A Fractional Order Model to Study the Effectiveness of Government Measures and Public Behaviours in COVID-19 Pandemic

Citation: Das, M.; Samanta, G.; De la Sen, M. A Fractional Order Model to Study the Effectiveness of Government Measures and Public Behaviours in COVID-19 Pandemic. Mathematics 2022,10, 3020. https:// doi.org/10.3390/math10163020 Academic Editor: Sergey A. Lashin Received: 26 July 2022 Accepted: 19 August 2022 Published: 22 August 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). mathematics Article A Fractional Order Model to Study the Effectiveness of Government Measures and Public Behaviours in COVID-19 Pandemic Meghadri Das 1, Guruprasad Samanta 1and Manuel De la Sen 2,* 1Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, India 2Institute of Research and Development of Processes, University of the Basque Country, 48940 Leioa, Spain *Correspondence: [email protected] Abstract: In this work, we emphasise the dynamical study of spreading COVID-19 in Bangladesh. Considering the uncertainty caused by the limited coronavirus (COVID-19) information, we have taken the modified Susceptible-Asymptomatic-Infectious-Hospitalised-Recovered (SAIHR) compartmental model in a Caputo fractional order system. We have also introduced public behavioural and government policy dynamics in our model. The dynamical nature of the solutions of the system is analysed and we have also calculated the sensitivity index of different parameters. It has been observed that public behaviour and government measures play an important role in controlling the pandemic situation. The government measures (social distance, vaccination, hospitalisation, awareness programme) are more helpful than only public responses to the eradication of the COVID-19 pandemic. Keywords: Caputo fractional differential equation; COVID-19; SAIHR compartmental model; stability ; sensitivity index MSC: 9208; 26A33; 37C75 1. Introduction Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) is the infectious agent that causes Coronavirus Disease 2019 (COVID-19), which was initially discovered in China in early December 2019. Since then, it has spread worldwide, destroying the health, economy, and lives of billions of people. This has made it clear how important it is to accurately represent infectious illnesses. In reality, statistical studies are generally based on nonlinear mathematical models, which deal with epidemiology, and mostly determine worldwide government policies. The COVID-19 pandemic was first confirmed in Bangladesh on 8 March 2020. The Health and Family Welfare department, Government of the People’s Republic of Bangladesh, has confirmed a total of 1,962,213 COVID-19 positive cases and 29,135 deaths from 3 January 2022 to 23 June 2022 [ 1 ]. A total of 274,923,522 vaccine doses have been administrated. The government of Bangladesh took actions, such as social distancing, mask-wearing, travelling restrictions, lockdowns, vaccination, and hospitalisation, to control the COVID-19 situation [2]. Fractional calculus is a parallel branch of calculus that cannot be considered a generalised version of integer order calculus [ 3 , 4 ]. Fractional order systems are more appropriate than integer order systems in many fields and can express phenomena that are linked to memory and affected by hereditary properties [ 5 , 6 ]. In endemic and epidemic areas, people’s awareness of infection will reduce the rate of contact between various compartments, such as between humans and mosquitoes in the dengue SIR-SI model [ 7 ], whereas in the Mathematics 2022,10, 3020. https://doi.org/10.3390/math10163020 https://www.mdpi.com/journal/mathematics Mathematics 2022,10, 3020 