Forecasting COVID-19 pandemic: A data-driven analysis

Abstract

In this paper, a new Susceptible-Exposed-Symptomatic Infectious-Asymptomatic Infectious-Quarantined-Hospitalized-Recovered-Dead (SEI_DI_UQHRD) deterministic compartmental model has been proposed and calibrated for interpreting the transmission dynamics of the novel coronavirus disease (COVID-19). The purpose of this study is to give tentative predictions of the epidemic peak for Russia, Brazil, India and Bangladesh which could become the next COVID-19 hotspots in no time by using a newly developed algorithm based on well-known Trust-region-reflective (TRR) algorithm, which is one of the robust real-time optimization techniques. Based on the publicly available epidemiological data from late January until 10 May, it has been estimated that the number of daily new symptomatic infectious cases for the above mentioned countries could reach the peak around the middle of June with the peak size of  ∼ 15, 774 (95% CI, 12,814–16,734) symptomatic infectious cases in Russia,  ∼ 26, 449 (95% CI, 25,489–31,409) cases in Brazil,  ∼ 9, 504 (95% CI, 8,378–13,630) cases in India and  ∼ 2, 209 (95% CI, 2,078–2,840) cases in Bangladesh if current epidemic trends hold. As of May 11, 2020, incorporating the infectiousness capability of asymptomatic carriers, our analysis estimates the value of the basic reproductive number (R₀) was found to be  ∼ 4.234 (95% CI, 3.764–4.7) in Russia,  ∼ 5.347 (95% CI, 4.737–5.95) in Brazil,  ∼ 5.218 (95% CI, 4.56–5.81) in India,  ∼ 4.649 (95% CI, 4.17–5.12) in the United Kingdom and  ∼ 3.53 (95% CI, 3.12–3.94) in Bangladesh. Moreover, Latin hypercube sampling-partial rank correlation coefficient (LHS-PRCC) which is a global sensitivity analysis (GSA) method has been applied to quantify the uncertainty of our model mechanisms, which elucidates that for Russia, the recovery rate of undetected asymptomatic carriers, the rate of getting home-quarantined or self-quarantined and the transition rate from quarantined class to susceptible class are the most influential parameters, whereas the rate of getting home-quarantined or self-quarantined and the inverse of the COVID-19 incubation period are highly sensitive parameters in Brazil, India, Bangladesh and the United Kingdom which could significantly affect the transmission dynamics of the novel coronavirus disease (COVID-19). Our analysis also suggests that relaxing social distancing restrictions too quickly could exacerbate the epidemic outbreak in the above-mentioned countries.

1. Introduction

The novel coronavirus disease 2019 (COVID-19) has evolved as a global public health emergency affecting 212 countries and territories around the world as of May 12, 2020 [30]. COVID-19 is one kind of respiratory disease caused by the novel coronavirus (SARS-CoV-2) that was first spotted around late December 2019, in Wuhan, Hubei province, China [3], [7]. This novel virus started transmitting around the world rapidly, and WHO declared the outbreak as a Public Health Emergency of International Concern (PHEIC). Later, WHO Director-General announced COVID-19 as a global pandemic on January 30, 2020. As of May 11, 2020, the outbreak of COVID-19 has resulted in 4,271,689 confirmed cumulative cases with reported deaths of 287,613 worldwide [30]. In most of the countries, infected patients are struggling to get the proper treatment due to highly transmissible and virulent nature of the virus. Nevertheless, numerous mitigation strategies have been promoted so far such as quarantine, isolation, promoting the wearing of face masks, travel restrictions, and lockdowns with a view to controlling the rapid community transmission of the disease.

In the absence of any clinically established treatment or an effective vaccine, the already fragile health care systems of different developed and developing countries could be overburdened due to the continuous surge of infections in the coming months, provided that the outbreak is not controlled. This pandemic is giving rise to various socio-economic and public health concerns and has highlighted the significance of unearthing the evolution of the disease and forecasting of the disease future dynamics, which will contribute to the disease prevention and control strategies, sustainable public health policies and economical activity guidelines. Different mathematical paradigm has always played a notable role in providing deeper understanding of the transmission mechanisms of a disease outbreak, contributing considerable insights for controlling the disease outbreak. One of the familiar models for human-to-human transmission which is reasonably predictive as Susceptible-Infectious-Removed (SIR) epidemic model proposed by Kermack-Mckendrick in 1927 [10], [14]. Afterwards, the SIR epidemic model has been extended to Susceptible-Exposed-Infectious-Removed (SEIR) model and many of its variants to explore the risk factors of a disease or predict the future dynamics of a disease outbreak [9], [10], [11], [12], [13], [14]. In population based model, it is always really challenging to incorporate certain real-world complexities. In fact, analysis and prediction could go wrong in the absence of adequate historical real data. On the other hand, various agent-based often stochastic models, where individuals interact on a network structure and get infected stochastically, have been treated as useful tools for tracing fine-grained effects of heterogeneous intervention policies in diverse disease outbreaks [4], [15], [16], [17]. However, accuracy of this approach can be a vital issue due to the time-varying nature of network-structure.

