key: cord-0813557-wqfuzg10 authors: Nazir, Ghazala; Zeb, Anwar; Shah, Kamal; Saeed, Tareq; Khan, Rahmat Ali; Ullah Khan, Sheikh Irfan title: Study of COVID-19 Mathematical Model of Fractional Order Via Modified Euler Method date: 2021-04-18 journal: nan DOI: 10.1016/j.aej.2021.04.032 sha: 45976dcdb54593513706811418bdd793f2a64a2d doc_id: 813557 cord_uid: wqfuzg10 Our main goal is to develop some results for transmission of COVID-19 disease through Bats-Hosts-Reservoir-People (BHRP) mathematical model under the Caputo fractional order derivative (CFOD). In first step, derived the feasible region and bounded ness of the model. Also, we derived the disease free equilibrium points (DFE) and basic reproductive number for the model. Next, we establish theoretical results for the considered model via fixed point theory. Further, the condition for Hyers-Ulam’s (H-U) type stability for the approximate solution is also established. Then, we compute numerical solution for the concerned model by applying the modified Euler’s method (MEM). For the demonstration of our proposed method, we provide graphical representation of the concerned results using some real values for the parameters involve in our considered model. For numerical illustration, we use Matlab. Recently a threatful disease which is called COVID-19 another form of SARS has started to spread in globe from Wuhan a big city of China during the end of 2019. Up-to date more than 0.8 millions people all around the world have been died. Further, more than fifteen million people have been infected around the globe. According to World Health Organization (WHO) in China a medical office was identified the cases of pneumonia of unknown etiology in Wuhan City of Hubei Province of China on 31 December 2019. WHO informed that a COVID-19 was detected. Further it was declared most dangerous virus by Chinese authorities on 7 January 2020 [1, 2] . Tiamen et al on 19 January 2020 developed a Bats-Hosts-Reservoir-People (BHRP) model for transmission from the infectious source to the human. They assumed that virus spread in the Bats population, then virus transmitted to an unknown wild animals (hosts). After hunting the hosts (defined as revivor), the virus spread in a seafood market which became cause of infection in some people. Also it has been considered that the source of COVID-19 is the transmission from animal to human. Some other researchers guaranteed that transmission also occurs from person to person. Therefore various countries implemented strict lock downed in their states and advised the public to keep social distance. Such policy have controlled the disease in some countries but this is not a permanent way to save people. Because such policies have very badly destroyed the economies of low income countries all over the world. Now, the proper vaccine has been prepared for the cure of the COVID-19. On the other hand, bioengineers, mathematicians and researchers are also trying to make such procedure which may reduced or controlled the spreading of disease in our society further. As it is well known that mathematical models are powerful tools to study the transmission of infectious disease. Also mathematical models of infectious disease have been studied in last few decades very well [3, 4, 24-28, 30, 31, 33] . In this regards very recently many researchers developed various models to investigate the transmission of COVID-19. Many researcher worked on the COVID-19 model using the data for different countries [38] [39] [40] [41] [42] . Therefore, BHRP model was also built to investigate the aforesaid model. The Bats-Hosts-Reservoir-People BHRP model under classical derivative was established in [5] as "where people divided into five different compartments including susceptible people S p , exposed people E p , symptomatic infected people I p , asymptomatic infected people A p and removed people R p including recovered and removed (died) people and W represents the reservoir of virus." It is well known fact that differential equations under CFOD having wide range of applications in various fields of science and technology [6, 7] . Therefore, in recent years, model involving CFOD have been given much attention because the biological models containing aforesaid derivative are more realistic and comprehensive as compared to the classical order models. In this regards, various aspects of the considered problems like qualitative theory, analytical and numerical solutions have been studied. For this purpose numerous techniques have been established to handle the problems. Integral transform when coupled with perturbations or decompositions techniques, we get hybrid method which have been increasingly used to handle linear and nonlinear problems of fractional order, for detail see [8-13, 17, 18, 23, 29, 31-37] . with initial conditions as Since it is important that to check wether a model of real problem exists or not. This thing is guaranteed by applying fixed point theory. Therefore, we establish existence theory for the considered model (2) provided that the integral on the right side is point-wise converges on (0, ∞). where m = [η] + 1 and [η] represents the integer part of η. Through out in this paper, we use CFOD. Definition 2.4. [19] The "generalized Taylor formula" for f (t) can be written as such that ξ ∈ [0, t], at all t ∈ (0, a], η ∈ (0, 1]. We establish MEM using (4). Lemma 2.5. The solution of the problem for 0 < η ≤ 1 is provided by We represent Banach space by Here first, we derive feasible region and bounded-ness of the model (1). Theorem 3.1. The boundedness and feasible region of solution to the proposed model (1) is given by . 