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

Nitu Kumari; Sumit Kumar; Sandeep Sharma; Fateh Singh; Rana Parshad

doi:10.3934/cpaa.2021170

Article Contents

2023, Volume 22, Issue 2: 417-440. Doi: 10.3934/cpaa.2021170

This issue Previous Article Transmission dynamics and high infectiousness of Coronavirus Disease 2019 Next Article Existence of Solutions for a Weighted Integral equation with Negative Power

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
^* Corresponding author

Received: December 2020

Revised: August 2021

Early access: October 2021

Published: February 2023

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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Abstract

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.

Keywords:

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

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Figure 1. Flow chart for the proposed model

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Figure 2. Model fitting for the cumulative number of COVID-19 cases for India from $ 11^{th} $ July 2020 to $ 15^{th} $ July 2021

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Figure 3. Model fitting for the cumulative number of COVID-19 cases for Brazil from $ 11^{th} $ July 2020 to $ 15^{th} $ July 2021

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Figure 4. Model fitting for the cumulative number of COVID-19 cases for Peru from $ 11^{th} $ July 2020 to $ 15^{th} $ July 2021

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Figure 5. Model fitting for the active number of COVID-19 cases for India from $ 11^{th} $ July 2020 to $ 30^{th} $ June 2021

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Figure 6. Model fitting for the active number of COVID-19 cases for Brazil from $ 11^{th} $ July 2020 to $ 30^{th} $ June 2021

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Figure 7. Model fitting for the active number of COVID-19 cases for Peru from $ 11^{th} $ July 2020 to $ 30^{th} $ June 2021

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Figure 8. Short-term forecasting of the pandemic for India, Peru and Brazil

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Figure 9. Surface plot showing simultaneous effects of $ \lambda $ and $ \beta $ on the basic reproduction number $ R_0 $ for India, Brazil and Peru

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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

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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

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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

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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

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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))} $

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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

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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

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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

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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

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