Since the outbreak of the virus, considering travels between major cities in China, Li et al. [3] proposed an SEIR-model incorporating a metapopulation structure for both reported and unreported infections. It has been unearthed in their studies that around 86% of cases went undetected in Wuhan before travel restrictions imposed on January 23, 2020. According to their estimation, on an individual basis, around 55% asymptomatic spreaders were contagious who were responsible for 79% of new infected cases. Other studies [4], [19] have solidified the significance of incorporating asymptomatic carriers with a view to understanding COVID-19 future dynamics more accurately. Later, Calafiore et al. [10] estimated that around 63% cases went under-reported in Italy by analyzing a modified SIR-model. Anastassopoulou et al. [13] applied the SIRD model to Chinese official statistics and predicted the transmission dynamics of the COVID-19 pandemic in Hubei province estimating the model parameters using linear regression. Nonetheless, long-term forecasting is highly debatable while using a simple mathematical model for outbreak analysis. Caccavo [20] and Turchin [21] independently applied modified SIRD models, in which parameters are allowed to change overtime following specific function forms. Parameters that govern these functions are estimated by minimizing the sum-of-square-error. However, the sum-of-square method can cause over-fitting and always favors a complex model, therefore it is not suitable to access model-fitting effectiveness. Moreover, fitting the SIRD model at an early stage of infection is somewhat questionable as well.

In light of the above shortcomings of several established mathematical models, a more rigorous Susceptible-Exposed-Symptomatic Infectious-Asymptomatic Infectious-Quarantined-Hospitalized-Recovered-Dead (SEI_DI_UQHRD) model has been proposed considering all possible interactions, which can give more accurate and robust short-term as well as long-term predictions of the COVID-19 future dynamics. This model could be considered as a generalization of SEIR-model, which is based on the introduction of asymptomatic infectious state, quarantined state in order to understand the effect of preventive actions and hospitalized (isolated) state. Some assumptions are taken into account considering similar studies performed over SEIR models [9], [10], [13]. As of May 11, it is conspicuous that how the early stage mathematical model parameters have changed drastically due to the unprecedented aggressive changes in the COVID-19 dynamics. However, the outbreak situation has improved satisfactorily in several countries due to massive scale testing strategy and strict confinement measures [31]. Nominal values of the model parameters have been considered understanding the characteristics of the coronavirus infection, which are quantitatively estimated in the literature or published by health organizations [2], [3], [9], [19]. In sequence, a newly developed optimization algorithm based on well-known Trust-region-reflective algorithm has been applied to determine the best-fitted parameter values of the model. Our analysis is based on the publicly available data of the daily confirmed COVID-19 cases until May 09, 2020, in Russia, Brazil, India and Bangladesh which could be future COVID-19 hotspots in the world [18]. Further, we have validated our model for the United Kingdom using the same time frame. Based on the released data, we have attempted to estimate the probable ranges of the crucial epidemiological parameters for COVID-19, such as the infectiousness factor for asymptotic carriers, isolation period, the estimation of the inflection point with probable date, recovery period for both symptomatic and asymptomatic individuals, case-fatality ratio and mortality rate in those potential hotspots. Perfect data-driven and curve-fitting methods for the forecasting of any disease outbreak have always been question of interest in epidemiological research. Trust-region-reflective algorithm is one of the robust least-squares optimization techniques that can promote a fine relation between model driving mechanisms and model responses. This real-time data-fitting approach could be efficient enough to provide considerable insights on disease outbreak dynamics in different countries and design worthwhile public health policies in curtailing the disease burden. Calibrating our model parameters by using the above algorithm, we have provided probable forecasts for the emerging COVID-19 hotspots.

This paper has been organized as follows. In Section 2, the mathematical model has been described and the background of choosing baseline parameter values for the model has been discussed. In Sections 3 and 4, the model has been analysed and model calibration technique has been discussed respectively. In Section 5, forecasing accuracy of the model has been illustrated by representing a graphical comparison between model responses and real-time data. In Section 6, one of the robust global sensitivity analysis techniques has been applied to quantify the most influential mechanisms in our model. This paper ends with some qualitative and quantitative observations and discussions.

2. Formulation of the mathematical model

In this work, a compartmental differential model has been used to understand the transmission dynamics of COVID-19 in the world. The spread of the infection usually starts with an introduction of a small group of infected individuals to a large population. In this model, we focus our study on eight components of the epidemic flow, i.e. {S(t), E(t), I_D(t), I_U(t), Q(t), H(t), R(t), D(t)} which represent the number of the susceptible individuals, exposed individuals (infected however not yet to be infectious, in an incubation period), symptomatic infectious individuals (confirmed with infectious capacity), asymptomatic infectious individuals (undetected but infectious), quarantined, hospitalized (under treatment), recovered cases (immuned) and death cases (or closed cases). The entire number of population in a certain region or country is, $N=S+E+I_D+I_U+Q+H+R+D$. The following assumptions are considered in the formulation of the model:

• •Emigration from the population and immigration into the population have not been taken into account in model formulation, as there is negligible proportion of individuals move in and out of the population at a specific time frame.
• •Births and natural deaths in the population are not considered.
• •The susceptible population are exposed to a latent class.
• •Hospitalized patients cannot spread disease while in isolation treatment after confirmed diagnosis.
• •Recovered individuals do not return to susceptible class as they develop certain immunity against the disease. Hence they cannot become re-infected again and cannot infect susceptible either.

