doi: 10.3934/cpaa.2021170
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Basic reproduction number estimation and forecasting of COVID-19: A case study of India, Brazil and Peru

1. 

School of Basic Sciences, Indian Institute of Technology Mandi, Mandi, Himachal Pradesh 175001, India

2. 

Department of Mathematics, DIT University, Dehradun, Uttarakhand, 248009, India

3. 

Department of Mathematics, Iowa State University, IA 50011, USA

* Corresponding author

Received  December 2020 Revised  August 2021 Early access October 2021

Fund Project: The research of the corresponding author (Nitu Kumari) is funded by Science and Engineering Research Board (SERB), under three separate grants with grant numbers MSC/2020/000369, MTR/2018/000727 and EMR/2017/005203

Since the start of COVID-19 pandemic, the definition of normal life has changed drastically. The number of cases of this pandemic is rising everyday across the globe. In this study, we propose a compartmental model, which considers the isolation factor of Coronavirus infected individuals. The model consists of five compartments: susceptible (S), exposed (E), Infected (I), Isolated (L) and recovered (R). We have estimated the parameters of the model system and the expression of the basic reproduction number $ R_0 $ using real data set. The exact value of the basic reproduction number is computed for India, Brazil and Peru. The local and global stability analysis of disease-free equilibrium and endemic equilibrium points is carried out. The forecasting of the pandemic is done using real data. It has been observed that to understand the pandemic the time frame has to be divided into small intervals as the parameters of the pandemic are changing with time. Within a time frame of approximately four months (i.e. from July to October 2020), the transmission rate of India has been reduced by approximately 84%. Whereas the transmission rate in Brazil and Peru has increased by 79% and 45% respectively. The sensitivity of various parameters involved in the model has been analyzed. We have presented a complete analysis to check the existence of backward bifurcation.

Citation: Nitu Kumari, Sumit Kumar, Sandeep Sharma, Fateh Singh, Rana Parshad. Basic reproduction number estimation and forecasting of COVID-19: A case study of India, Brazil and Peru. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021170
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C. Hou, J. Chen, Y. Zhou, L. Hua and J. Yuan et. al, The effectiveness of quarantine of Wuhan city against the Corona Virus Disease 2019 (COVID-19): A well-mixed SEIR model analysis, J. med. virol., 92 (2020), 841-848. Google Scholar

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B. IvorraM. R. FerrándezM. Vela-Pérez and A. M. Ramos, Mathematical modeling of the spread of the coronavirus disease 2019 (COVID-19) taking into account the undetected infections. The case of China, Commun. nonlinear sci. numer. simul., 88 (2020), 105303.  doi: 10.1016/j.cnsns.2020.105303.  Google Scholar

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S. A. Lauer, K. H. Grantz, Q. Bi and F. K. Jones et. al, The incubation period of coronavirus disease 2019 (COVID-19) from publicly reported confirmed cases: estimation and application, Ann. intern. med., 172 (2020), 577-582. Google Scholar

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T. LiuD. GongJ. XiaoJ. HuG. HeZ. Rong and W. Ma, Cluster infections play important roles in the rapid evolution of COVID-19 transmission: a systematic review, Int. J. Infect. Diseases, 139 (2020), 374-380.  doi: 10.1016/j.ijid.2020.07.073.  Google Scholar

[26]

S. MarimuthuM. JoyB. MalavikaA. NadarajE. S. Asirvatham and L. Jeyaseelan, Modelling of reproduction number for COVID-19 in India and high incidence states, Clinical Epidemiology and Global Health, 9 (2021), 57-61.   Google Scholar

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C. V. Munayco, A. Tariq and R. Rothenberg et. al, Early transmission dynamics of COVID-19 in a southern hemisphere setting: Lima-Peru: February 29th–March 30th, 2020, Infect. Disease Model., 5 (2020), 338–345. Google Scholar

