is the next generation matrix of the SEQIR model structure (7). So, as per [41] R 0 = (F V 1 ) where stands for spectral radius of the next-generation matrix F V 1 . Therefore, It is notable that A represents the number of susceptible individual at the DFE E 0 . Based on the each of the parameters of R 0 a sensitivity analysis is performed to check the sensitivity of the basic reproduction number. As indicated by Arriola and Hyman [43] we have computed the normalized forward sensitivity index with respect to each of the parameters. 10 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 29, 2020. . From the above calculations, it is obvious that the BRN R 0 is mainly sensitive to alters in . The value of R 0 will be enhanced if we raise the value of . On the contrary, the value of R 0 will be as well reduced in the same proportion if we diminish the value of . Again it is also noticed that the parameters 1 , 1 , d 1 , r 1 and 2 are related to R 0 inversely. Therefore, for any increasing value of any of this mentioned parameters will de…nitely reduce the value of R 0 . But the decreasing value of R 0 will be comparatively slighter. Since the e¤ect of parameters 1 , 1 , d 1 , r 1 and 2 are very small on R 0 ; it will be sensible to focus e¤orts on the reduction of (transmission rate at which the susceptible individual converted to exposed individual). Therefore, the sensitive analysis of the basic reproduction number emphasized that prevention is better than treatment. That is exertions to enhance prevention are further e¤ective in controlling the spread of COVID-19 disease than to enlarge the numbers of individuals accessing treatment, as yet there is no proper vaccine, which is medically proven this technique is very much e¤ective to control the spread of Coronavirus disease in country like India. Following the same concept, the Indian Government as well as all State Government of India have taken necessary action such as Lockdown, campaign against the disease etc. to protect the out break of this dangerous disease. This section is constructed with an aim to study the stability nature of the Diseases Free Equilibrium (DFE) E 0 with the assistance of subsequent Theorems: Theorem 3 The SEQIR model structure (7) at DFE E 0 ( A ; 0; 0; 0; 0) is locally asymptotically stable under the condition R 0 < 1 and became unstable if R 0 > 1. Proof. The Jacobian of the system (7) at DFE E 0 ( A ; 0; 0; 0; 0) is given by The characteristic equation of the matrix J E0 is given where is a eigenvalue of the matrix J E0 . Therefore, roots of the equation (9) i.e, eigenvalues of the matrix J E0 are 1 = A < 0, 2 = A B, 3 = C < 0, 4 = D < 0 as well as 5 = d 1 < 0. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 29, 2020. . https://doi.org/10.1101/2020.04. 25.20079111 doi: medRxiv preprint Therefore, Therefore the system (7) is locally asymptotically at the DFE E 0 ( A ; 0; 0; 0; 0) if 2 < 0. Now Hence, DEF E 0 ( A ; 0; 0; 0; 0) is locally asymptotically stable under the condition R 0 < 1. Theorem 4 The SEQIR model structure (7) at DFE E 0 ( A ; 0; 0; 0; 0) is globally asymptotically stable (GAS) under the condition R 0 < 1 and became unstable if R 0 > 1. Proof. We can rewrite the system of di¤erential equation (7) in the following form given below Obviously, b G(X; V ) 0 every time the state variables are within . It is also obvious that X = ( A ; 0) is a globally asymptotically stable equilibrium of the system dX dt = F (X; 0). Hence, the theorem is proved. This section re ‡ects the existence of endemic equilibrium as well as demonstrates its stability nature. To stumble on the endemic equilibrium E 1 (S ; E ; Q ; I ; R ) of our wished-for SEQIR structure (7) we deem the following: 12 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 29, 2020. . https://doi.org/10.1101/2020.04. 25.20079111 doi: medRxiv preprint as well as Solving the system of equation (10) we obtain Therefore, E has unique positive value if R 0 1 > 0 i.e., R 0 > 1. Shortening the above discussions and discussions in the previous sections we have reached the following theorem: Theorem 5 The SEQIR model structure (7) has a unique DFE E 0 ( A ; 0; 0; 0; 0) for all parameter values as well as the system (7) has also a unique endemic equilibrium E 1 (S ; E ; Q ; I ; R ) under the condition The next theorem establish the local stability nature of the endemic equilibrium E 1 (S ; E ; Q ; I ; R ) We assume = , as the bifurcation parameter, predominantly as it has been explained in (8) that R 0 is extra sensitive to alter in than the other parameters. If it is considered that R 0 = 1, then we have Now, the Jacobian of the linearized system (11) via (12) at DFE E 0 when = is as follows: J ( ) = 13 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 29, 2020. . https://doi.org/10.1101/2020.04. 25.20079111 doi: medRxiv preprint Eigenvalues of the matrix (13) are given by (0; A; C; D; d 1 ) T . We observe that the matrix (13) has simple zero eigenvalue, and the other eigenvalues are negative. We are now at the stage to apply center manifold theory [45] to analyze the dynamics of system (11) . Corresponding to zero eigenvalue the right eigenvector ! = (! 1 ; ! 2 ; ! 3 ; ! 4 ; ! 