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The Analytic Solutions of the Fractional-Order Model for the Spatial Epidemiology of the COVID-19 Infection

The Analytic Solutions of the Fractional-Order Model for the Spatial Epidemiology of the COVID-19... Hindawi Advances in Mathematical Physics Volume 2023, Article ID 5578900, 19 pages https://doi.org/10.1155/2023/5578900 Research Article The Analytic Solutions of the Fractional-Order Model for the Spatial Epidemiology of the COVID-19 Infection 1 2 3 1 Benedict Barnes , Martin Anokye, Mohammed Muniru Iddrisu, Bismark Gawu, and Emmanuel Afrifa Kwame Nkrumah University of Science and Technology, Department of Mathematics, Ghana University of Cape Coast, Department of Mathematics, Ghana C.K. Tedam University of Technology and Applied Sciences, School of Mathematical Sciences, Department of Mathematics, Ghana Correspondence should be addressed to Benedict Barnes; ewiekwamina@gmail.com Received 16 December 2022; Revised 19 March 2023; Accepted 28 March 2023; Published 4 May 2023 Academic Editor: Andrei Mironov Copyright © 2023 Benedict Barnes et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper provides a mathematical fractional-order model that accounts for the mindset of patients in the transmission of COVID-19 disease, the continuous inflow of foreigners into the country, immunization of population subjects, and temporary loss of immunity by recovered individuals. The analytic solutions, which are given as series solutions, are derived using the fractional power series method (FPSM) and the residual power series method (RPSM). In comparison, the series solution for the number of susceptible members, using the FPSM, is proportional to the series solution, using the RPSM for the first two terms, with a proportional constant of ψΓððnα +1Þ, where ψ is the natural birth rate of the baby into the susceptible population, Γ is the gamma function, n is the nth term of the series, and α is the fractional order as the initial number of susceptible individuals approaches the population size of Ghana. However, the variation in the two series solutions of the number of members who are susceptible to the COVID-19 disease begins at the third term and continues through the remaining terms. This is brought on by the nonlinear function present in the equation for the susceptible subgroup. The similar finding is made in the series solution of the number of exposed individuals. The series solutions for the number of deviant people, the number of nondeviant people, the number of people quarantined, and the number of people recovered using the FPSM are unquestionably almost identical to the series solutions for same subgroups using the RPSM, with the exception that these series solutions have initial conditions of the subgroup of the population size. It is observed that, in this paper, the series solutions of the nonlinear system of fractional partial differential equations (PDEs) provided by the RPSM are more in line with the field data than the series solutions provided by the FPSM. 1. Introduction is more appropriately classified as a pandemic disease than an epidemic disease. The primary factor in the global The development of a mathematical model for under- transmission of the COVID-19 disease is the movement standing and unravelling the underlying mechanisms of of exposed or infected individuals, who may or may not the epidemiology of the COVID-19 disease has garnered have the aim of coming into contact with the vulnerable interest from public health systems to academia in several individuals in the host country. The spatial spread of the different countries; the majority of these models focus on disease in the various countries was accounted for in the epidemics of the disease progression from one person to mathematical models developed by [5, 6] by taking into consideration the diffusing susceptible individuals, exposed another person, as described by [1–3]. However, the COVID-19 disease originated in Wuhan, China, and geo- individuals, and infected individuals. Despite this, these graphically spread to other parts of the world as a pan- models do not account for the vaccinations that the indi- viduals of the population received. demic disease [4]. In this sense, COVID-19 epidemiology 2 Advances in Mathematical Physics There are currently treatments available that are given to can only produce fixed-point solutions of differential equa- people all around the world regardless of their health condi- tions. The vast majority of nonstationary points are uncov- tions, such as the Johnson and Johnson vaccine and the ered by this method. More crucially, neither a quantitative AstraZeneca vaccine. Although the usefulness of these vac- nor a qualitative method provides the function that describes cines has been scientifically demonstrated, these immuniza- the theoretical foundation for describing the epidemiology tions do lose some of their efficacy over time. There is no of COVID-19 disease. assurance that a person receiving the COVID-19 vaccination Recently, it has been discovered that the integer differen- will be protected from getting the disease upon contact with tial equations suffer from several shortcomings when com- a person with the SARS virus. For example, see authors in pared to the differential equations of fractional order. The [7]. This observation makes it unclear which individuals fractional differential equation has memory and heredity are completely unprotected from the disease and which properties because of its nonlocal property for describing persons are temporarily protected for a short period of time COVID-19 pandemics. Any solution to the system of PDEs, following immunization. Only a few researchers have used regardless of order, may be easily obtained using fractional vaccinated subjects in their models without the inclusion of ordering. The theory of controls, infectious diseases, growth the spatial transmission of the disease. For example, have a of tumours, and feedback systems are examples of applied scientific problems where the differential equations of frac- look at the authors in [8–10]. All of these epidemiological models account for people moving from one subgroup of tional order have proven to be effective models. For example, the population to another subgroup of the population. see authors in [13] who applied the fractal fraction Adams- Although they captured vaccinated persons, they did not Bashforth method to search for the solution of fractal- incorporate the diffusing individuals who brought the fractional susceptible-infective-recovered model. Another COVID-19 disease into their respective countries. Also, for numerical approach for solving systems of differential equa- the mathematical models on the control of transmission of tions that are both linear and nonlinear is the Pade approx- COVID-19 disease, see [11]. The findings of these imation method. High-order approximations are necessary researchers, however, are not all inclusive since they when using this method. More crucially, given a nonlinear neglected to take into account an important observation of system of PDEs, there is no systematic procedure in selecting the progression of patient through the disease. Evidence the parameters in the Pade approximation method[14]. from numerous countries has demonstrated that infected Since Mittag-Leffler functions or their derivatives make up individuals (patients) either plan or do not intend to trans- the majority of the solutions to the system of fractional dif- mit the COVID-19 virus to susceptible individuals [4]. In ferential equations, rigorous mathematics is necessary to developing a mathematical model to describe the epidemiol- solve these equations. One of the methods for solving system ogy of the COVID-19 infection, the mindset of the spreaders of fractional differential equations is the RPSM which was was not captured in their models. Additionally, statistics first observed by [15] for solving fuzzy differential equation. from different countries have revealed that whether a person With this approach, a power series is assumed to exist for the takes medicine to treat COVID-19 or not, they still run the system of ODEs, and the coefficients of the power series are risk of getting the illness again if they come into contact with used to create a recurrence equation. When the residual an infected person. Thus, the recovery from the disease is for coefficients, from the power series, are equal to zero, an alge- a short period of time (see [12]). When creating a mathe- braic system of equations results, from which the values of matical model to describe the epidemiology of COVID-19, the series solution of the unknown coefficients can be all these issues were not taken into account. deduced. Nevertheless, while solving fractional-order PDE, The type of mathematical tools a researcher uses to say in two variables, this method assumes that one of the arrive at his or her conclusion(s) ultimately determines the independent variables has a representation in a fractional success of any mathematical analysis. Since the beginning power series, and the second independent variable is han- of the COVID-19 outbreak in China till now, researchers dled as a coefficient variable, which is roughly derived from have mainly relied heavily on either the use of the qualitative the variation in the given fractional-order PDE based on the method, the quantitative method, or both. These methods initial or boundary condition. For example, see authors in have significantly more drawbacks than advantages. A [16–19]. The same method was used by [20] to solve nonlin- numerical scheme is an example of a quantitative method ear fractional-order PDEs. In [21], the authors used the that always approximates the exact solution of the differen- Atangana-Baleanu fractional derivative to obtain asymptotic tial equation with some level of precision. This quantitative interval approximation solutions to the fractional differential method yields intolerable inaccuracies; in the worst situa- equation under various conditions. A nonlinear system of tion, its solution diverges from the exact solution of a differ- stiff fractional-order PDEs and the nonlinear system of frac- ential equation. As usual, even if the solution suggested by tional PDEs have not been solved using the RPSM. The kind the numerical scheme exists, one must perform a number of nonlinearity in a fractional PDE largely depends on the of iterations before reaching the desired solution. The qual- functional space which contains the solution of fractional itative method narrows down the information contained in differential PDE. The FPSM is another intriguing method the solution of the differential equation. The domain ele- which was first observed by [22]. The authors in [23, 24] ments of the function that describes the epidemiology of applied this method to obtain the solutions of fractional COVID-19 infection are revealed by this method of investi- PDEs. It is challenging to find analytic solutions to a nonlin- gation on a microscopic level. In light of this, this method ear system of fractional-order partial differential equations. Advances in Mathematical Physics 3 Additionally, researchers from all over the world have not RPSM-based analytical solutions of the nonlinear system of observed a comparison of the series solutions utilizing both fractional PDEs are presented as series solutions. the RPSM and the FPSM. More importantly, no information 3.1. Model Description. Despite the fact that the COVID-19 regarding comparing the series solutions obtained by these vaccination is given to country residents by the Ministry of methods with field data is provided in the literature. The series solution (analytic) method of the nonlinear system Health (MOH), neither the Johnson