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F. NdaïrouI. AreaJ. J. Nieto and D. F. M. Torres, Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan, Chaos, Solitons & Fractals, 135 (2020), 109846.  doi: 10.1016/j.chaos.2020.109846.  Google Scholar

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A. J. Rodriguez-Morales and V. Gallego et. al, COVID-19 in Latin America: The implications of the first confirmed case in Brazil, Travel med. infect. disease, 35 (2020), 101613. Google Scholar

[30]

P. SamuiJ. Mondal and S. Khajanchi, A mathematical model for COVID-19 transmission dynamics with a case study of India, Chaos, Solitons & Fractals, 140 (2020), 110173.  doi: 10.1016/j.chaos.2020.110173.  Google Scholar

[31]

K. SarkarS. Khajanchi and J. J. Nieto, Modeling and forecasting the COVID-19 pandemic in India, Chaos, Solitons & Fractals, 139 (2020), 110049.  doi: 10.1016/j.chaos.2020.110049.  Google Scholar

[32]

T. SardarS. S. NadimS. Rana and J Chattopadhyay, Healthcare Assessment of lockdown effect in some states and overall India: A predictive mathematical study on COVID-19 outbreak, Chaos, Solitons & Fractals, 139 (2020), 110078.  doi: 10.1016/j.chaos.2020.110078.  Google Scholar

[33]

S. Sharma and n. Kumari, Why to consider environmental pollution in cholera modeling?, Math. Methods Appl. Sci., 40 (2017), 6348-6370.  doi: 10.1002/mma.4461.  Google Scholar

[34]

P. Van den Driessche and J. Watmough, Further notes on the basic reproduction number, in Mathematical Epidemiology, Springer, Berlin, Heidelberg, 2008,159–178. doi: 10.1007/978-3-540-78911-6_6.  Google Scholar

[35]

Weekly operational update on covid-19, 9 september 2020, 2020. Google Scholar

[36]

WHO Director-General's remarks, 2020. Google Scholar

[37]

S. Zhao, Q. Lin, J. Ran, S. S. Musa and G. Yang et. al, Preliminary estimation of the basic reproduction number of novel coronavirus (2019-nCoV) in China, from 2019 to 2020: A datadriven analysis in the early phase of the outbreak, Int. j. infect. dis., 92 (2020), 214-217. Google Scholar

show all references

References:
[1]

S. AhmadS. OwyedA. H. Abdel-AtyE. E. MahmoudK. Shah and h. Alrabaiah, Mathematical analysis of COVID-19 via new mathematical model, Chaos, Solitons & Fractals, 143 (2021), 110585.  doi: 10.1016/j.chaos.2020.110585.  Google Scholar

[2]

R. M. AndersonH. HeesterbeekD. Klinkenberg and T. d. Hollingworth, How will country-based mitigation measures influence the course of the COVID-19 epidemic?, The lancet, 395 (2020), 931-934.   Google Scholar

[3]

F. Braur, Compartmental models in epidemiology, Math. epid., 5 (2008), 19-79.  doi: 10.1007/978-3-540-78911-6_2.  Google Scholar

[4]

D. D. S. Candido, A. Watts and L Abade et. al, Routes for COVID-19 importation in Brazil, J. Travel Med., 27 (2020), taaa042. Google Scholar

[5]

C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), 361-404.  doi: 10.3934/mbe.2004.1.361.  Google Scholar

[6]

C. Castillo-Chavez, S. Blower, P. Van den Driessche, D. Kirschner and A. Yakuba, Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory, Springer, New York, 2002. doi: 10.1007/978-1-4613-0065-6.  Google Scholar

[7]

K. ChatterjeeK. ChatterjeeA. Kumar and S. Shankar, Healthcare impact of COVID-19 epidemic in India: A stochastic mathematical model, Med. J. Arm. Force. India, 76 (2020), 147-155.   Google Scholar

[8]

N. ChitnisJ. M. Hyman and J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bullet. math. biol., 70 (2008), 1272-1296.  doi: 10.1007/s11538-008-9299-0.  Google Scholar