2 of 17 epidemic model with vaccination, people who have received vaccinations have a stronger tendency to be aware of previous epidemics than people who are susceptible. In order for specialists to gain the most knowledge from the available data before making major judgments, there should be a systematic way to combine the models and observations. This overview refers to some obstacles in these models and looks at some intriguing approaches focusing on the development of general structures for such models, and proposes an alternative approach, namely fractional calculus, whose main contrast to integer order models is where such effects are overlooked or difficult to integrate. The fractional derivative formalism of epidemic models provides a useful tool for incorporating memory and hereditary features of systems. Furthermore, the fractional models have one more degree of freedom than the integer order model for fitting data. Examining numerous papers on fractional epidemic models and count models based on dynamics with fractional order derivatives, we have proposed several mathematical models on epidemiology and suggest that developing numerical tools for fitting mathematical models to actual data will assist concerned authorities in avoiding or control infectious disease outbreaks. Aside from these benefits, there are some drawbacks to using the Caputo fractional system. 1. Finding analytical solutions is difficult for Caputo differential systems. 2. There are many concepts, such as bifurcation theory, parametric optimisation, persistence, etc., that have not yet been developed for Caputo fractional order systems. 3. The numerical algorithms for delayed systems and stochastic fractional systems have not yet been developed. Considering all of the above facts, we have constructed our model in the Caputo differential framework. In this context, the works of Das et al. [ 8 ] and Das and Samanta [9–11] on fractional order dynamics may be mentioned. Significant contributions have recently been made by several researchers to the various COVID-19 models in both integer and fractional order systems [12–18]. Fractional order modelling is a useful approach for studying the nature of diseases because it is an extension of the integer-order derivative. The fractional order system also adds an extra parameter that can be used to improve numerical simulations. In this model, we have considered a new infection function that includes the strength of the government action and the strength of public response. This infection function has not yet been used in previous works of fractional order systems. We have also studied the public response to the spreading of COVID-19 disease. Our main objective is to study the effect of public behaviour and governmental measures on disease spreading. We have modified the contemporary SAIHR model by constructing a new infection function and introducing a new state variable depicting social behavioural dynamics of public awareness. There are several models on COVID-19, but our model is truly different from the others and may answer new queries. We have taken Caputo fractional derivative because the proposed system is autonomous in nature, which has been made physically meaningful by dimensional homogeneity among the fractional order Caputo derivatives and the parameters used in the system. The Caputo system can be simulated easily using Adams devised technique (FDE12) for finding approximate solutions of fractional Caputo ordinary differential equations. In this work, a modified SAIHR model is formulated emphasising how the government measures and public behaviour control the disease spread. Section 2contains the proposed model on COVID-19 with non-negative initial conditions. Section 3describes the dynamical nature of solutions in two different scenarios. Section 4deals with the sensitivity indices of different parameters. Section 5shows numerical evidence of the dynamical nature of the proposed model supporting analytical results. The work ends with a brief conclusion. 