and Johnson nor the AstraZeneca vaccine is anticipated to provide COVID-19 of fractional-order partial differential equations has a solu- patients with a lifetime of immunity against the illness. tion, is the most dependable and efficient method as com- In Figure 1, the population size of Ghana is split into six pared to both the qualitative and the quantitative methods. distinct subgroups namely: the susceptible subgroup, Sðx, tÞ; In this paper, the infected group of the SEIQR model is exposed subgroup, Eðx, tÞ; deviant infected subgroup, Iðx, tÞ; further divided into two subgroups: the deviant infected sub- nondeviant infected subgroup, I ðx, tÞ; quarantined sub- group and nondeviant subgroup of the population in the group, QðtÞ; and the recovered subgroup, RðtÞ. A susceptible classical susceptible-exposed-infected-quarantined-recov- person is any member of the population who is capable of ered model with diffusion terms. Thus, the susceptible- catching the SARS virus from an infected COVID-19 exposed-deviant infected-nondeviant infected-quarantined- patient. An exposed person is someone who has caught the recovered (SEII QR) model with diffusion terms, and vacci- SARS virus, but for a brief while, he or she is unable to pass nated susceptible term is developed. In addition, the frac- tional form of this model is provided herein. Moreover, it on to a susceptible person. The waiting period is therefore in effect for this person. The deviant infected person is a both the FPSM and the RPSM are used to obtain the series patient who has chosen to purposefully spread the SARS solution (analytic) of the nonlinear system of fractional PDEs. The solutions that are yielded by these two methods virus to susceptible family members on the grounds that since they already have the illness, they must also experience are compared with field data accounting for the robustness the COVID-19 disease-related consequences. The nondevi- of the methods. ant individual, on the other hand, is an infected person who does not willingly spread the SARS virus to susceptible 2. Fundamental Concept in Fractional Calculus family members or friends because they do not want them to get the COVID-19 illness. The person under quarantine is a Definition 1. A real function uðx, tÞ, x ∈ I, t >0 is said to be COVID-19 patient who was deviant infected person or non- in the space C ðI × ℝ Þ, μ ∈ ℝ, if there exist a real number p + deviant infected person, or an exposed person, whose move- p > α such that uðx, tÞ = t f ðx, tÞ, where f ðx, tÞ ∈ CðI × ℝ Þ, n n ments are restricted in a specific place for an extended length and it is said to be in the space C ,if ∂ /∂t ∈ C , n ∈ ℕ α α of time. The recovered person is the person who either (see [25]). recovers naturally or receives treatment at the hospital for a period of time. After being exposed to the COVID-19 Definition 2. For n − 1< β < n, n ∈ ℕ. The Caputo fractional infection, this person is still at a significant risk of reacquir- derivative operator of the order β is define by (see [26]) ing the SARS virus. As a consequence of this, the immune system is unable to recuperate and is therefore vulnerable n−α−1 α ðÞ n to losing its immunity. To take into account the continuous ðÞ D u ðÞ t = u ðÞ ξðÞ t − ξ dξ, t >0: ð1Þ ΓðÞ n − α inflow of foreigners into the country, the subgroups of sus- ceptible, exposed, deviant infected person, and nondeviant mα infected person depend on the distance, x, as well as the pas- Theorem 3. The fractional power series (FPS)∑ a ðt − t Þ : m=0 m 0 sage of time. Additionally, ψ stands for the natural birth rate, β for (i) converges only for t = t , that is, the radius of conver- transmission rate, and μ for natural death rate, ν is the rate gence equal to zero for quarantining exposed individuals, and α is the rate at (ii) converges for all t ≥ t , that is; the radius of conver- which an exposed person intends to transmit the SARS virus gence equal to ∞ to susceptible members. The rate at which an exposed per- son has no intention to infect a susceptible member with (iii) converges for t ∈ ½t , t + RŠ , for some positive real 0 0 the SARS virus is α . The rates at which the exposed person, numbers R, and diverges for t > t + R. Here, R is the deviant infected person, and the nondeviant infected the radius of convergence for the FPS [27]. person are quarantined, respectively, are ν, ω , and ω . The 1 2 disease-induced death rates from the subgroups of deviants, nondeviants, and confined individuals are δ , δ , and δ , 1 2 3 3. Main Results respectively. The ϕ, γ , and γ are the rates of recoveries 1 2 In this section, for modelling the COVID-19 epidemiology from COVID-19 disease by the quarantined, the deviant, in Ghana, a mathematical model that takes into account and the nondeviant individuals, respectively. The η is the the mindset of the patients in spreading the COVID-19 dis- rate at which susceptible moves to the recovered compart- ease, temporary loss of immunity by recoveries, and the con- ment after receiving a vaccine, and κ is the rate at which tinuous influx of foreigners entering the country with or recovered person loses their immunity and becomes suscep- without the disease is needed. Therein, the FPSM- and tible again. The natural death rate from each subgroup of the 4 Advances in Mathematical Physics equations (2)–(7) together with the initial conditions in equation (8) are obtained in Hilbert space using the FPSM. In obtaining each solution of the system of equations (2)–(7) together with initial conditions, it is assumed that the unknown function defining the equation is in series form which converges to a known function. In addition, the proof of the existence of these series solution as well as its unique- ness is provided here. Setting kα Sx, t = 〠 S x t , ð9Þ ðÞ ðÞ Figure 1: shows the various subgroups of the population size for k=0 describing the epidemiology of COVID-19. population is denoted by μ. Due to the fact that the model takes into account diffusion of the foreigners into the sus- kα Ex, t = 〠 E x t , ð10Þ ðÞ ðÞ ceptible, exposed, deviant, and nondeviant subgroups, the k=0 rates of diffusion into the corresponding compartments are specified as follows: D represents the rate of diffusion into the susceptible compartment, D represents the rate of diffu- kα Ix, t = 〠 I x t , ð11Þ ðÞ ðÞ sion into the exposed compartment, D represents the rate of k=0 diffusion into the deviant subgroup, and D represents the rate of diffusion into the nondeviant subgroup. Based on above facts, the following nonlinear system of ∗ ∗ kα I x, t = 〠 I x t , ð12Þ ðÞ ðÞ fractional PDEs is obtained for describing the epidemiology k=0 of COVID-19 in Ghana. α 2α ∂ S βSI + I + Q ∂ S ðÞ ð2Þ kα = ψN + κR − − μ + η S + D , ðÞ 1 Qx, t = 〠 Q x t , ð13Þ α 2α ðÞ ðÞ ∂t N ∂x k=0 α 2α ∂ E βSIðÞ + I + Q ∂ E ð3Þ = −ðÞ α + α + μ + v E + D , 1 2 2 ∞ 2α ∂t N ∂x kα RxðÞ , t = 〠 R ðÞ x t , ð14Þ α 2α k=0 ∂ I ∂ I ð4Þ = α E −ðÞ γ + μ + δ + ω I + D , 1 1 1 3 α 2α ∂t ∂x we can see that α 2α ∗ ∗ ∂ I ∂ I ð5Þ = α E − γ + μ + δ + ω I + D , ðÞ 2 2 2 2 4 α 2α ∂t ∂x ΓðÞ kα +1 α ðÞ k−1 α DðÞ SxðÞ , t = 〠 S ðÞ x t , α k d Q ΓðÞ ðÞ k − 1 α +1 k=1 ð6Þ = vE + ω I + ω I − μ + ϕ + δ E, ðÞ 1 2 3 α ð15Þ dt 2α ∞ 2α ∂ S ∂ kα = 〠 S x t : ðÞ ∂ R k 2α 2α ∂x ∂x ð7Þ = ϕQ + γ I + γ I + ηS − μ + κ R, ðÞ k=0 1 2 ∂t together with the initial conditions Substituting equations (8), (9), (10), (11), (12), (13), and (14) into equation (2) yields Sx,0 = S x , ðÞ ðÞ Ex,0 = E x , ðÞ ðÞ ΓðÞ kα +1 k−1 α ðÞ 〠 S x t ðÞ Γ k − 1 α +1 Ix,0 = I x , ðÞ ðÞ ðÞ ðÞ 0 k=1 ð8Þ ∞ ∞ ∗ ∗ I x,0 = I x , ðÞ ðÞ kα kα = ψN + κ〠 R ðÞ x t − 〠 S ðÞ x t k k k=0 k=0 Q 0 =0, ðÞ ð16Þ ∞ ∞ ∞ kα ∗ kα kα R 0 =0: ðÞ Á 〠 I x t + 〠 I x t + 〠 Q x t ðÞ ðÞ ðÞ k k k k=0 k=0 k=0 3.2. Analytic Solutions of the System of Fractional Partial ∞ 2α ∞ kα kα Differential Equations Using the Fractional Power Series −ðÞ μ + η 〠 S ðÞ x t + D 〠 S ðÞ x t : k 1 k 2α ∂x k=0 k=0 Method. In this section, series solutions of the system of Advances in Mathematical Physics 5 Comparing the powers of t , we have ΓαðÞ +1 β E x = S x I x + S x I x ðÞ ðÞ ðÞ ðÞ ðÞ 2 0 1 0 ΓðÞ 2α +1 N 1 β + S x Q x + I x S x + S x I x + S x Q x ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ S ðÞ x = ψN + κR ðÞ x − S ðÞ x ðI ðÞ x 0 1 0 1 1 0 1 0 1 0 0 0 ΓαðÞ +1 N ) 2α ∂ E ðÞ x ) 1 − α + α + μ + ν E + D , 2α ðÞ 1 2 1 2 2α ∂x + I x + Q x − μ + η S x + D S x : ðÞ ðÞÞ ðÞ ðÞ ðÞ 0 0 0 1 0 ( 2α ∂x Γ 2α +1 β À ðÞ E ðÞ x = S ðÞ x I ðÞ x + S ðÞ x I ðÞ x 3 0 2 0 Γ 3α +1 N ð17Þ ðÞ + S x Q x + S x I x + S x I x ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ 0 2 1 1 1 Similarly, we observe the following results. For S ðxÞ, S 2 3 + S x Q x + S x I x + S x I x ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ 1 1 2 0 2 0 ðxÞ, ⋯, S ðxÞ, we have n ) 2α ∂ E ðÞ x ( + S ðÞ x Q ðÞ x −ðÞ α + α + μ + ν E + D , 2 0 1 2 2 2 2α ∂x ΓαðÞ +1 β S x = ψN + κR x − S x I x ðÞ ðÞ ðÞ ðÞ 2 1 0 1 Γ 2α +1 N ðÞ ( " n−1 + S ðÞ x I ðÞ x + S ðÞ x Q ðÞ x + S ðÞ x I ðÞ x 0 0 1 1 0 1 À ΓðÞ ðÞ n − 1 α +1 β E x = 〠 S x I x ∗ ðÞ ðÞ ðÞ n k n−1−k + S ðÞ x I ðÞ x + S ðÞ x Q ðÞ x −ðÞ μ + η S 1 1 0 1 Γ nα +1 N 0 ðÞ k=0 2α + D S , 1 1 2α + S ðÞ x I ðÞ x + S ðÞ x I ðÞ x ∂x k k n−1−k n−1−k 2α ΓðÞ 2α +1 β ∂ E x x ðÞðÞ n−1 S ðÞ x = ψN + κR ðÞ x − S ðÞ x I ðÞ x − α + α + μ + ν E x + D : 3 2 0 2 ðÞ ðÞ 1 2 n−1 2 2α ΓðÞ 3α +1 N ∂x + S x I x + S x Q x + S x I x ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ ð19Þ 0 2 0 2 1 1 + S x I x + S x Q x + S x I x ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ 1 1 1 1 2 0 ∗ Similarly, the series solutions for the number of the devi- + S x I x + S x Q x − μ + η S ðÞ ðÞ ðÞ ðÞ ðÞ 2 0 2 0 2 ant infected people, the number of nondeviant infected peo- 2α ple, the quarantined, and the number of recoveries are as ∂ S + D , 2α follows: ∂x I x = α E x − γ + μ + δ ðÞ ðÞ ( 1 1 0 1 1 ΓαðÞ +1 Γ n − 1 α +1 ðÞ ðÞ S ðÞ x = ψN + κR ðÞ x n n−1 2α ∂ I x ΓðÞ nα +1 ðÞ + ω ÞI ðÞ x + D , 1 0 3 2α ∂x n−1 − 〠 S x I x + S x I x ðÞ ðÞ ðÞ ðÞ i n−1−i i n−1−i N À ΓαðÞ +1 i=0 I x = α E x − γ + μ + δ ðÞ ðÞ 2 1 1 1 1 ΓðÞ 2α +1 + S ðÞ x Q ðÞ x −ðÞ μ + η S ðÞ x i n−1−i n−1 2α ∂ I ðÞ x + ω ÞI x + D , ðÞ 1 1 3 2α ∂x 2α ∂ S x ðÞ n−1 + D : 2α ∂x Γ 2α +1 ðÞ ð20Þ I ðÞ x = α E ðÞ x − γ + μ 3 1 2 Γ 3α +1 ðÞ ð18Þ 2α ∂ I ðÞ x + δ + ω ÞI x + D , ðÞ 1 1 2 3 2α Following the similar procedure above, the following ∂x results are obtained for E ðxÞ, E ðxÞ, E ðxÞ, ⋯, E ðxÞ: 1 2 3 n 1 β ∗ À ΓðÞ ðÞ n − 1 α +1 E x = S x I x + I x + Q x ðÞ ðÞðÞ ðÞ ðÞ ðÞ 1 0 0 0 0 I ðÞ x = α E ðÞ x − γ + μ n 1 n−1 Γα +1 N ðÞ ΓðÞ nα +1 2α 2α ∂ E x ðÞ Á ∂ I x 0 ðÞ n−1 −ðÞ α + α + μ + ν E + D , + δ + ω I ðÞ x + D , 1 2 0 2 1 1 n−1 3 2α 2α ∂x ∂x 6 Advances in Mathematical Physics The series solutions of the nonlinear system of fractional I ðÞ x = α E ðÞ x − γ + μ + δ PDEs order of nth term are given by 1 2 0 2 ΓαðÞ +1 2α ∗ ∞ ∞ ∂ I x ðÞ ΓðÞ ðÞ n − 1 α +1 ∗ 0 + ω ÞI x + D , ðÞ S x, t = 〠 ψN + 〠 κR x ðÞ ðÞ 2 0 4 2α n n−1 ∂x Γ nα +1 ðÞ n=1 n=1 n−1 ΓαðÞ +1 − 〠 S ðÞ x I ðÞ x + S ðÞ x I ðÞ x I ðÞ x = α E ðÞ x − γ + μ + δ k n−1−k k n−1−k 2 2 1 2 Γ 2α +1 ðÞ k=0 2α ∗ ∂ I x ðÞ ∗ 1 + S x Q x − μ + η S x ðÞ ðÞ ðÞ ðÞ + ω ÞI ðÞ x + D , k n−1−k n−1 2 4 2α ∂x 2α ∂ S ðÞ x n−1 kα ΓðÞ 2α +1 ð21Þ ∗ + D t , I x = α E x − γ + μ + δ 2α ðÞ ðÞ 3 2 2 2 2 ∂x Γ 3α +1 ðÞ ð24Þ 2α ∗ ∂ I x ðÞ ∗ 2 + ω ÞI ðÞ x + D , 2 4 2α ( " ∂x ∞ n−1 ΓðÞ ðÞ n − 1 α +1 β E ðÞ x, t = 〠 〠 S ðÞ x I ðÞ x n k n−1−k Γ nα +1 N ⋮ ðÞ n=1 k=0 Γ n − 1 α +1 À ðÞ ðÞ + S ðÞ x I ðÞ x + S ðÞ x I ðÞ x I x = α E x − γ ðÞ ðÞ k k n−1−k n−1−k n 2 n−1 2 Γ nα +1 ðÞ 2α 2α ∗ ∂ E ðÞ x ∂ I ðÞ x 2 kα ∗ n−1 − α + α + μ + ν E x + D t , ðÞ ðÞ + μ + δ + ω ÞI ðÞ x + D , 1 2 n−1 2 2 2 n−1 4 2α 2α ∂x ∂x ð25Þ Q ðÞ x = vE ðÞ x + ω I ðÞ x 1 0 1 0 ΓαðÞ +1 Γ n − 1 α +1 ðÞ ðÞ I ðÞ x, t = 〠 α E ðÞ x + ω I ðÞ x −ðÞ μ + ϕ + δ Q ðÞ x , n 1 n−1 2 0 3 0 ΓðÞ nα +1 n=1 Γα +1 È ðÞ 2α Q x = vE x + ω I x ðÞ ðÞ ðÞ ∂ I ðÞ x 2 1 1 1 n−1 kα Γ 2α +1 ðÞ − γ + μ + δ + ω I x + D t , ðÞ ðÞ 1 1 1 n−1 3 2α É ∂x + ω I ðÞ x −ðÞ μ + ϕ + δ Q ðÞ x , 2 1 3 1 ð26Þ Γ 2α +1 È ðÞ Q x = vE x + ω I x ð22Þ ( ðÞ ðÞ ðÞ 3 2 1 2 Γ 3α +1 ðÞ Γ n − 1 α +1 ðÞ ðÞ I ðÞ x, t = 〠 α E ðÞ x ∗ 2 n−1 + ω I ðÞ x −ðÞ μ + ϕ + δ Q ðÞ x , ΓðÞ nα +1 2 2 3 2 n=1 2α ∂ I ðÞ x ∗ n−1 kα − γ + μ + δ + ω I x + D t , ðÞ ðÞ 2 2 2 n−1 4 2α ∂x ΓðÞ ðÞ n − 1 α +1 Q x = vE + ω I x ðÞ ðÞ n n−1 1 n−1 Γ nα +1 ðÞ ð27Þ + ω I ðÞ x −ðÞ μ + ψ + δ Q ðÞ x , 2 n−1 3 n−1 Γ n − 1 α +1 ðÞ ðÞ Q ðÞ x, t = 〠 νE ðÞ x + ω I ðÞ x n n−1 1 n−1 ∗ ΓðÞ nα +1 R x = ϕQ + γ I + γ I + ηS − μ + κ R , n=1 ðÞ fg ðÞ 1 0 0 0 0 1 2 0 Γα +1 ðÞ É ∗ kα + ω I x − μ + ψ + δ Q x t , ðÞðÞ ðÞ 2 n−1 3 n−1 ΓαðÞ +1 R ðÞ x = fg ϕQ + γ I + γ I + ηS −ðÞ μ + κ R , 2 1 1 1 1 1 1 2 ð28Þ ΓðÞ 2α +1 ΓðÞ 2α +1 ∗ ΓðÞ ðÞ n − 1 α +1 R ðÞ x = fg ϕQ + γ I + γ I + ηS −ðÞ μ + κ R , 3 2 1 2 2 2 2 2 R ðÞ x, t = 〠 ϕQ ðÞ x + γ I ðÞ x Γ 3α +1 n n−1 n−1 ðÞ 1 ΓðÞ nα +1 n=1 ⋮ ∗ kα + γ I x + ηS x − μ + κ R x t : ðÞ ðÞ ðÞ ðÞ 2 n−1 n−1 n−1 ΓðÞ ðÞ n − 1 α +1 ð29Þ R ðÞ x = ϕQ ðÞ x + γ I ðÞ x n n−1 1 n−1 ΓðÞ nα +1 3.2.1. Existence and Uniqueness of the Series Solution of the + γ I x + ηS x − μ + κ R x : ðÞ ðÞ ðÞ ðÞ n−1 n−1 2 n−1 Nonsystem of Fractional PDEs. The proofs of the existence ð23Þ and uniqueness of the series solutions in equations Advances in Mathematical Physics 7 (24)–(29) of the nonlinear system of fractional PDEs are PðÞ x, t,stðÞ − P x, t, s ðÞ t 1 1 provided therein. 