5 ) T of the matrix (13) is given by with v 2 free. Therefore, we have Substituting the values of all the second-order derivatives calculated at DFE as well as = we obtain, a = 2 B 2 < 0 and b = A > 0 As a < 0 and b > 0 at = , hence, from Remark 1 of the Theorem 4.1 as declared in [44] , a transcritical bifurcation takes place at R 0 = 1 as well as the inimitable endemic equilibrium is locally asymptotically stable due to R 0 > 1. Hence, we conclude the proof. To …t our proposed COVID- 19 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 29, 2020. Initial densities as of 21th March 2020 is given in Table-3 Table 3 : Initial densities of the model system (2.6) for India In our proposed model system (6) most sensitive parameter is (transmission rate from susceptible population to infected but not detected by testing population). Therefore, we mainly aspire to see the e¤ect of on COVID-19 disease spreading. For di¤erent values of , the value of R 0 is presented in Table 4 15 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 29, 2020. From the above …gure we observe that as the values of increases R 0 . It is also observed that after certain value of , R 0 becomes greater than 1. Therefore, up to certain value of DFE is stable (Theorem 3), and beyond that value of DFE becomes unstable. In this current situation of the universe, we are interested about the infected population I(t) as days progress. Therefore, drawing the time series plot of infected population taking initial densities as given in Table 3 . Table 2 and Table 3 respectively. The …gure 3 is very interesting because we see that for = 2: . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 29, 2020. Table 2 and Table 3 respectively Table 2 and Table 3 respectively for one month period 17 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 29, 2020. Table 2 and Table 3 respectively for two months period Therefore, implementing Government restrictions mathematically, Fig 5 and Since, Government is changing its restricting measures after certain period of time policies, so the system parameters are also altering after that period of time. Therefore, in the current position we are not interested about infected population for long time of period. Now, we simulate the daily new con…rmed COVID-19 cases for the …ve states of India namely Maharashtra, Kerala, Uttar Pradesh, Delhi and West Bengal starting from 21 March, 2020. We try to …t the model (6) to day by day novel con…rmed COVID-19 cases in the …ve states of India. Here we also taken (transmission rate from susceptible population to infected but not detected by testing population) as before. Estimated initial densities as of 21 March, 2020 are given in Table 5 . Estimated Recruitment rates and infection tempted mortality rates for the …ve states of India are given in Table 6 . Lastly, rests of the estimated parameters are given in Table 7 . 18 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 29, 2020. . https://doi.org/10.1101/2020.04.25.20079111 doi: medRxiv preprint Using these estimated parameters, the …xed parameters and initial densities ( Table 5 to Table 7) we graphically present infected population for di¤erent states. Figure 7 and Figure 8 present the graphical presentation of infected population for the state Maharashtra Table 5 Table 6 and Table 7 for one month period from 22 nd March, 2020. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 29, 2020. Table 5 Table 6 and Table 7 for two months period from 22nd March, 2020 From Figure 7 , we monitor that as per our imposed restricting measures as given by Government of India, total number of infected individuals in Maharashtra reaches to 4100 in thirty days from 21 March, 2020. We also detect from Figure 7 that the actual number of infected persons also coincides with the infected COVID-19 individuals of our proposed system for 13 days starting from 21 March, 2020. Again Figure 8 infers that total number of infected population reaches to 260000 in two months. But in reality this situation may not happen as Government is continuously changing its restricting measures to stop the spreading of COVID-19. Now to illustrate the COVID-19 situation in Kerala based on values of parameters as given in Table 5 to Table 7 we draw the following two …gures 20 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 29, 2020. Table 5 Table 6 and Table 7 for one month period from 22nd March, 2020. Table 5 Table 6 and Table 7 for two months period from 22nd March, 2020 The Figure 9 shows that in 30 days the infected population of Kerala reached to 2100. Also we have observed real infected individual for 13 days overlap with our infected population curve. Figure 10 shows Table 5 Table 6 and Table 7 for one month period from 22nd March, 2020. Table 5 Table 6 and Table 7 for two months period from 22nd March, 2020 Figure 11 and Figure 12 depict that as per our construction infected population in 30 days and 60 days are 4400 and 620000 respectively starting from 21 March, 2020. Similarly, for the state Delhi we draw the following two …gures 22 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 29, 2020. Table 5 Table 6 and Table 7 respectively for one month period from 21 March, 2020. Table 5 Table 6 and Table 7 for two month period from 21 March, 2020. Table 5 Table 6 and Table 7 respectively for one month period from 21 March, 2020. Table 5 Table 6 and Table 7 for two months period from 21 March, 2020. As before Figure 15 and Figure 16 infer that as per our construction infected population in West Bengal during 30 days and 60 days are 1100 and 210000 respectively starting from 21 March, 2020. In this current paper we have formulated and studied an epidemic model of COVID-19 virus which is transferred from human to human. So far the daily con…rmed COVID-19 cases are increasing day by day 24 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 29, 2020. . https://doi.org/10.1101/2020.04. 