Fig. 1 is the flow diagram of the proposed model, where susceptible individuals (S) get infected at a baseline infectious contact rate β, via contact with either symptomatic infectious individuals (I_D) or at a rate βλ of asymptomatic infectious carriers and move to exposed (E) class. The relative infectiousness capacity of asymptomatic spreaders is λ (related to symptomatic infectious individuals). Susceptible individuals can be quarantined with contact tracing procedure and the rate of getting home-quarantined or self-quarantined of S is q, whereas κ is the rate of progression of symptoms of COVID-19 (hence $\frac{1}{\kappa }$is the mean incubation period of COVID-19). Detected symptomatic infected individuals are generated at a progression rate σ ₁ and undetected asymptomatic infectious cases are generated at a rate σ ₂ from the exposed class. In addition, individuals who are exposed to virus can be quarantined with contact tracing procedure at a rate $(1-{\sigma }_1-{\sigma }_2)$. γis the rate of confirmation and isolation after symptom onset in detected infectious individuals and 1/γ is the time period when the infection can spread. The proportion of individuals from quarantined to symptomatic infectious, susceptible and asymptomatic infectious are r ₂ η, r ₁ η and $(1-r_1-r_2)\eta$respectively. δ_U is the disease-induced death rate for undetected asymptomatic patients and δ_H is the death rate for hospitalized (highly critical) patients. ϕ_D, ϕ_U and ϕ_H are the recovery rates for the symptomatic patients, asymptomatic carriers and hospitalized patients respectively. In addition, 1/ϕ_D, 1/ϕ_U and 1/ϕ_H represent the mean period of isolation for the detected symptomatic infectious individuals, undetected asymptomatic carriers and hospitalized patients. Based on these assumptions, the basic model structure for the transmission dynamics of COVID-19 is illustrated by the following deterministic framework of nonlinear model Eq. (1).

$$\begin{array}{c}\hfill \{\begin{array}{cc}\frac{dS}{dt}\hfill & =-\beta S\frac{(I_D+\lambda I_U)}{N-D}+r_1\eta Q-qS,\hfill \\ \frac{dE}{dt}\hfill & =\beta S\frac{(I_D+\lambda I_U)}{N-D}-\kappa E,\hfill \\ \frac{d{I}_D}{dt}\hfill & ={\sigma }_1\kappa E-\gamma I_D-{\varphi }_D{I}_D+r_2\eta Q,\hfill \\ \frac{d{I}_U}{dt}\hfill & ={\sigma }_2\kappa E-{\varphi }_U{I}_U-{\delta }_U{I}_U+(1-r_1-r_2)\eta Q\hfill \\ \frac{dQ}{dt}\hfill & =(1-{\sigma }_1-{\sigma }_2)\kappa E-\eta Q+qS\hfill \\ \frac{dH}{dt}\hfill & =\gamma I_D-{\varphi }_{H}H-{\delta }_{H}H\hfill \\ \frac{dR}{dt}\hfill & ={\varphi }_D{I}_D+{\varphi }_U{I}_U+{\varphi }_{H}H\hfill \\ \frac{dD}{dt}\hfill & ={\delta }_U{I}_U+{\delta }_{H}H\hfill \end{array}\end{array}$$ (1)

Fig. 1.

A schematic diagram that illustrates the proposed COVID-19 model.

2.1. Baseline epidemiological parameters

In the study of [5], the average incubation period of COVID-19 was estimated to be 5.1 days and similar model-based studies also justified the above estimation [3], [6], [4]. Moreover, within 11.5 days (i.e. $\kappa =1/11.5{\text{day}}^{-1}$) of infection, the individuals who were exposed to the virus started developing symptoms [5], [19]and 20.0 days was the mean duration of viral shedding observed in COVID-19 survivors [7]. The time duration from symptoms onset to recovery was estimated to be 24.7 days, whereas the mean duration from onset of symptoms to death was estimated to be 17.8 days [19]. In previous modeling studies, Shen et al. [1], Read et al. [2], Li et al. [3], the novel coronavirus effective transmission rate, β ranges from around $0.25-1.5{\text{day}}^{-1},$which gradually follows a downward trend with time [6]. For this reason, this parameter has been considered as a time-varying parameter in our fits. Table 1 illustrates the list of baseline model parameters with brief description, probable ranges based on clinical studies and calibration, and default base value grounded on prior studies.

Table 1.

Model parameters with brief definition and probable ranges based on model calibration, relevant literature and clinical studies.

Parameter	Definition	Units	Likely range	Default value	References
β	infectious contact rate	${\text{day}}^{-1}$	0.2–1.5	0.5	[1], [2], [3]
λ	infectiousness factor for undetected infected carrier	−	0.4–0.6	0.5	[3], [4]
r1	fraction of quarantined that become susceptible	−	0.7–0.99	0.9	Calibrated
r2	fraction of quarantined that become detected symptomatic	−	0.005–0.1	0.02	Calibrated
η	transition from quarantined to either infectious or susceptible	${\text{day}}^{-1}$	0.005–0.3	0.01	Calibrated
κ	transition from exposed to infectious	${\text{day}}^{-1}$	$\frac{1}{14}-\frac{1}{3}$	1/5.1	[3], [5]
q	transition from susceptible to quarantined	${\text{day}}^{-1}$	0.001–0.5	0.01	Calibrated
σ1	fraction of infected individuals that become detected symptomatic	−	0.0001–0.1	0.0032	Calibrated
σ2	fraction of infected individuals that become undetected symptomatic	−	0.01–0.95	0.5	Calibrated
γ	transition from detected symptomatic to hospitalized	${\text{day}}^{-1}$	0.2–0.9	0.5	[7], [4]
ϕD	recovery rate, detected symptomatic	${\text{day}}^{-1}$	$\frac{1}{30}-\frac{1}{3}$	1/7	[6], [7]
ϕU	recovery rate, undetected symptomatic	${\text{day}}^{-1}$	$\frac{1}{30}-\frac{1}{3}$	1/7	[6], [7]
ϕH	recovery rate, hospitalized	${\text{day}}^{-1}$	$\frac{1}{30}-\frac{1}{3}$	1/7	[6], [7]
δU	disease induced death rate, undetected symptomatic	${\text{day}}^{-1}$	0.001–0.1	0.015	[4]
δH	disease induced death rate, hospitalized	${\text{day}}^{-1}$	0.001–0.1	0.015	[4]