2α β ΓðÞ ðÞ n − 1 α +1 ∂ ≤ l + l + l + μ + η + D ( 1 2 3 2α N Γ nα +1 ∂x ðÞ Γ n − 1 α +1 ðÞ ðÞ PðÞ x, t,stðÞ = ψN + κR ðÞ x 1 n−1 Á S x − S x , ΓðÞ nα +1 ðÞ ðÞ k k n−1 β À ′ ′ PðÞ x, t,stðÞ − P x, t, s ðÞ t ≤ λ S ðÞ x − S ðÞ x , 1 1 1 k k − 〠 S x I x ðÞ ðÞ k n−1−k k=0 ð30Þ ! ð31Þ + S ðÞ x I ðÞ x + S ðÞ x Q ðÞ x Þ k n−1−k k n−1−k where 2α ∂ S x β ΓðÞ ðÞ n − 1 α +1 ðÞ n−1 kα − μ + η S x + D t , λ = ðÞ l + l + l + l + l , ðÞ ðÞ 1 1 2 3 4 5 n−1 1 2α ∂x N ΓðÞ nα +1 n−1 n−1 l = 〠 I x , l = 〠 I x , ðÞ ðÞ ð32Þ 1 n−1−k 2 n−1−k k=0 k=0 n−1 2α Γ n − 1 α +1 ∂ ðÞ ðÞ P x, t, s t = ψN + κR x l = 〠 Q ðÞ x , l = μ + η, l = D , ðÞ ðÞ 3 n−1−k 4 5 1 1 n−1 2α ∂x ΓðÞ nα +1 k=0 n−1 ′ ′ with 0< λ ≤ 1. − 〠 S ðÞ x I ðÞ x + S ðÞ x I ðÞ x n−1−k 1 k k n−1−k k=0 This implies that the function is Lipschitz continuous on + + + the domain fðx, t, sðtÞÞjx ∈R , t ∈ ½0Š ∪R and sðtÞ ∈R g. ′ ′ + S ðÞ x Q ðÞ x −ðÞ μ + η S ðÞ x Following similar procedure above, the following contin- k n−1−k n−1 uous functions are obtained over the domain: 2α ∂ S ðÞ x n−1 kα + D t , 2α ′ P x, t,st − P x, t, s t ðÞ ðÞ ðÞ ∂x 2 2 ≤ λ E x − E , ðÞ 2 k k PðÞ x, t,stðÞ −Px, t, s P x, t,Ex − P x, t, E x n−1 ðÞ ðÞ ðÞ 2 2 Γ n − 1 α +1 β ðÞ ðÞ = − 〠 I ðÞ x ( n−1−k Γ nα +1 N n−1 ðÞ k=0 Γ n − 1 α +1 β ðÞ ðÞ ≤ 〠 I ðÞ x + I ðÞ x n−1−k n−1−k ′ ′ ΓðÞ nα +1 N Á S ðÞ x − S ðÞ x + I ðÞ x S ðÞ x − S ðÞ x k k k n−1−k k k=0 + Q x Þ + α + α + μ + ν ðÞðÞ n−1−k 1 2 + Q ðÞ x S ðÞ x − S ðÞ x n−1−k k 2α + D E x − E x , ðÞ ðÞ 2 n n 2α ′ ∂x − μ + η S x − S x jj ðÞ ðÞ n−1 n−1 2α ∂ S x − S x ðÞ ðÞ n−1 n−1 kα Px, t,Ex − P x, t, E x ðÞ ðÞ ðÞ + D t 2α ∂x ( ≤ λ E x − E x , ðÞ ðÞ 2 n n n−1 Γ n − 1 α β À ðÞ ðÞ ≤ 〠 I x ðÞ ð33Þ n−1−k ΓðÞ nα +1 N k=0 where + I ðÞ x + Q ðÞ x S ðÞ x − S ðÞ x n−1−k k n−1−k k β Γ n − 1 α +1 ðÞ ðÞ ! ) λ = l + l + l + l + l + l , ð34Þ ðÞ 2α 2 1 2 3 4 5 6 N Γ nα +1 ðÞ kα + ðÞ μ + η + D S ðÞ x − S ðÞ x t 1 n−1 n−1 2α ∂x ( " where l = jα + α + μ + νj and l + l + l and l have usual n−1 6 1 2 1 2 3 5 ΓðÞ ðÞ n − 1 α β meanings. ≤ 〠 I x + I x + Q x ðÞ ðÞ ðÞ ðÞ n−1−k n−1−k n−1−k ΓðÞ nα +1 N k=0 !# ) 2α ′ ′ PxðÞ , t,Ix ðÞ − P x, t, I ðÞ x ≤ λ I ðÞ x − I , ∂ 3 3 n−1 n−1 kα + μ + η + D S x − S x t , ðÞ ðÞ ðÞ 1 n−1 n−1 2α ∂x ð35Þ 8 Advances in Mathematical Physics where Substituting equation (37), (39), (40), (41), and (42) into equation (43) yields β Γ n − 1 α +1 ðÞ ðÞ λ = l + l + l + l + l + l , ð36Þ ðÞ 3 1 2 3 4 5 6 N ΓðÞ nα +1 α k ∂ S x ðÞ n nα Re s x, t = S x + 〠 t − ψN ðÞ ðÞ k o where, l = jα + α + μ + νj and l + l + l and l have usual ∂t ΓðÞ nα +1 6 1 2 1 2 3 5 n=1 meanings. R ðÞ x nα − κ R ðÞ x + 〠 t 3.3. Analytic Solutions of the System of Fractional Partial Γ nα +1 ðÞ n=1 Differential Equations Using the Residual Power Series Method. This section contains the series solutions of the β S x ðÞ n nα + S x + 〠 t ðÞ nonlinear system of equations (2)–(7), together with the ini- N ΓðÞ nα +1 n=1 tial conditions in equation (8). In using the RPSM, it is assumed that there are discrepancies between the terms on I x ðÞ n nα ∗ Á I ðÞ x + 〠 t + I ðÞ x the right hand sides and the left hand sides of the system Γ nα +1 ðÞ n=1 of equations (2)–(7). With this assumption, the approxima- k k tions of the dependent variable with respect to only one I ðÞ x Q ðÞ x nα n nα + 〠 t + Q ðÞ x + 〠 t independent variable are obtained depending on the given Γ nα +1 Γ nα +1 ðÞ ðÞ n=1 n=1 initial condition or boundary point condition. The other S x independent variable is automatically in fractional form ðÞ n nα +ðÞ μ + η S ðÞ x + 〠 t which converges to a point in the Holder’s spaces. In doing ΓðÞ nα +1 n=1 this, we set ! 2α k ∂ S ðÞ x n nα − D S x + 〠 t : ðÞ 1 o 2α ∂x Γ nα +1 ðÞ S x n=1 ðÞ n nα ð37Þ S x, t = S x + 〠 t , ðÞ ðÞ k o Γ nα +1 ðÞ ð44Þ n=1 E x ðÞ n nα ð38Þ EðÞ x, t = E ðÞ x + 〠 t , k o To obtain S ðxÞ, equation (43) is reduced to ΓðÞ nα +1 n=1 ∂ S ðÞ x 1 α I ðÞ x Re sðÞ x, t = S ðÞ x + t − ψN nα 1 o ð39Þ I ðÞ x, t = I ðÞ x + 〠 t , ∂t ΓαðÞ +1 k o ΓðÞ nα +1 n=1 R ðÞ x 1 α − κ R ðÞ x + t Γα +1 ðÞ β S ðÞ x 1 α + S x + t ðÞ k ∗ I x ðÞ N Γα +1 ðÞ ∗ ∗ n nα ð40Þ I ðÞ x, t = I ðÞ x + 〠 t , k o ΓðÞ nα +1 I x n=1 ðÞ 1 α ∗ Á I ðÞ x + t + I ðÞ x ð45Þ ΓαðÞ +1 I x Q x ðÞ ðÞ 1 α 1 α + t + Q x + t ðÞ Γα +1 Γα +1 ðÞ ðÞ Q x ðÞ n nα ð41Þ QðÞ x, t = Q ðÞ x + 〠 t , k o S x ðÞ ΓðÞ nα +1 1 α n=1 + μ + η S x + t ðÞ ðÞ ΓαðÞ +1 2α ∂ S ðÞ x 1 α − D S ðÞ x + t : R ðÞ x 1 o ∗ n nα 2α ∂x Γα +1 ðÞ ð42Þ R ðÞ x, t = R ðÞ x + 〠 t : k o Γ nα +1 ðÞ n=1 Setting Re sðx,0Þ =0, it implies that ∂ S Re sðÞ x, t = − ψN − κRxðÞ , t ∂t ∗ ∗ + Sx, t Ix, t + I x, t +Qx, t S ðÞ x = ψN + κR ðÞ x −fg S ðÞ xðÞ I ðÞ x + I ðÞ x + Q ðÞ x ðÞðÞ ðÞ ðÞ ðÞ 1 o o o o N N 2α 2α − μ + η S x + D S x : +ðÞ μ + η SxðÞ , t − D SxðÞ , t : ðÞ ðÞ ðÞ 1 o 1 o 2α 2α ∂x ∂x ð43Þ ð46Þ Advances in Mathematical Physics 9 Similarly, the S ðxÞ is obtained as 2 n − 1 + S x I x + I x + Q x +⋯ ðÞðÞ ðÞ ðÞ ðÞ n−1−r r r r α r ∂ S ðÞ x S ðÞ x 1 α 2 2α Re s x, t = S x + t + t ðÞ ðÞ 2 0 ∂t ΓαðÞ +1 ΓðÞ 2α +1 +S x I x + I x + Q x ðÞðÞ ðÞ ðÞ ðÞ R x R x ðÞ ðÞ 0 n−1 n−1 n−1 1 α 2 2α − ψN − κ R ðÞ x + t + t ΓαðÞ +1 ΓðÞ 2α +1 2α β S x S x ðÞ ðÞ 1 2 ∂ α 2α + S ðÞ x + t + t −ðÞ μ + η S ðÞ x + D S ðÞ x , N Γα +1 Γ 2α +1 n−1 1 n−1 ðÞ ðÞ 2α ∂x I ðÞ x I ðÞ x 1 α 2 2α Á I ðÞ x + t + t ΓαðÞ +1 ΓðÞ 2α +1 n−1 n − 1 ∗ ∗ β I x I x ðÞ ðÞ ∗ α 2α 1 2 S ðÞ x = κR ðÞ x − 〠 S ðÞ x I ðÞ x + I ðÞ x + t + t n n−1 n−1−r r Γα +1 Γ 2α +1 N ðÞ ðÞ r=0 Q ðÞ x Q ðÞ x 1 α 2 2α 2α + Q x + t + t ðÞ Á 0 ∂ ΓαðÞ +1 ΓðÞ 2α +1 + I ðÞ x + Q ðÞ x −ðÞ μ + η S ðÞ x + D S ðÞ x : r n−1 1 n−1 2α ∂x S x S x ðÞ ðÞ 1 α 2 2α +ðÞ μ + η S ðÞ x + t + t Γα +1 Γ 2α +1 ð48Þ ðÞ ðÞ 2α ∂ S ðÞ x S ðÞ x 1 α 2 2α − D S x + t + t , ðÞ 1 0 2α ∂x ΓαðÞ +1 ΓðÞ 2α +1 Similarly, we obtain the following results for EðxÞ as È À ∂ β Re s x,0 = S x − κR x + S x I x ðÞ ðÞ ðÞ ðÞ ðÞ 2 2 1 1 0 ∂t N Á À E x = S x I x + I x + Q x + I ðÞ x + Q ðÞ x + S ðÞ x I ðÞ x ðÞfg ðÞðÞ ðÞ ðÞ ðÞ 0 0 0 1 1 0 0 0 0 ÁÉ + I ðÞ x + Q ðÞ x +ðÞ μ + η S ðÞ x 1 1 1 2α 2α ∂ −ðÞ α + α + μ + ν E ðÞ x + D E ðÞ x , 1 2 0 2 0 2α − D S ðÞ x =0, ∂x 1 1 2α ∂x β ∗ S ðÞ x = κR ðÞ x − S ðÞ xðÞ I ðÞ x + I ðÞ x + Q ðÞ x E ðÞ x = S ðÞ xðÞ I ðÞ x + I ðÞ x + Q ðÞ x 2 1 1 0 0 0 2 1 0 0 0 + S x I x + I x + Q x ∗ ðÞðÞ ðÞ ðÞ ðÞ 0 1 1 + S ðÞ xðÞ I ðÞ x + I ðÞ x + Q ðÞ x 0 1 1 2α 2α − μ + η S x + D S x : ðÞ ðÞ ðÞ 1 1 1 ∂ 2α ∂x − α + α + μ + ν E x + D E x , ðÞ ðÞ ðÞ 1 2 1 2 1 2α ∂x ð47Þ β È E x = S x I x + I x + Q x ðÞ ðÞðÞ ðÞ ðÞ ðÞ 3 2 0 0 0 Similarly, the following results are obtained: +2S ðÞ xðÞ I ðÞ x + I ðÞ x + Q ðÞ x 1 1 1 1 + S ðÞ xðÞ I ðÞ x + I ðÞ x + Q ðÞ x 0 2 2 2 È À ∂ β Re s x,0 = S x − κR x + S x I x + I x 2α ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ 3 3 2 2 0 0 α ∂ ∂t N − α + α + μ + ν E x + D E x , Á ðÞ ðÞ ðÞ 1 2 2 2 2 2α ∂x + Q ðÞ x +2S ðÞ xðÞ I ðÞ x + I ðÞ x + Q ðÞ x 0 1 1 1 + S x I x + I x + Q x ðÞðÞ ðÞ ðÞ ðÞ 0 2 2 2 2α n−1 n − 1 +ðÞ μ + η S ðÞ x − D S ðÞ x =0, E x = 〠 S x I x + I x + Q x 2 1 2 ðÞ ðÞðÞ ðÞ ðÞ ðÞ 2α n n−1−r r r r ∂x r=0 r 2α S ðÞ x = κR ðÞ x − S ðÞ xðÞ I ðÞ x + I ðÞ x + Q ðÞ x ∂ 3 2 2 0 0 0 N −ðÞ α + α + μ + ν E ðÞ x + D E ðÞ x : 1 2 n−1 2 n−1 2α ∂x +2S x I x + I x + Q x ðÞðÞ ðÞ ðÞ ðÞ 1 1 1 1 É ð49Þ + S x I x + I x + Q x ðÞðÞ ðÞ ðÞ ðÞ 0 2 2 2 2α Following similar procedure, the series solutions of the −ðÞ μ + η S ðÞ x + D S ðÞ x 2 1 2 2α fractional PDEs for describing the number of deviant infec- ∂x tives, number of non-deviant infectives, number of quaran- tined persons, and number of recoveries from the COVID- 19 disease are obtained as follows: 2α S ðÞ x = κR ðÞ x − S ðÞ xðÞ I ðÞ x + I ðÞ x + Q ðÞ x n n−1 n−1 0 0 N I x = α E x − γ + μ + δ + ω I x + D I x , ðÞ ðÞðÞ ðÞ ðÞ 1 1 0 1 1 1 0 3 0 2α ∂x n − 1 2α + S x I x + I x + Q x +⋯ ðÞðÞ ðÞ ðÞ ðÞ I x = α E x − γ + μ + δ + ω I x + D I x , ðÞ ðÞðÞ ðÞ ðÞ n−2 1 1 1 2 1 1 1 1 1 1 3 1 2α ∂x 1 10 Advances in Mathematical Physics 2α infectives, quarantined persons, and recovered persons: I ðÞ x = α E ðÞ x −ðÞ γ + μ + δ + ω I ðÞ x + D I ðÞ x , 3 1 2 1 1 2 3 2 2α ∂x ( ! k n−1 n − 1 1 β E x, t = E x + 〠 〠 ðÞ ðÞ k 0 Γ nα +1 N ðÞ n=1 r=0 2α I ðÞ x = α E ðÞ x −ðÞ γ + μ + δ + ω I ðÞ x + D I ðÞ x , n 1 n−1 1 1 n−1 3 n−1 Á S x I x + I x + Q x 1 ðÞðÞ ðÞ ðÞ ðÞ 2α n−1−r r r r ∂x − α + α + μ + ν E x ðÞ ðÞ 2α 1 2 n−1 ∗ ∗ ∗ 2α I ðÞ x = α E ðÞ x −ðÞ γ + μ + δ + ω I ðÞ x + D I ðÞ x , 1 2 0 2 2 0 4 0 2 ∂ 2α nα ∂x + D E x gt , ðÞ 2 n−1 2α ∂x ð52Þ 2α ∂ ( ∗ ∗ ∗ I ðÞ x = α E ðÞ x −ðÞ γ + μ + δ + ω I ðÞ x + D I ðÞ x , 2 2 1 2 2 1 4 1 2 1 2α ∂x I ðÞ x, t = I ðÞ x + 〠 α E ðÞ x k 0 1 n−1 ΓðÞ nα +1 2α n=1 ∗ ∗ ∗ I ðÞ x = α E ðÞ x −ðÞ γ + μ + δ + ω I ðÞ x + D I ðÞ x , 3 2 2 2 2 2 2 4 2 2α − γ + μ + δ + ω I x ðÞ ðÞ ∂x 1 1 1 n−1 2α nα + D I x gt , ðÞ 3 n−1 2α ∂x 2α ∗ ∗ ∗ I ðÞ x = α E ðÞ x −ðÞ γ + μ + δ + ω I ðÞ x + D I ðÞ x , 2 n−1 2 2 4 ( n 2 n−1 n−1 2α ∂x ∗ ∗ I ðÞ x, t = I ðÞ x + 〠 α E ðÞ x k 0 2 n−1 Q x = vE x + ω I x + ω I x − μ + ϕ + δ I x , ðÞ ðÞ ðÞ ðÞðÞ ðÞ Γ nα +1 1 0 1 0 2 0 3 0 ðÞ n=1 Q ðÞ x = vE ðÞ x + ω I ðÞ x + ω I ðÞ x −ðÞ μ + ϕ + δ I ðÞ x , − γ + μ + δ + ω I x ð53Þ 2 1 1 1 2 1 3 1 ðÞ ðÞ 2 2 2 n−1 ∗ 2α Q ðÞ x = vE ðÞ x + ω I ðÞ x + ω I ðÞ x −ðÞ μ + ϕ + δ I ðÞ x , ∂ 3 2 1 2 2 3 2 2 ∗ nα + D I x t , ðÞ 4 n−1 2α ∂x Q x = vE x + ω I x + ω I x − μ + ϕ + δ I x , ðÞ ðÞ ðÞ ðÞðÞ ðÞ QðÞ x, t = Q ðÞ x + 〠 fvE ðÞ x n n−1 1 n−1 2 n−1 3 n−1 k 0 n−1 ΓðÞ nα +1 n=1 ∗ ð54Þ R ðÞ x = ϕQ ðÞ x + γ I ðÞ x + γ I ðÞ x + ηS ðÞ x −ðÞ μ + κ R ðÞ x , 1 0 1 0 2 0 0 0 + ω I ðÞ x + ω I ðÞ x −ðÞ μ + ϕ + δ 1 n−1 2 n−1 3 nα R x = ϕQ x + γ I x + γ I x + ηS x − μ + κ R x , ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ 2 1 1 1 1 Á I ðÞ x gt , 1 2 1 n−1 R x = ϕQ x + γ I x + γ I x + ηS x − μ + κ R x , ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ 3 2 1 2 2 2 2 2 ⋮ R x, t = R x + 〠 ϕQ x + γ I x ðÞ ðÞ f ðÞ ðÞ k 0 n−1 1 n−1 Γ nα +1 ðÞ n=1 R ðÞ x = ϕQ ðÞ x + γ I ðÞ x + γ I ðÞ x + ηS ðÞ x ∗ nα n n−1 n−1 n−1 n−1 1 2 + γ I x + ηS x − μ + κ R x t : ðÞ ðÞ ðÞ ðÞg ð50Þ 2 n−1 n−1 n−1 −ðÞ μ + κ R ðÞ x : n−1 ð55Þ In order to obtain the series solution for the number of susceptible individuals, we substitute the last equation of 3.4. Numerical Results. In this section, the three- the system of equations (48) into equation (37) which dimensional plots, as well as the two-dimensional plots, yields are provided here. The initial condition for each subgroup was taken from [28]. 