25.20079111 doi: medRxiv preprint worldwide. Therefore, prediction about infected individual is very much important for health concern arrangement of the citizens. It is also important to control spread rate of the COVID-19 virus with restricted supply. Our mathematical study is based on COVID-19 virus spread in India. We try to …t our model system to COVID-19 disease in India as per as limited data are available. The basic reproduction number R 0 is calculated of our anticipated model. It is observed that when R 0 < 1 then DFE E 0 is globally asymptotically stable. Again, from sensitivity analysis of R 0 we observe that the most sensitive parameter of our model structure is (transmission rate from susceptible population to infected but not detected by testing population). Also the endemic equilibrium E 1 exist and stable if R 0 > 1. In this study our main aim is to predict mathematically the number of infected individual due COVID-19 virus in India. To ful…ll our aim, we perform a detail numerical simulation of our proposed model in Section 6. We …rst simulate the COVID-19 situation in India, and thereafter situation of …ve di¤erent states (Maharashtra, Kerala, Uttar Pradesh, Delhi and West Bengal) are predicted numerically through graphical approach using Matlab software. We estimated the values of the parameter and the initial condition taken as per the information accessible on 21 March 2020. We see that is most sensitive parameter of our model system. So for di¤erent values of infected population curve is presented in Figure 3 . We observe that for = 2:5 10 10 , the infected population curve is best …tted since it coincides with the actual infected individuals for 13 days starting from 21 March, 2020. Long term behavior of the infection is observed in …gure 4. This …gure shows us that as per the model construction as well as data available, the peak of infection is reached about 120 days after 21 march, 2020. Therefore, as per our prediction, the estimated date to reach the peak of infection is 21 July 2020 if the same restriction of measures is followed by the Indian Government as declared on 21 March 2020. Short term prediction about infected population is given via Figure 5 and Figure 6 respectively. Accordingly two of these …gures predicted number of infected individual in India of COVID-19 patients in 30 and 60 days are 32000 and 3000000 respectively starting from 21 March 2020. Now, we look upon the situation of di¤erent states. Figures 7 and 8 present that the number of infected population reaches to 4100 and 260000 (for = 2:5 10 9 ) in next 30 and 60 days respectively starting from 21 March, 2020. For Kerala this number goes to 2100 and 50000 (for = 0:0000000164) respectively (see Figure 9 and Figure 10 ). In Uttar Pradesh, as per Figure 11 and Figure 12 (for = 0:000000020) predicted numbers of infected individuals are 4400 and 620000. Total numbers of 4600 and 400000 people are infected in 30 and 60 days respectively in Delhi (look at Figure 13 and Figure 14) . Finally, in West Bengal 1100 and 210000 people may be infected due to COVID -19 virus in 30 and 60 days respectively (see Figure 15 and Figure 16 ). Therefore, analyzing all results of our proposed model and observing the situation of di¤erent countries, . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted April 29, 2020. . https://doi.org/10.1101/2020.04. 25.20079111 doi: medRxiv preprint we may conclude that India may be in a big trouble in very near future due to COVID-19 virus. To avoid this big trouble, Indian Government should take stricter measures other than quarantine, lock down etc. So far the Indian Government are continuously changing its policy to protect India from COVID-19 virus. Recently, all districts of India are classi…ed by Indian Government into three zones namely Red zones (hotspots), Orange zones (non hotspots) and Green zones (save zones). Preliminarily 170 districts of India are hotspots zones, where rapid testing facility is available for the public. As the time progress more strategies are applied by the Government of India as well as all state Governments to stop the spread of COVID-19 virus in India. Therefore, we may assume that if the Indian Government take proper step time to time then the infected number of population will be di¤er from our predicted number as time progress, and India will recover from this virus in recent future. Lastly, we say that public of India should help the Indian Government to …ght against this dangerous COVID-19 as per the proposed model. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted April 29, 2020. . https://doi.org/10.1101/2020.04. 25.20079111 doi: medRxiv preprint Clinical features of patients infected with 2019 novel coronavirus in wuhan, china. 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