Projections and fractional dynamics of COVID-19 with optimal control strategies.

When the entire world is eagerly waiting for a safe, effective and widely available COVID-19 vaccine, unprecedented spikes of new cases are evident in numerous countries. To gain a deeper understanding about the future dynamics of COVID-19, a compartmental mathematical model has been proposed in this paper incorporating all possible non-pharmaceutical intervention strategies. Model parameters have been calibrated using sophisticated trust-region-reflective algorithm and short-term projection results have been illustrated for Bangladesh and India. Control reproduction numbers ( R c ) have been calculated in order to get insights about the current epidemic scenario in the above-mentioned countries. Forecasting results depict that the aforesaid countries are having downward trends in daily COVID-19 cases. Nevertheless, as the pandemic is not over in any country, it is highly recommended to use efficacious face coverings and maintain strict physical distancing in public gatherings. All necessary graphical simulations have been performed with the help of Caputo-Fabrizio fractional derivatives. In addition, optimal control strategies for fractional system have been designed and the existence of unique solution has also been showed using Picard-Lindelof technique. Finally, unconditional stability of the fractional numerical technique has been proved.

Introduction

Mass-vaccination campaigns have already been launched in several countries amid coronavirus surges. However, many scientists have expressed their concerns regarding the efficacy of approved vaccines potential emergence of virus variants. As worldwide distribution of COVID-19 vaccines is indeed a tedious process, non-pharmaceutical intervention strategies are the realistic and effective solutions to control the new spikes of cases. Generally, an effective vaccine would take years, if not decades, to develop. Lack of transparency could be a vital issue in upcoming days and a false sense of security could evolve among general people if the vaccine does not work effectively.

Mathematical models can always provide considerable insights of the transmission dynamics and complexities of any infectious diseases, which eventually help government officials design overall epidemic planning. Importantly, mathematical analysis always plays a notable role in making vital public health decisions, resource allocation and implementation of social distancing measures and other non-pharmaceutical interventions. From the beginning of the COVID-19 outbreak, mathematicians and researchers are working relentlessly and have already done tremendous contributions in limiting the spread of the coronavirus in different parts of the world [1], [2], [3], [4], [5]. In an early contribution, Ferguson et al. [1] showed the impact of different non-pharmaceutical intervention strategies on COVID-19 mortality by developing an agent-based model. In another study, Ngonghala et al. showed that effective and comprehensive usage of face coverings can significantly limit the spread of the virus and reduce the COVID-induced mortality in different states of the USA in general in the absence of community lockdown measures and stringent social distancing practice. On the other hand, Nabi [6] projected the future dynamics of COVID-19 for various COVID-19 hotspots by proposing a compartmental mathematical model and concluded that early relaxation of lockdown measures and social distancing could bring a second wave in no time. As a matter of reality, inhabitants in several countries compelled to violate containment measures due to prolonged lockdown measures and severe economic recession [3]. Netherlands having one of the best health care systems in the world, is grappling with continuous spikes in daily cases due to aversion to masks.

In this study, in the absence of a safe, effective and world-wide approved vaccine, a new compartmental mathematical COVID-19 model has been designed incorporating all possible non-pharmaceutical intervention strategies such as wearing face coverings, social distancing, home or self-quarantine and self or institutional isolation. In addition, the impacts of different interventions scenarios have been analysed rigorously. The aim of this work is to project the future dynamics of COVID-19 outbreak in two countries namely Bangladesh and India, which are one of the worst-hit countries in the world. Estimation of parameters has been performed by using real-time data, followed by a projection of the evolution of the disease. For fractional simulations, we used the well known non-integer order derivative called Caputo–Fabrizio (CF) fractional derivative [7]. Since last few decades, there are so many epidemic models have been solved by non-integer order derivatives [8]. Recently some applications of non-integer order derivatives in mathematical epidemiology can be seen from [9], [10], [11]. There are so many research papers have been come to study the outbreaks of coronavirus, in which some are [12], [13], [14], [15], [16]. We performed the optimal control problem in CF derivative sense and provided the existence of unique solution by well-known technique named as Picard–Lindelof technique. We also proved the unconditionally stability of the given fractional numerical technique. We used the numerical data of the two given countries and perform the all necessary graphical simulations.