3.4.1. The Numerical Results for the Series Solutions of the S x, t = S x + 〠 κR x ðÞ ðÞ ðÞ Nonlinear System of Fractional PDEs Using the FPSM. This k 0 n−1 Γ nα +1 ðÞ n=1 section contains the plots for the series solutions to the n−1 n − 1 system of equations (22)–(27) for α =0:2000. Both three- − 〠 S ðÞ x ðI ðÞ x + I ðÞ x dimensional and two-dimensional plots for each series n−1−r r r r=0 solution of these equations are depicted in Figures 2 and 2α 3, respectively. The initial conditions for each subgroup nα + Q ðÞ x Þ −ðÞ μ + η S ðÞ x + D S ðÞ x t : of the population were estimated using the available data r n−1 1 n−1 2α ∂x in [28]. The number of susceptible decreases rapidly as they come into contact with the infectives, both locals ð51Þ and foreigners, as shown in Figure 2(a). In contrast to the number of nondeviant infective persons (see Similarly, the following results are obtained for the Figure 3(a)), the number of deviant infectives unexpect- number of exposed persons, deviant infectives, nondeviant edly grows to its peak for a long time before it starts to Ti T me (t) T Ti ime me (t) (t) Ti Time me (t) Time (t) T Ti ime (t) me (t) Ti Tim me e (t) (t) Advances in Mathematical Physics 11 S (x, t) E (x, t) ×10 4 30 2 15 0 0 1 1 1 1 0 0.8 .8 0.8 0.8 0.5 0.5 0.5 0.5 0.6 0.6 0.6 0.6 0. 0.4 4 0.4 02 0.2 0.2 0 0 0 0 (a) (b) I (x, t) ⁎ I (x, t) 0.8 0.8 0.8 0.5 0.6 0.5 0.5 0.6 0.6 0.4 0.4 0.4 0.2 0.2 02 (c) (d) Q (x, t) R (x, t) ×10 8 8 6 6 4 4 2 2 0 0 1 1 0.8 0.8 0 0.8 .8 0. 0.5 5 0.5 0.5 0. 0.6 6 0. 0.6 6 0. 0.4 4 0.4 0.4 0 0.2 2 02 0.2 0 0 0 0 (e) (f) Figure 2: Three-dimensional plots for the number of susceptible individuals, exposed persons, deviant infectives, nondeviant infectives, quarantined persons, and recovered persons for α =0:2000, using the FPSM. Distance (x) (x) Distance (x) Distance (x) Distance (x) ) Distance (x) (x) Distance (x) (x) 12 Advances in Mathematical Physics 7 E × 10 S 3.5 2.5 1.5 10 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (t) Time (t) (a) (b) I ⁎ 1.2 1.18 1.16 1.14 1.12 1.1 1.08 1.06 1.04 1.02 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (t) Time (t) 1 I (c) (d) Q 5 × 10 R 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (t) Time (t) 1 R 3 R (e) (f) Figure 3: Two-dimensional plots for the number of susceptible individuals, exposed persons, deviant infectives, nondeviant infectives, quarantined persons, and recovered persons α =0:2000, using the FPSM. S (t, ·) I (·, t) Q (t, ·) R (t, ·) E (t, ·) Advances in Mathematical Physics 13 × 10 S 3.5 22 2.5 1.5 0.5 0 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (t) Time (t) (a) (b) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (t) Time (t) (c) (d) × 10 R 2.5 1.5 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (t) Time (t) (e) (f) Figure 4: Two-dimensional plots for the various subgroups of the population size of Ghana for α =0:2000, using the FPSM. decline and eventually becomes asymptotically stable on tance is suppressed and time is varied. The first three series the t-axis. This implies that patients with the intention solutions for the system of equations (51)-(55) were of transmitting COVID-19 disease to the susceptible employed to show trends in the numbers of susceptible indi- members are the primary disease spreaders in Ghana. viduals, exposed individuals, deviant infectives, nondeviant For the number of susceptible, exposed, deviant, nonde- infectives, quarantined individuals, and recovered individ- viant, quarantined, and recovered subgroups, spatial dis- uals. The plots were repeated for the first fifteen terms of Q (t, ·) 14 Advances in Mathematical Physics × 10 S 5.5 3.5 2.5 4.5 1.5 3.5 0.5 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (t) Time (t) (a) (b) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (t) Time (t) (c) (d) Q × 10 R 1.5 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (t) Time (t) (e) (f) Figure 5: Two-dimensional plots for the various subgroups of the population size of Ghana for α =0:2000, using the RPSM. the series solutions of system of equations (51)-(55), shown idly, the curve for nondeviant infected individuals declines in Figure 4. sharply. This indicates that the majority of nondeviant To take into account the impact of each series as the carriers of the COVID-19 virus are not infecting suscepti- number of terms rises, plots for the first 80 terms of the ble members who are at risk of becoming infected. On the series solutions of the system of equations (51)-(55) were other hand, a large number of COVID-19 patients are reproduced as displayed in Figure 5. While the curves being isolated at numerous facilities across the country, for the quarantined and recovered populations climb rap- and these people are making progress every day. The fact Advances in Mathematical Physics 15 × 10 S 3.5 6 2.5 1.5 0.5 0 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Distance (x) Distance (x) S E 1 1 S E 2 2 (a) (b) 1.2 0.8 0.6 0.4 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Distance (x) Distance (x) 1 I 2 I 3 I (c) (d) Figure 6: Two-dimensional plots for the various subgroups of the population size using the FPSM. that the deviant subpopulation is still growing suggests irrespective of their home country will not ensue to the epi- that patients have been spreading the SARS virus for a demiology of COVID-19 disease in Ghana. However, there considerable amount of time with the intention of infect- is a slight positive nonlinear relationship and fairly negative ing a sizable number of vulnerable individuals with the nonlinear relationship between the nondeviant subpopula- COVID-19 infection. The plot for the exposed individuals tion and the distance. increases to a peak and then declines and asymptotically moves toward the t-axis, showing that those who are vul- 3.4.2. The Numerical Results for the Series Solutions of the nerable to contracting COVID-19 disease are at high risk. Nonlinear System of Fractional PDEs Using the RPSM. The SARS virus starts spreading to anybody who come Figure 7 displays the first three terms of the series solutions, into contact with it after a brief time of incubation. From equations (51)-(55), with α =0:2000. Every subgroup of the day zero, the susceptible curve declines and becomes population size in Ghana corresponds with the epidemiolog- asymptotically toward the t-axis, signalling the end of the ical pattern of the COVID-19 disease. disease. In Figure 6, there is relatively positive nonlinear relation- ship between the susceptible subgroups. This indicates that 3.5. Comparison of the FPSM and the RPSM. This section susceptible (foreigners) continuous inflow into the country uses quantitative results of the FPSM to compare the I (x, ·) S (x, ·) E (x, ·) 16 Advances in Mathematical Physics 7 E × 10 S 3.5 2.5 1.5 0.5 0 1 0 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time Time (a) (b) 0 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time Time 3 I (c) (d) Q 5 × 10 R 1000 5 800 4 600 3 400 2 200 1 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Time Q R 1 1 Q R 2 2 Q R 3 3 (e) (f) Figure 7: Two-dimensional plots for the various subgroups of the population size using the RPSM. Q (·, t) I (·, t) S (·, t) R (·, t) E (·, t) Advances in Mathematical Physics 17 15000 6000 20 4000 10 2000 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 data method 1 method 2 (a) 4 E × 10 15 2 20 4000 1 10 2000 0 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 data method 1 method 2 (b) Figure 8: Plots of the field data on the number of exposed persons and the series solutions using both the FPSM and the RPSM. quantitative results of the RPSM which are then superim- On the other hand, using the RPSM is more consistent posed on the quantitative results from the field data. with the field data as the COVID-19 infection is present in Figure 8 displays the series solutions of the first three the population subjects over a long period of time in com- terms using both the FPSM and the RPSM. The FPSM parison to the FPSM’s series solution. This is indicated by shows that the solution rises from the starting point to the deep red plot. A similar observation was made in the peak and then falls, showing the loss of the suscepti- Figure 9 when comparing the two series solutions using both ble members to the exposed subgroup and the exposed the FPSM and the RPSM. subgroup’s loss of members to both the deviant and non- deviant subgroups. On the other hand, the length of the 4. Discussion series solution obtained using the RPSM increases starting with the beginning of the COVID-19 pandemic and takes In contrast, the FPSM series solution for the number of a lot of time. It begins to decline in the direction of the t susceptible individuals is proportional to the RPSM series -axis, showing that the susceptible individuals who con- solution for the number of susceptible individuals, with a tract the SARS virus also persistently infect the popula- proportional constant of ψΓððn − 1Þα +1Þ. The series solu- tion subjects. Despite this, the series solution using the tion of the vulnerable members utilizing the RPSM then FPSM is more consistent with the field data at the start reduced quickly in comparison to the series solution using of the disease outbreak than the series solution using the FPSM. For instance, the third term of the series solu- the RPSM. tion using the RPSM is reduced by S ðxÞðI ðxÞ + I ðxÞ + 1 1 data data method 1 method 1 method 2 method 2 18 Advances in Mathematical Physics 10000 1200 5000 60 600 0 40 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 data method 1 method 2 (a) 10000 60 900 5000 50 0 40 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 data method 1 method 2 (b) Figure 9: Plots of the field data on the number of infected persons and the series solutions using both the FPSM and the RPSM. Q ðxÞÞ when compared to the series solution using the uals. The power of SðxÞðIðxÞ + I ðxÞ + QðxÞÞ is linear for FPSM. In comparison, using the RPSM in obtaining the the first and second terms and then increases in the pat- series solutions of the nonlinear system of equations, tern of natural numbers, that is, having positive integer (2)–(7), together with the initial conditions in equation binomial powers, when employing the RPSM. Because (8), are more consistent with the field data as compared the product of this nonlinear term is differentiated, this to the series solutions given by the FPSM, as displayed occurs. The nonlinear term, however, increases linearly in Figures 8 and 9. when the FPSM is used to find the series solutions of the system of equations (2)–(7) along with the initial con- ditions in equation (8). Interestingly, series solutions of the 5. Conclusion nonlinear system of the equations using the RPSM were observed to be more consistent as compared to series solu- There are more terms for the total number of susceptible tions given by FPSM. This is due to the fact that the members and exposed individuals in the series of solutions RPSM utilizes the variations in the nonlinear system of of the nonlinear system of fractional PDEs provided by the equations unlike the FPSM. However, both the RPSM RPSM than the series of solutions yielded by the FPSM. and the FPSM yield the same series solutions of the linear The nonlinear term SðxÞðIðxÞ + I ðxÞ + QðxÞÞ is what system of equations, as indicated by equations (24)–(29) causes the difference between the two series solutions for and system of equations (51)–(55). the number of susceptible members and exposed individ- I data data I method 1 method 1 method 2 I method 2 Advances in Mathematical Physics 19 Data Availability severity, hospitalization, and mortality: a systematic review,” Human Vaccines and Immunotherapeutics, vol. 18, no. 1, arti- The data is freely available at [28] https://ourworldindata cle 2027160, 2022. .org/coronavirus. [13] M. Shah, M. Arfan, I. Mahariq, A. 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Rodés-Guirao et al., “Coronavirus [12] I. Mohammed, A. Nauman, P. Paul et al., “The efficacy and pandemic (COVID-19),” 2020, https://ourworldindata.org/ effectiveness of the COVID-19 vaccines in reducing infection, coronavirus. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Mathematical Physics Hindawi Publishing Corporation