2. Model Construction A five compartmental model under a Caputo fractional order framework has been constructed, and the model is composed of susceptible (S) , asymptomatically infected (Ia) , symptomatically infected (Is) , hospitalised (H) and recovered (R) classes. In the context Mathematics 2022,10, 3020 3 of 17 of COVID-19, the infected class is divided into two sub-classes, namely asymptomatic and symptomatic, they are denoted by Ia and Is , respectively. According to some reports, recovery from the disease does not guarantee permanent recovery, so some of the recovered people revert back to the susceptible class at a constant rate ζ[19]. The model is given by C t0Dε tS(t) = Λε−Φ−µεS+ζεR, C t0Dε tIa(t) = Φ−(µε+σε)Ia, C t0Dε tIs(t) = σεIa−(µε 1+ρε 1+ρε 2)Is, C t0Dε tH(t) = ρε 1Is−(µε 2+γε)H, C t0Dε tR(t) = ρε 2Is+γεH−µεR−ζεR, C t0Dε tQ(t) = dρε 2Is−λεQ (1) where we have defined an infection function Φas follows: Φ= (1−α)[βε 1SIs(1−Q)κ+βε 2SIa] (2) In this function, α represents the strength of government action, and κ represents the strength of public response. It is worth noting that Q is a new state variable that represents the social behavioural dynamics. The term d represents the strength of the public perception of risk, λ−1 is the mean period of public response, and the model takes into account the fact that public reaction will increase as more people become infected and will naturally decrease over time. C t0Dε t denotes the Caputo fractional derivative with initial time t0 . Here all equations of system (1) are balanced with respect to the time dimension. For the sake of simplicity, we discard all the powers ε from the parameters. All parameters containing ε as power trace are taken into account for an impact in numerical analysis. We have discarded the power ε for analytical purposes only. The descriptions of all parameters are given in Table 1. Table 1. Description of biological interpretation of model parameters. Parameter Interpretation Values (Range) Reference Λrecruitment rate of the human population 0.001 [20] β1rate of infection per unit of time by the symptomatic infected Is0.35 (0.005–0.34) [21] β2reduction factor of infected population by the Iaclass compared to Isclass 0.32 (0.005–0.34) [21] σrate at which asymptomatic becomes symptomatic 0.025 (0.02–0.1) [21] ρ1rate at which the symptomatic infected individuals are hospitalised 0.07 Assumed ρ2rate of recovery of the symptomatic infected individuals 0.14 Assumed µ1rate of mortality of symptomatic infected individuals 0.05 (0.05–0.1) [20] ζrate of retreat from recovered class to susceptible class 0.1 [21] µ2rate of mortality of hospitalised individuals 0.07 [20] γrate of transfer of hospitalised individuals to recovered class 0.05 Assumed εorder of fractional derivative 0.95 (0–1) Assumed Mathematics 2022,10, 3020 4 of 17 3. Basic Nonlinear Analysis Definition 1 ([ 4 ]) . The Caputo fractional derivative operator of order ε for an absolutely continuous function g ∈Cn([0, ∞+),IR)is defined as: C 0Dε tg(t) =            1 Γ(n−ε)Zt 0 g(n)(s) (t−s)ε−n+1ds,ε∈(n−1, n),n∈N dn dtng(t),ε=n. where Γ(·)is the Gamma function, t ≥0, and n is a natural number. In particular, for ε∈(0, 1): C 0Dε tg(t) = 1 Γ(1−ε)Zt 0 g0(s) (t−s)εds Theorem 1 ([4]).Consider: C 0Dε tx(t) = Ψ(x), with ε∈( 0, 1 ) , x∈Rn . The equilibrium points (of this system) are solutions of the equation Ψ(x) = 0. If for all eigenvalues ( λi ) of the Jacobian matrix J , |arg(λi)|>επ 2 ,the equilibrium is locally asymptotically stable, where J =∂Ψ ∂xis calculated at the equilibrium point. 