The entire chapter is organized as follows. Materials and methods are presented in Section 2. Section 3 is solely devoted to properties of solutions and asymptotic stability of the proposed model. In Section 4, estimation of model parameters and projection results have been discussed using daily COVID-19 data of Bangladesh and India. In Section 5, numerical and graphical simulations have been illustrated using Caputo–Fabrizio fractional derivatives. Later, optimal control problem has been designed in fractional sense in Section 6. The chapter ends with some insightful findings and strategies, which could significantly control the transmission dynamics of COVID-19.

Materials and methods

Model formulation

A compartmental mathematical has been proposed describing the transmission dynamics of the novel coronavirus incorporating all possible real-life interactions and social behavior. Considering different infection status, the entire human population (denoted by at time ) has been stratified into nine mutually-exclusive compartments of susceptible individuals early-exposed individuals pre-symptomatic individuals symptomatically-infectious asymptomatically-infectious or infectious individuals with mild-symptoms quarantined infectious hospitalised or isolated individuals recovered individuals disease-induced death cases . Hence,

The schematic diagram of the proposed model is illustrated in Fig. 1 , where susceptible individuals can become infected by an effective contact with individuals in the pre-symptomatic symptomatically-infectious asymptomatically-infectious quarantined-infectious and isolated-infectious . Effective contact rates are and respectively and the expressions are defined in (2). Importantly, the compartment consists of early-infected individuals who are still not infectious, whereas the individuals in pre-symptomatic cohort have the capability of transmitting coronavirus before the end of the disease incubation period. A proportion of individuals in newly-exposed compartment progress to pre-symptomatic class at a rate . After the completion of disease mean incubation period, at a rate a fraction of individuals who have clear clinical symptoms of COVID-19 progress to compartment. Individuals in class who do not have any clear symptoms progress to class at a rate . Pre-symptomatic individuals are assumed to be self-quarantined at a rate . With the help of diagnostic or surveillance testing approaches, symptomatically-infectious individuals and asymptomatically-infectious individuals are brought under institutional or home isolation at rates and respectively. Moreover, the parameter represents the recovery rate for individuals in the class. Finally, the disease-induced mortality rate for individuals in the compartment is defined by the parameter . Considering all the above-mentioned interactions, the transmission dynamics of COVID-19 can be described by the following system of nonlinear ordinary differential equations.where the forces of infection are defined belowThe parameters are described in Table 1 .

Fig. 1

Flow diagram of the COVID-19 transmission dynamics.

Table 1

Model parameters and meaning.

Parameter	Description
β_I (β_A) (β_Q) (β_L) (β_E_2)	Effective contact rate (a measure of physical or social distancing)
m	Proportion of individuals who use face coverings or surgical masks
ζ	Efficacy of face coverings at reducing outward transmission by infected individuals as well as preventing acquisition
κ_1	Rate of progression from early-exposed class (E_1(t)) to pre-symptomatic class (E_2(t))
ρκ_2	Rate of progression from pre-symptomatic class (E_2(t)) to symptomatically-infectious class (I(t))
(1−ρ)κ_2	Rate of progression from pre-symptomatic class (E_2(t)) to asymptomatically-infectious class (A(t))
q	Confinement efficacy
τ_I	Rate of self or institutional isolation for symptomatically-infectious patients
τ_A	Rate of isolation for asymptomatically-infectious patients
γ_I	Recovery rate for symptomatically-infectious patients
γ_A	Recovery rate for asymptomatically-infectious individuals
γ_Q	Recovery rate for quarantined-infectious individuals
γ_L	Recovery rate for isolated or hospitalised individuals
δ_I	Disease-induced death rate for symptomatically-infectious individuals
δ_L	Disease-induced death rate for isolated individuals
δ_Q	Disease-induced death rate for quarantined-infectious

We set the vector of state variable, Let the the right hand side of system (1), which is a continuously differentiable function on . According to [17, Theorem III.10.VI], for any initial condition in a unique solution of (1) exists, at least locally, and remains in for its maximal interval of existence [17, Theorem III.10.XVI]. Hence, model (1) is biologically well-defined.