The Analytic Solutions of the Fractional-Order Model for the Spatial Epidemiology of the COVID-19 Infection

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Hindawi Advances in Mathematical Physics Volume 2023, Article ID 5578900, 19 pages https://doi.org/10.1155/2023/5578900 Research Article The Analytic Solutions of the Fractional-Order Model for the Spatial Epidemiology of the COVID-19 Infection 1 2 3 1 Benedict Barnes , Martin Anokye, Mohammed Muniru Iddrisu, Bismark Gawu, and Emmanuel Afrifa Kwame Nkrumah University of Science and Technology, Department of Mathematics, Ghana University of Cape Coast, Department of Mathematics, Ghana C.K. Tedam University of Technology and Applied Sciences, School of Mathematical Sciences, Department of Mathematics, Ghana Correspondence should be addressed to Benedict Barnes; ewiekwamina@gmail.com Received 16 December 2022; Revised 19 March 2023; Accepted 28 March 2023; Published 4 May 2023 Academic Editor: Andrei Mironov Copyright © 2023 Benedict Barnes et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper provides a mathematical fractional-order model that accounts for the mindset of patients in the transmission of COVID-19 disease, the continuous inflow of foreigners into the country, immunization of population subjects, and temporary loss of immunity by recovered individuals. The analytic solutions, which are given as series solutions, are derived using the fractional power series method (FPSM) and the residual power series method (RPSM). In comparison, the series solution for the number of susceptible members, using the FPSM, is proportional to the series solution, using the RPSM for the first two terms, with a proportional constant of ψΓððnα +1Þ, where ψ is the natural birth rate of the baby into the susceptible population, Γ is the gamma function, n is the nth term of the series, and α is the fractional order as the initial number of susceptible individuals approaches the population size of Ghana. However, the variation in the two series solutions of the number of members who are susceptible to the COVID-19 disease begins at the third term and continues through the remaining terms. This is brought on by the nonlinear function present in the equation for the susceptible subgroup. The similar finding is made in the series solution of the number of exposed individuals. The series solutions for the number of deviant people, the number of nondeviant people, the number of people quarantined, and the number of people recovered using the FPSM are unquestionably almost identical to the series solutions for same subgroups using the RPSM, with the exception that these series solutions have initial conditions of the subgroup of the population size. It is observed that, in this paper, the series solutions of the nonlinear system of fractional partial differential equations (PDEs) provided by the RPSM are more in line with the field data than the series solutions provided by the FPSM. 1. Introduction is more appropriately classified as a pandemic disease than an epidemic disease. The primary factor in the global The development of a mathematical model for under- transmission of the COVID-19 disease is the movement standing and unravelling the underlying mechanisms of of exposed or infected individuals, who may or may not the epidemiology of the COVID-19 disease has garnered have the aim of coming into contact with the vulnerable interest from public health systems to academia in several individuals in the host country. The spatial spread of the different countries; the majority of these models focus on disease in the various countries was accounted for in the epidemics of the disease progression from one person to mathematical models developed by [5, 6] by taking into consideration the diffusing susceptible individuals, exposed another person, as described by [1–3]. However, the COVID-19 disease originated in Wuhan, China, and geo- individuals, and infected individuals. Despite this, these graphically spread to other parts of the world as a pan- models do not account for the vaccinations that the indi- viduals of the population received. demic disease [4]. In this sense, COVID-19 epidemiology 2 Advances in Mathematical Physics There are currently treatments available that are given to can only produce fixed-point solutions of differential equa- people all around the world regardless of their health condi- tions. The vast majority of nonstationary points are uncov- tions, such as the Johnson and Johnson vaccine and the ered by this method. More crucially, neither a quantitative AstraZeneca vaccine. Although the usefulness of these vac- nor a qualitative method provides the function that describes cines has been scientifically demonstrated, these immuniza- the theoretical foundation for describing the epidemiology tions do lose some of their efficacy over time. There is no of COVID-19 disease. assurance that a person receiving the COVID-19 vaccination Recently, it has been discovered that the integer differen- will be protected from getting the disease upon contact with tial equations suffer from several shortcomings when com- a person with the SARS virus. For example, see authors in pared to the differential equations of fractional order. The [7]. This observation makes it unclear which individuals fractional differential equation has memory and heredity are completely unprotected from the disease and which properties because of its nonlocal property for describing persons are temporarily protected for a short period of time COVID-19 pandemics. Any solution to the system of PDEs, following immunization. Only a few researchers have used regardless of order, may be easily obtained using fractional vaccinated subjects in their models without the inclusion of ordering. The theory of controls, infectious diseases, growth the spatial transmission of the disease. For example, have a of tumours, and feedback systems are examples of applied scientific problems where the differential equations of frac- look at the authors in [8–10]. All of these epidemiological models account for people moving from one subgroup of tional order have proven to be effective models. For example, the population to another subgroup of the population. see authors in [13] who applied the fractal fraction Adams- Although they captured vaccinated persons, they did not Bashforth method to search for the solution of fractal- incorporate the diffusing individuals who brought the fractional susceptible-infective-recovered model. Another COVID-19 disease into their respective countries. Also, for numerical approach for solving systems of differential equa- the mathematical models on the control of transmission of tions that are both linear and nonlinear is the Pade approx- COVID-19 disease, see [11]. The findings of these imation method. High-order approximations are necessary researchers, however, are not all inclusive since they when using this method. More crucially, given a nonlinear neglected to take into account an important observation of system of PDEs, there is no systematic procedure in selecting the progression of patient through the disease. Evidence the parameters in the Pade approximation method[14]. from numerous countries has demonstrated that infected Since Mittag-Leffler functions or their derivatives make up individuals (patients) either plan or do not intend to trans- the majority of the solutions to the system of fractional dif- mit the COVID-19 virus to susceptible individuals [4]. In ferential equations, rigorous mathematics is necessary to developing a mathematical model to describe the epidemiol- solve these equations. One of the methods for solving system ogy of the COVID-19 infection, the mindset of the spreaders of fractional differential equations is the RPSM which was was not captured in their models. Additionally, statistics first observed by [15] for solving fuzzy differential equation. from different countries have revealed that whether a person With this approach, a power series is assumed to exist for the takes medicine to treat COVID-19 or not, they still run the system of ODEs, and the coefficients of the power series are risk of getting the illness again if they come into contact with used to create a recurrence equation. When the residual an infected person. Thus, the recovery from the disease is for coefficients, from the power series, are equal to zero, an alge- a short period of time (see [12]). When creating a mathe- braic system of equations results, from which the values of matical model to describe the epidemiology of COVID-19, the series solution of the unknown coefficients can be all these issues were not taken into account. deduced. Nevertheless, while solving fractional-order PDE, The type of mathematical tools a researcher uses to say in two variables, this method assumes that one of the arrive at his or her conclusion(s) ultimately determines the independent variables has a representation in a fractional success of any mathematical analysis. Since the beginning power series, and the second independent variable is han- of the COVID-19 outbreak in China till now, researchers dled as a coefficient variable, which is roughly derived from have mainly relied heavily on either the use of the qualitative the variation in the given fractional-order PDE based on the method, the quantitative method, or both. These methods initial or boundary condition. For example, see authors in have significantly more drawbacks than advantages. A [16–19]. The same method was used by [20] to solve nonlin- numerical scheme is an example of a quantitative method ear fractional-order PDEs. In [21], the authors used the that always approximates the exact solution of the differen- Atangana-Baleanu fractional derivative to obtain asymptotic tial equation with some level of precision. This quantitative interval approximation solutions to the fractional differential method yields intolerable inaccuracies; in the worst situa- equation under various conditions. A nonlinear system of tion, its solution diverges from the exact solution of a differ- stiff fractional-order PDEs and the nonlinear system of frac- ential equation. As usual, even if the solution suggested by tional PDEs have not been solved using the RPSM. The kind the numerical scheme exists, one must perform a number of nonlinearity in a fractional PDE largely depends on the of iterations before reaching the desired solution. The qual- functional space which contains the solution of fractional itative method narrows down the information contained in differential PDE. The FPSM is another intriguing method the solution of the differential equation. The domain ele- which was first observed by [22]. The authors in [23, 24] ments of the function that describes the epidemiology of applied this method to obtain the solutions of fractional COVID-19 infection are revealed by this method of investi- PDEs. It is challenging to find analytic solutions to a nonlin- gation on a microscopic level. In light of this, this method ear system of fractional-order partial differential equations. Advances in Mathematical Physics 3 Additionally, researchers from all over the world have not RPSM-based analytical solutions of the nonlinear system of observed a comparison of the series solutions utilizing both fractional PDEs are presented as series solutions. the RPSM and the FPSM. More importantly, no information 3.1. Model Description. Despite the fact that the COVID-19 regarding comparing the series solutions obtained by these vaccination is given to country residents by the Ministry of methods with field data is provided in the literature. The series solution (analytic) method of the