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 On solving, we get where C is the constant of integration. Since from (7), one has, when t → ∞ which is our required result. Now, we are going to compute disease free equilibrium points (DFE) and reproductive number of the model (1) . For the equilibrium points of the model (1), we have Theorem 3.2. The reproduction number of the model (1) is Proof. We take the 2 nd , 3 rd and 6 th equations of the model (1) to finding the reproduction number. Where necessary computations for the F and V matrices are given as 5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 where F is the nonlinear term and V is the linear term of the model (1) . Now taking the jacobian of the F and V matrices as Next we have to find the generation matrix as and Spectral radius at disease free equilibrium point (DFE) from the (9) Hence the required result is proved. If R 0 < 1 ,then the model (1) is locally asymptotically stable (out break will go to end). If R 0 > 1, then the model (1) is unstable(outbreak will spread). (2) The existence of solution to a physical problem is verified by using fixed point approach. We use the theorem [17, 18] to derive the intended results. The right sides of model (2) can be expressed 6 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 as: Using (15), Model (2) becomes In view of Lemma 2.5, (16) yields where To derive required results, some assumptions need to be hold: (A 1 ) There exists constants K Ψ > 0, such that for each V (t),V (t) with (A 2 ) There exists constants C Ψ > 0 and M Ψ > 0, such that To show the operator S in (18) is contraction, let V, V ∈ Y, we have 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 From which we have Which shows that S is contraction if K Ψ < 1. Next S is compact and continuous operator. To get this goal, consider Hence S is bounded in (19) . Let t 1 < t 2 ∈ J , one has Since if t 2 → t 1 , then right side of (20) goes to zero. Hence t 2 → t 1 , led us that Hence S is equi-continuous, so S is compact continuous. Therefore S is completely continues operator. Thus all the condition of Theorem 4.1 are satisfied so the Model (2) has at least one solution. Theorem 4.2. Under the continuity of Θ i , for i = 1, 2, 3, 4, 5, 6, and if the condition KΨ Γ(η+1) < 1 holds, then the system (2) has a unique solution. Proof. Let S : Y → Y, be the operator defined by Let V,V ∈ Y, then we have 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 Hence the model has unique solution by using Banach contraction theorem. Here we derive H-U type stability for the (16) which lead us to the stability of (2). Consider a small perturbation φ ∈ C(J ) with φ(0) = 0. solution obeys Proof. The proof is easy. Proof. LetV ∈ Ω be a unique solution and V ∈ Ω be any solution of (17), then From which we have From (25), we can write In this part of the paper, we have to evaluate approximate solutions of the model (2) under CFOD. Then the numerical simulations are acquired via the suggested scheme. To this aim, we employ the CFOD to establish a numerical procedure for the simulation of our considered model (2). Here we extend the numerical method of Euler [22] for our considered model (2) . The aforesaid considered model can be written as Let J be the interval of solution for (27) . We subdivide the intervalJ into j subintervals [t q , t q+1 ] with uniform width h = T /m via using the nodes t q = qh, for q = 0, 1, · · · m. Let up to higher order are continuous on J . Applying the MEM about t = t 0 = 0 to the considered model expressed in (27) and for each value t take value a, the expression for t 1 , one has Let the step size h is chosen small enough, then we may neglect the second-order term involving h 2η and get the results from (28)as , , Proceeding on aforesaid fashion, a general formula at t q+1 = t q + h is established as , where q = 0, 1, 2, · · · , m − 1. Here in this subsection, graphical interpretation of numerical results to the concerned model is given. For this aim, we use the adopted scheme for the numerical simulation. Here, we choose some appropriate values for the parameters used in the model that is given in the table 1 (see [34] ). Graphical presentations are given in Figures 1 − 6 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 0 20 40 60 80 100 120 140 160 180 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 0 20 40 60 80 100 120 140 160 180 increase the values of the η, we see that the solution tends to integers order solution. The growing and decaying rate of various classes