key: cord-0890011-provgipc authors: Yadav, Ram Prasad; Renu title: A numerical simulation of Fractional order mathematical modeling of COVID-19 disease in case of Wuhan China date: 2020-07-16 journal: Chaos Solitons Fractals DOI: 10.1016/j.chaos.2020.110124 sha: 8bc90e77e3b2260e5c7a2c6cf66a52588be798f8 doc_id: 890011 cord_uid: provgipc The novel Covid-19 was identified in Wuhan China in December, 2019 and has created medical emergency world wise and distorted many life in the couple of month, it is being burned challenging situation for the medical scientist and virologists. Fractional order derivative based modeling is quite important to understand the real world problems and to analyse realistic situation of the proposed model. In the present investigation a fractional model based on Caputo-Fabrizio fractional derivative has been developed for the transmission of CORONA VIRUS (COVID-19) in Wuhan China. The existence and uniqueness solutions of the fractional order derivative has been investigated with the help of fixed point theory. Adamas- Bashforth numerical scheme has been used in the numerical simulation of the Caputo-Fabrizio fractional order derivative. The analysis of susceptible population, exposed population, infected population, recovered population and concentration of the virus of COVID-19 in the surrounding environment with respect to time for different values of fractional order derivative has been shown by means of graph. The comparative analysis has also been performed from classical model and fractional model along with the certified experimental data. In the past December 2019, an outbreak of a mysterious pneumonia symbolized by dry cough along with fever, fatigue and occasional gastrointestinal symptoms identified in a seafood wholesale wet market in Hubei, Wuhan, China. This disease is officially announced as Coronavirus Disease (COVID-19, by WHO on February 11, 2020) . Nowadays this pandemic is spreading across the globe and causes a severe outbreak of viral pneumonia, catching the eyes of the world. Due to high rate of spreading of COVID-19 is rapidly at an unprecedented scale across continents and has emerged as the single biggest risk in the world has faced in the present time. It is an infectious disease due to acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Researchers and scientists from US, Europe, China, Japan, WHO show there is no certain conclusion to origin 01 rpmaths.ism@gmail.com 02 renuismmaths@gmail.com of COVID- 19 . China currently has more 81,600 cases of the novel coronavirus that first broke out in December last year in Wuhan city. In the current century, Covid-19 is the third emerging human to human transmission disease after the outbreak acute respiratory syndrome coronavirus (SARS-CoV) in 2002 that spread to 37 countries and the Middle East respiratory syndrome corona virus (MERS-CoV) in 2012 that spread to 27 countries. The symptoms of COVID-19 is serious bilateral lung infiltration including dry cough, fever, difficulty in breathing, fatigue and similar to those symptom caused by SARS-CoV and MERS-CoV infections [8] . The virus of this disease spread from one to another person through the respiratory droplets produced during coughing. It is also found that this is also spread from touching contaminated surfaces and then touching one's face. There is no vaccine or specific antiviral treatment for COVID-19 therefore it is a challenging problem for medical scientist and virologist. However, there are many ongoing clinical trials evaluating potential treatments. The best way to prevent and slow down transmission is be well informed about the COVID-19 virus, the disease it causes and how it spreads. Protect yourself and others from infection by washing your hands or using an alcohol based rub frequently and not touching your face. A number of fractional order modeling studied [1, 2, 3, 4] and recently A. Atangana et al., [1, 5, 6] developed a fractional order model for the COVID-19 pandemic. Wu et al. [7] developed SEIR model to study the transmission of the Covid-19 and reported the basic reproductive number for validated data recoded from December 31, 2019 to January 28 2020. The SEIR model based on Poisson-distributed daily time increments have been investigated by Read et. al., [9] and estimated the basic reproductive number. Tang et al., [10] developed mathematical model based on deterministic compartmental and analysed the clinical progression of the disease. Imai et al., [11] investigated computational modeling of potential epidemic trajectories to evaluate the disease outbreak in Wuhan. Gao et al., [12] investigated an algorithm to study novel corona virus and predict its potential hosts. They observed that bats and minks may be two animal hosts of this virus. New definition of fractional order derivative given by Caputo and Fabrizio [13] . León [14] proved lemma to