[9]

Covid-19 india: New cases vs cured, 2020, 2020-09-30. Google Scholar

[10]

COVID-19 Pandemic Data/Peru Medical Cases, July 2021, Available from: https://en.wikipedia.org/wiki/Template:COVID-19_pandemic_data/Peru_medical_cases. Google Scholar

[11]

Coronavirus Cases, July 2021, Available from: https://www.worldometers.info/coronavirus/#countries. Google Scholar

[12]

W. M. de Souza, L. F. Buss, D. da Silva Candido et. al, Epidemiological and clinical characteristics of the early phase of the COVID-19 epidemic in Brazil, medRxiv. Google Scholar

[13]

O. DiekmannJ. A. P. Heesterbeek and M. G. Roberts, The construction of next-generation matrices for compartmental epidemic models, J. Roy. Soc. Inter., 7 (2010), 873-885.   Google Scholar

[14]

A. DinY. LiT. Khan and G. Zaman, Mathematical analysis of spread and control of the novel corona virus (COVID-19) in China, Chaos, Solitons & Fractals, 141 (2020), 110286.  doi: 10.1016/j.chaos.2020.110286.  Google Scholar

[15]

C. FraserS. Riley and R. M. Anderson, Factors that make an infectious disease outbreak controllable, Proceed. Nation. Acad. Sci., 101 (2004), 6146-6151.   Google Scholar

[16]

T. GanyaniC. KremerD. ChenA. TorneriC. FaesJ. Wallinga and N. Hens, Estimating the generation interval for coronavirus disease (COVID-19) based on symptom onset data, March 2020, Eurosurveillance, 25 (2020), 2000257.   Google Scholar

[17]

M. GattoE. BertuzzoL. MariS. MiccoliL. Carraror. Casarandi and A. Rinaldo, Spread and dynamics of the COVID-19 epidemic in Italy: Effects of emergency containment measures, Proceed. Nation. Acad. Sci., 17 (2020), 10484-10491.   Google Scholar

[18]

S. HeY. Peng and K. Sun, SEIR modeling of the COVID-19 and its dynamics, Nonlinear dynam., 101 (2020), 1667-1680.   Google Scholar

[19]

J. Hellewell, S. Abbott, A. Gimma, N. I. Bosse, C. I. Jarvis and T. W. Russell et. al, Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts, The Lancet Global Health, 8 (2020), e488–e496. Google Scholar

[20]

C. Hou, J. Chen, Y. Zhou, L. Hua and J. Yuan et. al, The effectiveness of quarantine of Wuhan city against the Corona Virus Disease 2019 (COVID-19): A well-mixed SEIR model analysis, J. med. virol., 92 (2020), 841-848. Google Scholar

[21]

C. Huang, Y. Wang, X. Li, L. Ren, J. Zhao, Y. Hu, L. Zhang, G. Fan and J. Xu et. al, Clinical features of patients infected with 2019 novel coronavirus in Wuhan, China, The lancet, 395 (2020), 497–506. doi: 10.3934/mbe.2020148.  Google Scholar

[22]

B. IvorraM. R. FerrándezM. Vela-Pérez and A. M. Ramos, Mathematical modeling of the spread of the coronavirus disease 2019 (COVID-19) taking into account the undetected infections. The case of China, Commun. nonlinear sci. numer. simul., 88 (2020), 105303.  doi: 10.1016/j.cnsns.2020.105303.  Google Scholar

[23]

N. Kumari and S. Sharma, Modeling the dynamics of infectious disease under the influence of environmental pollution, Int. J. Appl. Comput. Math., 4 (2018), 1-24.  doi: 10.1007/s40819-018-0514-x.  Google Scholar

[24]

S. A. Lauer, K. H. Grantz, Q. Bi and F. K. Jones et. al, The incubation period of coronavirus disease 2019 (COVID-19) from publicly reported confirmed cases: estimation and application, Ann. intern. med., 172 (2020), 577-582. Google Scholar