Data sources

From the beginning of the COVID-19 outbreak, Center of Disease Control and Prevention (CDC) is providing authoritative and genuine data of daily confirmed COVID-19 cases continuously and it is indeed a trustworthy data repository source. Daily reported data of Bangladesh and India have been compiled using the source. Johns Hopkins University Center for Systems Science and Engineering (JHU CSSE) is carefully maintaining the data repository supported by ESRI Living Atlas Team and the Johns Hopkins University Applied Physics Lab (JHU APL). The repository is really convenient to use and publicly available [18].

Mathematical analysis

Positivity and boundedness of solutions

Here, we prove that all state variables of model (1) are non-negative for all time, i.e, solutions of the model (1) with positive initial data remain positive for all time . The following result can be obtained.

Solutions of COVID-19 model(1)with positive initial conditions are positive for all.

Assume that is a solution of (1) with positive initial conditions. Let us consider for . It follows from the third equation of system (1) thatSince it follows that for . We proceed with the same for and . □

Solutions of COVID-19 model(1)with positive initial conditions are bounded by the total population.

Let . Thus, in absence of disease, we havewhere is equal to the total population. It follows that for all we have and with . □

In what follows, we study the model (1) in the following setwhich is positively-invariant and attracting region for the model (1).

Asymptotic stability of disease-free equilibria

The disease-free equilibrium point denoted by can be defined as follows:

Using notations in Riessche and Watmough [19], matrices and for the new infection terms and the remaining transfer terms are, respectively, given byThen, the control reproduction ratio is defined, following [19], [20], as the spectral radius of the next generation matrix, :where,where represents the spectral radius operator.

The formula for control reproduction number has been formulated. Indeed, the insightful epidemic threshold, calculates the average number of new secondary COVID-19 cases generated by a COVID-19 positive individual in a population a portion susceptible people are using effective face coverings. Different non-pharmaceutical measures are acting as control measures which lead to bring down under unity [19]. Hence, we claim the following result followed by a direct consequence of the next generation operator method [19, Theorem 2]. where represents the spectral radius operator. The insightful epidemic threshold, calculates the average number of new secondary COVID-19 infections generated by an COVID-19 positive patient in a completely susceptible population. The control of COVID-19 pandemic passes by the application of some control measures which contribute to decrease until less than one [19]. Hence we claim the following result.

The COVID-19 transmission dynamics is influenced by the basic reproduction numberas follows:

Ifthen a sufficiently small flow of infected individuals will not generate an outbreak of the COVID-19,i.e the disease-equilibriumis locally asymptotically stable on.

Ifthen a sufficiently small flow of infected individuals will generate an outbreak of the COVID-19, and the disease-equilibriumis unstable.

Ifthe disease-free equilibriumis locally asymptotically stable and unstable if.

Lemma 3 implies that if then a sufficiently small flow of infected individuals will not generate an outbreak of COVID-19, whereas for epidemic curve reaches a peak by growing exponentially and then decreases to zero as .

The better control of the COVID-19 can be established by the fact that the DFE is globally asymptotically stable (GAS). In this context, we claim the following result.

ifthen the manifold,of disease-free equilibrium points of the model(1)is GAS in.