3.1. Case 1: Model without Control In this case, we can omit the last equation of system (1), and the infection function is taken as: Φ= [β1SIs+β2SIa] (3) The equilibrium points of the system (1) are mentioned below. 1. Disease-free equilibrium: E0=Λ µ, 0, 0, 0, 0 2. Endemic equilibrium: E1= (S∗,I∗ a,I∗ s,H∗,R∗). Here S∗=(µ1+ρ1+ρ2)(µ+σ) β1σ+β2(µ1+ρ1+ρ2) I∗ a=(µ1+ρ1+ρ2) σI∗ s I∗ s= µ(µ+σ)(µ1+ρ1+ρ2) β2(µ1+ρ1+ρ2) + σβ1 (R0−1) (µ+σ)(µ1+ρ1+ρ2) σ− ζρ2+γρ1 µ2+γ µ+ζ H∗=ρ1 µ2+γI∗ s R∗=ρ2+γρ1 µ2+γ1 µ+ζI∗ s, (4) where R0=λ µ(µ+σ)hβ+σβ1 µ1+ρ1+ρ2i.(5) Mathematics 2022,10, 3020 5 of 17 The necessary and sufficient conditions for the existence of E1 , in the feasible region in R5 , are as follows: (µ+σ)(µ1+ρ1+ρ2)(µ+ζ)>ζσρ2+γρ1 µ2+γ and R0> 1, or, (µ+σ)(µ1+ρ1+ρ2)(µ+ζ)<ζσρ2+γρ1 µ2+γ and R0< 1 . Here, R0 is the reproduction number for the uncontrolled scenario calculated at disease-free equilibrium by the nextgeneration matrix method. The next-generation matrix FV−1 at disease-free equilibrium E0is given as follows [22]: F=       β2Λ µβ1Λ µ 0 0       V=    µ+σ0 −σ µ1+ρ1+ρ2     Thus, we get R0=λ µ(µ+σ)hβ+σβ1 µ1+ρ1+ρ2i Theorem 2. The disease-free equilibrium E0=Λ µ, 0, 0, 0, 0 is asymptotically stable if the roots (η) of the following equation satisfy |arg(η)|>επ 2: µη2+ηc1+c2=0, c1= [2µ2+ρ1µ+ρ2µ−β2Λ−σµ], c2=µ(µ+σ)(µ+ρ1+ρ2)−β2Λ(µ+ρ1+ρ2+σ). (6) Proof. To study the local stability of disease-free equilibrium point E0=Λ µ, 0, 0, 0, 0 , we have to compute Jacobian matrix Jat E0. J(E0) =        −µ−β2Λ µ−β1Λ µ0ζ 0β2Λ µ−µ−σ β1Λ µ0 0 0σ−(µ1+ρ1+ρ2)0 0 0 0 ρ1(µ2+γ)0 0 0 ρ2γ−(µ+γ)        The characteristic equation of J(E0)is (λ+µ)(λ+µ+ζ)(λ+µ2+γ)Υ(λ) = 0, (7) where Υ(λ) = µλ2+λc1+c2, c1= [2µ2+ρ1µ+ρ2µ−β2Λ−σµ], c2=µ(µ+σ)(µ+ρ1+ρ2)−β2Λ(µ+ρ1+ρ2+σ). (8) The three roots of the characteristic Equation (6) are −µ,−(µ+ζ),−(µ2+γ), the diseasefree equilibrium is stable if |arg(λi)|>επ 2 , where λi , i= 1, 2 are the roots of Υ(λ) = 0. Mathematics 2022,10, 3020 6 of 17 To analyse the local stability of endemic equilibrium E∗ 1, we need the following: Definition 2 ([ 23 ]) . The discriminant ∇(f) of a polynomial f(x) = xn+α1xn−1+α2xn−2+ ... +αnis defined as: ∇(f) = (−1) n(n−1) 2|Sn(f,f0)|, Sn(f , g) is the Sylvester matrix of f(x) and g(x) of order (n+l)×(n+l) , where g(x) = xl+β1xl−1+β2xl−2+... +βl. For n=3, we have f(x) = x3+α1x2+α2x+α3and f0(x) = 3x2+2α1x+α2. |S3(f,f0)|=  1α1α2α30 0 1 α1α2α3 3 2α1α20 0 0 3 2α1α20 0 0 3 2α1α2  =−18α1α2α3−(α1α2)2+4α2 1α3+4α2 2+27α2 3 Hence ∇(f) = −|S3(f,f0)|=18α1α2α3+ (α1α2)2−4α2 1α3−4α2 2−27α2 3(9) Theorem 3 ([ 24 ]) . If ∇(P) is the discriminant of the characteristic equation P(λ)≡λ3+a1λ2+ a2λ+a3= 0of the Jacobian matrix of system ( 1 ) evaluated at the endemic equilibrium point E1= (S∗,I∗ a,I∗ s,H∗,R∗), where a1=µ1+ρ1+ρ2+−β2S∗+2µ+σ+β1I∗ s+β2I∗ a a2= (µ1+ρ1+ρ2)(µ+σ−β2S∗)−σβ1S∗+β1β2I∗ sS∗+β2 2I∗ aS∗ +(µ1+ρ1+ρ2−β2S∗+µ+σ)(µ+β1I∗ s+β2I∗ a) a3= (µ+σ+β1I∗ s+β2I∗ a)[(µ1+ρ1+ρ2)(µ+σ−β2S∗)−σβ1S∗] + (β1I∗ s+β2I∗ a)(µ1+ρ1+ρ2+σβ1S∗) then the system is asymptotically stable if any of the following conditions hold: 1. ∇(P)>0, a1>0, a3>0and a1a2>a3 2. ∇(P)<0, a1≥0, α2≥0, a3>0and α<2 3 3. ∇(P)<0, a1>0, a2>0, a1a2=a3and α∈(0, 1). 