of model is different at different fractional order. Therefore fractional calculus can be helpful in understanding the transmission dynamics of COVID-19. Here we, remark that at smaller fractional order the decay process is faster while the growth rate is slow. Increasing the fractional order the process of decay may become slow while,the grow rate goes on raising. Further, the fractional order has great impact on the transmission dynamics of the proposed model. Also, it helps in better understanding of physical behaviour of spreading of infection in a community. Moreover, the adopted numerical method can be used as a fruitful technique to achieve computational results for such type nonlinear problems. The concerned growth or decay process of various compartments is faster slightly at lower fractional order as compared to greater value of η. We have established some qualitative results for the mathematical model (2) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 we observed that the increase or decrease in different compartments is faster at higher fractional order of the derivative and we see that fractional calculus has the ability to explain the papulation dynamics more comprehensively. The presented results may be fruitful for the existing outbreak in a better way and can be used in taking defensive techniques to decrease the infection. In future the proposed scheme can be utilized to investigate more nonlinear problems of FODEs involving CFOD . 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Corona-virus World Health Organization Pneumonia of unknown aetiology in Wuhan, China: potential for international spread via commercial air travel Risk of imported Ebola virus disease in China Incidence dynamics and investigation of key interventions in a dengue outbreak in Ningbo City, China A mathematical model for simulating the phase-based transmissibility of a novel coronavirus Fractional Differential Equations, Mathematics in Science and Engineering Theory and application of fractional differential equations North Holland Mathematics Studies Study of HIV mathematical model under nonsingular kernel type derivative of fractional order Stability analysis for nonlinear fractional-order systems based on comparison principle Theory of Fractional Dynamic Systems Numerical treatment for traveling wave solutions of fractional Whitham-Broer-Kaup equations Application of the multistep generalized differential transform method to solve a time-fractional enzyme kinetics A kind of approximate solution technique which does not depend upon small parameters: a special example Numerical treatment of fractional order Cauchy reaction diffusion equations Prevention strategies for mathematical model Mers-Corona Virus with stability Analysis andoptimal control China Population Fractional dynamical analysis of measles spread model under vaccination corresponding to nonsingular fractional order derivative Existence theory and novel iterative method for dynamical system of infectious diseases Generalized Taylor's formula Novel dynamical structures of COVID-19 with nonlocal operator via new computational technique Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan Fractional modeling dynamics of HIV and CD4+ T-cells during primary infection Analysis of some generalized ABCfractional logistic models A Study of behaviour for immune and tumor cells in immunogentic tumor model with non-singular fractional derivative Chaotic behaviour of fractional predator-prey dynamical system A choas study of tumor and effector cells in fractional tumor immune model for cancer treatment Analysis of an epidemic spreading model with exponential decay law Modelling and analysis of CoVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan Fractional Logistic models in the frame of fractional operators generated by conformable derivatives Stability analysis and numerical computational of the fractional predator-prey model with the harvesting rate On class of ordinary differential equations in the frame of Atangana-Baleanu fractional derivative A mathematical model of Coronavirus disease (COVID-19) containing asymptomatic and symptomatic classes Choatic dynamics of fractional order HIV-1 model involving AIDS-related cancer cells Modelling the dynamics of noval coronavirus ( 2019-ncov) with fractional derivative Nonlocal cauchy problem via a fractional operator involving power kernal in Banach spaces Afractional model for propogation of classical optical solitons by using nonsingular derivative Novel solution methods for initial boundary value problems of fractional order with conformable differentiation Mathematical model for COVID-19 pandemic: A comparative analysis Mathematical Model to estimate and optimal control strategy Mathematical modelling of the dynamics and containment of COVID-19 in Ukraine Mathematical modelling for coronavirus disease (COVID-19) in predicting future behaviour and sensitivity analysis A mathematical model of COVID-19 using fractional derivative: outbreak in India with dynamics of transmission and control t S p (t)| t=a t 2η Γ(2η + 1) ,,