A report on COVID-19 epidemic in Pakistan using SEIR fractional model

Recently, novel coronavirus is a serious global issue and having a negative impact on the economy of the whole world. Like other countries, it also effected the economy and people of Pakistan. According to the publicly reported data, the first case of novel corona virus in Pakistan was reported on 27th February 2020. The aim of the present study is to describe the mathematical model and dynamics of COVID-19 in Pakistan. To investigate the spread of coronavirus in Pakistan, we develop the SEIR time fractional model with newly, developed fractional operator of Atangana–Baleanu. We present briefly the analysis of the given model and discuss its applications using world health organization (WHO) reported data for Pakistan. We consider the available infection cases from 19th March 2020, till 31st March 2020 and accordingly, various parameters are fitted or estimated. It is worth noting that we have calculated the basic reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{R}}_{0} \approx 2.30748$$\end{document}R0≈2.30748 which shows that virus is spreading rapidly. Furthermore, stability analysis of the model at disease free equilibrium DFE and endemic equilibriums EE is performed to observe the dynamics and transmission of the model. Finally, the AB fractional model is solved numerically. To show the effect of the various embedded parameters like fractional parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}α on the model, various graphs are plotted. It is worth noting that the base of our investigation, we have predicted the spread of disease for next 200 days.


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| (2020) 10:22268 | https://doi.org/10.1038/s41598-020-79405-9 www.nature.com/scientificreports/ epidemic disease. But the cases of corona virus spread in the whole world and then it became pandemic disease. Furthermore, it was found that symptoms of corona virus in human being takes 2-10 days 4 . As this deadly virus spread rapidly in China and in the whole world, like Europe, North America, Germany and Italy and caused many deaths. The number of deaths due to COVID-19 was record in thousands. This virus also affects Asian countries specially Pakistan. In Pakistan the first case of Corona virus was detected in 27th February 2020 5 . From the 27th February 2020 till to 31st March 2020, the number of cases of COVID-19 increased day by day. According to the data collected by WHO 5 the number of total infected cases were 1865 and total reported deaths were 25 till 31st March 2020.
Fractional order modeling is quite important to understand the realistic situations of the dynamical models. Different researchers use different fractional operators to fractionalize their models. Like, Zhang 6 modeled the dynamics of COVID-19 in five different compartments and extended the model to the fractional order differentiation to capture different memories using fractional operators of singular and non-singular kernels. Shaikh et al. 7 investigated the spread of the deadly virus COVID-19 in India and fractionalized the classical model with Caputo-Fabrizio (CF) fractional operator. Baleanu et al. 8 reported the numerical solutions of CF time fractional model of COVID-19 with different compartments by using homotopy analysis transform method. Yadav and Verma 9 examined the spread of COVID-19 in Wuhan, China through a dynamical CF fractional model with different classes and shown the effect of concentration of virus in the surrounding environment on these classes. Lu et al. 10 investigated the dynamics of COVID-19 in different high concentrated cities of China using Caputo fractional model with six different compartments. Rajagopal et al. 11 investigated the dynamics of COVID-19 in Italy using Caputo fractional model. They examined that results achieved through fractional model provides a closer forecast with the real data. Higazy 12 studied the spread of COVID-19 pandemic using SIDATHE fractional model with Caputo fractional operator. Besides this, fractional calculus is useful in various aspects of real-world problems such as Ozarslan et al. 13  The aim of present study is to investigate the dynamics and transmission of COVID-19 through SEIR model. Furthermore, we transformed the classical model by newly developed Atangana-Baleanu time fractional model. The purpose of using AB time fractional derivative is that it has non-singular and non-local kernel and it may predict the spread of COVID-19 accurately. Qureshi et al. 15 investigated the dynamics of diarrhea by considering AB fractional operator. They found that the diarrhea model under investigation estimates the real statistical data well enough when considered with the AB fractional order operator which has non-local and non-singular kernel. Bas and Ozarslan 16 investigated the analytical solutions of some dynamical models such as logistic equation, population growth/decay equation, blood alcohol model by considering AB fractional derivative. They concluded that AB fractional operator gives more accurate results to the derivative with exponential kernel. Bas et al. 17 analyzed the solutions of heating and cooling of building models by using the Laplace transform technique. They generalized their models by considering Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional operators and by comparing their results, they noticed that AB-fractional derivative provides more precise results for the given models. They concluded that CF and AB fractional operators can better describes the dynamics of complex systems as compared to classical operators.
Recently, many researchers have chosen AB fractional model for dynamics of different infectious diseases, like the dynamics of HIV 18 , chicken pox 19 , hepatitis E 20 , dengue fever 21 , rubella disease 22 , measles 23 , tuberculosis 24 . Besides this, AB fractional operator is also used by many researchers to investigate the dynamics of COVID-19. Gao et al. 25 reported the investigation of dynamics of COVID-19 in Wuhan, China using AB fractional model. Atangana and Araz 26 reported a comprehensive study of spread of COVID-19 in South Africa and Turkey using Atangana-Seda numerical scheme. They generalized their classical model with AB fractional model and compared the upcoming predictions of COVID-19 in Turkey and South Africa. Mohammad and Trounev 27 reported the numerical solutions of dynamical system of COVID-19 involving AB fractional operator. The applications of AB fractional derivative are not only limited to study the dynamics of infectious diseases but it has many applications in modern sciences and technologies e.g. nanofluids 28 , biofluids 29 , solar collectors 30 , chaotic processes 31 , electric circuits 32 etc.
More precisely, in the present research, we have taken the data of COVID-19 from 19th March 2020 to 31st March 2020 and parameterized the given model to the real data. Positivity and boundedness of the present model is also discussed. Fixed points and basic reproduction number is estimated. Stability analysis and some basic properties have been proved for the given model. A numerical scheme is developed for the given fractional model and displayed its results for the fractional parameter α through graphs. Some figures are plotted with different initial conditions for global asymptotical stability (GAS).