2.2. Dataset

Daily new COVID-19 cases in five different countries named Russia, Brazil, India, Bangladesh and the United Kingdom were accumulated from authoritative and genuine sources, which are Center of Disease Control and Prevention (CDC) and the COVID Tracking Project (testing and hospitalizations). The data repository is handled by the Johns Hopkins University Center for Systems Science and Engineering (JHU CSSE) and supported by ESRI Living Atlas Team and the Johns Hopkins University Applied Physics Lab (JHU APL). The repository is publicly available and easy to compile [18].

3. Analysis of the model

3.1. Basic reproduction number for proposed model

Using the next generation operator method [24] the local stability of disease free equilibrium (DFE ) has been investigated. According to the notation in Diekmann et al. [25], the associated non-negative matrix, $\mathcal{F}$represents new infection terms, and the non-singular matrix, $\mathcal{V}$denotes remaining transfer termswhich can be described as follows:

$$\mathcal{F}=\left(\begin{array}{ccc}0& \beta & \beta \lambda \\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$$

and

$$\mathcal{V}=\left(\begin{array}{ccc}k& 0& 0\\ -{\sigma }_{1}k& \gamma +{\varphi }_D& 0\\ -{\sigma }_{2}k& 0& {\varphi }_U+{\delta }_U\end{array}\right)$$

The associated basic reproduction number, denoted by ${\mathcal{R}}_0$is then given by,

${\mathcal{R}}_0=\rho \left(\mathcal{F}{\mathcal{V}}^{-1}\right),$where ρ is the spectral radius of $\mathcal{F}{\mathcal{V}}^{-1}$. It follows that,

$${\mathcal{R}}_0=\frac{\beta [{\delta }_U{\sigma }_1+(\gamma +{\varphi }_D)\lambda {\sigma }_2+{\sigma }_1{\varphi }_U]}{(\gamma +{\varphi }_D)({\delta }_U+{\varphi }_U)}$$ (2)

Thus, by Theorem 2 of [23], the following result is established.

Lemma 1

The DFE, of the system(1), is locally-asymptotically stable (LAS) if${\mathcal{R}}_0<1,$and unstable if${\mathcal{R}}_0\ge 1$.

The epidemic threshold quantity, ${\mathcal{R}}_0,$estimates the mean number of secondary cases generated by a single infected individual in an entirely susceptible human population [22]. The above result implies that a small influx of infected individuals would not generate large outbreaks if ${\mathcal{R}}_0<1,$and the disease will persist (be endemic) in the population if ${\mathcal{R}}_0>1$.

4. Model calibration

The proposed epidemic model (1) is a continuous-time non-linear system of differential equations together with a suitable set of initial conditions. In this study, a newly developed algorithm based on well-known Trust-region-reflective (TRR) algorithm has been used to determine the best-fitted parameters for our proposed model. A function named lsqcurvefit in MATLAB has been used to implement the algorithm for the proposed model. Basically, this Optimization Toolbox in MATLAB is highly used for solving non-linear least squares problem. The optimization process can be expressed as follows:

$${\mathit{\theta }}^{*}=arg\underset{\mathit{\theta }}{\min }:={\parallel SE{I}_D{I}_{U}QHRD(\mathit{\theta },\mathit{time})-\left(\begin{array}{c}{I}_D\\ R\\ D\end{array}\right)\parallel }^2$$

where θ* is the set of parameters of dynamically calibrated model, {I_D, R, D} is a set of the detected symptomatic infectious individuals, recovered and disease-induced death cases from the real-time data and $\mathit{\theta }=\{\beta ,\lambda ,r_1,r_2,\eta ,k,{\sigma }_1,{\sigma }_2,\gamma ,q,{\varphi }_D,{\varphi }_U,{\varphi }_H,{\delta }_U,{\delta }_H\}$is the initial set of parameters of the proposed model and SEI_DI_UQHRD(.) represents our proposed model. The “TRR algorithm” also necessitates an initial guess for each parameter which is described as “TRR input” in the following section.

5. Numerical experiments and forecasting

In this section, we have used our dynamically calibrated compartmental model for real-time analysis and real-time prediction of COVID-19 future dynamics for five different countries which are Russia, Brazil, India, Bangladesh and the United Kingdom. Figs. 2 , 4 , 6 , 8 , and 10 illustrate the best real-time data fitting results using the baseline parameters from Table 1 as initial inputs for our proposed model. The ranges of the parameters have been set compatible with the clinical studies and illustrated in Table 2, Table 3, Table 4, Table 5, Table 6 along with the parameters estimations from the calibration (“TRR output”). To ensure a huge number of susceptible individuals at the initial stage of an outbreak, 0.9 N has been set as a lower bound for S.

Fig. 2.

Fitting performance of calibrated SEI_DI_UQHRD model for Russia from February 1 to May 08, 2020.

Fig. 4.

Fitting performance of calibrated SEI_DI_UQHRD model for Brazil from 26 February to 8 May, 2020.

Fig. 6.

Fitting performance of calibrated SEI_DI_UQHRD model for India from 8 March to early May, 2020.

Fig. 8.