[25]

T. LiuD. GongJ. XiaoJ. HuG. HeZ. Rong and W. Ma, Cluster infections play important roles in the rapid evolution of COVID-19 transmission: a systematic review, Int. J. Infect. Diseases, 139 (2020), 374-380.  doi: 10.1016/j.ijid.2020.07.073.  Google Scholar

[26]

S. MarimuthuM. JoyB. MalavikaA. NadarajE. S. Asirvatham and L. Jeyaseelan, Modelling of reproduction number for COVID-19 in India and high incidence states, Clinical Epidemiology and Global Health, 9 (2021), 57-61.   Google Scholar

[27]

C. V. Munayco, A. Tariq and R. Rothenberg et. al, Early transmission dynamics of COVID-19 in a southern hemisphere setting: Lima-Peru: February 29th–March 30th, 2020, Infect. Disease Model., 5 (2020), 338–345. Google Scholar

[28]

F. NdaïrouI. AreaJ. J. Nieto and D. F. M. Torres, Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan, Chaos, Solitons & Fractals, 135 (2020), 109846.  doi: 10.1016/j.chaos.2020.109846.  Google Scholar

[29]

A. J. Rodriguez-Morales and V. Gallego et. al, COVID-19 in Latin America: The implications of the first confirmed case in Brazil, Travel med. infect. disease, 35 (2020), 101613. Google Scholar

[30]

P. SamuiJ. Mondal and S. Khajanchi, A mathematical model for COVID-19 transmission dynamics with a case study of India, Chaos, Solitons & Fractals, 140 (2020), 110173.  doi: 10.1016/j.chaos.2020.110173.  Google Scholar

[31]

K. SarkarS. Khajanchi and J. J. Nieto, Modeling and forecasting the COVID-19 pandemic in India, Chaos, Solitons & Fractals, 139 (2020), 110049.  doi: 10.1016/j.chaos.2020.110049.  Google Scholar

[32]

T. SardarS. S. NadimS. Rana and J Chattopadhyay, Healthcare Assessment of lockdown effect in some states and overall India: A predictive mathematical study on COVID-19 outbreak, Chaos, Solitons & Fractals, 139 (2020), 110078.  doi: 10.1016/j.chaos.2020.110078.  Google Scholar

[33]

S. Sharma and n. Kumari, Why to consider environmental pollution in cholera modeling?, Math. Methods Appl. Sci., 40 (2017), 6348-6370.  doi: 10.1002/mma.4461.  Google Scholar

[34]

P. Van den Driessche and J. Watmough, Further notes on the basic reproduction number, in Mathematical Epidemiology, Springer, Berlin, Heidelberg, 2008,159–178. doi: 10.1007/978-3-540-78911-6_6.  Google Scholar

[35]

Weekly operational update on covid-19, 9 september 2020, 2020. Google Scholar

[36]

WHO Director-General's remarks, 2020. Google Scholar

[37]

S. Zhao, Q. Lin, J. Ran, S. S. Musa and G. Yang et. al, Preliminary estimation of the basic reproduction number of novel coronavirus (2019-nCoV) in China, from 2019 to 2020: A datadriven analysis in the early phase of the outbreak, Int. j. infect. dis., 92 (2020), 214-217. Google Scholar