In the absence of use of face coverings, i.e. converges to the basic reproduction number, . Now, we will study the global stability of the disease-free equilibrium whenever the basic reproduction number is less than one (). For this, we use the following Lyapunov functionBy deriving this function along the trajectories of the system (1), we obtainSince we obtainWe choose such that coefficients of and become zero. That iswhich the non-zero solution is given byPlugging (9) into (7) givesSetting we finally obtainFrom (11), it follows that if and if and only if . Therefore, is a Lyapunov function for system (1). Moreover, the maximal invariant set contained in is the continuum of the disease-free equilibrium . Thus, from Lyapunov theory, we deduce that the disease-free equilibrium is GAS if . Hence, it follows, by the LaSalle’s Invariance Principal, that the continuum of disease-free equilibria of the model (1) is a stable global attractor in whenever The previous analysis can be summarize as follows:

Ifthen the disease-free equilibriumis globally asymptotically stable on.

Model calibration and forecasting

The model (1) calibration has been performed using a newly developed optimization algorithm based on trust-region-reflective (TRR) algorithm, which can be regarded as an evolution of Levenberg-Marquardt algorithm [6]. This robust optimization procedure can be used effectively for solving nonlinear least-squares problems. This algorithm has been implemented using the lsqcurvefit function, which is available in the Optimization Toolbox in MATLAB. Necessary model parameters have been estimated using this optimization technique. Daily infected cases data have been collected from a trusted data repository, which is available online. A 7-day moving average of the daily reported cases has been used for our model calibration due to moderate volatile nature of real data. It has been observed that the number of daily testing in Bangladesh and India have been really inconsistent. With an aim to capture the real outbreak scenario, the 7-day moving average has been used in this regard.

Bangladesh

Due to prolonged lockdown measures and severe economic recession, inhabitants of Bangladesh have started violating safety measures like wearing face coverings and maintaining physical distancing. Figs. 2 and 3 illustrate the model fitting performance with observed data from early March 2020 to early January 2021 for Bangladesh. The estimated error is found to be hovering around for daily new cases. The actual outbreak scenario in Bangladesh is still a puzzle to be solved for the health officials due to scant COVID-19 testing program. The control reproduction number is estimated to be as of January 05, 2021 and prior established findings for this metric go well with the estimation [6], [21]. Figs. 4 and 5 enlighten that the tally of cumulative infected cases is projected to reach around March 31, 2021 and the estimated total death cases could reach 11,400 by the end of March, 2021. Table 2 illustrates the key features used to calibrate this scenario, which have been justified in prior clinical studies and relevant literature.

Fig. 2

Fitting performance of the model for daily infected cases in Bangladesh from March 08, 2020 to January 05, 2021.

Fig. 3

Fitting performance of the model for cumulative infected cases in Bangladesh from March 08 to January 05, 2021.

Fig. 4

Projection results for daily new confirmed cases for Bangladesh from early March 2020 to late March 2021.

Fig. 5

Projection results for cumulative cases for Bangladesh early March 2020 to late March 2021.

Table 2

Calibrated parameters of the proposed model (1) using trust-region-reflective algorithm and daily COVID-19 cases data of Bangladesh.

Parameter	Range (Unit)	Baseline value	TRR output	Reference
β_I	0.1–1.5 day^−1	0.55	0.15	[6], [22]
β_A	0.1–0.9 day^−1	0.3	0.1	[6], [22]
β_Q	0.1–0.9 day^−1	0.5	0.1	[6], [22]
β_L	0.1–0.9 day^−1	0.3	0.12	[5], [6]
β_E_2	0.05–0.3 day^−1	0.3	0.11	[5]
m	0.01–0.3 (dimensionless)	0.1	0.3	[5]
ζ	0.5 (dimensionless)	0.5	0.5	[5]
κ_1	14 day^−1	14	14	[2], [5]
κ_2	1 day^−1	1	1	[2]
q	0.1–0.6 day^−1	0.3	0.47	Estimated
ρ	0.6–0.7 (dimensionless)	0.65	0.65	[5]
τ_I	114−15 day^−1	1/10	1/8	[5], [6]
τ_A	114−15 day^−1	1/10	1/8	[5], [6]
γ_I	114−17 day^−1	1/7	1/12	[23], [24]
γ_A	110−17 day^−1	1/7	1/10	[23], [24]
γ_Q	121−110 day^−1	1/21	0.071	[23], [24]
γ_L	121−110 day^−1	1/21	0.071	[23], [24]
δ_I	0.0001–0.01 day^−1	0.001	0.0004	[1], [6]
δ_L	0.0001–0.01 day^−1	0.001	0.0009	[1], [6]
δ_Q	0.0001–0.01 day^−1	0.001	0.0007	[1], [6]