Caputo fractional derivatives of Lyapunov functions and shown the uniform asymptotic stability of some epidemic systems. Later on several mathematical modeling based on fraction order derivative developed by scientist and researchers to analyse to real world's scenario of spreading outbreak of epidemic. It is well known fact that the virus causing of many epidemic as dengue Influenza, Ebola and Covid-19. Influenza is an infectious disease caused by an influenza virus. Parra et al., [15] was developed a fractional order derivative based mathematical model of outbreaks of influenza. They examined that in the case of fractional order model observed that the next state depends not only upon its current state but also upon all of its historical states. Therefore they found that the fractional model is more general mathematical model compare to classical epidemic modeling. The fractional order derivative based mathematical modeling for the Ebola epidemic had been provided by Area et. al., [16] . They studied classical model (susceptible, exposed, infections, removed) and fractional order SEIR Ebola epidemic model. Then they showed comparison study with real data examined by World Health Organization (WHO). Latha et. al., [17] investigated the fractional-order model with time-delay to report the transition of Ebola virus infection. They examined that the fractional-order derivative based model with time-delay can more realistic for analysis of stability condition of fractional-order infection model. Area et. al., [18] [20] . They showed that model has non-negative solutions, as preferred in any population dynamics and also point out analysis on the stability of equilibrium in a detailed. Arshad et. al., [21] was also investigated a fractional order derivative model and obtained numerical simulation for immunogenic tumours. They studied the model based on fractional derivative growing tumour cell population and also observed that growth rate in death of immune cells has significant role in tumour dynamical and system consisting saddle-node and transcritical bifurcation analysis. A mathematical model based on fractional order derivative for the HIV /AIDS epidemic had been investigated by Zafar et. al. [22] . They performed numerical simulations to study the influence of the parameter involving mathematical model for outbreak of the disease. A non linear dynamics and chaos analysis in mathematical model based on fractional order derivative for the HIV was obtained by Ye and Ding [23] . Later on several mathematician focused on to developing the mathematical modeling based on fractional order derivative for the pandemic. Ucar et. al., [24] modeled fractional order derivative based model for immune cells influenced by cancer cells. Sardar et. al., [25] provided a mathematical model for dengue transmission along with memory. They proposed new compartmental mathematical model of dengue transmission with memory between human to mosquito and mosquito to human. Kilicman [26] also investigated mathematical model based on fractional order for dengue dengue transmission. He showed that his proposed model is well validated by published weekly dengue cases in Malaysia and found that the proposed model provides more understanding analysis to study the dynamic of dengue disease. Later on Qureshi and Atangana [27] was also modeled mathematical modeling for dengue fever outbreak based on fractional order derivative operators. They showed that efficacy rate of obtained results in the case of fractional order modeling is more high compare to the case of classical model. Chitnis et. al., [28] was studied the bifurcation analysis of a mathematical model for malaria transmission. A novel mathematical model based on fractional derivative TB investigated by Khan et. al., [29] . They showed that fractional order derivative bestow more realistic situation and deeper knowledge about the insolubility of the dynamics of TB model with relapse. Recently Baleanu et al. [30] modeled fractional order derivative based model for human liver. They showed existence of a unique solution of the proposed model and a comparative study is made between the predicted values by the model and the certified clinical data. Mathematical models in epidemiology are used widely in order to understand better the dynamics of an infectious disease. Fractional order models are more reliable and helpful in the real phenomena than the classical models due to hereditary properties and the description of memory [31, 32] therefore Fractional order derivative based modeling is used to a important tool for analyzed real world problem to find better ways of understanding and providing more realistic situation and deeper information about the complexity of the proposed model. Also, in the explanation of real world scenario, the dynamics between two different points is not able to explored by classical