key: cord-0553249-wdyirrj7 authors: Okyere, Samuel; Bonyah, Ebenezer; Prah, Joseph Ackora title: A Fractional Differential Equation Modeling of SARS-CoV-2 (COVID-19) Disease in Ghana date: 2022-01-18 journal: nan DOI: nan sha: 56f21b921597ba0dd1dd6fd7ae65653a6163a456 doc_id: 553249 cord_uid: wdyirrj7 The coronavirus (COVID-19) has spread through almost 224 countries and has caused over 5 million deaths. In this paper, we propose a model to study the transmission dynamics of COVID-19 in Ghana using fractional-derivatives. The fractional-derivative is defined in the Atangana Beleanu Caputo (ABC) sense. This model considers seven (7) classes namely; Susceptible individuals, Exposed, Asymptomatic population, Symptomatic, Vaccinated, Quarantined, and Recovered population. The equilibrium points, stability analysis, and the basic reproduction number of the model have been determined. The existence and uniqueness of the solution and Ulam Hyers stability are established. The model is tested using Ghana demographical and COVID-19 data. Further, two preventive control measures are incorporated into the model. The numerical analysis reveals the impact of the fractional-order derivative on the various classes of the disease model as one can get reliable information at any integer or non-integer value of the fractional operator. The results of the simulation predict the COVID-19 cases in Ghana. Analysis of the optimal control reveals social distancing leads to an increase in the susceptible population, whereas vaccination reduces the number of susceptible individuals. Both vaccination and social distancing lead to a decline in COVID-19 infections. It was established that the fractional-order derivatives could influence the behavior of all classes in the proposed COVID-19 disease model. 3 fractionalorder can potentially describe more complex dynamics than the integer model and easily include memory effects present in many realworld phenomena. In this work, we propose fractional derivative to study the COVID-19 transmission in Ghana. In this section, we study the dynamic transmission of SARS-CoV-2 using fractional derivatives. Vaccinated (V), Quarantined (Q) and Recovered population (R). The population is homogeneously mixing, with no restriction on age, mobility or other social factors. The susceptibles are recruited into the population at a rate  . These individuals get exposed to the disease when they come in contact with the asymptomatic and symptomatic at a rate  . After being exposed to the covid -19 disease, they either progress to the asymptomatic class at the rate (1 )  − or the symptomatic class at the rate  . Both asymptomatic and symptomatic get quarantined at the rate  and  respectively. Those vaccinated according to this model, don't get infected but may join the susceptible class at a rate  . The parameter μ and  , are the natural and the disease-induced death rate respectively. The parameters , and  are the rate of recovery for the asymptomatic, symptomatic, and quarantined class, respectively. All newborns are susceptible (no inherited immunity. The model diagram is displayed in Fig. 2 . The following fractional derivatives describe the model. This section consists of the relevant definitions of Atanga -Baleanu derivative and integration in Caputo sense taken from [6] Definition 1 The Liouville-Caputo (LC) of fractional derivative of order  as defined in [14] is The Atangana -Baleanu definition in Liouville-Caputo sense as in [6] is is the normalized function. The corresponding fractional integral Atangana -Baleanu -Caputo derivative is defined as in [6] is 6 They found that when 0 =  , they recovered the initial function and when 1 =  , the ordinary integral is obtained. The Laplace transform of equation (4) gives the following results holds [9] : Furthermore, the ABC derivatives fulfill the Lipschitz condition in [9] : We denote a banach space by ) , containing real valued continuous function with sup norm and ) and the given norm ( ) R V Q I I E S R V Q I I E S S A S A + + + + + + = , , , , , , , where Using the ABC integral operator on the system (1), we have 0, Now from Definition 1, we have The function Similarly, Hence Lipschitz condition holds. Now taking system (9) in a reiterative manner gives . Taking into consideration equations (12) - (13) and considering ), holds. Since In a recurring manner, (16) reaches . Incorporating the triangular inequality and for any j , system (17) yields Hence there exists unique solution for system (1) The ABC fractional system given by equation (9) is said to be Hyers Ulam stable if for every satisfying: This section looks at the steady state of system (1). We have the disease -free steady state and the endemic steady state. The diseasefree equilibrium is the steady state solution where there is no infection in the population. The diseasefree equilibrium ( 0 ) is given as (20) 15 We now calculate the basic reproductive number ( 0 ) of system (1) . The basic reproductive number is the number of secondary cases produced, in a totally susceptible population, by a single infective individual during the time span of the infection [24] . Using the next generation operator method [24] , denote F and V, respectively, as matrices for the new infections generated and the transition terms we obtain Now the basic reproductive number is given as the spectra radius of the matrix Represents the reproduction number for system (1) The endemic equilibrium of system (1) is represented by ) , , , , , , ( The Jacobian matrix of system (1) is given as The Jacobian matrix evaluated at the disease-free equilibrium point is given as We need to show that all eigenvalues of system (27) The characteristic equation of system (28) is given as We apply the theorem in [8] . We start by dividing system (1) into two submodels, namely, the infectious class and noninfectious class. We denote the infectious class by V and the noninfectious class by F . The variable V and F are written as The reduce form of the system ) 0 , (F T dt dF = is given as , , is a globally asymptotically stable equilibrium point for the reduce system ) 0 , (F T dt dF = . (32) 19 Solving the third equation of system (33) gives . Solving the second equation of system (33) gives as . Solving the first equation of system (33) gives Hence, the convergence of system (1) , satisfies the following conditions given in [8] . satisfy the conditions above. , then the pandemic equilibrium ) ( * E is locally asymptotically