Figure 1.  Flow chart for the proposed model
Figure 2.  Model fitting for the cumulative number of COVID-19 cases for India from $ 11^{th} $ July 2020 to $ 15^{th} $ July 2021
Figure 3.  Model fitting for the cumulative number of COVID-19 cases for Brazil from $ 11^{th} $ July 2020 to $ 15^{th} $ July 2021
Figure 4.  Model fitting for the cumulative number of COVID-19 cases for Peru from $ 11^{th} $ July 2020 to $ 15^{th} $ July 2021
Figure 5.  Model fitting for the active number of COVID-19 cases for India from $ 11^{th} $ July 2020 to $ 30^{th} $ June 2021
Figure 6.  Model fitting for the active number of COVID-19 cases for Brazil from $ 11^{th} $ July 2020 to $ 30^{th} $ June 2021
Figure 7.  Model fitting for the active number of COVID-19 cases for Peru from $ 11^{th} $ July 2020 to $ 30^{th} $ June 2021
Figure 8.  Short-term forecasting of the pandemic for India, Peru and Brazil
Figure 9.  Surface plot showing simultaneous effects of $ \lambda $ and $ \beta $ on the basic reproduction number $ R_0 $ for India, Brazil and Peru
Table 1.  Parameters involved in the model
Parameter Meaning
$ A $ Constant recruitment rate
$ \beta $ Disease transmission rate from infected individuals to susceptible population
$ \lambda $ Parameter used to reduce contact rate of isolated individuals
$ \mu $ Natural death rate
$ \alpha $ Rate at which the exposed individuals join the infected class
$ \alpha_1 $ Rate at which exposed individuals provided the treatment or admitted to hospitals
$ \alpha_2 $ Recovery rate of exposed individuals join the recovered class
$ \theta $ Disease induced death rate
$ \gamma $ Rate at which infected individuals provided the treatment or admitted to hospitals
$ \gamma_1 $ Rate or recovery of Infected individuals
$ \delta $ Rate at which isolated individuals join the recovered class
Parameter Meaning
$ A $ Constant recruitment rate
$ \beta $ Disease transmission rate from infected individuals to susceptible population
$ \lambda $ Parameter used to reduce contact rate of isolated individuals
$ \mu $ Natural death rate
$ \alpha $ Rate at which the exposed individuals join the infected class
$ \alpha_1 $ Rate at which exposed individuals provided the treatment or admitted to hospitals
$ \alpha_2 $ Recovery rate of exposed individuals join the recovered class
$ \theta $ Disease induced death rate
$ \gamma $ Rate at which infected individuals provided the treatment or admitted to hospitals
$ \gamma_1 $ Rate or recovery of Infected individuals
$ \delta $ Rate at which isolated individuals join the recovered class
Table 2.  Estimated parameter values of the model for Phase-1
Parameter India Brazil Peru
$ \beta $ 0.6453 0.67743 0.3627
$ \lambda $ 0.84929 0.20308 0.89999