India

When India is celebrating a busy festival season, the tally of fresh COVID-19 cases continued to rise. Relaxation in protective and social-distancing measures could result in a significant upsurge in daily cases in upcoming days. Fig. 6 illustrates the fact that India is witnessing a downward trend after having peak. According to our projection results, India could reel under the second wave of infection unless non-pharmaceutical interventions strategies are followed comprehensively. As of January 09, 2021 the country’s caseload now stands at 10,448,134 and it’s death toll has mounted to 151,000. Figs. 6 and 7 illustrate the fitting performance of our proposed model for India from late January 2020 to early January 2021. Historical data from January 30, 2020 to January 05, 2021 have been considered to calibrate the model parameters. As we can see from the figures, model-fitting is exceptionally well for the historical observed data. Based on our projection results from Fig. 8 , the number of daily cases could be brought under 1000 cases if mass-level efficacious face coverings is strictly maintained. The control reproduction number is estimated to be as of January 09, 2021 complementing the prior studied observations [6], [21]. Fig. 9 depicts that the tally of cumulative infected cases is projected to reach by the end of March 2021 if current trend continues. It has also been enlightened in a recent study [25], [26] that, comprehensive usage of efficacious face coverings could be the most influential strategy to control the COVID-19 airborne transmission. In addition, country’s death toll could mount to over the period. Table 3 illustrates the key features used to calibrate this scenario.

Fig. 6

Fitting performance of the model for daily infected cases in India from January 30 to January 05, 2021.

Fig. 7

Fitting performance of the model for cumulative infected cases in India from January 30 to January 05, 2021.

Fig. 8

Projection results for daily new confirmed cases for India from late January 2020 to late March 2021.

Fig. 9

Projection results of cumulative cases for India from late January 2020 to late March 2021.

Table 3

Calibrated parameters of the proposed model (1) using trust-region-reflective algorithm and daily COVID-19 cases data of India.

Parameter	Range (Unit)	Baseline value	TRR output	Reference
β_I	0.1–1.5 day^−1	0.55	0.18	[6], [22]
β_A	0.1–0.9 day^−1	0.3	0.13	[6], [22]
β_Q	0.1–0.9 day^−1	0.5	0.2	[6], [22]
β_L	0.1–0.9 day^−1	0.3	0.22	[5], [6]
β_E_2	0.05–0.3 day^−1	0.3	0.15	[5]
m	0.01–0.3 (dimensionless)	0.1	0.23	[5]
ζ	0.5 (dimensionless)	0.5	0.5	[5]
κ_1	14 day^−1	14	14	[2], [5]
κ_2	1 day^−1	1	1	[2]
q	0.1–0.6 day^−1	0.3	0.25	Estimated
ρ	0.6–0.7 (dimensionless)	0.65	0.65	[5]
τ_I	114−15 day^−1	1/10	1/8	[5], [6]
τ_A	114−15 day^−1	1/10	1/9	[5], [6]
γ_I	114−17 day^−1	1/7	1/12	[23], [24]
γ_A	110−17 day^−1	1/7	1/10	[23], [24]
γ_Q	121−110 day^−1	1/14	1/16	[23], [24]
γ_L	121−110 day^−1	1/14	1/18	[23], [24]
δ_I	0.0001–0.01 day^−1	0.001	0.0002	[1], [6]
δ_L	0.0001–0.01 day^−1	0.001	0.00032	[1], [6]
δ_Q	0.0001–0.01 day^−1	0.001	0.0003	[1], [6]

Solution of the model in Caputo–Fabrizio fractional derivative sense

Preliminaries

Here we recall the definitions of Caputo and Caputo–Fabrizio fractional derivatives.