derivative. To analyze such types of failure in the classical dynamical system, fractional order derivative concept have been investigated. Fractional order models render a better fit to the real data for different diseases and other experimental work in the field of modeling and simulation. Nowadays, fractional order derivative is widely used in the mathematical modeling and have noticeable importance [33] . Several fractional differential operators like Riemann-Liouville, Hilfer, Caputo, etc. are mostly used in the modeling of physical problems. However, these fractional derivative possess a power law kernel and have own limitations, and reduce the field of application of fractional derivative. To deal with such type of difficulty, Caputo and Fabrizio [13] have developed an alternate fractional differential operator having a non-singular kernel with exponential decay. The Caputo-Fabrizio (C-F) operator has attracted many research scholars due to the fact that it has a non-singular kernel and to be found most appropriate for modeling some class of real world problem. Some researcher [34, 35, 36, 37, 38, 39] have been used in the modeling and have received tremendous success. Keeping this idea to mind Qureshi and Yusuf [40] modeled a fractional order mathematical model for a chickenpox disease andÖzturk andÖzkosc [41] examine stability analysis of fractional order mathematical model of tumor immune system interaction. After this study, recently Fanelli and Piazza [42] analyzed transmission of coronavirus disease -2019 which is spreding in China, France and Italy. The transmission of the COVID-19 virus from human to human has been a real worry within the community of modelers as this virus has destroyed many life in the last past years. Only science and technology in the hands of capable scientists and innovators can come to our rescue in developing innovative but effective solutions as we prepare for a future, which in the very short term looks increasingly uncertain. As per the authors knowledge no attempt has been made till date to develop a fractional order modeling for the COVID-19. The present investigation focused to develop a fraction order based mathematical model for Covid-19 disease in Wuhan China. The Caputo-Fabrizio fractional derivative concept has been used in the development of this fractional order mathematical model. Two step Adams-Bashforth numerical scheme has been used in the numerical simulation of Caputo-Fabrizio fractional order derivative. The analysis of susceptible population, exposed population, infected population, recovered and concentration of virus with respect to time for different value of fractional order has been shown by means of graph. The comparative analysis has also been performed from classical model and different order fractional along with the certified experimental data. In this section, we present some basic definitions, theorems and results related to Riemann-Liouville and Caputo-Fabrizio fractional order derivatives which are commonly used in the formulation of fractional order mathematical model. Definition 2.1. [43, 31] The integrability of a function f (θ) for any arbitrary real order Θ > 0 in the Riemann-Liouville sense is defined by the integral in the following form: (2.1) Definition 2.2. [43, 31] The Caputo-fabrizio fractional order derivative of any absolutely con- [13] The Caputo-Fabrizio derivative of a function f (θ) of order Θ > 0 is defined as; where W (Θ) is noted as normalization function such that W (0) = W (1) = 1 and f ∈ H 1 (0, T ), T > 0. where, Θ is the order of fractional integral, such that 0 < Θ < 1. In the present part of this work, we are considered to investigate a classical mathematical model developed by Yang and Wang [44] , for pandemic COVID-19, which was firstly reported in Wuhan, China in December 2019 and then spread out quickly across the glob. In the considered transmission model total human population in Wuhan China is divided into five sub compartments, where S c denotes susceptible population, which represent the section of human population who are susceptible to contact the virus and become infectious if exposed. E c denotes exposed population, which represent human population who are infected but not infectious so far. I c denotes infected population, who have fully developed the symptoms of COVID-19 and can spread the virus through contact with the susceptible population. The populations section who have fully recovered after the treatment and they have no symptoms of disease (free from the disease) is denoted by R c and V c represents the concentration of the virus of COVID-19 in the surrounding environment. The transmission model for COVID-19 pandemic is given by following system of ordinary differential equations; where the initial conditions for the considered transmission model of COVID-19 is given as The parameter Π c define as