3.2. Case 2: Model with Effects of Governmental Action and Additional Control Let us consider the combined effects of government action along with public perceptivity of risk regarding severe and critical cases. The variable Q is added to the model (system 1), which represents the public perception of risk. The value of Q increases when more infection occurs and decreases naturally. The intensity of perception of risk ( d ) is connected to the intensity of population (public) response κ . The infection function is mentioned in (2). Mathematics 2022,10, 3020 7 of 17 Theorem 4. If the intensity of public perception has no impact, the steady-state endemic state is as follows: S∗∗ =(µ+σ)(µ1+ρ1+ρ2) (1−α)[σβ1+β2(µ1+ρ1+ρ2)] I∗∗ a=(µ1+ρ1+ρ2) σI∗∗ s I∗∗ s= µ(µ+σ)(µ1+ρ1+ρ2) (1−α)[β2(µ1+ρ1+ρ2) + σβ1](R0−1) (µ+σ)(µ1+ρ1+ρ2) σ− ζρ2+γρ1 µ2+γ µ+ζ , provided (µ+σ)(µ1+ρ1+ρ2)(µ+ζ)>ζσ(ρ2+γρ1 µ2+γ),R0>1 or (µ+σ)(µ1+ρ1+ρ2)(µ+ζ)<ζσ(ρ2+γρ1 µ2+γ),R0<1 H∗∗ =ρ1 µ2+γI∗∗ s R∗∗ =ρ2+γρ1 µ2+γ1 µ+ζI∗∗ s, Q∗∗ =dρ2I∗∗ s λ Proof. The steady-state conditions lead to S∗∗(1−α)σβ11−dρ2I∗∗ s λ+β2(µ1+ρ1+ρ2)= (µ+σ)(µ1+ρ1+ρ2) (10) (1−α)[β1S∗∗ I∗∗ s(1−Q∗∗)κ+β2S∗∗ I∗∗ a]= (µ+σ)µ1+ρ1+ρ2 σI∗∗ s(11) I∗∗ a=(µ1+ρ1+ρ2) σI∗∗ s H∗∗ =ρ1 µ2+γI∗∗ s R∗∗ =ρ2+γρ1 µ2+γ1 µ+ζI∗∗ s, Q∗∗ =dρ2I∗∗ s λ (12) Substituting the value of S∗∗ into Equations (10) and (11), we arrive at a transcendental equation that would not lead to an explicit expression of I∗∗ s . In this context, we limit the analysis for κ=0. In this situation, we have found the following steady-state: Mathematics 2022,10, 3020 8 of 17 S∗∗ =(µ+σ)(µ1+ρ1+ρ2) (1−α)[σβ1+β2(µ1+ρ1+ρ2)] I∗∗ a=(µ1+ρ1+ρ2) σI∗∗ s I∗∗ s= µ(µ+σ)(µ1+ρ1+ρ2) (1−α)[β2(µ1+ρ1+ρ2) + σβ1](R0−1) (µ+σ)(µ1+ρ1+ρ2) σ− ζρ2+γρ1 µ2+γ µ+ζ H∗∗ =ρ1 µ2+γI∗∗ s R∗∗ =ρ2+γρ1 µ2+γ1 µ+ζI∗∗ s, Q∗∗ =dρ2I∗∗ s λ, (13) provided (µ+σ)(µ1+ρ1+ρ2)(µ+ζ)>ζσ(ρ2+γρ1 µ2+γ),R0>1 or (µ+σ)(µ1+ρ1+ρ2)(µ+ζ)<ζσ(ρ2+γρ1 µ2+γ),R0<1 4. Sensitivity Analysis To examine the sensitivity of R0 to any parameter (say, θ ), a normalised forward sensitivity index with respect to each parameter has been computed as follows [22,25]: ΩR0 θ=∂R0 ∂θ θ R0 In a numerical (or other) model, sensitivity analysis (SA) is a technique that measures how the effects of uncertainties in one or more input variables can lead to uncertainties in the output variables. The values of sensitivity indexes for the parameters Λ , ρ1 , ρ2 , σ , β1 , β2 corresponding to Table 1is given in Table 2. From Table 2, it is clear that β2 is more sensitive than β1apart from Λ, and σis more sensitive than ρ1,ρ2. Table 2. Sensitivity indices of different parameters of system (1) corresponding to Table 1. Parameters Sensitivity Index Λ+1 σ−0.5502 ρ1−0.0127 ρ2−0.0141 β1+0.0748 β2+0.9252 Mathematics 2022,10, 3020 9 of 17 5. Numerical Simulations In the numerical study, the predictor-corrector PECE method (mentioned in Appendix A) for fractional differential equations has been applied in the MATLAB software platform [ 26 ]. Table 3depicts the scenario of Bangladesh as a result of COVID-19 from 1 March to 10 June 2022. We ran numerical simulations to compare our model’s results to real data from various reports published by the WHO [ 1 ] and worldometer [ 27 ]. The value of the basic reproduction number is 1.2610, according to Table 1. Considering the demography of Bangladesh [ 28 ] and present covid situation, we have assumed S( 0 ) = 6,000,000, Ia(0) = 600,000. Is( 0 ) = 90,000, H( 0 ) = 80,000, R( 0 ) = 80,000, Q( 0 ) = 0.1. Initially, we performed a simulation of system (1) without government measures ( α= 0 but κ6= 0), depicted in Figures 1and 2 . We have taken κ= 2000 for simulations. It is found that the symptomatically infected population will be drastically increased if no action is taken by the government (Figure 1). Therefore, a large proportion of the population needs to be hospitalised, and this will create a massacre in the Health department of Bangladesh (Figure 1). It is also observed that the social behaviour variable Q is