Mathematical modeling
In the present mathematical model, we have considered the spread of COVID-19 in human. The total population is denoted by N h (τ ) which is further subdivided into four sub-classes. The four subclasses are susceptible, exposed, infected and recovered people which are denoted by S h (τ ), E h (τ ), I h (τ ) and R h (τ ) respectively. The interaction among these subclasses has shown in the flow chart which is Fig. 1.
In the given model, the recruitment rate of susceptible population is h which represents birth rate,η h represents death rate of each subclass, h represents interaction rate between susceptible and infected population www.nature.com/scientificreports/ with the route h S h I h N , ψ h represents the rate at which exposed class completed their incubation period and enter into the infected class, γ h is the removal or recovery rate of infected population.

Non-negativity and boundedness of the model
This section provided to prove the boundedness and positivity of the solutions of system (1). To show the positivity of the model (1), we state the following lemma: where Proof By rewriting the system (1), we have: We noted that According to Lemma 3.1, R 4 + is invariant set.

Preposition 3.2 The system (1) is bounded in the region:
(1) Figure 1. Flow chart of the given SEIR model. www.nature.com/scientificreports/ Proof Boundedness of the problem (1) can be verified by adding all equations of system (1), we obtained: The solution of Eq. (8) takes the form: It can be clearly seen from Eq. (9) that if τ → ∞ then N h (τ ) ≤ � h η h which means that the feasible region for the given model will be: Hence the solution of the system (1) is bounded.

Equilibrium points, basic reproduction number and local stability analysis
This section includes the possible fixed points of model (1). There exists two possible equilibrium points are calculated, i.e. Disease-free equilibrium (DFE) and endemic equilibrium (EE). Furthermore, basic reproduction number is calculated by next generation technique and discuss the local stable analysis of these equilibrium points. The steady state solution of model is given below by considering the rate of change w.r.t time becomes zero: Using Eq. (11), model (1) becomes: From steady state system (12), DFE can be obtained by assuming E h = I h = R h = 0 and is denoted by Similarly, EE of model (1) is obtained from system (12) and is given by The basic reproduction number R 0 is calculated by next generation technique 33 . The F and V matrices at DFE 0 is given as follows: www.nature.com/scientificreports/ According to 33 the spectral radius ρ(FV −1 ) is the required reproduction number R 0 which is calculated in the form: Scientific Reports | (2020) 10:22268 | https://doi.org/10.1038/s41598-020-79405-9 www.nature.com/scientificreports/ here, α is fractional parameter and AB 0 ℘ α τ (.) is the AB time fractional derivative. All these variables and initial conditions for model (25) are positive.
By solving the integration terms in the above equations, Eqs. (35)-(38) gets the following iterative shape:  Table 1. The given model is fitted for α = 1 . We have estimated the basic reproduction number R 0 ≈ 2.30748 using different parameter values given Table 1.