influx of population, µ c denotes the natural death rate of population. The quarantine period of the infected population is denoted by parameter (α c ) −1 , whereas γ c denote the rate of recovery and ψ 1c , ψ 2c denotes the exposed and infected population which contributing the coronavirus in the surrounding environment, ω c and τ c denotes the disease-induced death rate and removal rate of the COVID-19 virus from surrounding environment respectively. In the dynamical system of equations the rate of human to human transmission of disease between the exposed and susceptible individuals are given by the function β Ec and the rate of human to human transmission of disease between the infected and susceptible individuals are denoted by the function β Ic . The rate of transmission disease due to environmental contact to human individuals is denoted by the function β Vc . In the serve mathematical model we are considered all functions β Ec , β Ic and β Vc are nonnegative and non-increasing. Although sometime integer derivative based mathematical models can not replicate accurately the real world problem,of course due to that fact that the translation from real world observed problem to mathematical formula is not really accurate due to lack of information, or lack of accuracy to converting reality to mathematical formula. However, these models have been used in the last past decades with great success, thus their use is important to mankind for prediction. These prediction help human you have an idea of what could happen in near future, such that they can take some control measures to avoid worst case scenario. Therefore in the present section, we devolved a Caputo-Fabrizio fractional derivative based mathematical model that could be used to predict the spread of covid-19 for case of Wuhan china. For the purpose of above, Now applying the Caputo-Fabrizio fractional derivative [13, 34, 35, 36, 37, 38, 39] in the classical mathematical model (3.1). The fractional order transmission model for COVID-19 dynamics is given by the following system of non-linear fractional differential equation; 2) is the complete fractional order mathematical model for the transmission of covid-19 disease in the case of Wuhan China, which can portray the real world problem to a desired level of accuracy, offering valuable predictions. Such a prediction can lead to assess the forthcoming situations thereby, making one to adopt controlling measures well before time in order to avoid worst case scenario. We present below the solution analysis of the mathematical model, that includes the first testing of positivity of the solution, existence of the equilibrium points, reproductive number, and local stability analysis. Since the propose mathematical model (3.2), predict the characteristic of real world problem of covid-19 disease. Therefore for analysis of positivity of the solution of the model (3.2), we used mean value theorem given as in [51] , and we consider , for increasing and decreasing function u(ς), respectively. The solution lies in R 5 + , which is associated with a biological meaningful and defined by Now, we can modify the condition in model (3.2) with α = 1 and can be write in the matrix form as and Q ∈ C ∞ (R 5 ) denotes infinitely differentiable functions. The equilibrium states of fractional model (3.2) are acquired from solving CF To explore the infected state of the disease, investigation of Disease free and endemic equilibrium of an epidemiological models are prominent. The exploration of steady state of the fractional order model system of Equations (3.2) can be written as; The disease free equilibrium condition of Equation (3.8) is given as, For the analysis of endemic equilibrium point, first we need have to the expression for a threshold quality called the basic reproduction number R 0 . The infection components in the model systems are E c , I c and V c . The new infection matrix F and the transition matrix v are given by The basic reproduction number of fractional mathematical model (3.2) is defined as in the term of spectral radius of the next generation matrix F v −1 [50] and calculated as In the fractal fractional mathematical model Equation (3.8) , the second equilibrium point are endemic equilibrium (EE) is given as It view of the first two equations of (3.8) that S c can be represented by a function of I c , namely, (3.10) With the help of second equation of (3.8) and Equation (3.9), we have (3.12) Now, we assume that the curves S * c = Σ(I c ), I c ≥ 0 and S * c = Λ(I c ), I c ≥ 0. In particular, the intersections of these two curves in R 2 + examine the non-DFE equilibria. Clearly, Σ(I) is strictly decreasing, whereas Λ(I) is increasing since β Ec Thus we have (i) If the basic reproduction number R 0 > 1 then these two curves have a unique intersection lying in the interior of R 2 + , since Λ(0) < Σ(0) and Λ(I c1 ) ≥ Λ(0) > 0 = Σ(I c1 ). Moreover, at this intersection point, Equation (3.9) gives a unique endemic equilibrium (EE) If the basic reproduction number R 0 ≤ 1then the two curves have no intersection in the interior of R 2 + as Λ(0) ≥ Σ(0). Therefore, by Eq (3.8) we find that the model (3.2) admits a unique equilibrium, the DFE X 0 , if R 0 ≤ 1 and it admits two equilibria, the DFE Q f 0 and the EE Q * f , if R 0 > 1. In the present section of this study, we will investigate uniqueness and existence of the solution of fractional order differential equation with the help of fixed point theory and fraction derivative [35, 36, 48] on the fractional order differential Equation (3.2), we obtain as Now using the fractional order derivative concept given in [48] , we have After simplifying the above equation, we can be written as Proof: Let us consider functions S c and S c1 and proceed from A 1 , then we applying the following processure; Therefore, the Equation (4.3) implies that A 1 is satisfied Lipschitz condition, however the condition 0 ≤ ((β Ec + β Ic + β Vc ) Φ + µ c ) < 1 provides the condition of contraction. Similarly, we can show that the Lipschitz condition for other cases as; S c (t) = S c (0) + 2(1 − Θ) (2 − Θ) W (Θ) A 1 (t, S c ) + 2Θ (2 − Θ) W (Θ) t 0 (A 1 (y, S c )) dy, E c (t) = E c (0) + 2(1 − Θ) (2 − Θ) W (Θ) A 2 (t, E c ) + 2Θ (2 − Θ) W (Θ) t 0 (A 2 (y, E c )) dy, I c (t) = I c (0) + 2(1 − Θ) (2 − Θ) W (Θ) A 3 (t, I c ) + 2Θ (2 − Θ) W (Θ) t 0 (A 3 (y, I c )) dy, R c (t) = R c (0) + 2(1 − Θ) (2 − Θ) W (Θ) A 4 (t, R c ) + 2Θ (2 − Θ) W (Θ) t 0 (A 4 (y, R c )) dy, V c (t) = V c (0) + 2(1 − Θ) (2 − Θ) W (Θ) A 5 (t, V c ) + 2Θ (2 − Θ) W (Θ) t 0 (A 5 (y, V c )) dy. . Furthermore, applying the difference of successive terms, we find out as It is worth to be noted that Estimating with the same processure, we have With the help of triangle inequality, Equation (4.7) converted as Similarly, The existence of the solution is showed by operating the results stated in Equation (4.8). That is we can find t 0 such that where S c (t), E c (t), I c (t), R c (t) and V c (t) are bounded functions. Therefore, the Lipschitz condition is satisfied by Kernels as, Now, follow the following processure Therefore, we have We followed as Then at t 0 , we have (4.14) We can obtain from Equation (4.14) as, Similarly, we have, This state that the proposed fractional order mathematical system (3.2) has a solution. For the uniqueness of the solution of system (3.2), on the contrary, we suppose that S c1 (t), E c1 (t), I c1 (t), R c1 (t), V c1 (t) is another solution of system (3.2), then (4.16) With the help of norm Equation (4.16) takes the form (4.17) Further, Lipschitz condition of kernel gives Therefore, we have Proof: If condition (4.20) is true then in view of condition (4.19) implies that S c (t) − S c1 (t) = 0. Therefore, we get S c (t) = S c1 (t). Furthermore, we have Hence, the proposed fractional order mathematical model (3.2) is unique. In the present section of this investigation, we have described a new numerical technique (Agangana and Owolabi [48] ) by using the new Caputo-Fabrizio fractional order derivative for the discretization of fractional differential equation. Agangana and Owolabi [48] assumed the given fractional differential equation. In the view of Equation (5.1 ) and from the fundamental theorem of analysis, we have According to above, and In view of Equations (5.3) and (5.4), we have obtained the following system of equations. Therefore, from Equation (5.5), we get g(t n−1 , z n−1 ), (5.6) which indicates that Hence, The Equation (5.8) is to correspond two-step Adamas-Bashforth numerical scheme for the Caputo-Fabrizio fractional order derivative. Theorem 5.1. Let us suppose that z(t) be a solution of fractional order differential equation CF 0 D Θ t z(t) = g (t, z(t)) and g is a continuous bounded function for the Caputo-Fabrizio fractional order derivative [48] , then where G n Θ ≤ M . In the present analysis, we utilize the newly developed numerical scheme invented by [35, 36, 48] for simulation of new Caputo-Fabrizio fractional derivative in the proposed model system (3.2) for fractional order COVID-19 disease. For approximate solution of the proposed model system with the help of numerical iteration of this scheme, firstly we use the fundamental theorem of calculus to rearrange the model system (3.2) in the following fractional equation. Therefore, In view of Equations (6.1) and (6.2), the following equation system is obtained. From Equation (6.5), In the view of a Theorem (5.1), we obtain (n + 1)! h n+1 M, i = 1, 2, 3, 4, 5. β Ec the rate of human to human transmission of disease between the exposed and susceptible individuals 6.11 × 10 −8 [10] β Ic the rate of human to human transmission of disease between the infected and susceptible individuals 2.62 × 10 −8 [10] β Vc the rate of transmission disease due to environmental