Fitting performance of calibrated SEI_DI_UQHRD model for the United Kingdom from 31 January to early May, 2020.

Fig. 10.

Fitting performance of calibrated SEI_DI_UQHRD model for Bangladesh from 8 March to early May, 2020.

Table 2.

Necessary parameters for trust-region-reflective algorithm and for the Fig. 3 calibrated response.

Parameter	TRR input	lb	ub	TRR output	References
β	0.4	0.2	1.5	0.4852	[1], [3]
λ	0.45	0.4	0.6	0.5249	[3], [4]
r1	0.76	0.1	0.999	0.9810	Estimated
r2	0.03	0.01	0.9	0.0224	Estimated
η	0.05	0.01	0.9	0.0779	Estimated
κ	0.196	1/14	1/3	0.2443	[3], [5]
σ1	0.01	0.0001	0.9	0.0067	[3], [4]
σ2	0.5	0.01	0.95	0.6894	Estimated
γ	0.3	0.1	0.9	0.7297	[7], [4]
ϕD	0.1428	1/30	1/3	0.3332	[6], [7]
ϕU	0.1428	1/30	1/3	0.0401	[6], [7]
ϕH	0.1428	1/30	1/3	0.0433	[6], [7]
δU	0.09	0.001	0.01	0.0013	[4]
δH	0.09	0.001	0.01	0.0095	[4]
q	0.015	0.01	0.5	0.0107	Estimated

Table 3.

Necessary parameters for trust-region-reflective algorithm and for the Fig. 5 calibrated response.

Parameter	TRR input	lb	ub	TRR output	References
β	0.3	0.2	1.5	0.5155	[1], [3]
λ	0.45	0.4	0.6	0.5855	[3], [4]
r1	0.76	0.1	0.999	0.9855	Estimated
r2	0.03	0.01	0.9	0.0159	Estimated
η	0.05	0.01	0.9	0.0897	Estimated
κ	0.196	1/14	1/3	0.1867	[3], [5]
σ1	0.01	0.0001	0.9	0.0072	[19], [8]
σ2	0.5	0.1	0.95	0.7308	Estimated
γ	0.3	0.1	0.9	0.7	[7], [4]
ϕD	0.1428	1/30	1/3	0.3333	[6], [7]
ϕU	0.1428	1/30	1/3	0.0356	[6], [7]
ϕH	0.1428	1/30	1/3	0.0431	[6], [7]
δU	0.09	0.001	0.01	0.001	[4]
δH	0.09	0.001	0.01	0.009	[4]
q	0.015	0.01	0.5	0.04	Estimated

Table 4.

Necessary parameters for trust-region-reflective algorithm and for the Fig. 7 calibrated response.

Parameter	TRR input	lb	ub	TRR output	References
β	0.3	0.2	1.5	0.3995	[1], [3]
λ	0.4	0.4	0.6	0.5995	[3], [4]
r1	0.76	0.1	0.999	0.9911	Estimated
r2	0.03	0.01	0.9	0.01	Estimated
η	0.05	0.01	0.9	0.0997	Estimated
κ	0.196	1/14	1/3	0.3296	[3], [5]
σ1	0.01	0.0001	0.1	0.00046	[19], [8]
σ2	0.5	0.1	0.95	0.7495	Estimated
γ	0.3	0.1	0.9	0.8	[7], [4]
ϕD	0.1428	1/30	1/3	0.3333	[6], [7]
ϕU	0.1428	1/30	1/3	0.0334	[6], [7]
ϕH	0.1428	1/30	1/3	0.3335	[6], [7]
δU	0.09	0.001	0.01	0.001	[4]
δH	0.09	0.001	0.01	0.01	[4]
q	0.015	0.01	0.5	0.0507	Estimated

Table 5.

Necessary parameters for trust-region-reflective algorithm and for the Fig. 9 calibrated response.

Parameter	TRR input	lb	ub	TRR output	References
β	0.3	0.2	1.5	0.73	[1], [3]
λ	0.4	0.4	0.6	0.59	[3], [4]
r1	0.76	0.1	0.999	0.97	Estimated
r2	0.03	0.01	0.9	0.024	Estimated
η	0.05	0.9	0.1	0.09	Estimated
κ	0.196	1/14	1/3	0.3333	[3], [5]
σ1	0.01	0.0001	0.01	0.0044	[19], [8]
σ2	0.5	0.1	0.95	0.37	Estimated
γ	0.3	0.1	0.9	0.79	[7], [4]
ϕD	0.1428	1/30	1/3	0.3333	[6], [7]
ϕU	0.1428	1/30	1/3	0.0542	[6], [7]
ϕH	0.1428	1/30	1/3	0.0333	[6], [7]
δU	0.09	0.001	0.01	0.001	[4]
δH	0.09	0.001	0.01	0.001	[4]
q	0.015	0.01	0.5	0.015	Estimated

Table 6.

Necessary parameters for trust-region-reflective algorithm and for the Fig. 11 calibrated response.