stable. The Jacobian matrix of system (1) evaluated at the endemic equilibrium point is given as Let us consider the first equation of system (1) )), Applying the fundamental theorem of fractional calculus to equation (37), we obtain is a normalised function and at 1 + n t we have, Equation (40) is replaced with equation (39) and by performing the steps given in [21] , we obtain To obtain high stability, we replace the step size h in equation (41) with The new scheme which is called the nonstandard two -step Lagrange interpolation method (NS2LIM) is given as: Similarly, In this section, we validate the COVID-19 model by using COVID-19 confirmed cases data from Ghana Health Service for the period March -September, 2020 [12] . We also estimate (42) (43) 23 the parameters of the COVID-19 model and test the effect of the fractional order derivative on the various classes of the model. After formulating a model, one important thing is to validate the model to see if it will stand the test of time. Model validation is the process of determining the degree to which a mathematical model is an accurate representation of the available data. Using matlab gaussfit, the cumulative data of confirmed COVID-19 cases for the period March -September, 2020 is depicted in Fig. 3 . And Fig.4 shows the residuals of the best fitted curve. The parameter values is given in Table 1 . The simulation is displayed in Figures 5 -11 , where Fig.5 -11 depicts the behaviour of the susceptible, exposed, asymptomatic, symptomatic, quarantine, vaccinated and recovered individuals respectively. We add two control functions, 1 u and 2 u into the system (1). Where control 1 u and 2 u are social distancing and vaccination respectively. We include the time-dependent controls into system (1) and we have The objective function for fixed time is given as f t Where 1 T and 2 T are the measure of relative cost of interventions associated with the controls 1 u and 2 u . We find optimal controls 1 u and 2 u that minimizes the cost function , where f t is the final time and with initial conditions To define the fractional optimal control, we consider the following modified cost function [22] : For the fractional optimal control, the Hamiltonian is ) ) , , , . The following are essential for the formulation of the fractional optimal control [22] . are the Lagrange multipliers. Equation (48)        , , , , , , , with the transversality conditions The differential equation characterized by the adjoint variables are obtained by considering the right hand side differentiation of system (51) determined at the optimal control. The adjoint equations derived are given as In this section, we analyze the numerical behavior of the optimal control model. Using the parameter values given in Table 1 , and the same initial conditions S(0)=30,800000, E(0)=100, A(0)=100, Q(0)=100, V(0)=0, R(0)=0, Figs. 12 -16 depicts the behaviour of the susceptible, exposed, asymptomatic, symptomatic and quarantine when the fractional derivative 1 =  and the control 25 . 0 1 = u . Fig. 12 depicts an increase in the number of susceptible individuals whilst there is a decline in the number of exposed, asymptomatic, symptomatic and quarantined individuals. Hence social distancing measures taken by the government rather increased the number of susceptible and also reduces infection Figs. 17 -21 depicts the behaviour of the the susceptibles, exposed, asymptomatic, symptomatic and quarantine when the fractional derivative 1 =  and the control 005 . 0 2 = u . There is a decline in the number of susceptibles, exposed, asymptomatic, symptomatic and quarantined individuals when there is a vaccination control. In this study, a COVID-19 model has been examined using the fractional ABC operator in the The data/information supporting the formulation of the mathematical model in this paper are/is from Ghana health service website: https://www.ghs.gov.gh/covid19/ which has been cited in the manuscript. No conflict of interest regarding the content of this article The research did not receive funding from any sources. Manuscript was submitted as a pre-print in the link https://arxiv.org/ftp/arxiv/papers/2201/2201.08689.pdf and has been referenced. Smell and taste Dysfunction in Patients with COVID-19: A systematic Review and Meta -analysis A mathematical model of coronavirus disease (COVID-19) containing asymptomatic and symptomatic classes Modeling the evolution of the coronavirus disease (COVID-19) in Saudi Arabia A mathematical COVID-19 model considering asymptomatic and symptomatic classes with waning immunity Modelling spatial variations of novel coronavirus disease (COVID-19): evidence from a global perspective New Fractional Derivatives with Non-Local and Non-Singular Kernel: Theory and Application to Heat Transfer Model Modeling fractional-order dynamics of Syphilis via Mittag-Leffler law Dynamical model of tuberculosis and their applications A dynamic model for a study of diabetes Two cases of coronavirus confirmed in Ghana European Centre for Disease Prevention and Control COVID-19 Updates|Ghana Mathematical modeling of the spread of the coronavirus disease 2019 (COVID-19) taking into account the undetected infections. The case of China. Communications in nonlinear science and numerical simulation An extended predictor-corrector algorithm for variable-order fractional delay differential equations Ghana Birth rate', 2021, www.indexmundi.com (accessed 11 th Fractional model of COVID-19 applied to Galicia, Spain, and Portugal Nonstandard finite difference methods: recent trends and further developments Mathematical modeling of epidemic diseases: A case study of the COVID-19 coronavirus Prevalence and characteristics of taste disorders in Cases of COVID-19: Systematic review and meta-analysis of 29349 patients A mathematical model of COVID-19 using fractional derivative: outbreak in India with dynamics of transmission and control Novel numerical method for solving variable-order fractional differential equations with power, exponential and Mittag-Leffler laws Optimal control for a fractional tuberculosis infection model including the impact of diabetes and resistant strains A fractional model for the dynamics of tuberculosis infection using Caputo-Fabrizio derivative Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission World Health Organization (WHO) Q&A on coronaviruses (COVID-19) A fractional differential equation modelling of SARS -CoV -2 (COVID-19) disease: A case study of Ghana , , ,