diminished around 20 days from 1 March 2022 (Figure 2). Figures 3and 4depict the model with the control (both government measures and public behaviour) scenario, which shows that the model fits well with real-world scenarios of the pandemic situation in Bangladesh. Figure 5 portrays the variation of the time series of S , Ia , Is , H , R , Q for different values of the order of derivatives (0.5, 0.6, 0.7, 0.8, 0.9). The curve of the symptomatically infected population fits well with real data for ε= 0.9. The time series for different values of governmental measures ( α= 0.3, 0.5, 0.7) are given in Figure 6, and it is revealed that the curve of Is is an approximation of active infected cases of the real scenario in Bangladesh for α= 0.7. The variation in the time series of state variables (with κ ) is given in Figure 7. The interesting observation in Figure 7is that the change in time series of all state variables, including Is , is negligible for values of κ ranging from 0 to 80,000 but increases for values greater than 80,000. The time series for the case κ= 0 (Figure 7) depicts the situation in which only government control is imposed, and no public behaviour is regulated. It is also observed that the number of symptomatic infected individuals ( Is ) will increase if we reduce the order of derivative closer to 0. The value of Q decreases slowly if the order of derivative ε is fixed in a higher range (Figure 5). Table 3. Number of active cases between 1 March 2022 and 10 June 2022. Day Active Cases 1 March 2022 93,206 11 March 2022 62,302 21 March 2022 50,030 31 March 2022 42,010 10 April 2022 25,650 20 April 2022 31,241 31 April 2022 27,005 10 May 2022 25,665 20 May 2022 21,616 30 May 2022 19,739 10 May 2022 20,011 Mathematics 2022,10, 3020 16 of 17 with X(t0) =         S(t0) Ia(t0) Is(t0) H(t0) R(t0) Q(t0)         , in this context, t0= 0. Then, system (A1) can be solved numerically by using the following scheme: G(tj) = hε Γ(ε+2)((j−1)ε+1−(j−ε−1)jε)G(X(t0)) + X(t0) +hε Γ(ε+2) j−1 ∑ i=1 ((j−i+1)ε+1−2(j−i)ε+1 +(j−i−1)ε+1)G(X(ti)) +hε Γ(ε+2)GX(tj−1) + hε Γ(ε+1)G(X(tj−1), (A2) with tj+1=tj+h, for j=0, 1.2....., N−1, we have used N=10,000, h=0.01. The above scheme is used in the FDE12 function, which is easily available in the MATLAB File exchange [29]. References 1. WHO COVID-19 Dashboard; World Health Organization: Geneva, Switzerland, 2020. Available online: https://covid19.who.int/ (accessed on 23 June 2022). [CrossRef] 2. Islam, K.; Ali, S.; Akanda, S.Z.R.; Rahman, S.; Kamruzzaman, A.H.M. COVID -19 Pandemic and Level of Responses in Bangladesh. Int. J. Rare Dis. Disord. 2020,3, 19. 3. Podlubny, I. Fractional Differential Equations; Academic Press: San Diego, CA, USA, 1999. 4. Petras, I. Fractional-Order Nonlinear Systems: Modeling Aanlysis and Simulation; Higher Education Press: Beijing, China, 2011. [CrossRef] [PubMed] 5. Du, M.; Wang, Z.; Hu, H. Measuring memory with the order of fractional derivative. Sci. Rep. 2013,3, 3431. 6. Podlubny, I. Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation. Fract. Calc. Appl. Anal. 2002,5, 367–386. [CrossRef] [PubMed] 7. Sardar, T.; Rana, S.; Bhattacharya, S.; Al-Khaled, K.; Chattopadhyay, J. A generic model for a single strain mosquito-transmitted disease with memory on the host and the vector. Math. Biosci. 2015,263, 18–36. [CrossRef] 8. Das, M.; Maiti, A.; Samanta, G.P. Stability analysis of a prey-predator fractional order model incorporating prey refuge. Ecol. Genet. Genom. 2018,7–8, 33–46. [CrossRef] 9. Das, M.; Samanta, G.P. A prey-predator fractional order model with fear effect and group defense. Int. J. Dyn. Control. 2020 ,9, 334–349. 