Parameter	TRR input	lb	ub	TRR output	Reference
β	0.3	0.2	1.5	0.3972	[1], [3]
λ	0.4	0.4	0.6	0.5498	[3], [4]
r1	0.76	0.1	0.999	0.9454	Estimated
r2	0.03	0.01	0.9	0.02	Estimated
η	0.05	0.01	0.9	0.06	Estimated
κ	0.196	1/14	1/3	0.0714	[3], [5]
σ1	0.01	0.0001	0.9	0.001	[19], [8]
σ2	0.5	0.3	0.98	0.95	Estimated
γ	0.3	0.1	0.9	0.5	[7], [4]
ϕD	0.1428	1/30	1/3	0.3333	[6], [7]
ϕU	0.1428	1/30	1/3	0.0333	[6], [7]
ϕH	0.1428	1/30	1/3	0.3275	[6], [7]
δU	0.09	0.001	0.01	0.0011	[4]
δH	0.09	0.001	0.01	0.01	[4]
q	0.015	0.01	0.5	0.012	Estimated

5.1. Analysis and prediction for Russia

With much of Europe now easing itself out of confinement, Russia could become the continent’s new COVID-19 hotspot according to our analysis. The model fitting and projection results for Russia from early February to late August are shown in Figs. 2 and 3. We collected real-time data from February 01 to May 08, 2020, to calibrate the model parameters.

Fig. 3.

Forecasting of the proposed SEI_DI_UQHRD model for Russia until September, 2020.

As we can see, the results from the proposed model match the real data very well. Based on the model forecasting results model, from Fig. 3, the number of daily detected symptomatic infectious cases in Russia could reach the peak at around May 27 with about 15.774K (95% CI: 12,814-16,734) cases. As time progresses, our estimated daily projected mean error drops to 10% for the cumulative cases and daily new cases, which represents the robustness of the model forecasting. The basic reproduction number is 4.234 as of May 08, which lies in prior established findings 2–7 for COVID-19 [26]. The number of cumulative infected cases is projected to reach 970K around August 30, and the estimated total death cases could reach 21.3K in the end of the outbreak. To date, Russia’s official death toll is 1827, which is relatively low because not all deaths of people who have contracted the virus are being counted as COVID-19 deaths. Table 2 illustrates the key features used to calibrate this scenario which are compatible with the previous clinical studies and relevant literature.

5.2. Analysis and prediction for Brazil

The coronavirus disease 2019 (COVID-19) pandemic headed toward Latin America later than other continents. On Feb 25, 2020, the first infected case was documented in Brazil. But now, Brazil has surpassed the records in Latin America in terms of deaths and new infected cases (155,939 cases and 10,627 deaths as of May 9, 2020). This is probably an underestimated scenario in comparison to real severity in Brazil. Our analysis projects that Brazil is emerging as one of the world’s next deadliest coronavirus hotspots and has strong possibility to take the leading position in terms of total infected cases. The model fitting and projection results for Brazil from late February to late August are shown in Figs. 4 and 5. We took real-time data from February 25 to May 08 to calibrate the model parameters. As we can see, the results of the proposed model fit the real-time data very well. Based on the proposed model, we project that from Fig. 5, the number of daily detected symptomatic infectious cases in Brazil could reach its peak around June 11 with about 26.449K cases (95% CI: 25,489-31,409). The basic reproduction number is estimated to be 5.3467 as of May 11, which is in between the observed basic reproduction number for COVID-19, estimated about 2–7 for COVID-19 [26]. This crucial epidemiological parameter could blow up in near term because of scant diagnostics and inconsistent non-pharmaceutical interventions. The case-fatality rate which is considered as a signifiant measure of disease severity, has been hovering around 9.3% according to our estimation as of May 11, 2020. According to our projection, this ratio could be doubled within one months. The number of cumulative infected cases is projected to reach 1800K around August 30 if current trend is held, and the estimated total death cases could reach 108K in the end. Table 3 illustrates the key features used to calibrate this scenario, which are compatible with the previous clinical studies and relevant literature.

Fig. 5.

Forecasting of the proposed SEI_DI_UQHRD model for Brazil until September, 2020.

5.3. Analysis and prediction for India

Some people might think about the possibilities of the presence of a less virulent strain of the virus in India, along with the possibility that its hot weather could diminish the contagion. Nevertheless, our mathematical analysis suggests that India could become the new COVID-19 hotspot in South Asia within one month. According to our projection, India could continue witnessing spikes in the number of cases as time progresses, despite strict lockdown measures and other disease mitigation policies. In India, the first COVID-19 case who has fresh travel history to China, was reported on January 30, 2020. At the moment, India has the most infected cases and deaths in South Asia (62,808 cases and 2101 deaths as of May 9), and these are probably substantial underestimates. Figs. 6 and 7 illustrate the fitting performance of the proposed model and prediction for India from late January to late August. We took real-time data from January 30 to May 8, to calibrate the model parameters. As we can see, model-fitting is exceptionally well for the historical real data. Based on our prediction from Fig. 7, the number of daily detected symptomatic infectious cases in India could reach its peak around June 15 with about 9.504K (95% CI: 8378-13,630) cases. The basic reproduction number is 5.218 as of May 11, which lies between the prior studied observations [26]. This estimation might be considered as an overestimation of this insightful parameter. Notwithstanding, this is owing to the infectiousness factor of the asymptomatic spreaders. Symptomatic infectious and the asymptomatic spreaders should be quarantined strictly to avoid the rapid transmission of the virus, with the critically symptomatic patients isolated in proper health care settings. By carrying out massive scale contract tracing, infectious individuals could be identified by means of their exposure. The number of cumulative infected cases is projected to reach 730K around August 30 if the current pattern is continued, and the estimated total death cases could reach 33.8K in the end. Table 4 illustrates the key features used to calibrate this scenario, which are compatible with the previous clinical studies and relevant literature. Interestingly, we have found in our analysis that India’s case-fatality rate is at 3.5% and the country’s recovery rate is at 33% as of May 11, which go well with the government reported statistics precisely. However, situation could exacerbate significantly within one month period in the absence of strict lockdown measures and required public health policies.