nonlinear system Health (MOH), neither the Johnson and Johnson nor the AstraZeneca vaccine is anticipated to provide COVID-19 of fractional-order partial differential equations has a solu- patients with a lifetime of immunity against the illness. tion, is the most dependable and efficient method as com- In Figure 1, the population size of Ghana is split into six pared to both the qualitative and the quantitative methods. distinct subgroups namely: the susceptible subgroup, Sðx, tÞ; In this paper, the infected group of the SEIQR model is exposed subgroup, Eðx, tÞ; deviant infected subgroup, Iðx, tÞ; further divided into two subgroups: the deviant infected sub- nondeviant infected subgroup, I ðx, tÞ; quarantined sub- group and nondeviant subgroup of the population in the group, QðtÞ; and the recovered subgroup, RðtÞ. A susceptible classical susceptible-exposed-infected-quarantined-recov- person is any member of the population who is capable of ered model with diffusion terms. Thus, the susceptible- catching the SARS virus from an infected COVID-19 exposed-deviant infected-nondeviant infected-quarantined- patient. An exposed person is someone who has caught the recovered (SEII QR) model with diffusion terms, and vacci- SARS virus, but for a brief while, he or she is unable to pass nated susceptible term is developed. In addition, the frac- tional form of this model is provided herein. Moreover, it on to a susceptible person. The waiting period is therefore in effect for this person. The deviant infected person is a both the FPSM and the RPSM are used to obtain the series patient who has chosen to purposefully spread the SARS solution (analytic) of the nonlinear system of fractional PDEs. The solutions that are yielded by these two methods virus to susceptible family members on the grounds that since they already have the illness, they must also experience are compared with field data accounting for the robustness the COVID-19 disease-related consequences. The nondevi- of the methods. ant individual, on the other hand, is an infected person who does not willingly spread the SARS virus to susceptible 2. Fundamental Concept in Fractional Calculus family members or friends because they do not want them to get the COVID-19 illness. The person under quarantine is a Definition 1. A real function uðx, tÞ, x ∈ I, t >0 is said to be COVID-19 patient who was deviant infected person or non- in the space C ðI × ℝ Þ, μ ∈ ℝ, if there exist a real number p + deviant infected person, or an exposed person, whose move- p > α such that uðx, tÞ = t f ðx, tÞ, where f ðx, tÞ ∈ CðI × ℝ Þ, n n ments are restricted in a specific place for an extended length and it is said to be in the space C ,if ∂ /∂t ∈ C , n ∈ ℕ α α of time. The recovered person is the person who either (see [25]). recovers naturally or receives treatment at the hospital for a period of time. After being exposed to the COVID-19 Definition 2. For n − 1< β < n, n ∈ ℕ. The Caputo fractional infection, this person is still at a significant risk of reacquir- derivative operator of the order β is define by (see [26]) ing the SARS virus. As a consequence of this, the immune system is unable to recuperate and is therefore vulnerable n−α−1 α ðÞ n to losing its immunity. To take into account the continuous ðÞ D u ðÞ t = u ðÞ ξðÞ t − ξ dξ, t >0: ð1Þ ΓðÞ n − α inflow of foreigners into the country, the subgroups of sus- ceptible, exposed, deviant infected person, and nondeviant mα infected person depend on the distance, x, as well as the pas- Theorem 3. The fractional power series (FPS)∑ a ðt − t Þ : m=0 m 0 sage of time. Additionally, ψ stands for the natural birth rate, β for (i) converges only for t = t , that is, the radius of conver- transmission rate, and μ for natural death rate, ν is the rate gence equal to zero for quarantining exposed individuals, and α is the rate at (ii) converges for all t ≥ t , that is; the radius of conver- which an exposed person intends to transmit the SARS virus gence equal to ∞ to susceptible members. The rate at which an exposed per- son has no intention to infect a susceptible member with (iii) converges for t ∈ ½t , t + RŠ , for some positive real 0 0 the SARS virus is α . The rates at which the exposed person, numbers R, and diverges for t > t + R. Here, R is the deviant infected person, and the nondeviant infected the radius of convergence for the FPS [27]. person are quarantined, respectively, are ν, ω , and ω . The 1 2 disease-induced death rates from the subgroups of deviants, nondeviants, and confined individuals are δ , δ , and δ , 1 2 3 3. Main Results respectively. The ϕ, γ , and γ are the rates of recoveries 1 2 In this section, for modelling the COVID-19 epidemiology from COVID-19 disease by the quarantined, the deviant, in Ghana, a mathematical model that takes into account and the nondeviant individuals, respectively. The η is the the mindset of the patients in spreading the COVID-19 dis- rate at which susceptible moves to the recovered compart- ease, temporary loss of immunity by recoveries, and the con- ment after receiving a vaccine, and κ is the rate at which tinuous influx of foreigners entering the country with or recovered person loses their immunity and becomes suscep- without the disease is needed. Therein, the FPSM- and tible again. The natural death rate from each subgroup of the 4 Advances in Mathematical Physics equations (2)–(7) together with the initial conditions in equation (8) are obtained in Hilbert space using the FPSM. In obtaining each solution of the system of equations (2)–(7) together with initial conditions, it is assumed that the unknown function defining the equation is in series form which converges to a known function. In addition, the proof of the existence of these series solution as well as its unique- ness is provided here. Setting kα Sx, t = 〠 S x t , ð9Þ ðÞ ðÞ Figure 1: shows the various subgroups of the population size for k=0 describing the epidemiology of COVID-19. population is denoted by μ. Due to the fact that the model takes into account diffusion of the foreigners into the sus- kα Ex, t = 〠 E x t , ð10Þ ðÞ ðÞ ceptible, exposed, deviant, and nondeviant subgroups, the k=0 rates of diffusion into the corresponding compartments are specified as follows: D represents the rate of diffusion into the susceptible compartment, D represents the rate of diffu- kα Ix, t = 〠 I x t , ð11Þ ðÞ ðÞ sion into the exposed compartment, D represents the rate of k=0 diffusion into the deviant subgroup, and D represents the rate of diffusion into the nondeviant subgroup. Based on above facts, the following nonlinear system of ∗ ∗ kα I x, t = 〠 I x t , ð12Þ ðÞ ðÞ fractional PDEs is obtained for describing the epidemiology k=0 of COVID-19 in Ghana. α 2α ∂ S βSI + I + Q ∂ S ðÞ ð2Þ kα = ψN + κR − − μ + η S + D , ðÞ 1 Qx, t = 〠 Q x t , ð13Þ α 2α ðÞ ðÞ ∂t N ∂x k=0 α 2α ∂ E βSIðÞ + I + Q ∂ E ð3Þ = −ðÞ α + α + μ + v E + D , 1 2 2 ∞ 2α ∂t N ∂x kα RxðÞ , t = 〠 R ðÞ x t , ð14Þ α 2α k=0 ∂ I ∂ I ð4Þ = α E −ðÞ γ + μ + δ + ω I + D , 1 1 1 3 α 2α ∂t ∂x we can see that α 2α ∗ ∗ ∂ I ∂ I ð5Þ = α E − γ + μ + δ + ω I + D , ðÞ 2 2 2 2 4 α 2α ∂t ∂x ΓðÞ kα +1 α ðÞ k−1 α DðÞ SxðÞ , t = 〠 S ðÞ x t , α k d Q ΓðÞ ðÞ k − 1 α +1 k=1 ð6Þ = vE + ω I + ω I − μ + ϕ + δ E, ðÞ 1 2 3 α ð15Þ dt 2α ∞ 2α ∂ S ∂ kα = 〠 S x t : ðÞ ∂ R k 2α 2α ∂x ∂x ð7Þ = ϕQ + γ I + γ I + ηS − μ + κ R, ðÞ k=0 1 2 ∂t together with the initial conditions Substituting equations (8), (9), (10), (11), (12), (13), and (14) into equation (2) yields Sx,0 = S x , ðÞ ðÞ Ex,0 = E x , ðÞ ðÞ ΓðÞ kα +1 k−1 α ðÞ 〠 S x t ðÞ Γ k − 1 α +1 Ix,0 = I x , ðÞ ðÞ ðÞ ðÞ 0 k=1 ð8Þ ∞ ∞ ∗ ∗ I x,0 = I x , ðÞ ðÞ kα kα = ψN + κ〠 R ðÞ x t − 〠 S ðÞ x t k k k=0 k=0 Q 0 =0, ðÞ ð16Þ ∞ ∞ ∞ kα ∗ kα kα R 0 =0: ðÞ Á 〠 I x t + 〠 I x t + 〠 Q x t ðÞ ðÞ ðÞ k k k k=0 k=0 k=0 3.2. Analytic Solutions of the System of Fractional Partial ∞ 2α ∞ kα kα Differential Equations Using the Fractional Power Series −ðÞ μ + η 〠 S ðÞ x t + D 〠 S ðÞ x t : k 1 k 2α ∂x k=0 k=0 Method. In this section, series solutions of the system of Advances in Mathematical Physics 5 Comparing the powers of t , we have ΓαðÞ +1 β E x = S x I x + S x I x ðÞ ðÞ ðÞ ðÞ ðÞ 2 0 1 0 ΓðÞ 2α +1 N 1 β + S x Q x + I x S x + S x I x + S x Q x ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ S ðÞ x = ψN + κR ðÞ x − S ðÞ x ðI ðÞ x 0 1 0 1 1 0 1 0 1 0 0 0 ΓαðÞ +1 N ) 2α ∂ E ðÞ x ) 1 − α + α + μ + ν E + D , 2α ðÞ 1 2 1 2 2α ∂x + I x + Q x − μ + η S x + D S x : ðÞ ðÞÞ ðÞ ðÞ ðÞ 0 0 0 1 0 ( 2α ∂x Γ 2α +1 β À ðÞ E ðÞ x = S ðÞ x I ðÞ x + S ðÞ x I ðÞ x 3 0 2 0 Γ 3α +1 N ð17Þ ðÞ + S x Q x + S x I x + S x I x ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ 0 2 1 1 1 Similarly, we observe the following results. For S ðxÞ, S 2 3 + S x Q x + S x I x + S x I x ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ 1 1 2 0 2 0 ðxÞ, ⋯, S ðxÞ, we have n ) 2α ∂ E ðÞ x ( + S ðÞ x Q ðÞ x −ðÞ α + α + μ + ν E + D , 2 0 1 2 2 2 2α ∂x ΓαðÞ +1 β S x = ψN + κR x − S x I x ðÞ ðÞ ðÞ ðÞ 2 1 0 1 Γ 2α +1 N ðÞ ( " n−1 + S ðÞ x I ðÞ x + S ðÞ x Q ðÞ x + S ðÞ x I ðÞ x 0 0 1 1 0 1 À ΓðÞ ðÞ n − 1 α +1 β E x = 〠 S x I x ∗ ðÞ ðÞ ðÞ n k n−1−k + S ðÞ x I ðÞ x + S ðÞ x Q ðÞ x −ðÞ μ + η S 1 1 0 1 Γ nα +1 N 0 ðÞ k=0 2α + D S , 1 1 2α + S ðÞ x I ðÞ x + S ðÞ x I ðÞ x ∂x k k n−1−k n−1−k 2α ΓðÞ 2α +1 β ∂ E x x ðÞðÞ n−1 S ðÞ x = ψN + κR ðÞ x − S ðÞ x I ðÞ x − α + α + μ + ν E x + D : 3 2 0 2 ðÞ ðÞ 1 2 n−1 2 2α ΓðÞ 3α +1 N ∂x + S x I x + S x Q x + S x I x ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ ð19Þ 0 2 0 2 1 1 + S x I x + S x Q x + S x I x ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ 1 1 1 1 2 0 ∗ Similarly, the series solutions for the number of the devi- + S x I x + S x Q x − μ + η S ðÞ ðÞ ðÞ ðÞ ðÞ 2 0 2 0 2 ant infected people, the number of nondeviant infected peo- 2α ple, the quarantined, and the number of recoveries are as ∂ S + D , 2α follows: ∂x I x = α E x − γ + μ + δ ðÞ ðÞ ( 1 1 0 1 1 ΓαðÞ +1 Γ n − 1 α +1 ðÞ ðÞ S ðÞ x = ψN + κR ðÞ x n n−1 2α ∂ I x ΓðÞ nα +1 ðÞ + ω ÞI ðÞ x + D , 1 0 3 2α ∂x n−1 − 〠 S x I x + S x I x ðÞ ðÞ ðÞ ðÞ i n−1−i i n−1−i N À ΓαðÞ +1 i=0 I x = α E x − γ + μ + δ ðÞ ðÞ 2 1 1 1 1 ΓðÞ 2α +1 + S ðÞ x Q ðÞ x −ðÞ μ + η S ðÞ x i n−1−i n−1 2α ∂ I ðÞ x + ω ÞI x + D , ðÞ 1 1 3 2α ∂x 2α ∂ S x ðÞ n−1 + D : 2α ∂x Γ 2α +1 ðÞ ð20Þ I ðÞ x = α E ðÞ x − γ + μ 3 1 2 Γ 3α +1 ðÞ ð18Þ 2α ∂ I ðÞ x + δ + ω ÞI x + D , ðÞ 1 1 2 3 2α Following the similar procedure above, the following ∂x results are obtained for E ðxÞ, E ðxÞ, E ðxÞ, ⋯, E ðxÞ: 1 2 3 n 1 β ∗ À ΓðÞ ðÞ n − 1 α +1 E x = S x I x + I x + Q x ðÞ ðÞðÞ ðÞ ðÞ ðÞ 1 0 0 0 0 I ðÞ x = α E ðÞ x − γ + μ n 1 n−1 Γα +1 N ðÞ ΓðÞ nα +1 2α 2α ∂ E x ðÞ Á ∂ I x 0 ðÞ n−1 −ðÞ α + α + μ + ν E + D , + δ + ω I ðÞ x + D , 1 2 0 2 1 1 n−1 3 2α 2α ∂x ∂x 6 Advances in Mathematical Physics The series solutions of the nonlinear system of fractional I ðÞ x = α E ðÞ x − γ + μ + δ PDEs order of nth term are given by 1 2 0 2 ΓαðÞ +1 2α ∗ ∞ ∞ ∂ I x ðÞ ΓðÞ ðÞ n − 1 α +1 ∗ 0 + ω ÞI x + D , ðÞ S x, t = 〠 ψN + 〠 κR x ðÞ ðÞ 2 0 4 2α n n−1 ∂x Γ nα +1 ðÞ n=1 n=1 n−1 ΓαðÞ +1 − 〠 S ðÞ x I ðÞ x + S ðÞ x I ðÞ x I ðÞ x = α E ðÞ x − γ + μ + δ k n−1−k k n−1−k 2 2 1 2 Γ 2α +1 ðÞ k=0 2α ∗ ∂ I x ðÞ ∗ 1 + S x Q x − μ + η S x ðÞ ðÞ ðÞ ðÞ + ω ÞI ðÞ x + D , k n−1−k n−1 2 4 2α ∂x 2α ∂ S ðÞ x n−1 kα ΓðÞ 2α +1 ð21Þ ∗ + D t , I x = α E x − γ + μ + δ 2α ðÞ ðÞ 3 2 2 2 2 ∂x Γ 3α +1 ðÞ ð24Þ 2α ∗ ∂ I x ðÞ ∗ 2 + ω ÞI ðÞ x + D , 2 4 2α ( " ∂x ∞ n−1 ΓðÞ ðÞ n − 1 α +1 β E ðÞ x, t = 〠 〠 S ðÞ x I ðÞ x n k n−1−k Γ nα +1 N ⋮ ðÞ n=1 k=0 Γ n − 1 α +1 À ðÞ ðÞ + S ðÞ x I ðÞ x + S ðÞ x I ðÞ x I x = α E x − γ ðÞ ðÞ k k n−1−k n−1−k n 2 n−1 2 Γ nα +1 ðÞ 2α 2α ∗ ∂ E ðÞ x ∂ I ðÞ x 2 kα ∗ n−1 − α + α + μ + ν E x + D t , ðÞ ðÞ + μ + δ + ω ÞI ðÞ x + D , 1 2 n−1 2 2 2 n−1 4 2α 2α ∂x ∂x ð25Þ Q ðÞ x = vE ðÞ x + ω I ðÞ x 1 0 1 0 ΓαðÞ +1 Γ n − 1 α +1 ðÞ ðÞ I ðÞ x, t = 〠 α E ðÞ x + ω I ðÞ x −ðÞ μ + ϕ + δ Q ðÞ x , n 1 n−1 2 0 3 0 ΓðÞ nα +1 n=1 Γα +1 È ðÞ 2α Q x = vE x + ω I x ðÞ ðÞ ðÞ ∂ I ðÞ x 2 1 1 1 n−1 kα Γ 2α +1 ðÞ − γ + μ + δ + ω I x + D t , ðÞ ðÞ 1 1 1 n−1 3 2α É ∂x + ω I ðÞ x −ðÞ μ + ϕ + δ Q ðÞ x , 2 1 3 1 ð26Þ Γ 2α +1 È ðÞ Q x = vE x + ω I x ð22Þ ( ðÞ ðÞ ðÞ 3 2 1 2 Γ 3α +1 ðÞ Γ n − 1 α +1 ðÞ ðÞ I ðÞ x, t = 〠 α E ðÞ x ∗ 2 n−1 + ω I ðÞ x −ðÞ μ + ϕ + δ Q ðÞ x , ΓðÞ nα +1 2 2 3 2 n=1 2α ∂ I ðÞ x ∗ n−1 kα − γ + μ + δ + ω I x + D t , ðÞ ðÞ 2 2 2 n−1 4 2α ∂x ΓðÞ ðÞ n − 1 α +1 Q x = vE + ω I x ðÞ ðÞ n n−1 1 n−1 Γ nα +1 ðÞ ð27Þ + ω I ðÞ x −ðÞ μ + ψ + δ Q ðÞ x , 2 n−1 3 n−1 Γ n − 1 α +1 ðÞ ðÞ Q ðÞ x, t = 〠 νE ðÞ x + ω I ðÞ x n n−1 1 n−1 ∗ ΓðÞ nα +1 R x = ϕQ + γ I + γ I + ηS − μ + κ R , n=1 ðÞ fg ðÞ 1 0 0 0 0 1 2 0 Γα +1 ðÞ É ∗ kα + ω I x − μ + ψ + δ Q x t , ðÞðÞ ðÞ 2 n−1 3 n−1 ΓαðÞ +1 R ðÞ x = fg ϕQ + γ I + γ I + ηS −ðÞ μ + κ R , 2 1 1 1 1 1 1 2 ð28Þ ΓðÞ 2α +1 ΓðÞ 2α +1 ∗ ΓðÞ ðÞ n − 1 α +1 R ðÞ x = fg ϕQ + γ I + γ I + ηS −ðÞ μ + κ R , 3 2 1 2 2 2 2 2 R ðÞ x, t = 〠 ϕQ ðÞ x + γ I ðÞ x Γ 3α +1 n n−1 n−1 ðÞ 1 ΓðÞ nα +1 n=1 ⋮ ∗ kα + γ I x + ηS x − μ + κ R x t : ðÞ ðÞ ðÞ ðÞ 2 n−1 n−1 n−1 ΓðÞ ðÞ n − 1 α +1 ð29Þ R ðÞ x = ϕQ ðÞ x + γ I ðÞ x n n−1 1 n−1 ΓðÞ nα +1 3.2.1. Existence and Uniqueness of the Series Solution of the + γ I x + ηS x − μ + κ R x : ðÞ ðÞ ðÞ ðÞ n−1 n−1 2 n−1 Nonsystem of Fractional PDEs. The proofs of the existence ð23Þ and uniqueness of the series solutions in equations Advances in Mathematical Physics 7 (24)–(29) of the nonlinear system of fractional PDEs are PðÞ x, t,stðÞ − P x, t, s ðÞ t 1 1 provided therein. 