10. Das, M.; Samanta, G.P. A delayed fractional order food chain model with fear effect and prey refuge. Math. Comput. Simul. 2020 , 178, 218–245. [CrossRef] 11. Das, M.; Samanta, G.P. Optimal Control of Fractional Order COVID-19 Epidemic Spreading in Japan and India 2020. Biophys. Rev. Lett. 2020,15, 207–236. [CrossRef] 12. Das, M.; Samanta, G.P. Stability analysis of a fractional ordered COVID-19 model. Comput. Math. Biophys. 2021 ,9, 22–45. [CrossRef] 13. Das, M.; Samanta, G.P.; De la Sen, M. A Fractional Ordered COVID-19 Model Incorporating Comorbidity and Vaccination. Mathematics 2021,9, 2806. [CrossRef] 14. Khan, M.A.; Atangana, A. Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative. Alex. Eng. J. 2020 , 59, 2379–2389. [CrossRef] [PubMed] 15. Liu, Z.; Magal, P.; Seyd, O.; Webb, G. A COVID-19 epidemic model with latency period. Infect. Dis. Model. 2020 ,5, 323–337. [CrossRef] [PubMed] 16. Shaikh, A.S.; Shaikh, I.N.; Nisar, K.S. A mathematical model of COVID-19 using fractional derivative: Outbreak in India with dynamics of transmission and control. Adv. Differ. Equ. 2020,373, 1–19. [CrossRef] Mathematics 2022,10, 3020 17 of 17 17. Xie, Y.; Wang, Z.; Meng, B.; Xia Huang, X. Dynamical analysis for a fractional-order prey–predator model with Holling III type functional response and discontinuous harvest. Appl. Math. Lett. 2020,106, 106342. [CrossRef] 18. Wang, X.; Wang, Z.; Xia, J. Stability and bifurcation control of a delayed fractional-order eco-epidemiological model with incommensurate orders. J. Frankl. Inst. 2019,356, 8278–8295. [CrossRef] 19. Shen, M.; Peng, Z.; Xiao, Y.; Zhang, L. Modelling the epidemic trend of the 2019 novel coronavirus outbreak in China. bioRxiv 2020 . 20. Wikipedia Contributors. Statistics of the COVID-19 Pandemic in Bangladesh. In Wikipedia, The Free Encyclopedia. Retrieved 19:19. 12 June 2022. Available online: https://en.wikipedia.org/wiki/Statistics_of_the_COVID-19_pandemic_in_Bangladesh (accessed on 16 June 2022). [CrossRef] [PubMed] 21. Kwuimy, C.A.K.; Nazari, F.; Jiao, X.; Rohani, P.; Nataraj, C. Nonlinear dynamic analysis of an epidemiological model for COVID-19 including public behavior and government action. Nonlinear Dyn. 2020,101, 1545–1559. [CrossRef] 22. Van den Driessche, P.; Watmough, J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 2002,180, 29–48. 23. Gelf, I.M.; Kapranov, M.M.; Zelevinsky, A.V. Discriminants, Resultants, and Multidimensional Determinants; Birkhäuser: Boston, MA, USA, 1994; ISBN 978-0-8176-3660-9. [CrossRef] 24. Ahmed, E.; El-Sayed, A.M.A.; El-Saka, H. On some Routh–Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems. Phys. Lett. A 2006,358, 1–4. [CrossRef] 25. Arriola, L.; Hyman, J. Sensitivity Analysis for Uncertainty Quantification in Mathematical Models. In Mathematical and Statistical Estimation Approaches in Epidemiology; Chowell, G., Hyman, J.M., Bettencourt, L.M.A., Castillo-Chavez, C., Eds.; Springer, Dordrecht, The Netherland, 2009. [CrossRef] 26. Garrappa, R. On linear stability of predictor-corrector algorithms for fractional differential Equations. Internat. J. Comput. Math. 2010,87, 2281–2290. 27. COVID-19 Coronavirus Pandemic. 2022. Available online: https://www.worldometers.info/coronavirus/country/ italy (accessed on 3 September 2020). 28. World-Population. Bangladesh. Available online: https://www.worldometers.info/world-population/bangladesh-population (accessed on 26 June 2022). 29. Garrappa, R. Predictor-Corrector PECE Method for Fractional Differential Equations. MATLAB Central File Exchange. Available online: https://www.mathworks.com/matlabcentral/fileexchange/32918-predictor-corrector-pece-method-for-fractionaldifferential-equations (accessed on 16 August 2022).