contact to human individuals 3.03 × 10 −8 fitted µ c the natural death rate of population 3.01 × 10 −2 [46] α c the quarantine period of the infected population 0.143 [49] w c the disease-induced death rate 0.01 [46] γ c the rate of recovery 0.67 [49] ψ 1c the exposed population which contributing the coronavirus in the surrounding environment Figure 1 . It is to be noted that the value of infected population is very high and lowest value is the recovered population. It is also to be emphasized that initially high rate of infected population observed but after certain time rate of infected population density become slower and same result also obtained for the case of recovered population. The impact of fractional order parameter in fractional order mathematical model of covid-19 disease in Wuhan China for exposed population density with respect to time has been depicted in Figure 2 . From the examination of Figure 2 , it is revealed that, the value of fractional order parameter in the favour of exposed population density. It is also remarkable that when the value of fractional order parameter tending to unity the variation of exposed population found very smooth and apparent. Figure 3 shows the variation of infected population against time for different value of fraction order parameter in fractional order mathematical model for Covid-19 disease in Wuhan China. The observed variation of infected population we have found that fractional order parameter in the favor of infected population and growth rate is observed very smooth when the value of fractional order parameter closed to unity, therefore the fractional order mathematical model reflect the more realistic situation of this disease in Wuhan China. The effect fractional order parameter on the mathematical model for recovered population and concentration of virus in environments exhibited in Figure 4 and 5 respectively. It has Figure 6 -10. Figure 6 shows the comparative analysis between the classical model and fractional order model for the exposed population. In Figure 6 solid line correspond to fractional model system for the Covid-19 disease in Wuhan China and dotted line for the case of classical model. Further, it is reported from all the figures (Figure 6 -9 ) that the growth of exposed population, infected population and recovered population in the case To comparatively analyze population density in classical model and fractional order model for the Covid-19 disease in Wuhan China has been depicted in Figure 10 . It has been found from the observation of Figure 10 that infected population in the classical model is very high compare to other population (exposed, infected, recovered ) and the same results also obtained in the case of fractional order mathematical model. In addition of this it is also obtained that difference The present investigation focused to develop a fraction order based mathematical model for Covid-19 disease in Wuhan China. The Caputo-Fabrizio fractional derivative concept has been used in the development of this fractional order mathematical model. Two step Adamas-Bashforth numerical scheme has been used in the numerical simulation of Caputo-Fabrizio fractional derivative. The analysis of susceptible population, exposed population, infected population, recovered population and concentration of virus with respect to time for different value of fractional order has been shown by means of graph. The comparative analysis has also been performed from classical model and different order fractional along with the certified experimental data. The major outcomes of the present study can be encapsulated as follows: 1. In the graphical analysis it is reported that at initially the infected population is very high after the certain time the growth of infected population become slow, therefore this study reflect that at initially social distancing and lockdown is very effective on controlling of Covid-19. 2. Through a comparative examination of population density in the case of fractional order mathematical model, the growth of infected population is high compare to recovered population and the same results was also obtained in the classical mathematical model. 3. In the fractional order mathematical model growth of recovered population is less than exposed population for Covid-19 disease in Wuhan China. 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Dr. Renu carried out discussion and conclusion of the manuscript *Credit Author Statement The detailed examination of growth of exposed population, infected population and recovered population against time in the fractional order mathematical model for Covid-19 disease in Wuhan China may help to future in detail deep study of the dangerous pandemic Covid-19 disease. The present analysis of the graphical illustration may contribute to help in the analysis of control and data analysis for medical science.