$ \alpha_1 $ 0.21455 0.89942 0.13065
$ \alpha_2 $ 0.10222 0.017178 0.010001
$ \gamma $ 0.23304 0.059289 0.37737
$ \delta $ 0.37832 0.62475 0.345
$ \alpha $ 0.18673 0.11303 0.113
$ \theta $ 0.020635 0.003614 0.001001
$ \gamma_1 $ 0.10105 0.012809 0.010012
Parameter India Brazil Peru
$ \beta $ 0.6453 0.67743 0.3627
$ \lambda $ 0.84929 0.20308 0.89999
$ \alpha_1 $ 0.21455 0.89942 0.13065
$ \alpha_2 $ 0.10222 0.017178 0.010001
$ \gamma $ 0.23304 0.059289 0.37737
$ \delta $ 0.37832 0.62475 0.345
$ \alpha $ 0.18673 0.11303 0.113
$ \theta $ 0.020635 0.003614 0.001001
$ \gamma_1 $ 0.10105 0.012809 0.010012
Table 3.  Estimated parameter values of the model for Phase-2
Parameter India Brazil Peru
$ \beta $ 0.1 1.2193 0.52871
$ \lambda $ 10.8843 8.5936 0.9
$ \alpha_1 $ 0.027703 0.83587 1.8109
$ \alpha_2 $ 0.010006 1.535 0.62839
$ \gamma $ 0.010002 0.010022 0.01
$ \delta $ 0.79152 1.1143 0.53075
$ \alpha $ 0.113 0.69971 0.77369
$ \theta $ 0.00979 0.039429 0.001059
$ \gamma_1 $ 0.067068 0.039667 0.22347
Parameter India Brazil Peru
$ \beta $ 0.1 1.2193 0.52871
$ \lambda $ 10.8843 8.5936 0.9
$ \alpha_1 $ 0.027703 0.83587 1.8109
$ \alpha_2 $ 0.010006 1.535 0.62839
$ \gamma $ 0.010002 0.010022 0.01
$ \delta $ 0.79152 1.1143 0.53075
$ \alpha $ 0.113 0.69971 0.77369
$ \theta $ 0.00979 0.039429 0.001059
$ \gamma_1 $ 0.067068 0.039667 0.22347
Table 4.  Estimated parameter values of the model for Phase-3
Parameter India Brazil Peru
$ \beta $ 0.15682 2.1393 1.0947
$ \lambda $ 5.4021 10.8995 0.38317
$ \alpha_1 $ 0.37634 0.020001 0.091451
$ \alpha_2 $ 0.11997 1.1891 0.112276
$ \gamma $ 0.010782 0.13319 0.31172
$ \delta $ 0.34508 0.34502 0.45806
$ \alpha $ 0.11305 0.113 0.17682
$ \theta $ 0.0010774 0.0010022 0.083306
$ \gamma_1 $ 0.4946 0.010003 0.30467
Parameter India Brazil Peru
$ \beta $ 0.15682 2.1393 1.0947
$ \lambda $ 5.4021 10.8995 0.38317
$ \alpha_1 $ 0.37634 0.020001 0.091451
$ \alpha_2 $ 0.11997 1.1891 0.112276
$ \gamma $ 0.010782 0.13319 0.31172
$ \delta $ 0.34508 0.34502 0.45806
$ \alpha $ 0.11305 0.113 0.17682
$ \theta $ 0.0010774 0.0010022 0.083306
$ \gamma_1 $ 0.4946 0.010003 0.30467
Table 5.  Estimated values of $ R_0 $ for India, Brazil and Peru
Sr. No. Country $ R_0 $(Phase-1) $ R_0 $(Phase-2) $ R_0 $(Phase-3)
1. India 2.19036 1.02192 1.77117
2. Brazil 0.866587 3.09619 56.3508
3. Peru 3.96803 0.67683 3.20376
Sr. No. Country $ R_0 $(Phase-1) $ R_0 $(Phase-2) $ R_0 $(Phase-3)
1. India 2.19036 1.02192 1.77117
2. Brazil 0.866587 3.09619 56.3508
3. Peru 3.96803 0.67683 3.20376
Table 6.  Sensitivity expressions of various parameters involved in the model
Parameter Expression of the sensitivity index
$ \beta $ 1