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Numerical results. This section provides graphical results of the given study. Figure 1 describes the transmission of COVID-19 between different compartments of the total population with the evolution of different parameters which are associated with the dynamics of virus. It is important to study the spread of COVID-19 with the influence of these parameters to predict the spread of this deadly virus. Figure 2 depicts WHO reported cases of infected individuals for COVID-19. Figure 3 displays number of deaths confirmed by WHO due to COVID-19. Figure 4 highlights the total number of recovered individuals of COVID-19 in the selected period of time. Comparative bar chart and line plot of reported deaths and recovered individuals of COVID-19 has been shown in Fig. 5. Figure 6 represents WHO reported cases for COVID-19. Figure 7 shows the comparison between real data from WHO verses present considered model. From the figure, it can be noticed that our model shows a strong agreement with real data collected by WHO. Finally, it is necessary to mention here that from Figs. 2, 3, 4, 5, 6 and 7, all these figures have been plotted for the data collected by WHO from 19th March 2020 till 31st March 2020. From the graphical analysis, one can observe that the rate of spread of COVID-19 in Pakistan in the above mentioned interval is very high as compared to the initial days. Moreover, to obtain the graphical results from the given model we have simulated the given iterative scheme through MATLAB software. Time unit is considered in days for the present model of COVID-19.
Fractional models are important to capture the different memories, decay and crossover behavior. Fractional operator can also demonstrate some upcoming situations for the dynamical models. Figure 8   www.nature.com/scientificreports/    www.nature.com/scientificreports/    www.nature.com/scientificreports/ dynamics and transmission of COVID-19 for the next 200 days. Finally, Fig. 9 is obtained by considering the different initial values which provides the global asymptotical stability of our model equilibrium.

Concluding remarks
To understand the spread different infectious diseases including COVID-19 among plants, human or any other animals, mathematicians implement the concept of differentiation and integration to model the dynamics of that disease in the form of ODEs or PDEs that could be used to predict the transmission of that disease within a specific population. For this task, they divide the total population into different compartments. The solution of these systems may deliver an indication to health and other sectors that how severe the spread can be and what parameter is needed to control the spread of virus or disease under consideration. In the present article, we have taken SEIR model for the dynamics and transmission of COVID-19. We have also proved that the given model is bounded and invariant. Two equilibrium points DFE and EE are calculated for the steady state of the presented model. Furthermore, the basic reproduction number R 0 is calculated using next generation technique. Through stability analysis, we have concluded that the given model is locally asymptotically stable when R 0 < 1 . It means that by taking R 0 < 1 the total population tends to DFE and the disease will die out from the population. Additionally, we have generalized the classical SEIR model by applying the recently developed AB fractional model. The real data have been fitted for the classical model i.e. α = 1 . For the fitted values given in Table 1, we have approximated the basic reproduction number R 0 ≈ 2.30748 . A numerical scheme is adopted for the solutions of AB fractional model. After simulation of numerical scheme different graphical results have been obtained. We have considered the unit of time in days. Furthermore, the dynamics and transmission of COVID-19 is predicted for the next 200 days. A decrease is noticed in the infected class by decreasing the values of AB fractional parameter α . As the infectious disease in not yet control in Pakistan as well as worldwide. We all know that the researchers and scientist are trying to find proper medical treatment or vaccination to control www.nature.com/scientificreports/ the spread of COVID-19. In future one may add the quarantine or vaccination or both the classes to the given model and show the effect of these classes on the spread of this deadly infectious disease. www.nature.com/scientificreports/ Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/.