2α β ΓðÞ ðÞ n − 1 α +1 ∂ ≤ l + l + l + μ + η + D ( 1 2 3 2α N Γ nα +1 ∂x ðÞ Γ n − 1 α +1 ðÞ ðÞ PðÞ x, t,stðÞ = ψN + κR ðÞ x 1 n−1 Á S x − S x , ΓðÞ nα +1 ðÞ ðÞ k k n−1 β À ′ ′ PðÞ x, t,stðÞ − P x, t, s ðÞ t ≤ λ S ðÞ x − S ðÞ x , 1 1 1 k k − 〠 S x I x ðÞ ðÞ k n−1−k k=0 ð30Þ ! ð31Þ + S ðÞ x I ðÞ x + S ðÞ x Q ðÞ x Þ k n−1−k k n−1−k where 2α ∂ S x β ΓðÞ ðÞ n − 1 α +1 ðÞ n−1 kα − μ + η S x + D t , λ = ðÞ l + l + l + l + l , ðÞ ðÞ 1 1 2 3 4 5 n−1 1 2α ∂x N ΓðÞ nα +1 n−1 n−1 l = 〠 I x , l = 〠 I x , ðÞ ðÞ ð32Þ 1 n−1−k 2 n−1−k k=0 k=0 n−1 2α Γ n − 1 α +1 ∂ ðÞ ðÞ P x, t, s t = ψN + κR x l = 〠 Q ðÞ x , l = μ + η, l = D , ðÞ ðÞ 3 n−1−k 4 5 1 1 n−1 2α ∂x ΓðÞ nα +1 k=0 n−1 ′ ′ with 0< λ ≤ 1. − 〠 S ðÞ x I ðÞ x + S ðÞ x I ðÞ x n−1−k 1 k k n−1−k k=0 This implies that the function is Lipschitz continuous on + + + the domain fðx, t, sðtÞÞjx ∈R , t ∈ ½0Š ∪R and sðtÞ ∈R g. ′ ′ + S ðÞ x Q ðÞ x −ðÞ μ + η S ðÞ x Following similar procedure above, the following contin- k n−1−k n−1 uous functions are obtained over the domain: 2α ∂ S ðÞ x n−1 kα + D t , 2α ′ P x, t,st − P x, t, s t ðÞ ðÞ ðÞ ∂x 2 2 ≤ λ E x − E , ðÞ 2 k k PðÞ x, t,stðÞ −Px, t, s P x, t,Ex − P x, t, E x n−1 ðÞ ðÞ ðÞ 2 2 Γ n − 1 α +1 β ðÞ ðÞ = − 〠 I ðÞ x ( n−1−k Γ nα +1 N n−1 ðÞ k=0 Γ n − 1 α +1 β ðÞ ðÞ ≤ 〠 I ðÞ x + I ðÞ x n−1−k n−1−k ′ ′ ΓðÞ nα +1 N Á S ðÞ x − S ðÞ x + I ðÞ x S ðÞ x − S ðÞ x k k k n−1−k k k=0 + Q x Þ + α + α + μ + ν ðÞðÞ n−1−k 1 2 + Q ðÞ x S ðÞ x − S ðÞ x n−1−k k 2α + D E x − E x , ðÞ ðÞ 2 n n 2α ′ ∂x − μ + η S x − S x jj ðÞ ðÞ n−1 n−1 2α ∂ S x − S x ðÞ ðÞ n−1 n−1 kα Px, t,Ex − P x, t, E x ðÞ ðÞ ðÞ + D t 2α ∂x ( ≤ λ E x − E x , ðÞ ðÞ 2 n n n−1 Γ n − 1 α β À ðÞ ðÞ ≤ 〠 I x ðÞ ð33Þ n−1−k ΓðÞ nα +1 N k=0 where + I ðÞ x + Q ðÞ x S ðÞ x − S ðÞ x n−1−k k n−1−k k β Γ n − 1 α +1 ðÞ ðÞ ! ) λ = l + l + l + l + l + l , ð34Þ ðÞ 2α 2 1 2 3 4 5 6 N Γ nα +1 ðÞ kα + ðÞ μ + η + D S ðÞ x − S ðÞ x t 1 n−1 n−1 2α ∂x ( " where l = jα + α + μ + νj and l + l + l and l have usual n−1 6 1 2 1 2 3 5 ΓðÞ ðÞ n − 1 α β meanings. ≤ 〠 I x + I x + Q x ðÞ ðÞ ðÞ ðÞ n−1−k n−1−k n−1−k ΓðÞ nα +1 N k=0 !# ) 2α ′ ′ PxðÞ , t,Ix ðÞ − P x, t, I ðÞ x ≤ λ I ðÞ x − I , ∂ 3 3 n−1 n−1 kα + μ + η + D S x − S x t , ðÞ ðÞ ðÞ 1 n−1 n−1 2α ∂x ð35Þ 8 Advances in Mathematical Physics where Substituting equation (37), (39), (40), (41), and (42) into equation (43) yields β Γ n − 1 α +1 ðÞ ðÞ λ = l + l + l + l + l + l , ð36Þ ðÞ 3 1 2 3 4 5 6 N ΓðÞ nα +1 α k ∂ S x ðÞ n nα Re s x, t = S x + 〠 t − ψN ðÞ ðÞ k o where, l = jα + α + μ + νj and l + l + l and l have usual ∂t ΓðÞ nα +1 6 1 2 1 2 3 5 n=1 meanings. R ðÞ x nα − κ R ðÞ x + 〠 t 3.3. Analytic Solutions of the System of Fractional Partial Γ nα +1 ðÞ n=1 Differential Equations Using the Residual Power Series Method. This section contains the series solutions of the β S x ðÞ n nα + S x + 〠 t ðÞ nonlinear system of equations (2)–(7), together with the ini- N ΓðÞ nα +1 n=1 tial conditions in equation (8). In using the RPSM, it is assumed that there are discrepancies between the terms on I x ðÞ n nα ∗ Á I ðÞ x + 〠 t + I ðÞ x the right hand sides and the left hand sides of the system Γ nα +1 ðÞ n=1 of equations (2)–(7). With this assumption, the approxima- k k tions of the dependent variable with respect to only one I ðÞ x Q ðÞ x nα n nα + 〠 t + Q ðÞ x + 〠 t independent variable are obtained depending on the given Γ nα +1 Γ nα +1 ðÞ ðÞ n=1 n=1 initial condition or boundary point condition. The other S x independent variable is automatically in fractional form ðÞ n nα +ðÞ μ + η S ðÞ x + 〠 t which converges to a point in the Holder’s spaces. In doing ΓðÞ nα +1 n=1 this, we set ! 2α k ∂ S ðÞ x n nα − D S x + 〠 t : ðÞ 1 o 2α ∂x Γ nα +1 ðÞ S x n=1 ðÞ n nα ð37Þ S x, t = S x + 〠 t , ðÞ ðÞ k o Γ nα +1 ðÞ ð44Þ n=1 E x ðÞ n nα ð38Þ EðÞ x, t = E ðÞ x + 〠 t , k o To obtain S ðxÞ, equation (43) is reduced to ΓðÞ nα +1 n=1 ∂ S ðÞ x 1 α I ðÞ x Re sðÞ x, t = S ðÞ x + t − ψN nα 1 o ð39Þ I ðÞ x, t = I ðÞ x + 〠 t , ∂t ΓαðÞ +1 k o ΓðÞ nα +1 n=1 R ðÞ x 1 α − κ R ðÞ x + t Γα +1 ðÞ β S ðÞ x 1 α + S x + t ðÞ k ∗ I x ðÞ N Γα +1 ðÞ ∗ ∗ n nα ð40Þ I ðÞ x, t = I ðÞ x + 〠 t , k o ΓðÞ nα +1 I x n=1 ðÞ 1 α ∗ Á I ðÞ x + t + I ðÞ x ð45Þ ΓαðÞ +1 I x Q x ðÞ ðÞ 1 α 1 α + t + Q x + t ðÞ Γα +1 Γα +1 ðÞ ðÞ Q x ðÞ n nα ð41Þ QðÞ x, t = Q ðÞ x + 〠 t , k o S x ðÞ ΓðÞ nα +1 1 α n=1 + μ + η S x + t ðÞ ðÞ ΓαðÞ +1 2α ∂ S ðÞ x 1 α − D S ðÞ x + t : R ðÞ x 1 o ∗ n nα 2α ∂x Γα +1 ðÞ ð42Þ R ðÞ x, t = R ðÞ x + 〠 t : k o Γ nα +1 ðÞ n=1 Setting Re sðx,0Þ =0, it implies that ∂ S Re sðÞ x, t = − ψN − κRxðÞ , t ∂t ∗ ∗ + Sx, t Ix, t + I x, t +Qx, t S ðÞ x = ψN + κR ðÞ x −fg S ðÞ xðÞ I ðÞ x + I ðÞ x + Q ðÞ x ðÞðÞ ðÞ ðÞ ðÞ 1 o o o o N N 2α 2α − μ + η S x + D S x : +ðÞ μ + η SxðÞ , t − D SxðÞ , t : ðÞ ðÞ ðÞ 1 o 1 o 2α 2α ∂x ∂x ð43Þ ð46Þ Advances in Mathematical Physics 9 Similarly, the S ðxÞ is obtained as 2 n − 1 + S x I x + I x + Q x +⋯ ðÞðÞ ðÞ ðÞ ðÞ n−1−r r r r α r ∂ S ðÞ x S ðÞ x 1 α 2 2α Re s x, t = S x + t + t ðÞ ðÞ 2 0 ∂t ΓαðÞ +1 ΓðÞ 2α +1 +S x I x + I x + Q x ðÞðÞ ðÞ ðÞ ðÞ R x R x ðÞ ðÞ 0 n−1 n−1 n−1 1 α 2 2α − ψN − κ R ðÞ x + t + t ΓαðÞ +1 ΓðÞ 2α +1 2α β S x S x ðÞ ðÞ 1 2 ∂ α 2α + S ðÞ x + t + t −ðÞ μ + η S ðÞ x + D S ðÞ x , N Γα +1 Γ 2α +1 n−1 1 n−1 ðÞ ðÞ 2α ∂x I ðÞ x I ðÞ x 1 α 2 2α Á I ðÞ x + t + t ΓαðÞ +1 ΓðÞ 2α +1 n−1 n − 1 ∗ ∗ β I x I x ðÞ ðÞ ∗ α 2α 1 2 S ðÞ x = κR ðÞ x − 〠 S ðÞ x I ðÞ x + I ðÞ x + t + t n n−1 n−1−r r Γα +1 Γ 2α +1 N ðÞ ðÞ r=0 Q ðÞ x Q ðÞ x 1 α 2 2α 2α + Q x + t + t ðÞ Á 0 ∂ ΓαðÞ +1 ΓðÞ 2α +1 + I ðÞ x + Q ðÞ x −ðÞ μ + η S ðÞ x + D S ðÞ x : r n−1 1 n−1 2α ∂x S x S x ðÞ ðÞ 1 α 2 2α +ðÞ μ + η S ðÞ x + t + t Γα +1 Γ 2α +1 ð48Þ ðÞ ðÞ 2α ∂ S ðÞ x S ðÞ x 1 α 2 2α − D S x + t + t , ðÞ 1 0 2α ∂x ΓαðÞ +1 ΓðÞ 2α +1 Similarly, we obtain the following results for EðxÞ as È À ∂ β Re s x,0 = S x − κR x + S x I x ðÞ ðÞ ðÞ ðÞ ðÞ 2 2 1 1 0 ∂t N Á À E x = S x I x + I x + Q x + I ðÞ x + Q ðÞ x + S ðÞ x I ðÞ x ðÞfg ðÞðÞ ðÞ ðÞ ðÞ 0 0 0 1 1 0 0 0 0 ÁÉ + I ðÞ x + Q ðÞ x +ðÞ μ + η S ðÞ x 1 1 1 2α 2α ∂ −ðÞ α + α + μ + ν E ðÞ x + D E ðÞ x , 1 2 0 2 0 2α − D S ðÞ x =0, ∂x 1 1 2α ∂x β ∗ S ðÞ x = κR ðÞ x − S ðÞ xðÞ I ðÞ x + I ðÞ x + Q ðÞ x E ðÞ x = S ðÞ xðÞ I ðÞ x + I ðÞ x + Q ðÞ x 2 1 1 0 0 0 2 1 0 0 0 + S x I x + I x + Q x ∗ ðÞðÞ ðÞ ðÞ ðÞ 0 1 1 + S ðÞ xðÞ I ðÞ x + I ðÞ x + Q ðÞ x 0 1 1 2α 2α − μ + η S x + D S x : ðÞ ðÞ ðÞ 1 1 1 ∂ 2α ∂x − α + α + μ + ν E x + D E x , ðÞ ðÞ ðÞ 1 2 1 2 1 2α ∂x ð47Þ β È E x = S x I x + I x + Q x ðÞ ðÞðÞ ðÞ ðÞ ðÞ 3 2 0 0 0 Similarly, the following results are obtained: +2S ðÞ xðÞ I ðÞ x + I ðÞ x + Q ðÞ x 1 1 1 1 + S ðÞ xðÞ I ðÞ x + I ðÞ x + Q ðÞ x 0 2 2 2 È À ∂ β Re s x,0 = S x − κR x + S x I x + I x 2α ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ 3 3 2 2 0 0 α ∂ ∂t N − α + α + μ + ν E x + D E x , Á ðÞ ðÞ ðÞ 1 2 2 2 2 2α ∂x + Q ðÞ x +2S ðÞ xðÞ I ðÞ x + I ðÞ x + Q ðÞ x 0 1 1 1 + S x I x + I x + Q x ðÞðÞ ðÞ ðÞ ðÞ 0 2 2 2 2α n−1 n − 1 +ðÞ μ + η S ðÞ x − D S ðÞ x =0, E x = 〠 S x I x + I x + Q x 2 1 2 ðÞ ðÞðÞ ðÞ ðÞ ðÞ 2α n n−1−r r r r ∂x r=0 r 2α S ðÞ x = κR ðÞ x − S ðÞ xðÞ I ðÞ x + I ðÞ x + Q ðÞ x ∂ 3 2 2 0 0 0 N −ðÞ α + α + μ + ν E ðÞ x + D E ðÞ x : 1 2 n−1 2 n−1 2α ∂x +2S x I x + I x + Q x ðÞðÞ ðÞ ðÞ ðÞ 1 1 1 1 É ð49Þ + S x I x + I x + Q x ðÞðÞ ðÞ ðÞ ðÞ 0 2 2 2 2α Following similar procedure, the series solutions of the −ðÞ μ + η S ðÞ x + D S ðÞ x 2 1 2 2α fractional PDEs for describing the number of deviant infec- ∂x tives, number of non-deviant infectives, number of quaran- tined persons, and number of recoveries from the COVID- 19 disease are obtained as follows: 2α S ðÞ x = κR ðÞ x − S ðÞ xðÞ I ðÞ x + I ðÞ x + Q ðÞ x n n−1 n−1 0 0 N I x = α E x − γ + μ + δ + ω I x + D I x , ðÞ ðÞðÞ ðÞ ðÞ 1 1 0 1 1 1 0 3 0 2α ∂x n − 1 2α + S x I x + I x + Q x +⋯ ðÞðÞ ðÞ ðÞ ðÞ I x = α E x − γ + μ + δ + ω I x + D I x , ðÞ ðÞðÞ ðÞ ðÞ n−2 1 1 1 2 1 1 1 1 1 1 3 1 2α ∂x 1 10 Advances in Mathematical Physics 2α infectives, quarantined persons, and recovered persons: I ðÞ x = α E ðÞ x −ðÞ γ + μ + δ + ω I ðÞ x + D I ðÞ x , 3 1 2 1 1 2 3 2 2α ∂x ( ! k n−1 n − 1 1 β E x, t = E x + 〠 〠 ðÞ ðÞ k 0 Γ nα +1 N ðÞ n=1 r=0 2α I ðÞ x = α E ðÞ x −ðÞ γ + μ + δ + ω I ðÞ x + D I ðÞ x , n 1 n−1 1 1 n−1 3 n−1 Á S x I x + I x + Q x 1 ðÞðÞ ðÞ ðÞ ðÞ 2α n−1−r r r r ∂x − α + α + μ + ν E x ðÞ ðÞ 2α 1 2 n−1 ∗ ∗ ∗ 2α I ðÞ x = α E ðÞ x −ðÞ γ + μ + δ + ω I ðÞ x + D I ðÞ x , 1 2 0 2 2 0 4 0 2 ∂ 2α nα ∂x + D E x gt , ðÞ 2 n−1 2α ∂x ð52Þ 2α ∂ ( ∗ ∗ ∗ I ðÞ x = α E ðÞ x −ðÞ γ + μ + δ + ω I ðÞ x + D I ðÞ x , 2 2 1 2 2 1 4 1 2 1 2α ∂x I ðÞ x, t = I ðÞ x + 〠 α E ðÞ x k 0 1 n−1 ΓðÞ nα +1 2α n=1 ∗ ∗ ∗ I ðÞ x = α E ðÞ x −ðÞ γ + μ + δ + ω I ðÞ x + D I ðÞ x , 3 2 2 2 2 2 2 4 2 2α − γ + μ + δ + ω I x ðÞ ðÞ ∂x 1 1 1 n−1 2α nα + D I x gt , ðÞ 3 n−1 2α ∂x 2α ∗ ∗ ∗ I ðÞ x = α E ðÞ x −ðÞ γ + μ + δ + ω I ðÞ x + D I ðÞ x , 2 n−1 2 2 4 ( n 2 n−1 n−1 2α ∂x ∗ ∗ I ðÞ x, t = I ðÞ x + 〠 α E ðÞ x k 0 2 n−1 Q x = vE x + ω I x + ω I x − μ + ϕ + δ I x , ðÞ ðÞ ðÞ ðÞðÞ ðÞ Γ nα +1 1 0 1 0 2 0 3 0 ðÞ n=1 Q ðÞ x = vE ðÞ x + ω I ðÞ x + ω I ðÞ x −ðÞ μ + ϕ + δ I ðÞ x , − γ + μ + δ + ω I x ð53Þ 2 1 1 1 2 1 3 1 ðÞ ðÞ 2 2 2 n−1 ∗ 2α Q ðÞ x = vE ðÞ x + ω I ðÞ x + ω I ðÞ x −ðÞ μ + ϕ + δ I ðÞ x , ∂ 3 2 1 2 2 3 2 2 ∗ nα + D I x t , ðÞ 4 n−1 2α ∂x Q x = vE x + ω I x + ω I x − μ + ϕ + δ I x , ðÞ ðÞ ðÞ ðÞðÞ ðÞ QðÞ x, t = Q ðÞ x + 〠 fvE ðÞ x n n−1 1 n−1 2 n−1 3 n−1 k 0 n−1 ΓðÞ nα +1 n=1 ∗ ð54Þ R ðÞ x = ϕQ ðÞ x + γ I ðÞ x + γ I ðÞ x + ηS ðÞ x −ðÞ μ + κ R ðÞ x , 1 0 1 0 2 0 0 0 + ω I ðÞ x + ω I ðÞ x −ðÞ μ + ϕ + δ 1 n−1 2 n−1 3 nα R x = ϕQ x + γ I x + γ I x + ηS x − μ + κ R x , ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ 2 1 1 1 1 Á I ðÞ x gt , 1 2 1 n−1 R x = ϕQ x + γ I x + γ I x + ηS x − μ + κ R x , ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ 3 2 1 2 2 2 2 2 ⋮ R x, t = R x + 〠 ϕQ x + γ I x ðÞ ðÞ f ðÞ ðÞ k 0 n−1 1 n−1 Γ nα +1 ðÞ n=1 R ðÞ x = ϕQ ðÞ x + γ I ðÞ x + γ I ðÞ x + ηS ðÞ x ∗ nα n n−1 n−1 n−1 n−1 1 2 + γ I x + ηS x − μ + κ R x t : ðÞ ðÞ ðÞ ðÞg ð50Þ 2 n−1 n−1 n−1 −ðÞ μ + κ R ðÞ x : n−1 ð55Þ In order to obtain the series solution for the number of susceptible individuals, we substitute the last equation of 3.4. Numerical Results. In this section, the three- the system of equations (48) into equation (37) which dimensional plots, as well as the two-dimensional plots, yields are provided here. The initial condition for each subgroup was taken from [28]. 