Fig. 7.

Forecasting of the proposed SEI_DI_UQHRD model for India until late August, 2020.

5.4. Analysis and prediction for the United Kingdom

Our analysis suggests that the United Kingdom could face the risk of a second wave of coronavirus infections as country's government gradually plans easing of nationwide lockdown. According to a recent report [30], as of May 11, there are 223,060 confirmed cases and 32,065 deaths with an overall rate of 465 deaths per million population. The outbreak in London has the highest number and highest rate of infections, while England and Wales are the UK countries with the highest recorded death rate per capita. Recently, the prime minister of the UK has expressed his plan to unveil a coronavirus warning system as an intervention strategy, while he was planning to ease the lockdown gradually in the UK. For the UK, the modeling and projection results from January 31 to August 31 are shown in Figs. 8 and 9. As we can see from Fig. 9, the number of daily detected symptomatic infectious cases reached the peak around April 10. Since then the curve has been maintaining a plateau, which is really an unusual scenario. This phenomenon has closely been captured by our proposed model. This case study was really important for the validation of our model. In fact, this guarantees the fact that this model is capable of providing more precise and realistic short-term predictions of COVID-19 dynamics. According to our calculation, the basic reproduction number is around 4.649 as of May 09, which satisfies the prior studies of COVID-19 [13], [26]. The number of cumulative infected cases is projected to reach 618K around August 30, and the estimated total death cases could reach 63.48K in the above mentioned period. Importantly, as of May 11, we have found in our analysis that the UK’s case-fatality rate is at 17.2%, which is the worst among our five studied cases. This could exacerbate as time progresses in the absence of proven effective therapy or a vaccine. In addition, our study suggests that relaxing social distancing too soon could result in thousands of additional death in the UK. Table 5 illustrates the key features used to calibrate this scenario which are pertinent to the previous clinical studies and literature.

Fig. 9.

Forecasting of the proposed SEI_DI_UQHRD model for the United Kingdom until late September, 2020.

5.5. Analysis and prediction for Bangladesh

The calibration and projection results from early March to late August for Bangladesh have been shown in Figs. 10 and 11. On March 22, in the wake of four deaths and 39 infections, Bangladesh government deployed a nationwide lockdown effective from March 26, with an aim to slow down the spread of the novel coronavirus. Evidently, there are some irregular jumps in the number of confirmed daily new infected cases data from April 16 to April 20, due to the increase of limited testing system (from around 2000 samples to 2700 samples per day) [32]. Bangladesh is still struggling (around 6700 samples per day) [32] to design a massive scale testing program as of May 11, 2020. Despite the fact, as we can see from Fig. 10, our proposed model fits well for the historical real-time data. As time progresses, the estimated error declines and is hovering around 10% for the cumulative cases and daily new cases according to our calculated daily projected mean error. The estimated case-fatality rate in Bangladesh is at 2.7%, which is kind of satisfactory. However, without massive scale testing this rate could surge in coming days. Moreover, the country’s estimated recovery rate is at 24%, which complements the real statistics precisely. Importantly, without an aggressive mass-testing program, it is difficult to portray the real outbreak scenario in Bangladesh. Table 6 illustrates the key features used to calibrate this scenario, which have been justified in prior clinical studies and relevant literature.

Fig. 11.

Forecasting of the proposed SEI_DI_UQHRD model for Bangladesh until late August, 2020.

On May 4, officials of Bangladesh decided to open up factories, shopping malls and logistics operations, as the country has already started facing catastrophic economic recession due to strict lockdown, which is going to be eased on May 16 [33]. This easing of lockdown could worsen the ongoing community transmission drastically. As we can see from Fig. 11, the daily detected symptomatic infected cases (confirmed) could reach its peak of 2209 (95% CI: 2078-2840) around June 11, and then the epidemic trend could decline. However, the probable peak time could occur in no time due to easing of lockdown measures too quickly and this could bring a second wave of infections during the post-peak period. The estimated effective reproduction number is hovering around 3.5 as of May 11, which again lies in prior established ranges [13], [26]. Our estimation is reasonably high due to the fact that the infectiousness factor of the asymptomatic carriers has been considered. Massive level testing is highly required and recommended to identify the asymptomatic spreaders quickly. Only a massive level test-and-isolate approach can control the disease burden of COVID-19 effectively, compared to other mitigation strategies such as lockdown measures, putting on face-masks in public places and maintaining physical distancing guidelines. Otherwise, this epidemic threshold could increase upto 5.7 within one month and inhabitants of Bangladesh could see a catastrophic COVID-19 outbreak in near-term. Estimation and projection could differ significantly due to various volatile factors such as efficacy of mitigation measures, public awareness and social behaviors of citizens.

6. Global sensitivity analysis

Partial rank correlation coefficient (PRCC) analysis is a global sensitivity analysis method that calculates the PRCC for the model inputs (sampled by Latin hypercube sampling method) and model responses [27], [28], [29]. The PRCC method assumes a monotonic relationship between the model input parameters and the model outputs.

The calculated PRCC values generally range from −1 to 1. Quantitative relationship between model inputs and model responses can be determined by calculating the PRCC values. A positive PRCC value depicts that the value of the model response can be increased by increasing the respective model input parameter, and the model output can be decreased by forcing down the relative input parameter. In addition, a negative PRCC value indicates a negative correlation between the model input and output. The magnitude of the PRCC index measures the significance of the model input in contributing to the change of respective model output.