Podlubny [27] The Caputo definition of non-integer order derivative of order of a function is defined bywhere and is the integer part of

Jajarmi and Baleanu [28], Losada and Nieto [29]

For and the Caputo–Fabrizio (CF) fractional derivative (FD) of order is defined bywhere

The CF non- integer order integral is defined as

Existence and uniqueness analysis

Now, we prove the existence of unique solution for the given COVID-19 model in the sense of Caputo–Fabrizio fractional derivative by the application of fixed-point theory. In this concern, the proposed system can be rewritten in the equivalent form as follows:By applying the CF non-integer order integral operator, the above system (13), reduces to the following integral equation of Volterra type of order Now, we get the subsequent iterative algorithmHere we assume that we can get the exact solution by taking the limit as n tends to infinity.

Existence analysis by using Picard–Lindelof approach

Let us considerwhereconsidering the Picard operator as

given as follows:where and Next we assume that the solution of the non-integer order model are bounded within a time period, where we demand that

Now by the application of fixed point theorem pertaining to Banach space along with the metric, we obtain

Now we havewith Since is a contraction, we have hence the given operator is also a contraction. Therefore, the model involving C-F derivative given in Eq. (13) has a unique set of solution.

Solution method in Caputo–Fabrizio operator

We now derive the solution method for equation of the system (13) and for the rest of the equations it will be similar. The corresponding Volterra integral equation for is as follows.We have the following estimations at and at Subtracting Eq. (21) from (20), we obtainThen by linear interpolation about and applying trapezoid rule for integration on the integral term, we can then writewhere . Hence, we have the numerical approximation for equation of as

The numerical approximation(24)is unconditionally stable if

Let be the solution of a differential equation as shown in (19) under CF non- integer order derivative operator sense. Then we have to evaluate the norm

For we have

Clearly, the second term of the above inequality approaches zero when . Now, if as we educe that the numerical solution is stable. □

Graphical simulations

After giving all necessary theoretical concerns, graphical simulations have been done by using CF fractional derivative. In our paper, we have used the real numerical data of COVID-19 for two different countries named as Bangladesh and India respectively. To perform numerical simulations for Bangladesh, we have used calibrated parameter values summarized in Table 2. In the family of Fig. 10 , plots of and have been analysed. We have observed that for different fractional order values peaks are well defined and when we decrease the fractional order then the peaks shifted towards the later time period. In the group of Figs. 11 and 12 , first we have illustrated the nature of and then analysed the plots of versus and . In the comparison of given classes with infected individuals, we have observed that the nature of infectious is same as for India, as when the population of infected individuals increases then asymptomatic infectious also increases with same nature. In sub-Figs. 11d and 12 b, we see that the fractional order does not play any big role because the nature of the classes is nearly same at all different fractional order values . Initial values of given classes for Bangladesh are and We have used the total population of the country for .

Fig. 10

Plots of and for Bangladesh data.

Fig. 11

Plots of and relationship of versus for Bangladesh data.

Fig. 12

versus for Bangladesh data.

We have done the graphical simulations for India which is the second highest populous country in the world and also the second worst-hit country by COVID-19. To study the outbreak of COVID-19 in India, in the family of Fig. 13, Fig. 14, Fig. 15 , we exemplified the all necessary graphs of given classes to observe the dynamics of COVID-19. To perform numerical simulations, we took the numerical values from the Table 3. In the family of Fig. 13, we analysed the plots of and . We observed that the nature of peaks is mostly same as for other above analysed data, for different fractional order values peaks are well defined and when we decrease the fractional order then the peaks sifted towards the later time period. In the collection of Figs. 14 and 15, first we show the nature of and then analysed the plots of versus and . When we compare the given classes with we again observed that when the population of infected individuals increases then asymptomatic infectious also increases with same nature. In sub-Figs. 14d and 15 b, we observed that at the different fractional order values the nature of the classes is nearly same. Initial values of given classes for India are and We have used the of the total population of India for .