$ \lambda $ $ \frac{\lambda [\alpha \gamma + \alpha_1(\theta+\gamma+\gamma_1+\mu)]}{(\delta+\theta+\mu)(\theta+\gamma+\gamma_1+\mu)+\lambda[\alpha\gamma+\alpha_1(\theta+\gamma+\gamma_1+\mu)} $
$ \alpha $ $ \frac{\alpha \lambda\gamma}{(\delta+\theta+\mu)(\theta+\gamma+\gamma_1+\mu)+\lambda\alpha\gamma+\lambda\alpha_1(\theta+\gamma+\gamma_1+\mu)} $
$ \alpha_1 $ $ \frac{\alpha_1 \times (\theta+\gamma+\gamma_1+\mu)\big( (\alpha+\alpha_1+\alpha-2+\mu)\lambda -(\delta+\theta+\mu)-\lambda \big)-\lambda\alpha\gamma}{[(\delta+\theta+\mu)(\theta+\gamma+\gamma_1+\mu)+\lambda\alpha\gamma+\lambda\alpha_1(\theta+\gamma+\gamma_1+\mu)] (\alpha+\alpha_1+\alpha_2+\mu)} $
$ \alpha_2 $ $ - \frac{\alpha_2}{(\alpha+\alpha_1+\alpha_2+\mu)^2} $
$ \theta $ $ - \frac{\theta \big[\lambda \alpha_1 (\theta+\gamma+\gamma_1+\mu)^2 +\lambda \alpha \gamma ( 2\theta +\gamma+\gamma_1+\delta+2\mu) \big]}{[(\delta+\theta+\mu)(\theta+\gamma+\gamma_1+\mu)+\lambda[\alpha\gamma+\alpha_1(\theta+\gamma+\gamma_1+\mu)] (\theta+\gamma+\gamma_1+\mu)(\delta+\theta+\mu)} $
$ \gamma $ $ \frac{\gamma \lambda \alpha (\theta+\gamma_1+\mu)}{[(\delta+\theta+\mu)(\theta+\gamma+\gamma_1+\mu)+\lambda[\alpha\gamma+\alpha_1(\theta+\gamma+\gamma_1+\mu)](\theta+\gamma+\gamma_1+\mu)} $
$ \gamma_1 $ $ - \frac{\lambda \alpha \gamma \gamma_1}{[(\delta+\theta+\mu)(\theta+\gamma+\gamma_1+\mu)+\lambda[\alpha\gamma+\alpha_1(\theta+\gamma+\gamma_1+\mu)](\theta+\gamma+\gamma_1+\mu))} $
$ \delta $ $ - \frac{\delta \lambda \big(\alpha \gamma + \alpha_1(\theta+\gamma+\gamma_1+\mu) \big)}{[(\delta+\theta+\mu)(\theta+\gamma+\gamma_1+\mu)+\lambda[\alpha\gamma+\alpha_1(\theta+\gamma+\gamma_1+\mu)](\delta+\theta+\mu))} $
Parameter Expression of the sensitivity index
$ \beta $ 1
$ \lambda $ $ \frac{\lambda [\alpha \gamma + \alpha_1(\theta+\gamma+\gamma_1+\mu)]}{(\delta+\theta+\mu)(\theta+\gamma+\gamma_1+\mu)+\lambda[\alpha\gamma+\alpha_1(\theta+\gamma+\gamma_1+\mu)} $
$ \alpha $ $ \frac{\alpha \lambda\gamma}{(\delta+\theta+\mu)(\theta+\gamma+\gamma_1+\mu)+\lambda\alpha\gamma+\lambda\alpha_1(\theta+\gamma+\gamma_1+\mu)} $
$ \alpha_1 $ $ \frac{\alpha_1 \times (\theta+\gamma+\gamma_1+\mu)\big( (\alpha+\alpha_1+\alpha-2+\mu)\lambda -(\delta+\theta+\mu)-\lambda \big)-\lambda\alpha\gamma}{[(\delta+\theta+\mu)(\theta+\gamma+\gamma_1+\mu)+\lambda\alpha\gamma+\lambda\alpha_1(\theta+\gamma+\gamma_1+\mu)] (\alpha+\alpha_1+\alpha_2+\mu)} $
$ \alpha_2 $ $ - \frac{\alpha_2}{(\alpha+\alpha_1+\alpha_2+\mu)^2} $
$ \theta $ $ - \frac{\theta \big[\lambda \alpha_1 (\theta+\gamma+\gamma_1+\mu)^2 +\lambda \alpha \gamma ( 2\theta +\gamma+\gamma_1+\delta+2\mu) \big]}{[(\delta+\theta+\mu)(\theta+\gamma+\gamma_1+\mu)+\lambda[\alpha\gamma+\alpha_1(\theta+\gamma+\gamma_1+\mu)] (\theta+\gamma+\gamma_1+\mu)(\delta+\theta+\mu)} $
