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Basic reproduction number estimation and forecasting of COVID-19: A case study of India, Brazil and Peru

  • * Corresponding author

    * Corresponding author 

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

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  • 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.

    Mathematics Subject Classification: Primary: 92B05, 93A30; Secondary: 34C23.

    Citation:

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  • 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
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    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
     | Show Table
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    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
     | Show Table
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    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
     | Show Table
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    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
     | Show Table
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    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))} $
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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