Fig. 13

Plots of and for India data.

Fig. 14

Plots of and relationship of versus for India data.

Fig. 15

versus for India data.

From the all above graphical observations we found that the Caputo–Fabrizio fractional derivative playing well to study the outbreaks of coronavirus in the aforesaid two countries.

Optimal control problem formulation

In this concern, our main aim is to decrease the number of infected individuals with COVID-19 at the same time decrease the cost associated with their strategies. For this purpose, we use a control function where is for introducing the public education or aware the public with health-care measures, is the control function for enhancement of the strength of treatment for the infected individuals, is the control function for the necessary suggestions of health care measures for those who are in asymptomatic infectious class and yet not admitted in the hospital.To define the optimal control problem (OCP), we are excluding the death equation because there is no significance of deaths in optimal controls. Now consider the state system given in (26) in with the set of admissible control function. So the objective functional is defined bywhere the constants and are a measure of associative cost with the controls and . Then we find the optimal controls and to minimize the cost function.subject to constraint, where and the given initial coordination are agreed

Now let us take the following modified cost functionwhere and Hence the Hamiltonian is defined as follows:where and from Eqs. (29) and (30), the necessary and sufficient conditions for the functional optimal control problem (FOCP) are given as:

Moreover, are the lagseuges multipliers Eqs. (31) and (32) express the necessary condition in terms of a Hamiltonian for the OCP defined above.

Optimality conditions for fractional order

Let us write the Hamiltonian function as follows:

where representing the lagragars multipliers called co-states.

If and are optimal controls of the given OCP if are corresponding optimal paths, then there exists co-state variables such that besides the given control system is satisfied, the following conditions are satisfied:

Co-state equations: with transversality conditions and optimality conditions given by

The adjoint system (34) i.e are obtained from the Hamiltonian as

with zero final time conditions or transversality conditions,

and and the characteristic of the fractional optimal control given by (35) is obtained by solving the Eqn on the interior of the control set and using the property of control space . □

Conclusions

Different mathematical paradigms can provide considerable insights and scientific evidences pertinent to any ongoing epidemic dynamics. Based on those valuable information, health officials and public health experts can set up potential control strategies to battle against any epidemic. From the emergence of the novel coronavirus in China, researchers and scientists are working relentlessly to develop various mathematical modeling approaches to gain a deeper understanding on the progression dynamics of COVID-19 in the world. In addition, in the absence of any safe, effective and widely available COVID-19 vaccine, different preventive measures are the most effective tool in combating against the virulent virus. On the basis of robust forecasting results of reliable epidemiological models, government officials can deploy different public health intervention strategies to control the rapid transmission of the virus. In this paper, a compartmental mathematical has been designed to describe the transmission dynamics of the COVID-19 incorporating all possible real-life interactions and effective non-pharmaceutical interventions. Disease-free equilibrium (DFE) of the proposed model is found to be globally asymptotically stable (GAS), whenever control reproduction number less than unity. In addition, advanced forecasting techniques have also been applied for Bangladesh and India to portray the future dynamics of the pandemic in near term. It has been enlightened in our study that mass-level using of highly effective face coverings could be a crucial factor in controlling the spread of coronavirus. Moreover, strict social-distancing measures and comprehensive contact-tracing are also effective strategies in battling against this pandemic. The public health implication of these insightful findings is government officials can undertake crucial clinical and public health decisions by analyzing all mathematical results and scientific evidences. Caputo–Fabrizio non-integer order derivative has been applied to solve the proposed mathematical model in fractional sense. We proved the existence of unique solution for the proposed fractional initial value problem. We proved the unconditional stability of the given technique. An important concern of fractional optimal control problem is given to suggest the heath care measures for reducing the transmissibility of COVID-19 infection in the population.

Declaration of Competing Interest

This work does not have any conflicts of interest.