As our proposed epidemic model contains a moderate number of empirical parameters, uncertainty analysis can give considerable insights regarding the quantitative relationship between model responses and model input mechanisms. However, in case of analyzing relatively complex-structured model, it is often really challenging to determine the desired relationship with sufficient accuracy. Importantly, we have got startling yet realistic results from our sensitivity analysis. As we can see from Fig. 12 , we have found nearly the same qualitative and significant quantitative relationship between the number of detected symptomatic infectious individuals (one of the crucial model responses) and three parameters, which are the rate of getting quarantined of susceptible individuals (q), the transition rate from exposed to infectious (κ) that can also be referred as the inverse of the average incubation period of COVID-19 and the recovery rate of undetected asymptomatic infectious carriers in our proposed model.

Fig. 12.

Sensitivity of the symptomatic infected cases while changing parameters in the proposed SEI_DI_UQRHD model as indicated by the PRCC index for five different countries.

In case of Russia, from Fig. 12a, the recovery rate of undetected asymptomatic carriers (ϕ_U) is the most negatively influential parameter, when the number of detected infectious individuals has been chosen as the model response. The PRCC index has been estimated to be $-0.585$. In addition, rate of entering into home-quarantine or self-quarantine of susceptible individuals (q) and the fraction of wrongly quarantined people who become susceptible after certain latent period (r ₁) are the other two influential empirical features with PRCC indexes are $-0.5672$and 0.567 respectively.

In case of Brazil, the rate of getting home-quarantined or self-quarantined of susceptible individuals (q), the fraction of quarantined people who become susceptible because of avoiding home-quarantine (r ₁), and the inverse of the COVID-19 mean latent period are the most influential parameters on the symptomatic infectious population size (I_D(t)). The corresponding PRCC indices are $-0.62,$0.53 and $-0.499$. Fig. 12b elucidates that a high quarantine rate of the susceptible individuals can curtail the number of symptomatic infected individuals. In broader view, high efficacy of home or self-quarantine and strict lockdown measures could flatten the I_D(t) curve in Brazil.

In case of India, the rate of getting home-quarantined or self-quarantined of susceptible individuals (q) and the inverse of the COVID-19 incubation period are the most influential parameters on the symptomatic infectious population size (I_D(t)) and the corresponding PRCC indexes are $-0.64$and $-0.62$. Both the parameters are negatively sensitive to the size of detected infected individuals, which is illustrated in Fig. 12c. This elucidates that a high quarantine rate of the susceptible individuals can reduce the number of symptomatic infected individuals. Therefore, it is obvious that early unlockdown in India could worsen the outbreak situation in the blink of an eye.

In case of the UK, rate of getting home-quarantined or self-quarantined of susceptible individuals (q) and the inverse of the COVID-19 incubation period are the most influential parameters on the symptomatic infectious population size (I_D(t)) and the corresponding PRCC indexes are $-0.58$and $-0.499$. Both the parameters are negatively sensitive to the size of detected infected individuals which is illustrated in Fig. 12d. This again enlightens the fact that a high quarantine rate of the susceptible individuals can force down the number of symptomatic infected individuals. Moreover, early easing of lockdown measures could bring a second wave of infection in the UK in no time.

For Bangladesh, it has been found in our analysis, home-quarantine rate is the most negatively sensitive parameter on the size of symptomatic infectious individuals and the corresponding PRCC index is, $-0.68$which is the highest among our five studied cases. The result in Fig. 12e unfolds that if the people in Bangladesh start neglecting social distancing restrictions extensively, then it would be difficult to control the disease outbreak in the country. As of May 11, when Bangladesh is witnessing continuous spikes in COVID-19 infections on daily basis, the government has already announced easing of lockdown with a view to reviving the country’s nearly crippled economy. Moreover, the length of the inverse of the latent period of COVID-19 is the another negatively sensitive parameter with PRCC index at $-0.43$.

7. Concluding remarks

A methodology has been proposed for the calibration of the key epidemiological parameters, and for the forecasting of the outbreak dynamics of COVID-19 pandemic in Russia, Brazil, India, Bangladesh and the UK, considering the publicly available data from late January 2020 to early May 2020, with an introduction of an real-time differential SEI_DI_UQHRD epidemic model, which can give more accurate, realistic and precise short-term predictions. Baseline parameter ranges are chosen from the recent clinical studies and relevant literature concerning the COVID-19 infection. A newly developed algorithm based on Trust-region-reflective algorithm, which is one of the robust least-squares numerical optimization techniques, has been deployed to calibrate the parameters of the proposed model. With an aim to determine the probable peak dates and sizes for the emerging COVID-19 hotspots named Russia, Brazil, India and Bangladesh, the publicly available COVID-19 data have been analyzed meticulously. Based on the projection results as of May 11, 2020, Russia could reach the peak in terms of daily infected cases and death cases around the end of May. Brazil, Bangladesh and India could reach the peak in terms of daily symptomatic infectious cases and death cases around the middle of June. In addition, the United Kingdom might face a second wave of infection provided that lockdown is lifted early. Global sensitivity analysis results depict that home-quarantine or self-quarantine is the most effective non-pharmaceutical measure for controlling the transmission dynamics of the novel coronavirus disease (COVID-19). To fade out the pandemic, it is compulsory to identify potential carriers and asymptomatic infectious spreaders through massive scale testing scheme. Without sufficient level of evidence of controlled low transmission rate of the COVID-19 in any certain country, we cannot afford lifting up stringent confinement measures and physical distancing regulations.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.