3.4.1. The Numerical Results for the Series Solutions of the S x, t = S x + 〠 κR x ðÞ ðÞ ðÞ Nonlinear System of Fractional PDEs Using the FPSM. This k 0 n−1 Γ nα +1 ðÞ n=1 section contains the plots for the series solutions to the n−1 n − 1 system of equations (22)–(27) for α =0:2000. Both three- − 〠 S ðÞ x ðI ðÞ x + I ðÞ x dimensional and two-dimensional plots for each series n−1−r r r r=0 solution of these equations are depicted in Figures 2 and 2α 3, respectively. The initial conditions for each subgroup nα + Q ðÞ x Þ −ðÞ μ + η S ðÞ x + D S ðÞ x t : of the population were estimated using the available data r n−1 1 n−1 2α ∂x in [28]. The number of susceptible decreases rapidly as they come into contact with the infectives, both locals ð51Þ and foreigners, as shown in Figure 2(a). In contrast to the number of nondeviant infective persons (see Similarly, the following results are obtained for the Figure 3(a)), the number of deviant infectives unexpect- number of exposed persons, deviant infectives, nondeviant edly grows to its peak for a long time before it starts to Ti T me (t) T Ti ime me (t) (t) Ti Time me (t) Time (t) T Ti ime (t) me (t) Ti Tim me e (t) (t) Advances in Mathematical Physics 11 S (x, t) E (x, t) ×10 4 30 2 15 0 0 1 1 1 1 0 0.8 .8 0.8 0.8 0.5 0.5 0.5 0.5 0.6 0.6 0.6 0.6 0. 0.4 4 0.4 02 0.2 0.2 0 0 0 0 (a) (b) I (x, t) ⁎ I (x, t) 0.8 0.8 0.8 0.5 0.6 0.5 0.5 0.6 0.6 0.4 0.4 0.4 0.2 0.2 02 (c) (d) Q (x, t) R (x, t) ×10 8 8 6 6 4 4 2 2 0 0 1 1 0.8 0.8 0 0.8 .8 0. 0.5 5 0.5 0.5 0. 0.6 6 0. 0.6 6 0. 0.4 4 0.4 0.4 0 0.2 2 02 0.2 0 0 0 0 (e) (f) Figure 2: Three-dimensional plots for the number of susceptible individuals, exposed persons, deviant infectives, nondeviant infectives, quarantined persons, and recovered persons for α =0:2000, using the FPSM. Distance (x) (x) Distance (x) Distance (x) Distance (x) ) Distance (x) (x) Distance (x) (x) 12 Advances in Mathematical Physics 7 E × 10 S 3.5 2.5 1.5 10 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (t) Time (t) (a) (b) I ⁎ 1.2 1.18 1.16 1.14 1.12 1.1 1.08 1.06 1.04 1.02 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (t) Time (t) 1 I (c) (d) Q 5 × 10 R 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (t) Time (t) 1 R 3 R (e) (f) Figure 3: Two-dimensional plots for the number of susceptible individuals, exposed persons, deviant infectives, nondeviant infectives, quarantined persons, and recovered persons α =0:2000, using the FPSM. S (t, ·) I (·, t) Q (t, ·) R (t, ·) E (t, ·) Advances in Mathematical Physics 13 × 10 S 3.5 22 2.5 1.5 0.5 0 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (t) Time (t) (a) (b) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (t) Time (t) (c) (d) × 10 R 2.5 1.5 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (t) Time (t) (e) (f) Figure 4: Two-dimensional plots for the various subgroups of the population size of Ghana for α =0:2000, using the FPSM. decline and eventually becomes asymptotically stable on tance is suppressed and time is varied. The first three series the t-axis. This implies that patients with the intention solutions for the system of equations (51)-(55) were of transmitting COVID-19 disease to the susceptible employed to show trends in the numbers of susceptible indi- members are the primary disease spreaders in Ghana. viduals, exposed individuals, deviant infectives, nondeviant For the number of susceptible, exposed, deviant, nonde- infectives, quarantined individuals, and recovered individ- viant, quarantined, and recovered subgroups, spatial dis- uals. The plots were repeated for the first fifteen terms of Q (t, ·) 14 Advances in Mathematical Physics × 10 S 5.5 3.5 2.5 4.5 1.5 3.5 0.5 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (t) Time (t) (a) (b) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (t) Time (t) (c) (d) Q × 10 R 1.5 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (t) Time (t) (e) (f) Figure 5: Two-dimensional plots for the various subgroups of the population size of Ghana for α =0:2000, using the RPSM. the series solutions of system of equations (51)-(55), shown idly, the curve for nondeviant infected individuals declines in Figure 4. sharply. This indicates that the majority of nondeviant To take into account the impact of each series as the carriers of the COVID-19 virus are not infecting suscepti- number of terms rises, plots for the first 80 terms of the ble members who are at risk of becoming infected. On the series solutions of the system of equations (51)-(55) were other hand, a large number of COVID-19 patients are reproduced as displayed in Figure 5. While the curves being isolated at numerous facilities across the country, for the quarantined and recovered populations climb rap- and these people are making progress every day. The fact Advances in Mathematical Physics 15 × 10 S 3.5 6 2.5 1.5 0.5 0 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Distance (x) Distance (x) S E 1 1 S E 2 2 (a) (b) 1.2 0.8 0.6 0.4 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Distance (x) Distance (x) 1 I 2 I 3 I (c) (d) Figure 6: Two-dimensional plots for the various subgroups of the population size using the FPSM. that the deviant subpopulation is still growing suggests irrespective of their home country will not ensue to the epi- that patients have been spreading the SARS virus for a demiology of COVID-19 disease in Ghana. However, there considerable amount of time with the intention of infect- is a slight positive nonlinear relationship and fairly negative ing a sizable number of vulnerable individuals with the nonlinear relationship between the nondeviant subpopula- COVID-19 infection. The plot for the exposed individuals tion and the distance. increases to a peak and then declines and asymptotically moves toward the t-axis, showing that those who are vul- 3.4.2. The Numerical Results for the Series Solutions of the nerable to contracting COVID-19 disease are at high risk. Nonlinear System of Fractional PDEs Using the RPSM. The SARS virus starts spreading to anybody who come Figure 7 displays the first three terms of the series solutions, into contact with it after a brief time of incubation. From equations (51)-(55), with α =0:2000. Every subgroup of the day zero, the susceptible curve declines and becomes population size in Ghana corresponds with the epidemiolog- asymptotically toward the t-axis, signalling the end of the ical pattern of the COVID-19 disease. disease. In Figure 6, there is relatively positive nonlinear relation- ship between the susceptible subgroups. This indicates that 3.5. Comparison of the FPSM and the RPSM. This section susceptible (foreigners) continuous inflow into the country uses quantitative results of the FPSM to compare the I (x, ·) S (x, ·) E (x, ·) 16 Advances in Mathematical Physics 7 E × 10 S 3.5 2.5 1.5 0.5 0 1 0 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time Time (a) (b) 0 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time Time 3 I (c) (d) Q 5 × 10 R 1000 5 800 4 600 3 400 2 200 1 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Time Q R 1 1 Q R 2 2 Q R 3 3 (e) (f) Figure 7: Two-dimensional plots for the various subgroups of the population size using the RPSM. Q (·, t) I (·, t) S (·, t) R (·, t) E (·, t) Advances in Mathematical Physics 17 15000 6000 20 4000 10 2000 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 data method 1 method 2 (a) 4 E × 10 15 2 20 4000 1 10 2000 0 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 data method 1 method 2 (b) Figure 8: Plots of the field data on the number of exposed persons and the series solutions using both the FPSM and the RPSM. quantitative results of the RPSM which are then superim- On the other hand, using the RPSM is more consistent posed on the quantitative results from the field data. with the field data as the COVID-19 infection is present in Figure 8 displays the series solutions of the first three the population subjects over a long period of time in com- terms using both the FPSM and the RPSM. The FPSM parison to the FPSM’s series solution. This is indicated by shows that the solution rises from the starting point to the deep red plot. A similar observation was made in the peak and then falls, showing the loss of the suscepti- Figure 9 when comparing the two series solutions using both ble members to the exposed subgroup and the exposed the FPSM and the RPSM. subgroup’s loss of members to both the deviant and non- deviant subgroups. On the other hand, the length of the 4. Discussion series solution obtained using the RPSM increases starting with the beginning of the COVID-19 pandemic and takes In contrast, the FPSM series solution for the number of a lot of time. It begins to decline in the direction of the t susceptible individuals is proportional to the RPSM series -axis, showing that the susceptible individuals who con- solution for the number of susceptible individuals, with a tract the SARS virus also persistently infect the popula- proportional constant of ψΓððn − 1Þα +1Þ. The series solu- tion subjects. Despite this, the series solution using the tion of the vulnerable members utilizing the RPSM then FPSM is more consistent with the field data at the start reduced quickly in comparison to the series solution using of the disease outbreak than the series solution using the FPSM. For instance, the third term of the series solu- the RPSM. tion using the RPSM is reduced by S ðxÞðI ðxÞ + I ðxÞ + 1 1 data data method 1 method 1 method 2 method 2 18 Advances in Mathematical Physics 10000 1200 5000 60 600 0 40 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 data method 1 method 2 (a) 10000 60 900 5000 50 0 40 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 data method 1 method 2 (b) Figure 9: Plots of the field data on the number of infected persons and the series solutions using both the FPSM and the RPSM. Q ðxÞÞ when compared to the series solution using the uals. The power of SðxÞðIðxÞ + I ðxÞ + QðxÞÞ is linear for FPSM. In comparison, using the RPSM in obtaining the the first and second terms and then increases in the pat- series solutions of the nonlinear system of equations, tern of natural numbers, that is, having positive integer (2)–(7), together with the initial conditions in equation binomial powers, when employing the RPSM. Because (8), are more consistent with the field data as compared the product of this nonlinear term is differentiated, this to the series solutions given by the FPSM, as displayed occurs. The nonlinear term, however, increases linearly in Figures 8 and 9. when the FPSM is used to find the series solutions of the system of equations (2)–(7) along with the initial con- ditions in equation (8). Interestingly, series solutions of the 5. Conclusion nonlinear system of the equations using the RPSM were observed to be more consistent as compared to series solu- There are more terms for the total number of susceptible tions given by FPSM. This is due to the fact that the members and exposed individuals in the series of solutions RPSM utilizes the variations in the nonlinear system of of the nonlinear system of fractional PDEs provided by the equations unlike the FPSM. However, both the RPSM RPSM than the series of solutions yielded by the FPSM. and the FPSM yield the same series solutions of the linear The nonlinear term SðxÞðIðxÞ + I ðxÞ + QðxÞÞ is what system of equations, as indicated by equations (24)–(29) causes the difference between the two series solutions for and system of equations (51)–(55). the number of susceptible members and exposed individ- I data data I method 1 method 1 method 2 I method 2 Advances in Mathematical Physics 19 Data Availability severity, hospitalization, and mortality: a systematic review,” Human Vaccines and Immunotherapeutics, vol. 18, no. 1, arti- The data is freely available at [28] https://ourworldindata cle 2027160, 2022. .org/coronavirus. [13] M. Shah, M. Arfan, I. Mahariq, A. 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Advances in Mathematical Physics – Hindawi Publishing Corporation

Published: May 4, 2023

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