$ \gamma $ $ \frac{\gamma \lambda \alpha (\theta+\gamma_1+\mu)}{[(\delta+\theta+\mu)(\theta+\gamma+\gamma_1+\mu)+\lambda[\alpha\gamma+\alpha_1(\theta+\gamma+\gamma_1+\mu)](\theta+\gamma+\gamma_1+\mu)} $
$ \gamma_1 $ $ - \frac{\lambda \alpha \gamma \gamma_1}{[(\delta+\theta+\mu)(\theta+\gamma+\gamma_1+\mu)+\lambda[\alpha\gamma+\alpha_1(\theta+\gamma+\gamma_1+\mu)](\theta+\gamma+\gamma_1+\mu))} $
$ \delta $ $ - \frac{\delta \lambda \big(\alpha \gamma + \alpha_1(\theta+\gamma+\gamma_1+\mu) \big)}{[(\delta+\theta+\mu)(\theta+\gamma+\gamma_1+\mu)+\lambda[\alpha\gamma+\alpha_1(\theta+\gamma+\gamma_1+\mu)](\delta+\theta+\mu))} $
Table 7.  Sensitivity index of $R_0$ for India, Brazil and Peru
Parameter India Brazil Peru
$\beta$ 1 1 1
$\lambda$ 0.4166 0.2415 0.2536
$\alpha$ 0.1511 0.0214 0.2149
$\alpha_1$ -0.5723 -0.6740 -0.3637
$\alpha_2$ -0.4008 -0.0162 -0.4823
$\theta$ -0.0307 -0.0024 -0.0010
$\gamma$ 0.0523 0.0049 0.0038
$\gamma_1$ -0.0429 -0.0036 -0.0031
$\delta$ -0.3936 -0.2397 -0.2522
Parameter India Brazil Peru
$\beta$ 1 1 1
$\lambda$ 0.4166 0.2415 0.2536
$\alpha$ 0.1511 0.0214 0.2149
$\alpha_1$ -0.5723 -0.6740 -0.3637
$\alpha_2$ -0.4008 -0.0162 -0.4823
$\theta$ -0.0307 -0.0024 -0.0010
$\gamma$ 0.0523 0.0049 0.0038
$\gamma_1$ -0.0429 -0.0036 -0.0031
$\delta$ -0.3936 -0.2397 -0.2522
Table 8.  Assumed parameter values for India, Brazil and Peru for parameter estimation of Phase-1
Parameter India Brazil Peru
$ S(0) $ 1000000000 200000000 31000000
$ E(0) $ 1000000 1000000 300000
$ I(0) $ 283407 555808 88831
$ L(0) $ 20000 100000 30000
$ R(0) $ 1000000 1000000 20000
Parameter India Brazil Peru
$ S(0) $ 1000000000 200000000 31000000
$ E(0) $ 1000000 1000000 300000
$ I(0) $ 283407 555808 88831
$ L(0) $ 20000 100000 30000
$ R(0) $ 1000000 1000000 20000
Table 9.  Assumed parameter values for India, Brazil and Peru for parameter estimation of Phase-2
Parameter India Brazil Peru
$ S(0) $ 1000000000 310000000 30000000
$ E(0) $ 1200000 500000 200000
$ I(0) $ 696212 752279 163914
$ L(0) $ 10000 35000 15000
$ R(0) $ 2277156 2670755 576067
Parameter India Brazil Peru
$ S(0) $ 1000000000 310000000 30000000
$ E(0) $ 1200000 500000 200000
$ I(0) $ 696212 752279 163914
$ L(0) $ 10000 35000 15000
$ R(0) $ 2277156 2670755 576067
Table 10.  Assumed parameter values for India, Brazil and Peru for parameter estimation of Phase-2
Parameter India Brazil Peru
$ S(0) $ 1000000000 310000000 30000000
$ E(0) $ 900000 4000000 200000
$ I(0) $ 168665 876672 49586
$ L(0) $ 10000 35000 15000
$ R(0) $ 2277156 9457100 1236668
Parameter India Brazil Peru
$ S(0) $ 1000000000 310000000 30000000
$ E(0) $ 900000 4000000 200000
$ I(0) $ 168665 876672 49586
$ L(0) $ 10000 35000 15000
$ R(0) $ 2277156 9457100 1236668
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